Dynamic analysis of flexible rotor-bearing systems using a modal approach

NASA Technical Reports Server (NTRS)

Choy, K. C.; Gunter, E. J.; Barrett, L. E.

1978-01-01

The generalized dynamic equations of motion were obtained by the direct stiffness method for multimass flexible rotor-bearing systems. The direct solution of the equations of motion is illustrated on a simple 3-mass system. For complex rotor-bearing systems, the direct solution of the equations becomes very difficult. The transformation of the equations of motion into modal coordinates can greatly simplify the computation for the solution. The use of undamped and damped system mode shapes in the transformation are discussed. A set of undamped critical speed modes is used to transform the equations of motion into a set of coupled modal equations of motion. A rapid procedure for computing stability, steady state unbalance response, and transient response of the rotor-bearing system is presented. Examples of the application of this modal approach are presented. The dynamics of the system is further investigated with frequency spectrum analysis of the transient response.

Derivation of Hamilton's equations of motion for mechanical systems with constraints on the basis of Pontriagin's maximum principle

NASA Astrophysics Data System (ADS)

Kovalev, A. M.

The problem of the motion of a mechanical system with constraints conforming to Hamilton's principle is stated as an optimum control problem, with equations of motion obtained on the basis of Pontriagin's principle. A Hamiltonian function in Rodrigues-Hamilton parameters for a gyrostat in a potential force field is obtained as an example. Equations describing the motion of a skate on a sloping surface and the motion of a disk on a horizontal plane are examined.

Equations of motion of a space station with emphasis on the effects of the gravity gradient

NASA Technical Reports Server (NTRS)

Tuell, L. P.

1987-01-01

The derivation of the equations of motion is based upon the principle of virtual work. As developed, these equations apply only to a space vehicle whose physical model consists of a rigid central carrier supporting several flexible appendages (not interconnected), smaller rigid bodies, and point masses. Clearly evident in the equations is the respect paid to the influence of the Earth's gravity field, considerably more than has been the custom in simulating vehicle motion. The effect of unpredictable crew motion is ignored.

Exact solutions to the Mo-Papas and Landau-Lifshitz equations

NASA Astrophysics Data System (ADS)

Rivera, R.; Villarroel, D.

2002-10-01

Two exact solutions of the Mo-Papas and Landau-Lifshitz equations for a point charge in classical electrodynamics are presented here. Both equations admit as an exact solution the motion of a charge rotating with constant speed in a circular orbit. These equations also admit as an exact solution the motion of two identical charges rotating with constant speed at the opposite ends of a diameter. These exact solutions allow one to obtain, starting from the equation of motion, a definite formula for the rate of radiation. In both cases the rate of radiation can also be obtained, with independence of the equation of motion, from the well known fields of a point charge, that is, from the Maxwell equations. The rate of radiation obtained from the Mo-Papas equation in the one-charge case coincides with the rate of radiation that comes from the Maxwell equations; but in the two-charge case the results do not coincide. On the other hand, the rate of radiation obtained from the Landau-Lifshitz equation differs from the one that follows from the Maxwell equations in both the one-charge and two-charge cases. This last result does not support a recent statement by Rohrlich in favor of considering the Landau-Lifshitz equation as the correct and exact equation of motion for a point charge in classical electrodynamics.

Newton's laws of motion in the form of a Riccati equation.

PubMed

Nowakowski, Marek; Rosu, Haret C

2002-04-01

We discuss two applications of a Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential V(r)=kr(epsilon). For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.

Numerical Simulation of Forced and Free-to-Roll Delta-Wing Motions

NASA Technical Reports Server (NTRS)

Chaderjian, Neal M.; Schiff, Lewis B.

1996-01-01

The three-dimensional, Reynolds-averaged, Navier-Stokes (RANS) equations are used to numerically simulate nonsteady vortical flow about a 65-deg sweep delta wing at 30-deg angle of attack. Two large-amplitude, high-rate, forced-roll motions, and a damped free-to-roll motion are presented. The free-to-roll motion is computed by coupling the time-dependent RANS equations to the flight dynamic equation of motion. The computed results are in good agreement with the forces, moments, and roll-angle time histories. Vortex breakdown is present in each case. Significant time lags in the vortex breakdown motions relative to the body motions strongly influence the dynamic forces and moments.

Spacecraft Constrained Maneuver Planning Using Positively Invariant Constraint Admissible Sets (Postprint)

DTIC Science & Technology

2013-08-14

Connectivity Graph; Graph Search; Bounded Disturbances; Linear Time-Varying (LTV); Clohessy - Wiltshire -Hill (CWH) 16. SECURITY CLASSIFICATION OF: 17...the linearization of the relative motion model given by the Hill- Clohessy - Wiltshire (CWH) equations is used [14]. A. Nonlinear equations of motion...equations can be used to describe the motion of the debris. B. Linearized HCW equations in discrete-time For δr << R, the linearized Hill- Clohessy

The dynamics and control of large flexible space structures. Part B: Development of continuum model and computer simulation

NASA Technical Reports Server (NTRS)

Bainum, P. M.; Kumar, V. K.; James, P. K.

1978-01-01

The equations of motion of an arbitrary flexible body in orbit were derived. The model includes the effects of gravity with all its higher harmonics. As a specific example, the motion of a long, slender, uniform beam in circular orbit was modelled. The example considers both the inplane and three dimensional motion of the beam in orbit. In the case of planar motion with only flexible vibrations, the pitch motion is not influenced by the elastic motion of the beam. For large values of the square of the ratio of the structural modal frequency to the orbital angular rate the elastic motion was decoupled from the pitch motion. However, for small values of the ratio and small amplitude pitch motion, the elastic motion was governed by a Hill's 3 term equation. Numerical simulation of the equation indicates the possibilities of instability for very low values of the square of the ratio of the modal frequency to the orbit angular rate. Also numerical simulations of the first order nonlinear equations of motion for a long flexible beam in orbit were performed. The effect of varying the initial conditions and the number of modes was demonstrated.

Dynamic characteristics of a variable-mass flexible missile

NASA Technical Reports Server (NTRS)

Meirovitch, L.; Bankovskis, J.

1970-01-01

The general motion of a variable mass flexible missile with internal flow and aerodynamic forces is considered. The resulting formulation comprises six ordinary differential equations for rigid body motion and three partial differential equations for elastic motion. The simultaneous differential equations are nonlinear and possess time-dependent coefficients. The differential equations are solved by a semi-analytical method leading to a set of purely ordinary differential equations which are then solved numerically. A computer program was developed for the numerical solution and results are presented for a given set of initial conditions.

The research of the coupled orbital-attitude controlled motion of celestial body in the neighborhood of the collinear libration point L1

NASA Astrophysics Data System (ADS)

Shmyrov, A.; Shmyrov, V.; Shymanchuk, D.

2017-10-01

This article considers the motion of a celestial body within the restricted three-body problem of the Sun-Earth system. The equations of controlled coupled attitude-orbit motion in the neighborhood of collinear libration point L1 are investigated. The translational orbital motion of a celestial body is described using Hill's equations of circular restricted three-body problem of the Sun-Earth system. Rotational orbital motion is described using Euler's dynamic equations and quaternion kinematic equation. We investigate the problem of stability of celestial body rotational orbital motion in relative equilibrium positions and stabilization of celestial body rotational orbital motion with proposed control laws in the neighborhood of collinear libration point L1. To study stabilization problem, Lyapunov function is constructed in the form of the sum of the kinetic energy and special "kinematic function" of the Rodriguez-Hamiltonian parameters. Numerical modeling of the controlled rotational motion of a celestial body at libration point L1 is carried out. The numerical characteristics of the control parameters and rotational motion are given.

Systematic generation of multibody equations of motion suitable for recursive and parallel manipulation

NASA Technical Reports Server (NTRS)

Nikravesh, Parviz E.; Gim, Gwanghum; Arabyan, Ara; Rein, Udo

1989-01-01

The formulation of a method known as the joint coordinate method for automatic generation of the equations of motion for multibody systems is summarized. For systems containing open or closed kinematic loops, the equations of motion can be reduced systematically to a minimum number of second order differential equations. The application of recursive and nonrecursive algorithms to this formulation, computational considerations and the feasibility of implementing this formulation on multiprocessor computers are discussed.

A computer program to generate equations of motion matrices, L217 (EOM). Volume 1: Engineering and usage

NASA Technical Reports Server (NTRS)

Kroll, R. I.; Clemmons, R. E.

1979-01-01

The equations of motion program L217 formulates the matrix coefficients for a set of second order linear differential equations that describe the motion of an airplane relative to its level equilibrium flight condition. Aerodynamic data from FLEXSTAB or Doublet Lattice (L216) programs can be used to derive the equations for quasi-steady or full unsteady aerodynamics. The data manipulation and the matrix coefficient formulation are described.

Stationary motion stability of monocycle on ice surface

NASA Astrophysics Data System (ADS)

Lebedev, Dmitri A.

2018-05-01

The problem of the one-wheeled crew motion on smooth horizontal ice is considered. The motion equations are worked out in quasicoordinates in the form of Euler-Lagrange's equations. The variety of stationary motions is defined. Stability of some stationary motions is investigated. Comparison of the results received for a similar model of one-wheeled crew at its motion on the horizontal plane without slipping is carried out.

Ground-Motion Prediction Equations (GMPEs) from a global dataset: the PEERPEER NGA equations

USGS Publications Warehouse

Boore, David M.; Akkar, Sinan; Gulkan, Polat; van Eck, Torild

2011-01-01

The PEER NGA ground-motion prediction equation s (GMPEs) were derived by five developer teams over several years, resulting in five sets of GMPEs. The teams used various subsets of a global database of ground motions and metadata from shallow earthquakes in tectonically active regions in the development of the equations. Since their publication, the predicted motions from these GMPEs have been compared with data from various parts of the world – data that largely were not used in the development of the GMPEs. The comparisons suggest that the NGA GMPEs are applicable globally for shallow earthquakes in tectonically active regions.

Mechanism test bed. Flexible body model report

NASA Technical Reports Server (NTRS)

Compton, Jimmy

1991-01-01

The Space Station Mechanism Test Bed is a six degree-of-freedom motion simulation facility used to evaluate docking and berthing hardware mechanisms. A generalized rigid body math model was developed which allowed the computation of vehicle relative motion in six DOF due to forces and moments from mechanism contact, attitude control systems, and gravity. No vehicle size limitations were imposed in the model. The equations of motion were based on Hill's equations for translational motion with respect to a nominal circular earth orbit and Newton-Euler equations for rotational motion. This rigid body model and supporting software were being refined.

Control of Supercavitation Flow and Stability of Supercavitating Motion of Bodies

DTIC Science & Technology

2001-02-01

sign opposite to a sign of angle Vf - accidental deflection of the model Sgn M = -Sgn i. 4.3. EQUATIONS OF THE SCM DYNAMICS The most effective method of...the motion stability in interactive regime "researcher - computer" [ 16]. The complete mathematical model of the SCM motion includes a set of equations ...of solid body dynamics, equations to calculate the unsteady cavity shape and relations to calculate the acting forces. A set of dynamic equations of

Model-based control strategies for systems with constraints of the program type

NASA Astrophysics Data System (ADS)

Jarzębowska, Elżbieta

2006-08-01

The paper presents a model-based tracking control strategy for constrained mechanical systems. Constraints we consider can be material and non-material ones referred to as program constraints. The program constraint equations represent tasks put upon system motions and they can be differential equations of orders higher than one or two, and be non-integrable. The tracking control strategy relies upon two dynamic models: a reference model, which is a dynamic model of a system with arbitrary order differential constraints and a dynamic control model. The reference model serves as a motion planner, which generates inputs to the dynamic control model. It is based upon a generalized program motion equations (GPME) method. The method enables to combine material and program constraints and merge them both into the motion equations. Lagrange's equations with multipliers are the peculiar case of the GPME, since they can be applied to systems with constraints of first orders. Our tracking strategy referred to as a model reference program motion tracking control strategy enables tracking of any program motion predefined by the program constraints. It extends the "trajectory tracking" to the "program motion tracking". We also demonstrate that our tracking strategy can be extended to a hybrid program motion/force tracking.

Spatiotemporal motion boundary detection and motion boundary velocity estimation for tracking moving objects with a moving camera: a level sets PDEs approach with concurrent camera motion compensation.

PubMed

Feghali, Rosario; Mitiche, Amar

2004-11-01

The purpose of this study is to investigate a method of tracking moving objects with a moving camera. This method estimates simultaneously the motion induced by camera movement. The problem is formulated as a Bayesian motion-based partitioning problem in the spatiotemporal domain of the image quence. An energy functional is derived from the Bayesian formulation. The Euler-Lagrange descent equations determine imultaneously an estimate of the image motion field induced by camera motion and an estimate of the spatiotemporal motion undary surface. The Euler-Lagrange equation corresponding to the surface is expressed as a level-set partial differential equation for topology independence and numerically stable implementation. The method can be initialized simply and can track multiple objects with nonsimultaneous motions. Velocities on motion boundaries can be estimated from geometrical properties of the motion boundary. Several examples of experimental verification are given using synthetic and real-image sequences.

f( R, L m ) gravity

NASA Astrophysics Data System (ADS)

Harko, Tiberiu; Lobo, Francisco S. N.

2010-11-01

We generalize the f( R) type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the matter Lagrangian L m . We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy density of the matter only. Generally, the motion is non-geodesic, and it takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert-Einstein Lagrange density are also derived.

Novel Approach for Solving the Equation of Motion of a Simple Harmonic Oscillator. Classroom Notes

ERIC Educational Resources Information Center

Gauthier, N.

2004-01-01

An elementary method, based on the use of complex variables, is proposed for solving the equation of motion of a simple harmonic oscillator. The method is first applied to the equation of motion for an undamped oscillator and it is then extended to the more important case of a damped oscillator. It is finally shown that the method can readily be…

Solving modal equations of motion with initial conditions using MSC/NASTRAN DMAP. Part 1: Implementing exact mode superposition

NASA Technical Reports Server (NTRS)

Abdallah, Ayman A.; Barnett, Alan R.; Ibrahim, Omar M.; Manella, Richard T.

1993-01-01

Within the MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) module TRD1, solving physical (coupled) or modal (uncoupled) transient equations of motion is performed using the Newmark-Beta or mode superposition algorithms, respectively. For equations of motion with initial conditions, only the Newmark-Beta integration routine has been available in MSC/NASTRAN solution sequences for solving physical systems and in custom DMAP sequences or alters for solving modal systems. In some cases, one difficulty with using the Newmark-Beta method is that the process of selecting suitable integration time steps for obtaining acceptable results is lengthy. In addition, when very small step sizes are required, a large amount of time can be spent integrating the equations of motion. For certain aerospace applications, a significant time savings can be realized when the equations of motion are solved using an exact integration routine instead of the Newmark-Beta numerical algorithm. In order to solve modal equations of motion with initial conditions and take advantage of efficiencies gained when using uncoupled solution algorithms (like that within TRD1), an exact mode superposition method using MSC/NASTRAN DMAP has been developed and successfully implemented as an enhancement to an existing coupled loads methodology at the NASA Lewis Research Center.

A vector-dyadic development of the equations of motion for N-coupled rigid bodies and point masses

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1974-01-01

The equations of motion are derived, in vector-dyadic format, for a topological tree of coupled rigid bodies, point masses, and symmetrical momentum wheels. These equations were programmed, and form the basis for the general-purpose digital computer program N-BOD. A complete derivation of the equations of motion is included along with a description of the methods used for kinematics, constraint elimination, and for the inclusion of nongyroscope forces and torques acting external or internal to the system.

Equations of motion for the variable mass flow-variable exhaust velocity rocket

NASA Technical Reports Server (NTRS)

Tempelman, W. H.

1972-01-01

An equation of motion for a one dimensional rocket is derived as a function of the mass flow rate into the acceleration chamber and the velocity distribution along the chamber, thereby including the transient flow changes in the chamber. The derivation of the mass density requires the introduction of the special time coordinate. The equation of motion is derived from both classical force and momentum approaches and is shown to be consistent with the standard equation expressed in terms of flow parameters at the exit to the acceleration chamber.

The equations of motion of a secularly precessing elliptical orbit

NASA Astrophysics Data System (ADS)

Casotto, S.; Bardella, M.

2013-01-01

The equations of motion of a secularly precessing ellipse are developed using time as the independent variable. The equations are useful when integrating numerically the perturbations about a reference trajectory which is subject to secular perturbations in the node, the argument of pericentre and the mean motion. Usually this is done in connection with Encke's method to ensure minimal rectification frequency. Similar equations are already available in the literature, but they are either given based on the true anomaly as the independent variable or in mixed mode with respect to time through the use of a supporting equation to track the anomaly. The equations developed here form a complete and independent set of six equations in time. Reformulations both of Escobal's and Kyner and Bennett's equations are also provided which lead to a more concise form.

Wigner distribution functions for complex dynamical systems: the emergence of the Wigner-Boltzmann equation.

PubMed

Sels, Dries; Brosens, Fons

2013-10-01

The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.

Dynamic characteristics of a two-stage variable-mass flexible missile with internal flow

NASA Technical Reports Server (NTRS)

Meirovitch, L.; Bankovskis, J.

1972-01-01

A general formulation of the dynamical problems associated with powered flight of a two stage flexible, variable-mass missile with internal flow, discrete masses, and aerodynamic forces is presented. The formulation comprises six ordinary differential equations for the rigid body motion, 3n ordinary differential equations for the n discrete masses and three partial differential equations with the appropriate boundary conditions for the elastic motion. This set of equations is modified to represent a single stage flexible, variable-mass missile with internal flow and aerodynamic forces. The rigid-body motion consists then of three translations and three rotations, whereas the elastic motion is defined by one longitudinal and two flexural displacements, the latter about two orthogonal transverse axes. The differential equations are nonlinear and, in addition, they possess time-dependent coefficients due to the mass variation.

On a class of exact solutions of the equations of motion of a viscous fluid

NASA Technical Reports Server (NTRS)

Yatseyev, V I

1953-01-01

The general solution is obtained of the equations of motion of a viscous fluid in which the velocity field is inversely proportional to the distance from a certain point. Some particular cases of such motion are investigated.

Nonclassical point of view of the Brownian motion generation via fractional deterministic model

NASA Astrophysics Data System (ADS)

Gilardi-Velázquez, H. E.; Campos-Cantón, E.

In this paper, we present a dynamical system based on the Langevin equation without stochastic term and using fractional derivatives that exhibit properties of Brownian motion, i.e. a deterministic model to generate Brownian motion is proposed. The stochastic process is replaced by considering an additional degree of freedom in the second-order Langevin equation. Thus, it is transformed into a system of three first-order linear differential equations, additionally α-fractional derivative are considered which allow us to obtain better statistical properties. Switching surfaces are established as a part of fluctuating acceleration. The final system of three α-order linear differential equations does not contain a stochastic term, so the system generates motion in a deterministic way. Nevertheless, from the time series analysis, we found that the behavior of the system exhibits statistics properties of Brownian motion, such as, a linear growth in time of mean square displacement, a Gaussian distribution. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.

Transient aging in fractional Brownian and Langevin-equation motion.

PubMed

Kursawe, Jochen; Schulz, Johannes; Metzler, Ralf

2013-12-01

Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t=0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on t(a) is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Han Yuecai; Hu Yaozhong; Song Jian, E-mail: jsong2@math.rutgers.edu

2013-04-15

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need tomore » develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.« less

Poincaré-MacMillan Equations of Motion for a Nonlinear Nonholonomic Dynamical System

NASA Astrophysics Data System (ADS)

Amjad, Hussain; Syed Tauseef, Mohyud-Din; Ahmet, Yildirim

2012-03-01

MacMillan's equations are extended to Poincaré's formalism, and MacMillan's equations for nonlinear nonholonomic systems are obtained in terms of Poincaré parameters. The equivalence of the results obtained here with other forms of equations of motion is demonstrated. An illustrative example of the theory is provided as well.

Subprograms for integrating the equations of motion of satellites. FORTRAN 4

NASA Technical Reports Server (NTRS)

Prokhorenko, V. I.

1980-01-01

The subprograms for the formation of the right members of the equations of motion of artificial Earth satellites (AES), integration of systems of differential equations by Adams' method, and the calculation of the values of various functions from the AES parameters of motion are described. These subprograms are written in the FORTRAN 4 language and constitute an essential part of the package of applied programs for the calculation of navigational parameters AES.

Equations of motion for a flexible spacecraft-lumped parameter idealization

NASA Technical Reports Server (NTRS)

Storch, Joel; Gates, Stephen

1982-01-01

The equations of motion for a flexible vehicle capable of arbitrary translational and rotational motions in inertial space accompanied by small elastic deformations are derived in an unabridged form. The vehicle is idealized as consisting of a single rigid body with an ensemble of mass particles interconnected by massless elastic structure. The internal elastic restoring forces are quantified in terms of a stiffness matrix. A transformation and truncation of elastic degrees of freedom is made in the interest of numerical integration efficiency. Deformation dependent terms are partitioned into a hierarchy of significance. The final set of motion equations are brought to a fully assembled first order form suitable for direct digital implementation. A FORTRAN program implementing the equations is given and its salient features described.

Test-particle motion in the nonsymmetric gravitation theory

NASA Astrophysics Data System (ADS)

Moffat, J. W.

1987-06-01

A derivation of the motion of test particles in the nonsymmetric gravitational theory (NGT) is given using the field equations in the presence of matter. The motion of the particle is governed by the Christoffel symbols, which are formed from the symmetric part of the fundamental tensor gμν, as well as by a tensorial piece determined by the skew part of the contracted curvature tensor Rμν. Given the energy-momentum tensor for a perfect fluid and the definition of a test particle in the NGT, the equations of motion follow from the conservation laws. The tensorial piece in the equations of motion describes a new force in nature that acts on the conserved charge in a body. Particles that carry this new charge do not follow geodesic world lines in the NGT, whereas photons do satisfy geodesic equations of motion and the equivalence principle of general relativity. Astronomical predictions, based on the exact static, spherically symmetric solution of the field equations in a vacuum and the test-particle equations of motion, are derived in detail. The maximally extended coordinates that remove the event-horizon singularities in the static, spherically symmetric solution are presented. It is shown how an inward radially falling test particle can be prevented from forming an event horizon for a value greater than a specified critical value of the source charge. If a test particle does fall through an event horizon, then it must continue to fall until it reaches the singularity at r=0.

On the local well-posedness of Lovelock and Horndeski theories

NASA Astrophysics Data System (ADS)

Papallo, Giuseppe; Reall, Harvey S.

2017-08-01

We investigate local well-posedness of the initial value problem for Lovelock and Horndeski theories of gravity. A necessary condition for local well-posedness is strong hyperbolicity of the equations of motion. Even weak hyperbolicity can fail for strong fields so we restrict to weak fields. The Einstein equation is known to be strongly hyperbolic in harmonic gauge so we study Lovelock theories in harmonic gauge. We show that the equation of motion is always weakly hyperbolic for weak fields but, in a generic weak-field background, it is not strongly hyperbolic. For Horndeski theories, we prove that, for weak fields, the equation of motion is always weakly hyperbolic in any generalized harmonic gauge. For some Horndeski theories there exists a generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a weak-field background. This includes "k-essence" like theories. However, for more general Horndeski theories, there is no generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a generic weak-field background. Our results show that the standard method used to establish local well-posedness of the Einstein equation does not extend to Lovelock or general Horndeski theories. This raises the possibility that these theories may not admit a well-posed initial value problem even for weak fields.

The Ffowcs Williams-Hawkings equation - Fifteen years of research

NASA Technical Reports Server (NTRS)

Farassat, F.

1986-01-01

The Ffowcs Williams-Hawkings equation governs the generation of sound in fluids in the presence of solid boundaries in motion. This equation is reviewed for situations where the linearization of the governing equations is allowed. In addition, research on the application of this equation to problems of aeroacoustic is briefly surveyed. Particular attention is given to the formulation of supersonic sources moving in uniform propeller-like motion.

Newton-Euler Dynamic Equations of Motion for a Multi-body Spacecraft

NASA Technical Reports Server (NTRS)

Stoneking, Eric

2007-01-01

The Magnetospheric MultiScale (MMS) mission employs a formation of spinning spacecraft with several flexible appendages and thruster-based control. To understand the complex dynamic interaction of thruster actuation, appendage motion, and spin dynamics, each spacecraft is modeled as a tree of rigid bodies connected by spherical or gimballed joints. The method presented facilitates assembling by inspection the exact, nonlinear dynamic equations of motion for a multibody spacecraft suitable for solution by numerical integration. The building block equations are derived by applying Newton's and Euler's equations of motion to an "element" consisting of two bodies and one joint (spherical and gimballed joints are considered separately). Patterns in the "mass" and L'force" matrices guide assembly by inspection of a general N-body tree-topology system. Straightforward linear algebra operations are employed to eliminate extraneous constraint equations, resulting in a minimum-dimension system of equations to solve. This method thus combines a straightforward, easily-extendable, easily-mechanized formulation with an efficient computer implementation.

The three dimensional motion and stability of a rotating space station: Cable-counterweight configuration

NASA Technical Reports Server (NTRS)

Bainum, P. M.; Evans, K. S.

1974-01-01

The three dimensional equations of motion for a cable connected space station--counterweight system are developed using a Lagrangian formulation. The system model employed allows for cable and end body damping and restoring effects. The equations are then linearized about the equilibrium motion and nondimensionalized. To first degree, the out-of-plane equations uncouple from the inplane equations. Therefore, the characteristic polynomials for the in-plane and out-of-plane equations are developed and treated separately. From the general in-plane characteristic equation, necessary conditions for stability are obtained. The Routh-Hurwitz necessary and sufficient conditions for stability are derived for the general out-of-plane characteristic equation. Special cases of the in-plane and out-of-plane equations (such as identical end masses, and when the cable is attached to the centers of mass of the two end bodies) are then examined for stability criteria.

Using "Tracker" to Prove the Simple Harmonic Motion Equation

ERIC Educational Resources Information Center

Kinchin, John

2016-01-01

Simple harmonic motion (SHM) is a common topic for many students to study. Using the free, though versatile, motion tracking software; "Tracker", we can extend the students experience and show that the general equation for SHM does lead to the correct period of a simple pendulum.

Active motion on curved surfaces

NASA Astrophysics Data System (ADS)

Castro-Villarreal, Pavel; Sevilla, Francisco J.

2018-05-01

A theoretical analysis of active motion on curved surfaces is presented in terms of a generalization of the telegrapher equation. Such a generalized equation is explicitly derived as the polar approximation of the hierarchy of equations obtained from the corresponding Fokker-Planck equation of active particles diffusing on curved surfaces. The general solution to the generalized telegrapher equation is given for a pulse with vanishing current as initial data. Expressions for the probability density and the mean squared geodesic displacement are given in the limit of weak curvature. As an explicit example of the formulated theory, the case of active motion on the sphere is presented, where oscillations observed in the mean squared geodesic displacement are explained.

A canonical form of the equation of motion of linear dynamical systems

NASA Astrophysics Data System (ADS)

Kawano, Daniel T.; Salsa, Rubens Goncalves; Ma, Fai; Morzfeld, Matthias

2018-03-01

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

Singularities of the Euler equation and hydrodynamic stability

NASA Technical Reports Server (NTRS)

Tanveer, S.; Speziale, Charles G.

1993-01-01

Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.

Singularities of the Euler equation and hydrodynamic stability

NASA Technical Reports Server (NTRS)

Tanveer, S.; Speziale, Charles G.

1992-01-01

Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.

SYMBOD - A computer program for the automatic generation of symbolic equations of motion for systems of hinge-connected rigid bodies

NASA Technical Reports Server (NTRS)

Macala, G. A.

1983-01-01

A computer program is described that can automatically generate symbolic equations of motion for systems of hinge-connected rigid bodies with tree topologies. The dynamical formulation underlying the program is outlined, and examples are given to show how a symbolic language is used to code the formulation. The program is applied to generate the equations of motion for a four-body model of the Galileo spacecraft. The resulting equations are shown to be a factor of three faster in execution time than conventional numerical subroutines.

The method of averages applied to the KS differential equations

NASA Technical Reports Server (NTRS)

Graf, O. F., Jr.; Mueller, A. C.; Starke, S. E.

1977-01-01

A new approach for the solution of artificial satellite trajectory problems is proposed. The basic idea is to apply an analytical solution method (the method of averages) to an appropriate formulation of the orbital mechanics equations of motion (the KS-element differential equations). The result is a set of transformed equations of motion that are more amenable to numerical solution.

Gyro-Landau fluid models for toroidal geometry

NASA Astrophysics Data System (ADS)

Waltz, R. E.; Dominguez, R. R.; Hammett, G. W.

1992-10-01

Gyro-Landau fluid model equations provide first-order time advancement for a limited number of moments of the gyrokinetic equation, while approximately preserving the effects of the gyroradius averaging and Landau damping. This paper extends the work of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for electrostatic motion parallel to the magnetic field and E×B motion to include the gyroaveraging linearly and the curvature drift motion. The equations are tested by comparing the ion-temperature-gradient mode linear growth rates for the model equations with those of the exact gyrokinetic theory over a full range of parameters.

Angular motion equations for a satellite with hinged flexible solar panel

NASA Astrophysics Data System (ADS)

Ovchinnikov, M. Yu.; Tkachev, S. S.; Roldugin, D. S.; Nuralieva, A. B.; Mashtakov, Y. V.

2016-11-01

Non-linear mathematical model for the satellite with hinged flexible solar panel is presented. Normal modes of flexible elements are used for motion description. Motion equations are derived using virtual work principle. A comparison of normal modes calculation between finite element method and developed model is presented.

Brownian microhydrodynamics of active filaments.

PubMed

Laskar, Abhrajit; Adhikari, R

2015-12-21

Slender bodies capable of spontaneous motion in the absence of external actuation in an otherwise quiescent fluid are common in biological, physical and technological contexts. The interplay between the spontaneous fluid flow, Brownian motion, and the elasticity of the body presents a challenging fluid-structure interaction problem. Here, we model this problem by approximating the slender body as an elastic filament that can impose non-equilibrium velocities or stresses at the fluid-structure interface. We derive equations of motion for such an active filament by enforcing momentum conservation in the fluid-structure interaction and assuming slow viscous flow in the fluid. The fluid-structure interaction is obtained, to any desired degree of accuracy, through the solution of an integral equation. A simplified form of the equations of motion, which allows for efficient numerical solutions, is obtained by applying the Kirkwood-Riseman superposition approximation to the integral equation. We use this form of equation of motion to study dynamical steady states in free and hinged minimally active filaments. Our model provides the foundation to study collective phenomena in momentum-conserving, Brownian, active filament suspensions.

Autonomous Navigation of a Satellite Cluster

DTIC Science & Technology

1990-12-01

satellite’s velocity are determined by the Clohessy - Wiltshire equations I (these equations will be introduced in the next section) and take the form: (8:80...transition matrix, is based upon the Clohessy - Wiltshire equations of motion. These equations describe "the relative motion of two satellites when one is in a...discovery warranted a re-examination of the solutions to the Clohessy - Wiltshire equations. If the solutions for satellite #1 and #2 are subtracted

Around the Sun in a Graphing Calculator.

ERIC Educational Resources Information Center

Demana, Franklin; Waits, Bert K.

1989-01-01

Discusses the use of graphing calculators for polar and parametric equations. Presents eight lines of the program for the graph of a parametric equation and 11 lines of the program for a graph of a polar equation. Illustrates the application of the programs for planetary motion and free-fall motion. (YP)

Modeling aerodynamic discontinuities and the onset of chaos in flight dynamical systems

NASA Technical Reports Server (NTRS)

Tobak, M.; Chapman, G. T.; Uenal, A.

1986-01-01

Various representations of the aerodynamic contribution to the aircraft's equation of motion are shown to be compatible within the common assumption of their Frechet differentiability. Three forms of invalidating Frechet differentiality are identified, and the mathematical model is amended to accommodate their occurrence. Some of the ways in which chaotic behavior may emerge are discussed, first at the level of the aerodynamic contribution to the equation of motion, and then at the level of the equations of motion themselves.

Modeling aerodynamic discontinuities and onset of chaos in flight dynamical systems

NASA Technical Reports Server (NTRS)

Tobak, M.; Chapman, G. T.; Unal, A.

1987-01-01

Various representations of the aerodynamic contribution to the aircraft's equation of motion are shown to be compatible within the common assumption of their Frechet differentiability. Three forms of invalidating Frechet differentiability are identified, and the mathematical model is amended to accommodate their occurrence. Some of the ways in which chaotic behavior may emerge are discussed, first at the level of the aerodynamic contribution to the equations of motion, and then at the level of the equations of motion themselves.

Space station rotational equations of motion

NASA Technical Reports Server (NTRS)

Rheinfurth, M. H.; Carroll, S. N.

1985-01-01

Dynamic equations of motion are developed which describe the rotational motion for a large space structure having rotating appendages. The presence of the appendages produce torque coupling terms which are dependent on the inertia properties of the appendages and the rotational rates for both the space structure and the appendages. These equations were formulated to incorporate into the Space Station Attitude Control and Stabilization Test Bed to accurately describe the influence rotating solar arrays and thermal radiators have on the dynamic behavior of the Space Station.

Correlation of Experimental and Theoretical Steady-State Spinning Motion for a Current Fighter Airplane Using Rotation-Balance Aerodynamic Data

DTIC Science & Technology

1978-07-01

were input into the computer program. The program was numerically intergrated with time by using a fourth-order Runge-Kutta integration algorithm with...equations of motion are numerically intergrated to provide time histories of the aircraft spinning motion. A.2 EQUATIONS DEFINING THE FORCE AND MOMENT...by Cy or Cn. 50 AE DC-TR-77-126 A . 4 where EQUATIONS FOR TRANSFERRING AERODYNAMIC DATA INPUTS TO THE PROPER HORIZONTAL CENTER OF GRAVITY

Approximate Solution to the Angular Speeds of a Nearly-Symmetric Mass-Varying Cylindrical Body

NASA Astrophysics Data System (ADS)

Nanjangud, Angadh; Eke, Fidelis

2017-06-01

This paper examines the rotational motion of a nearly axisymmetric rocket type system with uniform burn of its propellant. The asymmetry comes from a slight difference in the transverse principal moments of inertia of the system, which then results in a set of nonlinear equations of motion even when no external torque is applied to the system. It is often difficult, or even impossible, to generate analytic solutions for such equations; closed form solutions are even more difficult to obtain. In this paper, a perturbation-based approach is employed to linearize the equations of motion and generate analytic solutions. The solutions for the variables of transverse motion are analytic and a closed-form solution to the spin rate is suggested. The solutions are presented in a compact form that permits rapid computation. The approximate solutions are then applied to the torque-free motion of a typical solid rocket system and the results are found to agree with those obtained from the numerical solution of the full non-linear equations of motion of the mass varying system.

Chaotic Motion of Relativistic Electrons Driven by Whistler Waves

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Telnikhin, A. A.; Kronberg, Tatiana K.

2007-01-01

Canonical equations governing an electron motion in electromagnetic field of the whistler mode waves propagating along the direction of an ambient magnetic field are derived. The physical processes on which the equations of motion are based .are identified. It is shown that relativistic electrons interacting with these fields demonstrate chaotic motion, which is accompanied by the particle stochastic heating and significant pitch angle diffusion. Evolution of distribution functions is described by the Fokker-Planck-Kolmogorov equations. It is shown that the whistler mode waves could provide a viable mechanism for stochastic energization of electrons with energies up to 50 MeV in the Jovian magnetosphere.

Gravitational Agglomeration of Post-HCDA LMFBR Nonspherical Aerosols.

DTIC Science & Technology

1980-12-01

equations for two particle motions are developed . A computer program NGCEFF is constructed., the Navier-Stokes equation is solved by the finite difference...dynamic equations for two particle motions are developed . A computer program NGCEFF I is constructed, the Navier-Stokes equation is solved by the...spatial inhomogeneities for the aerosol. Thus, following an HCDA, an aerosol mixture of sodium compounds, fuel and core structural materials will

Three dimensional dynamics of a flexible Motorised Momentum Exchange Tether

NASA Astrophysics Data System (ADS)

Ismail, N. A.; Cartmell, M. P.

2016-03-01

This paper presents a new flexural model for the three dimensional dynamics of the Motorised Momentum Exchange Tether (MMET) concept. This study has uncovered the relationships between planar and nonplanar motions, and the effect of the coupling between these two parameters on pragmatic circular and elliptical orbits. The tether sub-spans are modelled as stiffened strings governed by partial differential equations of motion, with specific boundary conditions. The tether sub-spans are flexible and elastic, thereby allowing three dimensional displacements. The boundary conditions lead to a specific frequency equation and the eigenvalues from this provide the natural frequencies of the orbiting flexible motorised tether when static, accelerating in monotonic spin, and at terminal angular velocity. A rotation transformation matrix has been utilised to get the position vectors of the system's components in an assumed inertial frame. Spatio-temporal coordinates are transformed to modal coordinates before applying Lagrange's equations, and pre-selected linear modes are included to generate the equations of motion. The equations of motion contain inertial nonlinearities which are essentially of cubic order, and these show the potential for intricate intermodal coupling effects. A simulation of planar and non-planar motions has been undertaken and the differences in the modal responses, for both motions, and between the rigid body and flexible models are highlighted and discussed.

General Equations of Motion for a Damaged Asymmetric Aircraft

NASA Technical Reports Server (NTRS)

Bacon, Barton J.; Gregory, Irene M.

2007-01-01

There is a renewed interest in dynamic characteristics of damaged aircraft both in order to assess survivability and to develop control laws to enhance survivability. This paper presents a set of flight dynamics equations of motion for a rigid body not necessarily referenced to the body's center of mass. Such equations can be used when the body loses a portion of its mass and it is desired to track the motion of the body s previous center of mass/reference frame now that the mass center has moved to a new position. Furthermore, results for equations presented in this paper and equations in standard aircraft simulations are compared for a scenario involving a generic transport aircraft configuration subject to wing damage.

The centripetal force law and the equation of motion for a particle on a curved hypersurface

NASA Astrophysics Data System (ADS)

Hu, L. D.; Lian, D. K.; Liu, Q. H.

2016-12-01

It is pointed out that the current form of the extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version; for it is established without regard to the fact that the particle can never depart from the geodesics on the surface. Once this fact is taken into consideration, the equation takes the same form as that for the centripetal force law, provided that the symbols are re-interpreted so that the law is applicable for higher dimensions. The controversial issue of constructing operator forms of these equations is addressed, and our studies show the quantization of constrained system based on the extrinsic equation of motion is preferable.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field

NASA Astrophysics Data System (ADS)

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-01

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field.

PubMed

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-28

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

Equations of motion of slung load systems with results for dual lift

NASA Technical Reports Server (NTRS)

Cicolani, Luigi S.; Kanning, Gerd

1990-01-01

General simulation equations are derived for the rigid body motion of slung load systems. These systems are viewed as consisting of several rigid bodies connected by straight-line cables or links. The suspension can be assumed to be elastic or inelastic, both cases being of interest in simulation and control studies. Equations for the general system are obtained via D'Alembert's principle and the introduction of generalized velocity coordinates. Three forms are obtained. Two of these generalize previous case-specific results for single helicopter systems with elastic or inelastic suspensions. The third is a new formulation for inelastic suspensions. It is derived from the elastic suspension equations by choosing the generalized coordinates so as to separate motion due to cable stretching from motion with invariant cable lengths. The result is computationally more efficient than the conventional formulation, and is readily integrated with the elastic suspension formulation and readily applied to the complex dual lift and multilift systems. Equations are derived for dual lift systems. Three proposed suspension arrangements can be integrated in a single equation set. The equations are given in terms of the natural vectors and matrices of three-dimensional rigid body mechanics and are tractable for both analysis and programming.

Construction of Lagrangians and Hamiltonians from the Equation of Motion

ERIC Educational Resources Information Center

Yan, C. C.

1978-01-01

Demonstrates that infinitely many Lagrangians and Hamiltonians can be constructed from a given equation of motion. Points out the lack of an established criterion for making a proper selection. (Author/GA)

General relativity exactly described in terms of Newton's laws within curved geometries

NASA Astrophysics Data System (ADS)

Savickas, D.

2014-07-01

Many years ago Milne and McCrea showed in their well-known paper that the Hubble expansion occurring in general relativity could be exactly described by the use of Newtonian mechanics. It will be shown that a similar method can be extended to, and used within, curved geometries when Newton's second law is expressed within a four-dimensional curved spacetime. The second law will be shown to yield an equation that is exactly identical to the geodesic equation of motion of general relativity. This in itself yields no new information concerning relativity since the equation is mathematically identical to the relativistic equation. However, when the time in the second law is defined to have a constant direction as effectively occurs in Newtonian mechanics, and no longer acts as a fourth dimension as exists in relativity theory, it separates into a vector equation in a curved three-dimensional space and an additional second scalar equation that describes conservation of energy. It is shown that the curved Newtonian equations of motion define the metric coefficients which occur in the Schwarzschild solution and that they also define its equations of motion. Also, because the curved Newtonian equations developed here use masses as gravitational sources, as occurs in Newtonian mechanics, they make it possible to derive the solution for other kinds of mass distributions and are used here to find the metric equation for a thin mass-rod and the equation of motion for a mass particle orbiting it in its relativistic gravitational field.

Prediction and control of slender-wing rock

NASA Technical Reports Server (NTRS)

Kandil, Osama A.; Salman, Ahmed A.

1992-01-01

The unsteady Euler equations and the Euler equations of rigid-body dynamics, both written in the moving frame of reference, are sequentially solved to simulate the limit-cycle rock motion of slender delta wings. The governing equations of the fluid flow and the dynamics of the present multidisciplinary problem are solved using an implicit, approximately-factored, central-difference-like, finite-volume scheme and a four-stage Runge-Kutta scheme, respectively. For the control of wing-rock motion, leading-edge flaps are forced to oscillate anti-symmetrically at prescribed frequency and amplitude, which are tuned in order to suppress the rock motion. Since the computational grid deforms due to the leading-edge flaps motion, the grid is dynamically deformed using the Navier-displacement equations. Computational applications cover locally-conical and three-dimensional solutions for the wing-rock simulation and its control.

XTRAN2L - A PROGRAM FOR SOLVING THE GENERAL-FREQUENCY UNSTEADY TWO-DIMENSIONAL TRANSONIC SMALL-DISTURBANCE EQUATIONS

NASA Technical Reports Server (NTRS)

Seidel, D. A.

1994-01-01

The Program for Solving the General-Frequency Unsteady Two-Dimensional Transonic Small-Disturbance Equation, XTRAN2L, is used to calculate time-accurate, finite-difference solutions of the nonlinear, small-disturbance potential equation for two- dimensional transonic flow about airfoils. The code can treat forced harmonic, pulse, or aeroelastic transient type motions. XTRAN2L uses a transonic small-disturbance equation that incorporates a time accurate finite-difference scheme. Airfoil flow tangency boundary conditions are defined to include airfoil contour, chord deformation, nondimensional plunge displacement, pitch, and trailing edge control surface deflection. Forced harmonic motion can be based on: 1) coefficients of harmonics based on information from each quarter period of the last cycle of harmonic motion; or 2) Fourier analyses of the last cycle of motion. Pulse motion (an alternate to forced harmonic motion) in which the airfoil is given a small prescribed pulse in a given mode of motion, and the aerodynamic transients are calculated. An aeroelastic transient capability is available within XTRAN2L, wherein the structural equations of motion are coupled with the aerodynamic solution procedure for simultaneous time-integration. The wake is represented as a slit downstream of the airfoil trailing edge. XTRAN2L includes nonreflecting farfield boundary conditions. XTRAN2L was developed on a CDC CYBER mainframe running under NOS 2.4. It is written in FORTRAN 5 and uses overlays to minimize storage requirements. The program requires 120K of memory in overlayed form. XTRAN2L was developed in 1987.

On a Simple Formulation of the Golf Ball Paradox

ERIC Educational Resources Information Center

Pujol, O.; Perez, J. Ph.

2007-01-01

The motion of a ball rolling without slipping on the lateral section inside a fixed vertical cylinder is analysed in the Earth referential frame which is assumed to be Galilean. Equations of motion are rapidly obtained and the golf ball paradox is understood: these equations describe a motion consisting of a vertical harmonic oscillation related…

Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries.

PubMed

Jeon, Jae-Hyung; Metzler, Ralf

2010-02-01

Motivated by subdiffusive motion of biomolecules observed in living cells, we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time-averaged mean-squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular, we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single-particle trajectory data.

A computational procedure for large rotational motions in multibody dynamics

NASA Technical Reports Server (NTRS)

Park, K. C.; Chiou, J. C.

1987-01-01

A computational procedure suitable for the solution of equations of motion for multibody systems is presented. The present procedure adopts a differential partitioning of the translational motions and the rotational motions. The translational equations of motion are then treated by either a conventional explicit or an implicit direct integration method. A principle feature of this procedure is a nonlinearly implicit algorithm for updating rotations via the Euler four-parameter representation. This procedure is applied to the rolling of a sphere through a specific trajectory, which shows that it yields robust solutions.

Geometry of Lax pairs: Particle motion and Killing-Yano tensors

NASA Astrophysics Data System (ADS)

Cariglia, Marco; Frolov, Valeri P.; Krtouš, Pavel; Kubizňák, David

2013-01-01

A geometric formulation of the Lax pair equation on a curved manifold is studied using the phase-space formalism. The corresponding (covariantly conserved) Lax tensor is defined and the method of generation of constants of motion from it is discussed. It is shown that when the Hamilton equations of motion are used, the conservation of the Lax tensor translates directly to the well-known Lax pair equation, with one matrix identified with components of the Lax tensor and the other matrix constructed from the (metric) connection. A generalization to Clifford objects is also discussed. Nontrivial examples of Lax tensors for geodesic and charged particle motion are found in spacetimes admitting a hidden symmetry of Killing-Yano tensors.

Aeroelastic Analysis of a Trimmed Generic Hypersonic Vehicle

NASA Technical Reports Server (NTRS)

Nydick, I.; Friedmann, P. P.

1999-01-01

The aeroelastic equations of motion governing a hypersonic vehicle in free flight are derived. The equations of motion for a translating and rotating flexible body using Lagrange's equations in terms of quasi-coordinates are presented. These equations are simplified for the case of a vehicle with pitch and plunge rigid body degrees of freedom and small elastic displacements. The displacements are approximated by a truncated series of the unrestrained mode shapes, which are obtained using equivalent plate theory. Subsequently, the nonlinear equations of motion are linearized about the trim state, which is obtained using a rigid body trim model and steady hypersonic aerodynamics. The appropriate flutter derivatives are calculated from piston theory. Results describing mode shapes, trim behavior, and aeroelastic stability of a generic hypersonic vehicle are presented.

Computer program for the load and trajectory analysis of two DOF bodies connected by an elastic tether: Users manual

NASA Technical Reports Server (NTRS)

Doyle, G. R., Jr.; Burbick, J. W.

1973-01-01

The derivation of the differential equations of motion of a 3 Degrees of Freedom body joined to a 3 Degrees of Freedom body by an elastic tether. The tether is represented by a spring and dashpot in parallel. A computer program which integrates the equations of motion is also described. Although the derivation of the equations of motions are for a general system, the computer program is written for defining loads in large boosters recovered by parachutes.

Relating constrained motion to force through Newton's second law

NASA Astrophysics Data System (ADS)

Roithmayr, Carlos M.

When a mechanical system is subject to constraints its motion is in some way restricted. In accordance with Newton's second law, motion is a direct result of forces acting on a system; hence, constraint is inextricably linked to force. The presence of a constraint implies the application of particular forces needed to compel motion in accordance with the constraint; absence of a constraint implies the absence of such forces. The objective of this thesis is to formulate a comprehensive, consistent, and concise method for identifying a set of forces needed to constrain the behavior of a mechanical system modeled as a set of particles and rigid bodies. The goal is accomplished in large part by expressing constraint equations in vector form rather than entirely in terms of scalars. The method developed here can be applied whenever constraints can be described at the acceleration level by a set of independent equations that are linear in acceleration. Hence, the range of applicability extends to servo-constraints or program constraints described at the velocity level with relationships that are nonlinear in velocity. All configuration constraints, and an important class of classical motion constraints, can be expressed at the velocity level by using equations that are linear in velocity; therefore, the associated constraint equations are linear in acceleration when written at the acceleration level. Two new approaches are presented for deriving equations governing motion of a system subject to constraints expressed at the velocity level with equations that are nonlinear in velocity. By using partial accelerations instead of the partial velocities normally employed with Kane's method, it is possible to form dynamical equations that either do or do not contain evidence of the constraint forces, depending on the analyst's interests.

Numerical solution of equations governing longitudinal suspension line wave motion during the parachute unfurling process. Ph.D. Thesis - George Washington Univ., Washington, D. C.

NASA Technical Reports Server (NTRS)

Poole, L. R.

1973-01-01

Equations are presented which govern the dynamics of the lines-first parachute unfurling process, including wave motion in the parachute suspension lines. Techniques are developed for obtaining numerical solutions to the governing equations. Histories of tension at test data, and generally good agreement is observed. Errors in computed results are attributed to several areas of uncertainty, the most significant being a poorly defined boundary condition on the wave motion at the vehicle-suspension line boundary.

Dynamic characteristics of a hydrostatic gas bearing driven by oscillating exhaust pressure

NASA Technical Reports Server (NTRS)

Watkins, C. B.; Eronini, I. E.; Branch, H. D.

1984-01-01

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference aproximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

Thermohydrodynamic Analysis of Cryogenic Liquid Turbulent Flow Fluid Film Bearings

NASA Technical Reports Server (NTRS)

SanAndres, Luis

1996-01-01

Computational programs developed for the thermal analysis of tilting and flexure-pad hybrid bearings, and the unsteady flow and transient response of a point mass rotor supported on fluid film bearings are described. The motion of a cryogenic liquid on the thin film annular region of a fluid film bearing is described by a set of mass and momentum conservation, and energy transport equations for the turbulent bulk-flow velocities and pressure, and accompanied by thermophysical state equations for evaluation of the fluid material properties. Zeroth-order equations describe the fluid flow field for a journal static equilibrium position, while first-order (linear) equations govern the fluid flow for small amplitude-journal center translational motions. Solution to the zeroth-order flow field equations provides the bearing flow rate, load capacity, drag torque and temperature rise. Solution to the first-order equations determines the rotordynamic force coefficients due to journal radial motions.

Variational theorems for superimposed motions in elasticity, with application to beams

NASA Technical Reports Server (NTRS)

Doekmeci, M. C.

1976-01-01

Variational theorems are presented for a theory of small motions superimposed on large static deformations and governing equations for prestressed beams on the basis of 3-D theory of elastodynamics. First, the principle of virtual work is modified through Friedrichs's transformation so as to describe the initial stress problem of elastodynamics. Next, the modified principle together with a chosen displacement field is used to derive a set of 1-D macroscopic governing equations of prestressed beams. The resulting equations describe all the types of superimposed motions in elastic beams, and they include all the effects of transverse shear and normal strains, and the rotatory inertia. The instability of the governing equations is discussed briefly.

One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

NASA Astrophysics Data System (ADS)

Sakaguchi, Hidetsugu; Ishibashi, Kazuya

2018-06-01

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable ϕ. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.

Minimum-variance Brownian motion control of an optically trapped probe.

PubMed

Huang, Yanan; Zhang, Zhipeng; Menq, Chia-Hsiang

2009-10-20

This paper presents a theoretical and experimental investigation of the Brownian motion control of an optically trapped probe. The Langevin equation is employed to describe the motion of the probe experiencing random thermal force and optical trapping force. Since active feedback control is applied to suppress the probe's Brownian motion, actuator dynamics and measurement delay are included in the equation. The equation of motion is simplified to a first-order linear differential equation and transformed to a discrete model for the purpose of controller design and data analysis. The derived model is experimentally verified by comparing the model prediction to the measured response of a 1.87 microm trapped probe subject to proportional control. It is then employed to design the optimal controller that minimizes the variance of the probe's Brownian motion. Theoretical analysis is derived to evaluate the control performance of a specific optical trap. Both experiment and simulation are used to validate the design as well as theoretical analysis, and to illustrate the performance envelope of the active control. Moreover, adaptive minimum variance control is implemented to maintain the optimal performance in the case in which the system is time varying when operating the actively controlled optical trap in a complex environment.

Equations of motion for train derailment dynamics

DOT National Transportation Integrated Search

2007-09-11

This paper describes a planar or two-dimensional model to : examine the gross motions of rail cars in a generalized train : derailment. Three coupled, second-order differential equations : are derived from Newton's Laws to calculate rigid-body car : ...

A new fundamental model of moving particle for reinterpreting Schroedinger equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Umar, Muhamad Darwis

2012-06-20

The study of Schroedinger equation based on a hypothesis that every particle must move randomly in a quantum-sized volume has been done. In addition to random motion, every particle can do relative motion through the movement of its quantum-sized volume. On the other way these motions can coincide. In this proposed model, the random motion is one kind of intrinsic properties of the particle. The every change of both speed of randomly intrinsic motion and or the velocity of translational motion of a quantum-sized volume will represent a transition between two states, and the change of speed of randomly intrinsicmore » motion will generate diffusion process or Brownian motion perspectives. Diffusion process can take place in backward and forward processes and will represent a dissipative system. To derive Schroedinger equation from our hypothesis we use time operator introduced by Nelson. From a fundamental analysis, we find out that, naturally, we should view the means of Newton's Law F(vector sign) = ma(vector sign) as no an external force, but it is just to describe both the presence of intrinsic random motion and the change of the particle energy.« less

Steering particles by breaking symmetries

NASA Astrophysics Data System (ADS)

Bet, Bram; Samin, Sela; Georgiev, Rumen; Burak Eral, Huseyin; van Roij, René

2018-06-01

We derive general equations of motions for highly-confined particles that perform quasi-two-dimensional motion in Hele-Shaw channels, which we solve analytically, aiming to derive design principles for self-steering particles. Based on symmetry properties of a particle, its equations of motion can be simplified, where we retrieve an earlier-known equation of motion for the orientation of dimer particles consisting of disks (Uspal et al 2013 Nat. Commun. 4), but now in full generality. Subsequently, these solutions are compared with particle trajectories that are obtained numerically. For mirror-symmetric particles, excellent agreement between the analytical and numerical solutions is found. For particles lacking mirror symmetry, the analytic solutions provide means to classify the motion based on particle geometry, while we find that taking the side-wall interactions into account is important to accurately describe the trajectories.

Holonomicity analysis of electromechanical systems

NASA Astrophysics Data System (ADS)

Wcislik, Miroslaw; Suchenia, Karol

2017-12-01

Electromechanical systems are described using state variables that contain electrical and mechanical components. The equations of motion, both electrical and mechanical, describe the relationships between these components. These equations are obtained using Lagrange functions. On the basis of the function and Lagrange - d'Alembert equation the methodology of obtaining equations for electromechanical systems was presented, together with a discussion of the nonholonomicity of these systems. The electromechanical system in the form of a single-phase reluctance motor was used to verify the presented method. Mechanical system was built as a system, which can oscillate as the element of physical pendulum. On the base of the pendulum oscillation, parameters of the electromechanical system were defined. The identification of the motor electric parameters as a function of the rotation angle was carried out. In this paper the characteristics and motion equations parameters of the motor are presented. The parameters of the motion equations obtained from the experiment and from the second order Lagrange equations are compared.

Influence of tides in viscoelastic bodies of planet and satellite on the satellite's orbital motion

NASA Astrophysics Data System (ADS)

Emelyanov, N. V.

2018-06-01

The problem of influence of tidal friction in both planetary and satellite bodies upon satellite's orbital motion is considered. Using the differential equations in satellite's rectangular planetocentric coordinates, the differential equations describing the changes in semimajor axis and eccentricity are derived. The equations in rectangular coordinates were taken from earlier works on the problem. The calcultations carried out for a number of test examples prove that the averaged solutions of equations in coordinates and precise solutions of averaged equations in the Keplerian elements are identical. For the problem of tides raised on planet's body, it was found that, if satellite's mean motion n is equal to 11/18 Ω, where Ω is the planet's angular rotation rate, the orbital eccentricity does not change. This conclusion is in agreement with the results of other authors. It was also found that there is essential discrepancy between the equations in the elements obtained in this paper and analogous equations published by earlier researchers.

Hoph Bifurcation in Viscous, Low Speed Flows About an Airfoil with Structural Coupling

DTIC Science & Technology

1993-03-01

8 2.1 Equations of Motion ...... ..................... 8 2.2 Coordinate Transformation ....................... 13 2.3 Aerodynamic...a-frame) f - Apparent body forces applied in noninertial system fL - Explicit fourth-order numerical damping term Ai - Implicit fourth-order...resulting airfoil motion . The equations describing the airfoil motion are integrated in time using a fourth-order Runge-Kutta algorithm. The

Computer program for investigating effects of nonlinear suspension-system elastic properties on parachute inflation loads and motions

NASA Technical Reports Server (NTRS)

Poole, L. R.

1972-01-01

A computer program is presented by which the effects of nonlinear suspension-system elastic characteristics on parachute inflation loads and motions can be investigated. A mathematical elastic model of suspension-system geometry is coupled to the planar equations of motion of a general vehicle and canopy. Canopy geometry and aerodynamic drag characteristics and suspension-system elastic properties are tabular inputs. The equations of motion are numerically integrated by use of an equivalent fifth-order Runge-Kutta technique.

Empirical improvements for estimating earthquake response spectra with random‐vibration theory

USGS Publications Warehouse

Boore, David; Thompson, Eric M.

2012-01-01

The stochastic method of ground‐motion simulation is often used in combination with the random‐vibration theory to directly compute ground‐motion intensity measures, thereby bypassing the more computationally intensive time‐domain simulations. Key to the application of random‐vibration theory to simulate response spectra is determining the duration (Drms) used in computing the root‐mean‐square oscillator response. Boore and Joyner (1984) originally proposed an equation for Drms , which was improved upon by Liu and Pezeshk (1999). Though these equations are both substantial improvements over using the duration of the ground‐motion excitation for Drms , we document systematic differences between the ground‐motion intensity measures derived from the random‐vibration and time‐domain methods for both of these Drms equations. These differences are generally less than 10% for most magnitudes, distances, and periods of engineering interest. Given the systematic nature of the differences, however, we feel that improved equations are warranted. We empirically derive new equations from time‐domain simulations for eastern and western North America seismological models. The new equations improve the random‐vibration simulations over a wide range of magnitudes, distances, and oscillator periods.

Determination of constant-volume balloon capabilities for aeronautical research. [specifically measurement of atmospheric phenomena

NASA Technical Reports Server (NTRS)

Tatom, F. B.; King, R. L.

1977-01-01

The proper application of constant-volume balloons (CVB) for measurement of atmospheric phenomena was determined. And with the proper interpretation of the resulting data. A literature survey covering 176 references is included. the governing equations describing the three-dimensional motion of a CVB immersed in a flow field are developed. The flowfield model is periodic, three-dimensional, and nonhomogeneous, with mean translational motion. The balloon motion and flow field equations are cast into dimensionless form for greater generality, and certain significant dimensionless groups are identified. An alternate treatment of the balloon motion, based on first-order perturbation analysis, is also presented. A description of the digital computer program, BALLOON, used for numerically integrating the governing equations is provided.

On order and chaos in the mergers of galaxies

NASA Astrophysics Data System (ADS)

Vandervoort, Peter O.

2018-03-01

This paper describes a low-dimensional model of the merger of two galaxies. The governing equations are the complete sets of moment equations of the first and second orders derived from the collisionless Boltzmann equations representing the galaxies. The moment equations reduce to an equation governing the relative motion of the galaxies, tensor virial equations, and equations governing the kinetic energy tensors. We represent the galaxies as heterogeneous ellipsoids with Gaussian stratifications of their densities, and we represent the mean stellar motions in terms of velocity fields that sustain those densities consistently with the equation of continuity. We reduce and solve the governing equations for a head-on encounter of a dwarf galaxy with a giant galaxy. That reduction includes the effect of dynamical friction on the relative motion of the galaxies. Our criterion for chaotic behaviour is sensitivity of the motion to small changes in the initial conditions. In a survey of encounters and mergers of a dwarf galaxy with a giant galaxy, chaotic behaviour arises mainly in non-linear oscillations of the dwarf galaxy. The encounter disrupts the dwarf, excites chaotic oscillations of the dwarf, or excites regular oscillations. Dynamical friction can drive a merger to completion within a Hubble time only if the dwarf is sufficiently massive. The survey of encounters and mergers is the basis for a simple model of the evolution of a `Local Group' consisting of a giant galaxy and a population of dwarf galaxies bound to the giant as satellites on radial orbits.

Rigorous derivation of electromagnetic self-force

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gralla, Samuel E.; Harte, Abraham I.; Wald, Robert M.

2009-07-15

During the past century, there has been considerable discussion and analysis of the motion of a point charge in an external electromagnetic field in special relativity, taking into account 'self-force' effects due to the particle's own electromagnetic field. We analyze the issue of 'particle motion' in classical electromagnetism in a rigorous and systematic way by considering a one-parameter family of solutions to the coupled Maxwell and matter equations corresponding to having a body whose charge-current density J{sup a}({lambda}) and stress-energy tensor T{sub ab}({lambda}) scale to zero size in an asymptotically self-similar manner about a worldline {gamma} as {lambda}{yields}0. In thismore » limit, the charge, q, and total mass, m, of the body go to zero, and q/m goes to a well-defined limit. The Maxwell field F{sub ab}({lambda}) is assumed to be the retarded solution associated with J{sup a}({lambda}) plus a homogeneous solution (the 'external field') that varies smoothly with {lambda}. We prove that the worldline {gamma} must be a solution to the Lorentz force equations of motion in the external field F{sub ab}({lambda}=0). We then obtain self-force, dipole forces, and spin force as first-order perturbative corrections to the center-of-mass motion of the body. We believe that this is the first rigorous derivation of the complete first-order correction to Lorentz force motion. We also address the issue of obtaining a self-consistent perturbative equation of motion associated with our perturbative result, and argue that the self-force equations of motion that have previously been written down in conjunction with the 'reduction of order' procedure should provide accurate equations of motion for a sufficiently small charged body with negligible dipole moments and spin. (There is no corresponding justification for the non-reduced-order equations.) We restrict consideration in this paper to classical electrodynamics in flat spacetime, but there should be no difficulty in extending our results to the motion of a charged body in an arbitrary globally hyperbolic curved spacetime.« less

Quaternion regularization in celestial mechanics, astrodynamics, and trajectory motion control. III

NASA Astrophysics Data System (ADS)

Chelnokov, Yu. N.

2015-09-01

The present paper1 analyzes the basic problems arising in the solution of problems of the optimum control of spacecraft (SC) trajectory motion (including the Lyapunov instability of solutions of conjugate equations) using the principle of the maximum. The use of quaternion models of astrodynamics is shown to allow: (1) the elimination of singular points in the differential phase and conjugate equations and in their partial analytical solutions; (2) construction of the first integrals of the new quaternion; (3) a considerable decrease of the dimensions of systems of differential equations of boundary value optimization problems with their simultaneous simplification by using the new quaternion variables related with quaternion constants of motion by rotation transformations; (4) construction of general solutions of differential equations for phase and conjugate variables on the sections of SC passive motion in the simplest and most convenient form, which is important for the solution of optimum pulse SC transfers; (5) the extension of the possibilities of the analytical investigation of differential equations of boundary value problems with the purpose of identifying the basic laws of optimum control and motion of SC; (6) improvement of the computational stability of the solution of boundary value problems; (7) a decrease in the required volume of computation.

Equations of motion of slung-load systems, including multilift systems

NASA Technical Reports Server (NTRS)

Cicolani, Luigi S.; Kanning, Gerd

1992-01-01

General simulation equations are derived for the rigid body motion of slung-load systems. This work is motivated by an interest in trajectory control for slung loads carried by two or more helicopters. An approximation of these systems consists of several rigid bodies connected by straight-line cables or links. The suspension can be assumed elastic or inelastic. Equations for the general system are obtained from the Newton-Euler rigid-body equations with the introduction of generalized velocity coordinates. Three forms are obtained: two generalize previous case-specific results for single-helicopter systems with elastic and inelastic suspensions, respectively; and the third is a new formulation for inelastic suspensions. The latter is derived from the elastic suspension equations by choosing the generalized coordinates so that motion induced by cable stretching is separated from motion with invariant cable lengths, and by then nulling the stretching coordinates to get a relation for the suspension forces. The result is computationally more efficient than the conventional formulation, is readily integrated with the elastic suspension formulation, and is easily applied to the complex dual-lift and multilift systems. Results are given for two-helicopter systems; three configurations are included and these can be integrated in a single simulation. Equations are also given for some single-helicopter systems, for comparison with the previous literature, and for a multilift system. Equations for degenerate-body approximations (point masses, rigid rods) are also formulated and results are given for dual-lift and multilift systems. Finally, linearlized equations of motion are given for general slung-load systems are presented along with results for the two-helicopter system with a spreader bar.

Mathematics of thermal diffusion in an exponential temperature field

NASA Astrophysics Data System (ADS)

Zhang, Yaqi; Bai, Wenyu; Diebold, Gerald J.

2018-04-01

The Ludwig-Soret effect, also known as thermal diffusion, refers to the separation of gas, liquid, or solid mixtures in a temperature gradient. The motion of the components of the mixture is governed by a nonlinear, partial differential equation for the density fractions. Here solutions to the nonlinear differential equation for a binary mixture are discussed for an externally imposed, exponential temperature field. The equation of motion for the separation without the effects of mass diffusion is reduced to a Hamiltonian pair from which spatial distributions of the components of the mixture are found. Analytical calculations with boundary effects included show shock formation. The results of numerical calculations of the equation of motion that include both thermal and mass diffusion are given.

The rotational motion of an earth orbiting gyroscope according to the Einstein theory of general relativity

NASA Technical Reports Server (NTRS)

Hoots, F. R.; Fitzpatrick, P. M.

1979-01-01

The classical Poisson equations of rotational motion are used to study the attitude motions of an earth orbiting, rapidly spinning gyroscope perturbed by the effects of general relativity (Einstein theory). The center of mass of the gyroscope is assumed to move about a rotating oblate earth in an evolving elliptic orbit which includes all first-order oblateness effects produced by the earth. A method of averaging is used to obtain a transformation of variables, for the nonresonance case, which significantly simplifies the Poisson differential equations of motion of the gyroscope. Long-term solutions are obtained by an exact analytical integration of the simplified transformed equations. These solutions may be used to predict both the orientation of the gyroscope and the motion of its rotational angular momentum vector as viewed from its center of mass. The results are valid for all eccentricities and all inclinations not near the critical inclination.

Fractional calculus in hydrologic modeling: A numerical perspective

PubMed Central

Benson, David A.; Meerschaert, Mark M.; Revielle, Jordan

2013-01-01

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus. PMID:23524449

Complex double-mass dynamic model of rotor on thrust foil gas dynamic bearings

NASA Astrophysics Data System (ADS)

Sytin, A.; Babin, A.; Vasin, S.

2017-08-01

The present paper considers simulation of a rotor’s dynamics behaviour on thrust foil gas dynamic bearings based on simultaneous solution of gas dynamics differential equations, equations of theory of elasticity, motion equations and some additional equations. A double-mass dynamic system was considered during the rotor’s motion simulation which allows not only evaluation of rotor’s dynamic behaviour, but also to evaluate the influence of operational and load parameters on the dynamics of the rotor-bearing system.

Conformal field theory as microscopic dynamics of incompressible Euler and Navier-Stokes equations.

PubMed

Fouxon, Itzhak; Oz, Yaron

2008-12-31

We consider the hydrodynamics of relativistic conformal field theories at finite temperature. We show that the limit of slow motions of the ideal hydrodynamics leads to the nonrelativistic incompressible Euler equation. For viscous hydrodynamics we show that the limit of slow motions leads to the nonrelativistic incompressible Navier-Stokes equation. We explain the physical reasons for the reduction and discuss the implications. We propose that conformal field theories provide a fundamental microscopic viewpoint of the equations and the dynamics governed by them.

Satellite Formation Control Using Atmospheric Drag

DTIC Science & Technology

2007-03-01

of the formation. The linearized Clohessy - Wiltshire equations of motion are used to describe the motion of the two-satellite formation about an empty...control methods were applied to both the linear and nonlinear forms of the Clohessy - Wiltshire equations, and the performance of each control method was...r0δθ̈ = −2nδṙ + fθ (2.16) δz̈ = −n2δz + fz (2.17) These three equations are commonly known as Hill’s equations or the Clohessy - Wiltshire (CW

Modelling the Projectile Motion of a Cricket Ball.

ERIC Educational Resources Information Center

Coutis, Peter

1998-01-01

Presents the equations of motion governing the trajectory of a cricket ball subject to a linear drag force. Uses a perturbation expansion technique to solve the resulting trajectory equation for the range of a cricket ball struck into the outfield. (Author/ASK)

Computerized series solution of relativistic equations of motion.

NASA Technical Reports Server (NTRS)

Broucke, R.

1971-01-01

A method of solution of the equations of planetary motion is described. It consists of the use of numerical general perturbations in orbital elements and in rectangular coordinates. The solution is expanded in Fourier series in the mean anomaly with the aid of harmonic analysis and computerized series manipulation techniques. A detailed application to the relativistic motion of the planet Mercury is described both for Schwarzschild and isotropic coordinates.

Analytic theory of orbit contraction

NASA Technical Reports Server (NTRS)

Vinh, N. X.; Longuski, J. M.; Busemann, A.; Culp, R. D.

1977-01-01

The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by gravitational and aerodynamic forces. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. A rigorous analytic solution is used to integrate this equation with a high degree of accuracy, using Poincare's method of small parameters and Lagrange's expansion to explicitly express the major axis as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. Numerical solutions were generated to display the accuracy of the analytic theory.

Analysis of Three-Dimensional, Nonlinear Development of Wave-Like Structure in a Compressible Round Jet

NASA Technical Reports Server (NTRS)

Dahl, Milo D.; Mankbadi, Reda R.

2002-01-01

An analysis of the nonlinear development of the large-scale structures or instability waves in compressible round jets was conducted using the integral energy method. The equations of motion were decomposed into two sets of equations; one set governing the mean flow motion and the other set governing the large-scale structure motion. The equations in each set were then combined to derive kinetic energy equations that were integrated in the radial direction across the jet after the boundary-layer approximations were applied. Following the application of further assumptions regarding the radial shape of the mean flow and the large structures, equations were derived that govern the nonlinear, streamwise development of the large structures. Using numerically generated mean flows, calculations show the energy exchanges and the effects of the initial amplitude on the coherent structure development in the jet.

Modeling of aircraft unsteady aerodynamic characteristics. Part 1: Postulated models

NASA Technical Reports Server (NTRS)

Klein, Vladislav; Noderer, Keith D.

1994-01-01

A short theoretical study of aircraft aerodynamic model equations with unsteady effects is presented. The aerodynamic forces and moments are expressed in terms of indicial functions or internal state variables. The first representation leads to aircraft integro-differential equations of motion; the second preserves the state-space form of the model equations. The formulations of unsteady aerodynamics is applied in two examples. The first example deals with a one-degree-of-freedom harmonic motion about one of the aircraft body axes. In the second example, the equations for longitudinal short-period motion are developed. In these examples, only linear aerodynamic terms are considered. The indicial functions are postulated as simple exponentials and the internal state variables are governed by linear, time-invariant, first-order differential equations. It is shown that both approaches to the modeling of unsteady aerodynamics lead to identical models.

General Relativity Exactly Described by Use of Newton's Laws within a Curved Geometry

NASA Astrophysics Data System (ADS)

Savickas, David

2014-03-01

The connection between general relativity and Newtonian mechanics is shown to be much closer than generally recognized. When Newton's second law is written in a curved geometry by using the physical components of a vector as defined in tensor calculus, and by replacing distance within the momentum's velocity by the vector metric ds in a curved geometry, the second law can then be easily shown to be exactly identical to the geodesic equation of motion occurring in general relativity. By using a time whose vector direction is constant, as similarly occurs in Newtonian mechanics, this equation can be separated into two equations one of which is a curved three-dimensional equation of motion and the other is an equation for energy. For the gravitational field of an isolated particle, they yield the Schwarzschild equations. They can be used to describe gravitation for any array of masses for which the Newtonian gravitational potential is known, and is applied here to describe motion in the gravitational field of a thin mass-rod.

Stress stiffening and approximate equations in flexible multibody dynamics

NASA Technical Reports Server (NTRS)

Padilla, Carlos E.; Vonflotow, Andreas H.

1993-01-01

A useful model for open chains of flexible bodies undergoing large rigid body motions, but small elastic deformations, is one in which the equations of motion are linearized in the small elastic deformations and deformation rates. For slow rigid body motions, the correctly linearized, or consistent, set of equations can be compared to prematurely linearized, or inconsistent, equations and to 'oversimplified,' or ruthless, equations through the use of open loop dynamic simulations. It has been shown that the inconsistent model should never be used, while the ruthless model should be used whenever possible. The consistent and inconsistent models differ by stress stiffening terms. These are due to zeroth-order stresses effecting virtual work via nonlinear strain-displacement terms. In this paper we examine in detail the nature of these stress stiffening terms and conclude that they are significant only when the associated zeroth-order stresses approach 'buckling' stresses. Finally it is emphasized that when the stress stiffening terms are negligible the ruthlessly linearized equations should be used.

The wrinkle-like N-solitons for the thermophoretic motion equation through graphene sheets

NASA Astrophysics Data System (ADS)

Ma, Yu-Lan; Li, Bang-Qing

2018-03-01

The main work is focused on the thermophoretic motion equation, which was derived from wrinkle wave motions in substrate-supported graphene sheets. Via the bilinear method, a class of wrinkle-like N-soliton solutions is constructed. The one-soliton, two-soliton and three-soliton are observed graphically. The shape, amplitude, open direction and width of the N-solitons are controllable through certain parameters.

On the non-stationary generalized Langevin equation

NASA Astrophysics Data System (ADS)

Meyer, Hugues; Voigtmann, Thomas; Schilling, Tanja

2017-12-01

In molecular dynamics simulations and single molecule experiments, observables are usually measured along dynamic trajectories and then averaged over an ensemble ("bundle") of trajectories. Under stationary conditions, the time-evolution of such averages is described by the generalized Langevin equation. By contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable has. We employ the formalism of time-dependent projection operator techniques to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. The equation is similar in structure to the generalized Langevin equation but exhibits a time-dependent memory kernel as well as a fluctuating force that implicitly depends on the initial conditions of the process. We also derive a relation between this memory kernel and the autocorrelation function of the fluctuating force that has a structure similar to a fluctuation-dissipation relation. In addition, we show how the choice of the projection operator allows us to relate the Taylor expansion of the memory kernel to data that are accessible in MD simulations and experiments, thus allowing us to construct the equation of motion. As a numerical example, the procedure is applied to Brownian motion initialized in non-equilibrium conditions and is shown to be consistent with direct measurements from simulations.

Spectrum Analysis of Inertial and Subinertial Motions Based on Analyzed Winds and Wind-Driven Currents from a Primitive Equation General Ocean Circulation Model.

DTIC Science & Technology

1982-12-01

1Muter.Te Motions Based on Ana lyzed Winds and wind-driven December 1982 Currents from. a Primitive Squat ion General a.OW -love"*..* Oean Circulation...mew se"$ (comeS.... do oISN..u am ae~ 00do OWaor NUN Fourier and Rotary Spc , Analysis Modeled Inertial and Subinrtial Motion 4 Primitive Equation

Equations of motion for a spectrum-generating algebra: Lipkin Meshkov Glick model

NASA Astrophysics Data System (ADS)

Rosensteel, G.; Rowe, D. J.; Ho, S. Y.

2008-01-01

For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin-Meshkov-Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 106, computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point.

Nonlinear dynamics and chaotic motions in feedback-controlled two- and three-degree-of-freedom robots

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ravishankar, A.S. Ghosal, A.

1999-01-01

The dynamics of a feedback-controlled rigid robot is most commonly described by a set of nonlinear ordinary differential equations. In this paper, the authors analyze these equations, representing the feedback-controlled motion of two- and three-degrees-of-freedom rigid robots with revolute (R) and prismatic (P) joints in the absence of compliance, friction, and potential energy, for the possibility of chaotic motions. The authors first study the unforced or inertial motions of the robots, and show that when the Gaussian or Riemannian curvature of the configuration space of a robot is negative, the robot equations can exhibit chaos. If the curvature is zeromore » or positive, then the robot equations cannot exhibit chaos. The authors show that among the two-degrees-of-freedom robots, the PP and the PR robot have zero Gaussian curvature while the RP and RR robots have negative Gaussian curvatures. For the three-degrees-of-freedom robots, they analyze the two well-known RRP and RRR configurations of the Stanford arm and the PUMA manipulator, respectively, and derive the conditions for negative curvature and possible chaotic motions. The criteria of negative curvature cannot be used for the forced or feedback-controlled motions. For the forced motion, the authors resort to the well-known numerical techniques and compute chaos maps, Poincare maps, and bifurcation diagrams. Numerical results are presented for the two-degrees-of-freedom RP and RR robots, and the authors show that these robot equations can exhibit chaos for low controller gains and for large underestimated models. From the bifurcation diagrams, the route to chaos appears to be through period doubling.« less

Nonlinear Equations of Motion for Cantilever Rotor Blades in Hover with Pitch Link Flexibility, Twist, Precone, Droop, Sweep, Torque Offset, and Blade Root Offset

NASA Technical Reports Server (NTRS)

Hodges, D. H.

1976-01-01

Nonlinear equations of motion for a cantilever rotor blade are derived for the hovering flight condition. The blade is assumed to have twist, precone, droop, sweep, torque offset and blade root offset, and the elastic axis and the axes of center of mass, tension, and aerodynamic center coincident at the quarter chord. The blade is cantilevered in bending, but has a torsional root spring to simulate pitch link flexibility. Aerodynamic forces acting on the blade are derived from strip theory based on quasi-steady two-dimensional airfoil theory. The equations are hybrid, consisting of one integro-differential equation for root torsion and three integro-partial differential equations for flatwise and chordwise bending and elastic torsion. The equations are specialized for a uniform blade and reduced to nonlinear ordinary differential equations by Galerkin's method. They are linearized for small perturbation motions about the equilibrium operating condition. Modal analysis leads to formulation of a standard eigenvalue problem where the elements of the stability matrix depend on the solution of the equilibrium equations. Two different forms of the root torsion equation are derived that yield virtually identical numerical results. This provides a reasonable check for the accuracy of the equations.

A Hybrid Ground-Motion Prediction Equation for Earthquakes in Western Alberta

NASA Astrophysics Data System (ADS)

Spriggs, N.; Yenier, E.; Law, A.; Moores, A. O.

2015-12-01

Estimation of ground-motion amplitudes that may be produced by future earthquakes constitutes the foundation of seismic hazard assessment and earthquake-resistant structural design. This is typically done by using a prediction equation that quantifies amplitudes as a function of key seismological variables such as magnitude, distance and site condition. In this study, we develop a hybrid empirical prediction equation for earthquakes in western Alberta, where evaluation of seismic hazard associated with induced seismicity is of particular interest. We use peak ground motions and response spectra from recorded seismic events to model the regional source and attenuation attributes. The available empirical data is limited in the magnitude range of engineering interest (M>4). Therefore, we combine empirical data with a simulation-based model in order to obtain seismologically informed predictions for moderate-to-large magnitude events. The methodology is two-fold. First, we investigate the shape of geometrical spreading in Alberta. We supplement the seismic data with ground motions obtained from mining/quarry blasts, in order to gain insights into the regional attenuation over a wide distance range. A comparison of ground-motion amplitudes for earthquakes and mining/quarry blasts show that both event types decay at similar rates with distance and demonstrate a significant Moho-bounce effect. In the second stage, we calibrate the source and attenuation parameters of a simulation-based prediction equation to match the available amplitude data from seismic events. We model the geometrical spreading using a trilinear function with attenuation rates obtained from the first stage, and calculate coefficients of anelastic attenuation and site amplification via regression analysis. This provides a hybrid ground-motion prediction equation that is calibrated for observed motions in western Alberta and is applicable to moderate-to-large magnitude events.

Documentation of the Fourth Order Band Model

NASA Technical Reports Server (NTRS)

Kalnay-Rivas, E.; Hoitsma, D.

1979-01-01

A general circulation model is presented which uses quadratically conservative, fourth order horizontal space differences on an unstaggered grid and second order vertical space differences with a forward-backward or a smooth leap frog time scheme to solve the primitive equations of motion. The dynamic equations for motion, finite difference equations, a discussion of the structure and flow chart of the program code, a program listing, and three relevent papers are given.

Variational principles for relativistic smoothed particle hydrodynamics

NASA Astrophysics Data System (ADS)

Monaghan, J. J.; Price, D. J.

2001-12-01

In this paper we show how the equations of motion for the smoothed particle hydrodynamics (SPH) method may be derived from a variational principle for both non-relativistic and relativistic motion when there is no dissipation. Because the SPH density is a function of the coordinates the derivation of the equations of motion through variational principles is simpler than in the continuum case where the density is defined through the continuity equation. In particular, the derivation of the general relativistic equations is more direct and simpler than that of Fock. The symmetry properties of the Lagrangian lead immediately to the familiar additive conservation laws of linear and angular momentum and energy. In addition, we show that there is an approximately conserved quantity which, in the continuum limit, is the circulation.

Formulation of the aeroelastic stability and response problem of coupled rotor/support systems

NASA Technical Reports Server (NTRS)

Warmbrodt, W.; Friedmann, P.

1979-01-01

The consistent formulation of the governing nonlinear equations of motion for a coupled rotor/support system is presented. Rotor/support coupling is clearly documented by enforcing dynamic equilibrium between the rotor and the moving flexible support. The nonlinear periodic coefficient equations of motion are applicable to both coupled rotor/fuselage aeroelastic problems of helicopters in hover or forward flight and coupled rotor/tower dynamics of a large horizontal axis wind turbine (HAWT). Finally, the equations of motion are used to study the influence of flexible supports and nonlinear terms on rotor aeroelastic stability and response of a large two-bladed HAWT.

Deep circulations under simple classes of stratification

NASA Technical Reports Server (NTRS)

Salby, Murry L.

1989-01-01

Deep circulations where the motion field is vertically aligned over one or more scale heights are studied under barotropic and equivalent barotropic stratifications. The study uses two-dimensional equations reduced from the three-dimensional primitive equations in spherical geometry. A mapping is established between the full primitive equations and general shallow water behavior and the correspondence between variables describing deep atmospheric motion and those of shallow water behavior is established.

Remarks on the derivation of the governing equations for the dynamics of a nonlinear beam to a non ideal shaft coupling

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fenili, André; Lopes Rebello da Fonseca Brasil, Reyolando Manoel; Balthazar, José M., E-mail: jmbaltha@gmail.com

We derive nonlinear governing equations without assuming that the beam is inextensible. The derivation couples the equations that govern a weak electric motor, which is used to rotate the base of the beam, to those that govern the motion of the beam. The system is considered non-ideal in the sense that the response of the motor to an applied voltage and the motion of the beam must be obtained interactively. The moment that the motor exerts on the base of the beam cannot be determined without solving for the motion of the beam.

Free polar motion of a triaxial and elastic body in Hamiltonian formalism: Application to the Earth and Mars

NASA Astrophysics Data System (ADS)

Folgueira, M.; Souchay, J.

2005-03-01

The purpose of this paper is to show how to solve in Hamiltonian formalism the equations of the polar motion of any arbitrarily shaped elastic celestial body, i.e. the motion of its rotation axis (or angular momentum) with respect to its figure axis. With this aim, we deduce from canonical equations related to the rotational Hamiltonian of the body, the analytical solution for its free polar motion which depends both on the elasticity and on its moments of inertia. In particular, we study the influence of the phase angle δ, responsible for the dissipation, on the damping of the polar motion. In order to validate our analytical equations, we show that, to first order, they are in complete agreement with those obtained from the classical Liouville equations. Then we adapt our calculations to the real data obtained from the polar motion of the Earth (polhody). For that purpose, we characterize precisely the differences in radius J-χ and in angle l-θ between the polar coordinates (χ,θ) and (J,l) representing respectively the motion of the axis of rotation of the Earth and the motion of its angular momentum axis, with respect to an Earth-fixed reference frame, after showing the influence of the choice of the origin on these coordinates, and on the determination of the Chandler period as well. Then we show that the phase lag δ responsible for the damping for the selected time interval, between Feb. 1982 and Apr. 1990, might be of the order of δ ≈ 6 °, according to a numerical integration starting from our analytical equations. Moreover, we emphasize the presence in our calculations for both χ and θ, of an oscillation with a period TChandler/2, due to the triaxial shape of our planet, and generally not taken into account. In a last step, we apply our analytical formulation to the polar motion of Mars, thus showing the high dependence of its damping on the poorly known value of its Love number k. Moreover we emphasize the large oscillations of Mars' polar motion due to the triaxiality of this planet.

Complex methyl group and hydrogen-bonded proton motions in terms of the Arrhenius and Schrödinger equations.

PubMed

Latanowicz, L

2008-01-01

Equations for the temperature dependence of the spectral densities J(is)(m)(momega(I) +/-omega(T)), where m=1, 2, omega(I) and omega(T) are the resonance and tunnel splitting angular frequencies, in the presence of a complex motion, have been derived. The spin pairs of the protons or deuterons of the methyl group perform a complex motion consisting of three component motions. Two of them involve mass transportation over the barrier and through the barrier. They are characterized by k((H)) (Arrhenius) and k((T)) (Schrödinger) rate constants, respectively. The third motion causes fluctuations of the frequencies (nomega(I)+/-omega(T)) and it is related to the lifetime of the methyl spin at the energy level influenced by the rotor-bath interactions. These interactions induce rapid transitions, changing the symmetry of the torsional sublevels either from A to E or from E(a) to E(b). The correlation function for this third motion (k((omega)) rate constant) has been proposed by Müller-Warmuth et al. The spectral densities of the methyl group hindered rotation (k((H)), k((T)) and k((omega)) rate constants) differ from the spectral densities of the proton transfer (k((H)) and k((T)) rate constants) because three compound motions contribute to the complex motion of the methyl group. The recently derived equation [Formula: see text] , where [Formula: see text] and [Formula: see text] are the fraction and energy of particles with energies from zero to E(H), is taken into account in the calculations of the spectral densities. This equation follows from Maxwell's distribution of thermal energy. The spectral densities derived are applied to analyse the experimental temperature dependencies of proton and deuteron spin-lattice relaxation rate in solids containing the methyl group. A wide range of temperatures from zero Kelvin up to the melting point is considered. It has been established that the motion characterized by k((omega)) influences the spin-lattice relaxation up to the temperature T(tun) only. This temperature is directly determined by the equation C(p)T=E(H) (thermal energy=activation energy), where C(p) is the molar heat capacity. Probably the cessation of the third motion is a result of the de Broglie wavelength related to this motion becoming too short. As shown recently, the potential barrier can be an obstacle for the de Broglie wave. The theoretical equations derived in this paper are compared to those known in the literature.

Numerical simulation of rotating body movement in medium with various densities

NASA Astrophysics Data System (ADS)

Tenenev, Valentin A.; Korolev, Stanislav A.; Rusyak, Ivan G.

2016-10-01

The paper proposes an approach to calculate the motion of rotating bodies in resisting medium by solving the Kirchhoff equations of motion in a coordinate system moving with the body and in determination of aerodynamic characteristics of the body with a given geometry by solving the Navier-Stokes equations. We present the phase trajectories of the perturbed motion of a rotating projectile in media with different densities: gas and liquid.

Estimating the accuracy of the technique of reconstructing the rotational motion of a satellite based on the measurements of its angular velocity and the magnetic field of the Earth

NASA Astrophysics Data System (ADS)

Belyaev, M. Yu.; Volkov, O. N.; Monakhov, M. I.; Sazonov, V. V.

2017-09-01

The paper has studied the accuracy of the technique that allows the rotational motion of the Earth artificial satellites (AES) to be reconstructed based on the data of onboard measurements of angular velocity vectors and the strength of the Earth magnetic field (EMF). The technique is based on kinematic equations of the rotational motion of a rigid body. Both types of measurement data collected over some time interval have been processed jointly. The angular velocity measurements have been approximated using convenient formulas, which are substituted into the kinematic differential equations for the quaternion that specifies the transition from the body-fixed coordinate system of a satellite to the inertial coordinate system. Thus obtained equations represent a kinematic model of the rotational motion of a satellite. The solution of these equations, which approximate real motion, has been found by the least-square method from the condition of best fitting between the data of measurements of the EMF strength vector and its calculated values. The accuracy of the technique has been estimated by processing the data obtained from the board of the service module of the International Space Station ( ISS). The reconstruction of station motion using the aforementioned technique has been compared with the telemetry data on the actual motion of the station. The technique has allowed us to reconstruct the station motion in the orbital orientation mode with a maximum error less than 0.6° and the turns with a maximal error of less than 1.2°.

Hamilton-Jacobi modelling of relative motion for formation flying.

PubMed

Kolemen, Egemen; Kasdin, N Jeremy; Gurfil, Pini

2005-12-01

A precise analytic model for the relative motion of a group of satellites in slightly elliptic orbits is introduced. With this aim, we describe the relative motion of an object relative to a circular or slightly elliptic reference orbit in the rotating Hill frame via a low-order Hamiltonian, and solve the Hamilton-Jacobi equation. This results in a first-order solution to the relative motion identical to the Clohessy-Wiltshire approach; here, however, rather than using initial conditions as our constants of the motion, we utilize the canonical momenta and coordinates. This allows us to treat perturbations in an identical manner, as in the classical Delaunay formulation of the two-body problem. A precise analytical model for the base orbit is chosen with the included effect of zonal harmonics (J(2), J(3), J(4)). A Hamiltonian describing the real relative motion is formed and by differing this from the nominal Hamiltonian, the perturbing Hamiltonian is obtained. Using the Hamilton equations, the variational equations for the new constants are found. In a manner analogous to the center manifold reduction procedure, the non-periodic part of the motion is canceled through a magnitude analysis leading to simple boundedness conditions that cancel the drift terms due to the higher order perturbations. Using this condition, the variational equations are integrated to give periodic solutions that closely approximate the results from numerical integration (1 mm/per orbit for higher order and eccentricity perturbations and 30 cm/per orbit for zonal perturbations). This procedure provides a compact and insightful analytic description of the resulting relative motion.

Quadratic Damping

ERIC Educational Resources Information Center

Fay, Temple H.

2012-01-01

Quadratic friction involves a discontinuous damping term in equations of motion in order that the frictional force always opposes the direction of the motion. Perhaps for this reason this topic is usually omitted from beginning texts in differential equations and physics. However, quadratic damping is more realistic than viscous damping in many…

Notes on implementation of Coulomb friction in coupled dynamical simulations

NASA Technical Reports Server (NTRS)

Vandervoort, R. J.; Singh, R. P.

1987-01-01

A coupled dynamical system is defined as an assembly of rigid/flexible bodies that may be coupled by kinematic connections. The interfaces between bodies are modeled using hinges having 0 to 6 degrees of freedom. The equations of motion are presented for a mechanical system of n flexible bodies in a topological tree configuration. The Lagrange form of the D'Alembert principle was employed to derive the equations. The equations of motion are augmented by the kinematic constraint equations. This augmentation is accomplished via the method of singular value decomposition.

Investigation of the Observability of a Satellite Cluster in a Near Circular Orbit

DTIC Science & Technology

1989-12-01

investigation involved the use of dynamics based on the Clohessy - Wiltshire equa- tions and an on-board estimator based on the U-D covariance factorization...vectors were determined from the Clohessy - Wiltshire equations (10:80). These equations have the following * form: I I4 I I I sat i , ref pointI S iA lll ri... Clohessy - Wiltshire equations of motion, with 77 representing the mean motion (10:80). These equations are: 1 11I ;i - 2 )- 3 T12 X = 0 (27) +21 =0 (28

Maggi's equations of motion and the determination of constraint reactions

NASA Astrophysics Data System (ADS)

Papastavridis, John G.

1990-04-01

This paper presents a geometrical derivation of the constraint reaction-free equations of Maggi for mechanical systems subject to linear (first-order) nonholonomic and/or holonomic constraints. These results follow directly from the proper application of the concepts of virtual displacement and quasi-coordinates to the variational equation of motion, i.e., Lagrange's principle. The method also makes clear how to compute the constraint reactions (kinetostatics) without introducing Lagrangian multipliers.

Turbulent Fluid Motion 5: Fourier Analysis, the Spectral Form of the Continuum Equations, and Homogeneous Turbulence

NASA Technical Reports Server (NTRS)

Deissler, Robert G.

1996-01-01

Background material on Fourier analysis and on the spectral form of the continuum equations, both averaged and unaveraged, are given. The equations are applied to a number of cases of homogeneous turbulence with and without mean gradients. Spectral transfer of turbulent activity between scales of motion is studied in some detail. The effects of mean shear, heat transfer, normal strain, and buoyancy are included in the analyses.

Antigravity: Spin-gravity coupling in action

NASA Astrophysics Data System (ADS)

Plyatsko, Roman; Fenyk, Mykola

2016-08-01

The typical motions of a spinning test particle in Schwarzschild's background which show the strong repulsive action of the highly relativistic spin-gravity coupling are considered using the exact Mathisson-Papapetrou equations. An approximated approach to choice solutions of these equations which describe motions of the particle's proper center of mass is developed.

Motions, efforts and actuations in constrained dynamic systems: a multi-link open-chain example

NASA Astrophysics Data System (ADS)

Duke Perreira, N.

1999-08-01

The effort-motion method, which describes the dynamics of open- and closed-chain topologies of rigid bodies interconnected with revolute and prismatic pairs, is interpreted geometrically. Systems are identified for which the simultaneous control of forces and velocities is desirable, and a representative open-chain system is selected for use in the ensuing analysis. Gauge invariant transformations are used to recast the commonly used kinetic and kinematic equations into a dimensional gauge invariant form. Constraint elimination techniques based on singular value decompositions then recast the invariant equations into orthogonal and reciprocal sets of motion and effort equations written in state variable form. The ideal actuation is found that simultaneously achieves the obtainable portions of the desired constraining efforts and motions. The performance is then evaluated of using the actuation closest to the ideal actuation.

Computer program documentation for the dynamic analysis of a noncontacting mechanical face seal

NASA Technical Reports Server (NTRS)

Auer, B. M.; Etsion, I.

1980-01-01

A computer program is presented which achieves a numerical solution for the equations of motion of a noncontacting mechanical face seal. The flexibly-mounted primary seal ring motion is expressed by a set of second order differential equations for three degrees of freedom. These equations are reduced to a set of first order equations and the GEAR software package is used to solve the set of first order equations. Program input includes seal design parameters and seal operating conditions. Output from the program includes velocities and displacements of the seal ring about the axis of an inertial reference system. One example problem is described.

On the dynamics of a human body model.

NASA Technical Reports Server (NTRS)

Huston, R. L.; Passerello, C. E.

1971-01-01

Equations of motion for a model of the human body are developed. Basically, the model consists of an elliptical cylinder representing the torso, together with a system of frustrums of elliptical cones representing the limbs. They are connected to the main body and each other by hinges and ball and socket joints. Vector, tensor, and matrix methods provide a systematic organization of the geometry. The equations of motion are developed from the principles of classical mechanics. The solution of these equations then provide the displacement and rotation of the main body when the external forces and relative limb motions are specified. Three simple example motions are studied to illustrate the method. The first is an analysis and comparison of simple lifting on the earth and the moon. The second is an elementary approach to underwater swimming, including both viscous and inertia effects. The third is an analysis of kicking motion and its effect upon a vertically suspended man such as a parachutist.

A vectorized algorithm for 3D dynamics of a tethered satellite

NASA Technical Reports Server (NTRS)

Wilson, Howard B.

1989-01-01

Equations of motion characterizing the three dimensional motion of a tethered satellite during the retrieval phase are studied. The mathematical model involves an arbitrary number of point masses connected by weightless cords. Motion occurs in a gravity gradient field. The formulation presented accounts for general functions describing support point motion, rate of tether retrieval, and arbitrary forces applied to the point masses. The matrix oriented program language MATLAB is used to produce an efficient vectorized formulation for computing natural frequencies and mode shapes for small oscillations about the static equilibrium configuration; and for integrating the nonlinear differential equations governing large amplitude motions. An example of time response pertaining to the skip rope effect is investigated.

An exact solution to the relativistic equation of motion of a charged particle driven by a linearly polarized electromagnetic wave

NASA Technical Reports Server (NTRS)

Shebalin, John V.

1988-01-01

An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam..

BEM for wave equation with boundary in arbitrary motion and applications to compressible potential aerodynamics of airplanes and helicopters

NASA Technical Reports Server (NTRS)

Morino, Luigi; Bharadvaj, Bala K.; Freedman, Marvin I.; Tseng, Kadin

1988-01-01

The wave equation for an object in arbitrary motion is investigated analytically using a BEM approach, and practical applications to potential flows of compressible fluids around aircraft wings and helicopter rotors are considered. The treatment accounts for arbitrary combined rotational and translational motion of the reference frame and for the wake motion. The numerical implementation as a computer algorithm is demonstrated on problems with prescribed and free wakes, the former in compressible flows and the latter for incompressible flows; results are presented graphically and briefly characterized.

Corrigendum: New Form of Kane's Equations of Motion for Constrained Systems

NASA Technical Reports Server (NTRS)

Roithmayr, Carlos M.; Bajodah, Abdulrahman H.; Hodges, Dewey H.; Chen, Ye-Hwa

2007-01-01

A correction to the previously published article "New Form of Kane's Equations of Motion for Constrained Systems" is presented. Misuse of the transformation matrix between time rates of change of the generalized coordinates and generalized speeds (sometimes called motion variables) resulted in a false conclusion concerning the symmetry of the generalized inertia matrix. The generalized inertia matrix (sometimes referred to as the mass matrix) is in fact symmetric and usually positive definite when one forms nonminimal Kane's equations for holonomic or simple nonholonomic systems, systems subject to nonlinear nonholonomic constraints, and holonomic or simple nonholonomic systems subject to impulsive constraints according to Refs. 1, 2, and 3, respectively. The mass matrix is of course symmetric when one forms minimal equations for holonomic or simple nonholonomic systems using Kane s method as set forth in Ref. 4.

Evolution of nonlinear waves in a blood-filled artery with an aneurysm

NASA Astrophysics Data System (ADS)

Nikolova, E. V.; Jordanov, I. P.; Dimitrova, Z. I.; Vitanov, N. K.

2017-10-01

We discuss propagation of traveling waves in a blood-filled hyper-elastic artery with a local dilatation (an aneurysm). The processes in the injured artery are modeled by an equation of the motion of the arterial wall and by equations of the motion of the fluid (the blood). Taking into account the specific arterial geometry and applying the reductive perturbation method in long-wave approximation we reduce the model equations to a version of the perturbed Korteweg-de Vries kind equation with variable coefficients. Exact traveling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simplest equation. A particular case of the obtained exact solution is numerically simulated and discussed from the point of view of arterial disease mechanics.

The N-BOD2 user's and programmer's manual

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1978-01-01

A general purpose digital computer program was developed and designed to aid in the analysis of spacecraft attitude dynamics. The program provides the analyst with the capability of automatically deriving and numerically solving the equations of motion of any system that can be modeled as a topological tree of coupled rigid bodies, flexible bodies, point masses, and symmetrical momentum wheels. Two modes of output are available. The composite system equations of motion may be outputted on a line printer in a symbolic form that may be easily translated into common vector-dyadic notation, or the composite system equations of motion may be solved numerically and any desirable set of system state variables outputted as a function of time.

From quantum stochastic differential equations to Gisin-Percival state diffusion

NASA Astrophysics Data System (ADS)

Parthasarathy, K. R.; Usha Devi, A. R.

2017-08-01

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ (L2(R+ ) ⊗(Cn⊕Cn ) ) and the Hilbert space L2(μ ) , where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B (t ) ,t ≥0 } , we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

Coarse-grained forms for equations describing the microscopic motion of particles in a fluid.

PubMed

Das, Shankar P; Yoshimori, Akira

2013-10-01

Exact equations of motion for the microscopically defined collective density ρ(x,t) and the momentum density ĝ(x,t) of a fluid have been obtained in the past starting from the corresponding Langevin equations representing the dynamics of the fluid particles. In the present work we average these exact equations of microscopic dynamics over the local equilibrium distribution to obtain stochastic partial differential equations for the coarse-grained densities with smooth spatial and temporal dependence. In particular, we consider Dean's exact balance equation for the microscopic density of a system of interacting Brownian particles to obtain the basic equation of the dynamic density functional theory with noise. Our analysis demonstrates that on thermal averaging the dependence of the exact equations on the bare interaction potential is converted to dependence on the corresponding thermodynamic direct correlation functions in the coarse-grained equations.

Turbulent fluid motion 3: Basic continuum equations

NASA Technical Reports Server (NTRS)

Deissler, Robert G.

1991-01-01

A derivation of the continuum equations used for the analysis of turbulence is given. These equations include the continuity equation, the Navier-Stokes equations, and the heat transfer or energy equation. An experimental justification for using a continuum approach for the study of turbulence is given.

Aeroelastic equations of motion of a Darrieus vertical-axis wind-turbine blade

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.; Kvaternik, R. G.

1979-01-01

The second-degree nonlinear aeroelastic equations of motion for a slender, flexible, nonuniform, Darrieus vertical-axis wind turbine blade which is undergoing combined flatwise bending, edgewise bending, torsion, and extension are developed using Hamilton's principle. The blade aerodynamic loading is obtained from strip theory based on a quasi-steady approximation of two-dimensional incompressible unsteady airfoil theory. The derivation of the equations has its basis in the geometric nonlinear theory of elasticity and the resulting equations are consistent with the small deformation approximation in which the elongations and shears are negligible compared to unity. These equations are suitable for studying vibrations, static and dynamic aeroelastic instabilities, and dynamic response. Several possible methods of solution of the equations, which have periodic coefficients, are discussed.

An Argument Against Augmenting the Lagrangean for Nonholonomic Systems

NASA Technical Reports Server (NTRS)

Roithmayr, Carlos M.; Hodges, Dewey H.

2009-01-01

Although it is known that correct dynamical equations of motion for a nonholonomic system cannot be obtained from a Lagrangean that has been augmented with a sum of the nonholonomic constraint equations weighted with multipliers, previous publications suggest otherwise. An example has been proposed in support of augmentation and purportedly demonstrates that an accepted method fails to produce correct equations of motion whereas augmentation leads to correct equations; this paper shows that in fact the opposite is true. The correct equations, previously discounted on the basis of a flawed application of the Newton-Euler method, are verified by using Kane's method and a new approach to determining the directions of constraint forces. A correct application of the Newton-Euler method reproduces valid equations.

Toward higher order tests of the gravitational interaction

NASA Technical Reports Server (NTRS)

Nordtvedt, Ken

1989-01-01

Analyses and interpretations of experiments which test post-Newtonian gravity are usually done under the assumption that gravity is a metric field phenomenon - a manifestation of space-time geometry. This, however, is unnecessary and one can start at a more primitive level - that there simply exists a phenomenological, gravitational, many-body equation of motion which must be determined by a package of observations. In fact, over the last couple decades, a diverse collection of solar system interbody tracking observations, supplemented by data from the binary pulsar system PSR 1913 + 16, has completely mapped out the first post-Newtonian order. After the fact, using empirically determined equations of motion, along with some observed properties of nongravitational clocks and rulers and conservation laws for energy, momentum and angular momentum, a post-Newtonian Lagrangian can be constructed, a geometrical space-time metric field conceptual interpretation can be developed, Lorentz invariance of the equations of motion can be shown, and the equations of motion are found to agree with the predictions of Einstein's gravitational theory, General Relativity, within experimental accuracy.

Development of a numerical model for vehicle-bridge interaction analysis of railway bridges

NASA Astrophysics Data System (ADS)

Kim, Hee Ju; Cho, Eun Sang; Ham, Jun Su; Park, Ki Tae; Kim, Tae Heon

2016-04-01

In the field of civil engineering, analyzing dynamic response was main concern for a long time. These analysis methods can be divided into moving load analysis method and moving mass analysis method, and formulating each an equation of motion has recently been studied after dividing vehicles and bridges. In this study, the numerical method is presented, which can consider the various train types and can solve the equations of motion for a vehicle-bridge interaction analysis by non-iteration procedure through formulating the coupled equations for motion. Also, 3 dimensional accurate numerical models was developed by KTX-vehicle in order to analyze dynamic response characteristics. The equations of motion for the conventional trains are derived, and the numerical models of the conventional trains are idealized by a set of linear springs and dashpots with 18 degrees of freedom. The bridge models are simplified by the 3 dimensional space frame element which is based on the Euler-Bernoulli theory. The rail irregularities of vertical and lateral directions are generated by PSD functions of the Federal Railroad Administration (FRA).

G-DYN Multibody Dynamics Engine

NASA Technical Reports Server (NTRS)

Acikmese, Behcet; Blackmore, James C.; Broderick, Daniel

2011-01-01

G-DYN is a multi-body dynamic simulation software engine that automatically assembles and integrates equations of motion for arbitrarily connected multibody dynamic systems. The algorithm behind G-DYN is based on a primal-dual formulation of the dynamics that captures the position and velocity vectors (primal variables) of each body and the interaction forces (dual variables) between bodies, which are particularly useful for control and estimation analysis and synthesis. It also takes full advantage of the spare matrix structure resulting from the system dynamics to numerically integrate the equations of motion efficiently. Furthermore, the dynamic model for each body can easily be replaced without re-deriving the overall equations of motion, and the assembly of the equations of motion is done automatically. G-DYN proved an essential software tool in the simulation of spacecraft systems used for small celestial body surface sampling, specifically in simulating touch-and-go (TAG) maneuvers of a robotic sampling system from a comet and asteroid. It is used extensively in validating mission concepts for small body sample return, such as Comet Odyssey and Galahad New Frontiers proposals.

Stability of Dynamical Systems with Discontinuous Motions:

NASA Astrophysics Data System (ADS)

Michel, Anthony N.; Hou, Ling

In this paper we present a stability theory for discontinuous dynamical systems (DDS): continuous-time systems whose motions are not necessarily continuous with respect to time. We show that this theory is not only applicable in the analysis of DDS, but also in the analysis of continuous dynamical systems (continuous-time systems whose motions are continuous with respect to time), discrete-time dynamical systems (systems whose motions are defined at discrete points in time) and hybrid dynamical systems (HDS) (systems whose descriptions involve simultaneously continuous-time and discrete-time). We show that the stability results for DDS are in general less conservative than the corresponding well-known classical Lyapunov results for continuous dynamical systems and discrete-time dynamical systems. Although the DDS stability results are applicable to general dynamical systems defined on metric spaces (divorced from any kind of description by differential equations, or any other kinds of equations), we confine ourselves to finite-dimensional dynamical systems defined by ordinary differential equations and difference equations, to make this paper as widely accessible as possible. We present only sample results, namely, results for uniform asymptotic stability in the large.

Pendulum Motion and Differential Equations

ERIC Educational Resources Information Center

Reid, Thomas F.; King, Stephen C.

2009-01-01

A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…

Motion of a curved vortex filament with decaying vortical core and axial velocity

NASA Technical Reports Server (NTRS)

Callegari, A. J.; Ting, L.

1978-01-01

The motion and decay of a curved vortex filament having large axial and circumferential velocity components in a three-dimensional stream are analyzed by using the method of matched asymptotic expansions of the incompressible Navier-Stokes equations. The small parameter is the square root of the ratio of the kinematic viscosity to the circulation. The outer region is analyzed by the classical Biot-Savart law, and its solution is matched to that of the inner region, where viscous effects are important. Equations describing the coupling between the inner vortex structure and the motion of the vortex filament as well as the time evolution of the inner vortex structure are obtained. Equations are derived for the motion of the vortex filament and for the change and decay in time and space of the leading-order circumferential and axial velocity and vorticity components. Solutions are constructed for these components in terms of initial data.

DOE Office of Scientific and Technical Information (OSTI.GOV)

B.B. Rokhman

A two-dimensional stationary model of motion, heat and mass exchange, and chemical reaction of polydisperse coke and ash particles in ascending gas-suspension flow has been constructed with allowance for the turbulent and pseudo turbulent mechanisms of transfer in the dispersed phase. The system of equations that describes motion and heat transfer in the solid phase has been closed at the level of the equations for the second moments of velocity and temperature pulsations, whereas the momentum equations of the carrying medium have been closed using the equation for turbulent gas energy, which allows for the influence of the particles andmore » heterogeneous reactions.« less

Rocket/launcher structural dynamics

NASA Technical Reports Server (NTRS)

Ferragut, N. J.

1976-01-01

The equations of motion describing the interactions between a rocket and a launcher were derived using Lagrange's Equation. A rocket launching was simulated. The motions of both the rocket and the launcher can be considered in detail. The model contains flexible elements and rigid elements. The rigid elements (masses) were judiciously utilized to simplify the derivation of the equations. The advantages of simultaneous shoe release were illustrated. Also, the loading history of the interstage structure of a boosted configuration was determined. The equations shown in this analysis could be used as a design tool during the modification of old launchers and the design of new launchers.

Mixed variational formulations of finite element analysis of elastoacoustic/slosh fluid-structure interaction

NASA Technical Reports Server (NTRS)

Felippa, Carlos A.; Ohayon, Roger

1991-01-01

A general three-field variational principle is obtained for the motion of an acoustic fluid enclosed in a rigid or flexible container by the method of canonical decomposition applied to a modified form of the wave equation in the displacement potential. The general principle is specialized to a mixed two-field principle that contains the fluid displacement potential and pressure as independent fields. This principle contains a free parameter alpha. Semidiscrete finite-element equations of motion based on this principle are displayed and applied to the transient response and free-vibrations of the coupled fluid-structure problem. It is shown that a particular setting of alpha yields a rich set of formulations that can be customized to fit physical and computational requirements. The variational principle is then extended to handle slosh motions in a uniform gravity field, and used to derive semidiscrete equations of motion that account for such effects.

Master equation with quantized atomic motion including dipole-dipole interactions

NASA Astrophysics Data System (ADS)

Damanet, François; Braun, Daniel; Martin, John

2016-05-01

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and is relevant for experiments with ultracold trapped atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find analytical formulas for a number of relevant states (Gaussian states, Fock states and thermal states). In particular, we show that the dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. The effects predicted should be experimentally observable with Rydberg atoms. FD would like to thank the F.R.S.-FNRS for financial support. FD is a FRIA Grant holder of the Fonds de la Recherche Scientifique-FNRS.

Nonlinear ordinary difference equations

NASA Technical Reports Server (NTRS)

Caughey, T. K.

1979-01-01

Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.

RMS massless arm dynamics capability in the SVDS. [equations of motion

NASA Technical Reports Server (NTRS)

Flanders, H. A.

1977-01-01

The equations of motion for the remote manipulator system, assuming that the masses and inertias of the arm can be neglected, are developed for implementation into the space vehicle dynamics simulation (SVDS) program for the Orbiter payload system. The arm flexibility is incorporated into the equations by the computation of flexibility terms for use in the joint servo model. The approach developed in this report is based on using the Jacobian transformation matrix to transform force and velocity terms between the configuration space and the task space to simplify the form of the equations.

New predictive equations for Arias intensity from crustal earthquakes in New Zealand

NASA Astrophysics Data System (ADS)

Stafford, Peter J.; Berrill, John B.; Pettinga, Jarg R.

2009-01-01

Arias Intensity (Arias, MIT Press, Cambridge MA, pp 438-483, 1970) is an important measure of the strength of a ground motion, as it is able to simultaneously reflect multiple characteristics of the motion in question. Recently, the effectiveness of Arias Intensity as a predictor of the likelihood of damage to short-period structures has been demonstrated, reinforcing the utility of Arias Intensity for use in both structural and geotechnical applications. In light of this utility, Arias Intensity has begun to be considered as a ground-motion measure suitable for use in probabilistic seismic hazard analysis (PSHA) and earthquake loss estimation. It is therefore timely to develop predictive equations for this ground-motion measure. In this study, a suite of four predictive equations, each using a different functional form, is derived for the prediction of Arias Intensity from crustal earthquakes in New Zealand. The provision of a suite of models is included to allow for epistemic uncertainty to be considered within a PSHA framework. Coefficients are presented for four different horizontal-component definitions for each of the four models. The ground-motion dataset for which the equations are derived include records from New Zealand crustal earthquakes as well as near-field records from worldwide crustal earthquakes. The predictive equations may be used to estimate Arias Intensity for moment magnitudes between 5.1 and 7.5 and for distances (both rjb and rrup) up to 300 km.

Motion in a central field in the presence of a constant perturbing acceleration in a co-moving coordinate system

NASA Astrophysics Data System (ADS)

Sannikova, T. N.; Kholshevnikov, K. V.

2015-08-01

The motion of a point mass under the action of a gravitational force toward a central body and a perturbing acceleration P is considered. The magnitude of P is taken to be small compared to the main gravitational acceleration due to the central body, and the direction of P to be constant in a standard astronomical coordinate system with its origin at the central body and axes directed along the radius vector, the transversal, and the binormal. Consideration of a constant vector perturbing acceleration simplifies averaging of the Euler equations for the motion in osculating elements, making it straightforward to obtain evolutionary differential equations of motion in the mean elements, as was done earlier in a first small-parameter approximation. This paper is devoted to integration of the mean equations. The system is integratable by quadratures if at least one component of the perturbing acceleration is zero, and also if the orbit is initially circular. Moreover, all the quadratures can be expressed in terms of elementary functions and elliptical integrals of the first kind in Jacobi form. If all three components of P are non-zero, this problem reduces to a system of two first-order differential equations, which are apparently not integrable. Possible applications include the motion of natural and artificial satellites taking into account light pressure, the motion of a spacecraft with low thrust, and the motion of an asteroid subject to a thrust from an engine mounted on it or to a gravitational tractor designed, for example, to avoid a collision with Earth.

Actuation for simultaneous motions and constraining efforts: an open chain example

NASA Astrophysics Data System (ADS)

Perreira, N. Duke

1997-06-01

A brief discussion on systems where simultaneous control of forces and velocities are desirable is given and an example linkage with revolute and prismatic joint is selected for further analysis. The Newton-Euler approach for dynamic system analysis is applied to the example to provide a basis of comparison. Gauge invariant transformations are used to convert the dynamic equations into invariant form suitable for use in a new dynamic system analysis method known as the motion-effort approach. This approach uses constraint elimination techniques based on singular value decompositions to recast the invariant form of dynamic system equations into orthogonal sets of motion and effort equations. Desired motions and constraining efforts are partitioned into ideally obtainable and unobtainable portions which are then used to determine the required actuation. The method is applied to the example system and an analytic estimate to its success is made.

Electrokinetic motion of a rectangular nanoparticle in a nanochannel

NASA Astrophysics Data System (ADS)

Movahed, Saeid; Li, Dongqing

2012-08-01

This article presents a theoretical study of electrokinetic motion of a negatively charged cubic nanoparticle in a three-dimensional nanochannel with a circular cross-section. Effects of the electrophoretic and the hydrodynamic forces on the nanoparticle motion are examined. Because of the large applied electric field over the nanochannel, the impact of the Brownian force is negligible in comparison with the electrophoretic and the hydrodynamic forces. The conventional theories of electrokinetics such as the Poisson-Boltzmann equation and the Helmholtz-Smoluchowski slip velocity approach are no longer applicable in the small nanochannels. In this study, and at each time step, first, a set of highly coupled partial differential equations including the Poisson-Nernst-Plank equation, the Navier-Stokes equations, and the continuity equation was solved to find the electric potential, ionic concentration field, and the flow field around the nanoparticle. Then, the electrophoretic and hydrodynamic forces acting on the negatively charged nanoparticle were determined. Following that, the Newton second law was utilized to find the velocity of the nanoparticle. Using this model, effects of surface electric charge of the nanochannel, bulk ionic concentration, the size of the nanoparticle, and the radius of the nanochannel on the nanoparticle motion were investigated. Increasing the bulk ionic concentration or the surface charge of the nanochannel will increase the electroosmotic flow, and hence affect the particle's motion. It was also shown that, unlike microchannels with thin EDL, the change in nanochannel size will change the EDL field and the ionic concentration field in the nanochannel, affecting the particle's motion. If the nanochannel size is fixed, a larger particle will move faster than a smaller particle under the same conditions.

Statistical description and transport in stochastic magnetic fields

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vanden Eijnden, E.; Balescu, R.

1996-03-01

The statistical description of particle motion in a stochastic magnetic field is presented. Starting form the stochastic Liouville equation (or, hybrid kinetic equation) associated with the equations of motion of a test particle, the probability distribution function of the system is obtained for various magnetic fields and collisional processes. The influence of these two ingredients on the statistics of the particle dynamics is stressed. In all cases, transport properties of the system are discussed. {copyright} {ital 1996 American Institute of Physics.}

The coupled nonlinear dynamics of a lift system

DOE Office of Scientific and Technical Information (OSTI.GOV)

Crespo, Rafael Sánchez, E-mail: rafael.sanchezcrespo@northampton.ac.uk, E-mail: stefan.kaczmarczyk@northampton.ac.uk, E-mail: phil.picton@northampton.ac.uk, E-mail: huijuan.su@northampton.ac.uk; Kaczmarczyk, Stefan, E-mail: rafael.sanchezcrespo@northampton.ac.uk, E-mail: stefan.kaczmarczyk@northampton.ac.uk, E-mail: phil.picton@northampton.ac.uk, E-mail: huijuan.su@northampton.ac.uk; Picton, Phil, E-mail: rafael.sanchezcrespo@northampton.ac.uk, E-mail: stefan.kaczmarczyk@northampton.ac.uk, E-mail: phil.picton@northampton.ac.uk, E-mail: huijuan.su@northampton.ac.uk

2014-12-10

Coupled lateral and longitudinal vibrations of suspension and compensating ropes in a high-rise lift system are often induced by the building motions due to wind or seismic excitations. When the frequencies of the building become near the natural frequencies of the ropes, large resonance motions of the system may result. This leads to adverse coupled dynamic phenomena involving nonplanar motions of the ropes, impact loads between the ropes and the shaft walls, as well as vertical vibrations of the car, counterweight and compensating sheave. Such an adverse dynamic behaviour of the system endangers the safety of the installation. This papermore » presents two mathematical models describing the nonlinear responses of a suspension/ compensating rope system coupled with the elevator car / compensating sheave motions. The models accommodate the nonlinear couplings between the lateral and longitudinal modes, with and without longitudinal inertia of the ropes. The partial differential nonlinear equations of motion are derived using Hamilton Principle. Then, the Galerkin method is used to discretise the equations of motion and to develop a nonlinear ordinary differential equation model. Approximate numerical solutions are determined and the behaviour of the system is analysed.« less

Dynamic analysis of a system of hinge-connected rigid bodies with nonrigid appendages. [equations of motion

NASA Technical Reports Server (NTRS)

Likins, P. W.

1974-01-01

Equations of motion are derived for use in simulating a spacecraft or other complex electromechanical system amenable to idealization as a set of hinge-connected rigid bodies of tree topology, with rigid axisymmetric rotors and nonrigid appendages attached to each rigid body in the set. In conjunction with a previously published report on finite-element appendage vibration equations, this report provides a complete minimum-dimension formulation suitable for generic programming for digital computer numerical integration.

A vector-dyadic development of the equations of motion for N-coupled flexible bodies and point masses. [spacecraft trajectories

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1975-01-01

The equations of motion for a system of coupled flexible bodies, rigid bodies, point masses, and symmetric wheels were derived. The equations were cast into a partitioned matrix form in which certain partitions became nontrivial when the effects of flexibility were treated. The equations are shown to contract to the coupled rigid body equations or expand to the coupled flexible body equations all within the same basic framework. Furthermore, the coefficient matrix always has the computationally desirable property of symmetry. Making use of the derived equations, a comparison was made between the equations which described a flexible body model and those which described a rigid body model of the same elastic appendage attached to an arbitrary coupled body system. From the comparison, equivalence relations were developed which defined how the two modeling approaches described identical dynamic effects.

A revised ground-motion and intensity interpolation scheme for shakemap

USGS Publications Warehouse

Worden, C.B.; Wald, D.J.; Allen, T.I.; Lin, K.; Garcia, D.; Cua, G.

2010-01-01

We describe a weighted-average approach for incorporating various types of data (observed peak ground motions and intensities and estimates from groundmotion prediction equations) into the ShakeMap ground motion and intensity mapping framework. This approach represents a fundamental revision of our existing ShakeMap methodology. In addition, the increased availability of near-real-time macroseismic intensity data, the development of newrelationships between intensity and peak ground motions, and new relationships to directly predict intensity from earthquake source information have facilitated the inclusion of intensity measurements directly into ShakeMap computations. Our approach allows for the combination of (1) direct observations (ground-motion measurements or reported intensities), (2) observations converted from intensity to ground motion (or vice versa), and (3) estimated ground motions and intensities from prediction equations or numerical models. Critically, each of the aforementioned data types must include an estimate of its uncertainties, including those caused by scaling the influence of observations to surrounding grid points and those associated with estimates given an unknown fault geometry. The ShakeMap ground-motion and intensity estimates are an uncertainty-weighted combination of these various data and estimates. A natural by-product of this interpolation process is an estimate of total uncertainty at each point on the map, which can be vital for comprehensive inventory loss calculations. We perform a number of tests to validate this new methodology and find that it produces a substantial improvement in the accuracy of ground-motion predictions over empirical prediction equations alone.

Calculating Dynamics Of Helicopters And Slung Loads

NASA Technical Reports Server (NTRS)

Cicolani, Luigi; Kanning, Gerd

1991-01-01

General equations derived for numerical simulations of motions of multiple-lift, slung-load systems consisting of two or more lifting helicopters and loads slung from them by various combinations of spreader bars, cables, nets, and attaching hardware. Equations readily programmable for efficient computation of motions and lend themselves well to analysis and design of control strategies for stabilization and coordination.

Effective actions for bosonic topological defects

NASA Technical Reports Server (NTRS)

Gregory, Ruth

1990-01-01

A gauge field theory is considered which admits p-dimensional topological defects, expanding the equations of motion in powers of the defect thickness. In this way an effective action and effective equation of motion is derived for the defect in terms of the coordinates of the p-dimensional worldsurface defined by the history of the core of the defect.

Swimming motion of rod-shaped magnetotactic bacteria: the effects of shape and growing magnetic moment

PubMed Central

Kong, Dali; Lin, Wei; Pan, Yongxin; Zhang, Keke

2014-01-01

We investigate the swimming motion of rod-shaped magnetotactic bacteria affiliated with the Nitrospirae phylum in a viscous liquid under the influence of an externally imposed, time-dependent magnetic field. By assuming that fluid motion driven by the translation and rotation of a swimming bacterium is of the Stokes type and that inertial effects of the motion are negligible, we derive a new system of the twelve coupled equations that govern both the motion and orientation of a swimming rod-shaped magnetotactic bacterium with a growing magnetic moment in the laboratory frame of reference. It is revealed that the initial pattern of swimming motion can be strongly affected by the rate of the growing magnetic moment. It is also revealed, through comparing mathematical solutions of the twelve coupled equations to the swimming motion observed in our laboratory experiments with rod-shaped magnetotactic bacteria, that the laboratory trajectories of the swimming motion can be approximately reproduced using an appropriate set of the parameters in our theoretical model. PMID:24523716

Visualization in Mechanics: The Dynamics of an Unbalanced Roller

ERIC Educational Resources Information Center

Cumber, Peter S.

2017-01-01

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical…

Solving Modal Equations of Motion with Initial Conditions Using MSC/NASTRAN DMAP. Part 2; Coupled Versus Uncoupled Integration

NASA Technical Reports Server (NTRS)

Barnett, Alan R.; Ibrahim, Omar M.; Abdallah, Ayman A.; Sullivan, Timothy L.

1993-01-01

By utilizing MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) in an existing NASA Lewis Research Center coupled loads methodology, solving modal equations of motion with initial conditions is possible using either coupled (Newmark-Beta) or uncoupled (exact mode superposition) integration available within module TRD1. Both the coupled and newly developed exact mode superposition methods have been used to perform transient analyses of various space systems. However, experience has shown that in most cases, significant time savings are realized when the equations of motion are integrated using the uncoupled solver instead of the coupled solver. Through the results of a real-world engineering analysis, advantages of using the exact mode superposition methodology are illustrated.

Roller Coasters without Differential Equations--A Newtonian Approach to Constrained Motion

ERIC Educational Resources Information Center

Muller, Rainer

2010-01-01

Within the context of Newton's equation, we present a simple approach to the constrained motion of a body forced to move along a specified trajectory. Because the formalism uses a local frame of reference, it is simpler than other methods, making more complicated geometries accessible. No Lagrangian multipliers are necessary to determine the…

Global geometry of non-planar 3-body motions

NASA Astrophysics Data System (ADS)

Salehani, Mahdi Khajeh

2011-12-01

The aim of this paper is to study the global geometry of non-planar 3-body motions in the realms of equivariant Differential Geometry and Geometric Mechanics. This work was intended as an attempt at bringing together these two areas, in which geometric methods play the major role, in the study of the 3-body problem. It is shown that the Euler equations of a three-body system with non-planar motion introduce non-holonomic constraints into the Lagrangian formulation of mechanics. Applying the method of undetermined Lagrange multipliers to study the dynamics of three-body motions reduced to the level of moduli space {bar{M}} subject to the non-holonomic constraints yields the generalized Euler-Lagrange equations of non-planar three-body motions in {bar{M}} . As an application of the derived dynamical equations in the level of {bar{M}} , we completely settle the question posed by A. Wintner in his book [The analytical foundations of Celestial Mechanics, Sections 394-396, 435 and 436. Princeton University Press (1941)] on classifying the constant inclination solutions of the three-body problem.

Laminar Motion of the Incompressible Fluids in Self-Acting Thrust Bearings with Spiral Grooves

PubMed Central

Velescu, Cornel; Popa, Nicolae Calin

2014-01-01

We analyze the laminar motion of incompressible fluids in self-acting thrust bearings with spiral grooves with inner or external pumping. The purpose of the study is to find some mathematical relations useful to approach the theoretical functionality of these bearings having magnetic controllable fluids as incompressible fluids, in the presence of a controllable magnetic field. This theoretical study approaches the permanent motion regime. To validate the theoretical results, we compare them to some experimental results presented in previous papers. The laminar motion of incompressible fluids in bearings is described by the fundamental equations of fluid dynamics. We developed and particularized these equations by taking into consideration the geometrical and functional characteristics of these hydrodynamic bearings. Through the integration of the differential equation, we determined the pressure and speed distributions in bearings with length in the “pumping” direction. These pressure and speed distributions offer important information, both quantitative (concerning the bearing performances) and qualitative (evidence of the viscous-inertial effects, the fluid compressibility, etc.), for the laminar and permanent motion regime. PMID:24526896

Laminar motion of the incompressible fluids in self-acting thrust bearings with spiral grooves.

PubMed

Velescu, Cornel; Popa, Nicolae Calin

2014-01-01

We analyze the laminar motion of incompressible fluids in self-acting thrust bearings with spiral grooves with inner or external pumping. The purpose of the study is to find some mathematical relations useful to approach the theoretical functionality of these bearings having magnetic controllable fluids as incompressible fluids, in the presence of a controllable magnetic field. This theoretical study approaches the permanent motion regime. To validate the theoretical results, we compare them to some experimental results presented in previous papers. The laminar motion of incompressible fluids in bearings is described by the fundamental equations of fluid dynamics. We developed and particularized these equations by taking into consideration the geometrical and functional characteristics of these hydrodynamic bearings. Through the integration of the differential equation, we determined the pressure and speed distributions in bearings with length in the "pumping" direction. These pressure and speed distributions offer important information, both quantitative (concerning the bearing performances) and qualitative (evidence of the viscous-inertial effects, the fluid compressibility, etc.), for the laminar and permanent motion regime.

Experimental and analytical investigation of inertial propulsion mechanisms and motion simulation of rigid multi-body mechanical systems

NASA Astrophysics Data System (ADS)

Almesallmy, Mohammed

Methodologies are developed for dynamic analysis of mechanical systems with emphasis on inertial propulsion systems. This work adopted the Lagrangian methodology. Lagrangian methodology is the most efficient classical computational technique, which we call Equations of Motion Code (EOMC). The EOMC is applied to several simple dynamic mechanical systems for easier understanding of the method and to aid other investigators in developing equations of motion of any dynamic system. In addition, it is applied to a rigid multibody system, such as Thomson IPS [Thomson 1986]. Furthermore, a simple symbolic algorithm is developed using Maple software, which can be used to convert any nonlinear n-order ordinary differential equation (ODE) systems into 1st-order ODE system in ready format to be used in Matlab software. A side issue, but equally important, we have started corresponding with the U.S. Patent office to persuade them that patent applications, claiming gross linear motion based on inertial propulsion systems should be automatically rejected. The precedent is rejection of patent applications involving perpetual motion machines.

A linearized Euler analysis of unsteady flows in turbomachinery

NASA Technical Reports Server (NTRS)

Hall, Kenneth C.; Crawley, Edward F.

1987-01-01

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

NASA Technical Reports Server (NTRS)

Clark, William S.; Hall, Kenneth C.

1994-01-01

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

NASA Technical Reports Server (NTRS)

Liu, J. J. F.; Fitzpatrick, P. M.

1975-01-01

A mathematical model is developed for studying the effects of gravity gradient torque on the attitude stability of a tumbling triaxial rigid satellite. Poisson equations are used to investigate the rotation of the satellite (which is in elliptical orbit about an attracting point mass) about its center of mass. An averaging method is employed to obtain an intermediate set of differential equations for the nonresonant, secular behavior of the osculating elements which describe the rotational motions of the satellite, and the averaged equations are then integrated to obtain long-term secular solutions for the osculating elements.

Identification and control of structures in space

NASA Technical Reports Server (NTRS)

Meirovitch, L.; Quinn, R. D.; Norris, M. A.

1984-01-01

The derivation of the equations of motion for the Spacecraft Control Laboratory Experiment (SCOLE) is reported and the equations of motion of a similar structure orbiting the earth are also derived. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is proposed whereby the quantities defining the rigid-body maneuver are assumed to be relatively large, with the elastic deformations and deviations from the rigid-body maneuver being relatively small. The perturbation equations have the form of linear equations with time-dependent coefficients. An active control technique can then be formulated to permit maneuvering of the spacecraft and simultaneously suppressing the elastic vibration.

On the theory of Brownian motion with the Alder-Wainwright effect

NASA Astrophysics Data System (ADS)

Okabe, Yasunori

1986-12-01

The Stokes-Boussinesq-Langevin equation, which describes the time evolution of Brownian motion with the Alder-Wainwright effect, can be treated in the framework of the theory of KMO-Langevin equations which describe the time evolution of a real, stationary Gaussian process with T-positivity (reflection positivity) originating in axiomatic quantum field theory. After proving the fluctuation-dissipation theorems for KMO-Langevin equations, we obtain an explicit formula for the deviation from the classical Einstein relation that occurs in the Stokes-Boussinesq-Langevin equation with a white noise as its random force. We are interested in whether or not it can be measured experimentally.

GVE-Based Dynamics and Control for Formation Flying Spacecraft

NASA Technical Reports Server (NTRS)

Breger, Louis; How, Jonathan P.

2004-01-01

Formation flying is an enabling technology for many future space missions. This paper presents extensions to the equations of relative motion expressed in Keplerian orbital elements, including new initialization techniques for general formation configurations. A new linear time-varying form of the equations of relative motion is developed from Gauss Variational Equations and used in a model predictive controller. The linearizing assumptions for these equations are shown to be consistent with typical formation flying scenarios. Several linear, convex initialization techniques are presented, as well as a general, decentralized method for coordinating a tetrahedral formation using differential orbital elements. Control methods are validated using a commercial numerical propagator.

Transformation of Elastic Wave Energy to the Energy of Motion of Bodies

NASA Astrophysics Data System (ADS)

Vesnitskiĭ, A. I.; Lisenkova, E. E.

2002-01-01

The motion of a body along an elastic guide under the effect of an incident wave is considered. An equation describing the longitudinal motion of a body along an arbitrary guide is derived from the laws governing the energy and momentum variations for the case when the incident wave generates a single reflected wave. The equations that describe the motion of a body along a string and along a beam corresponding to the Bernoulli-Euler model are considered as examples. The process of the body acceleration along a beam of the aforementioned type is investigated. For the subcritical velocities, the law governing the motion of the body and the ratio of the kinetic energy variation to the energy supplied to the body are determined.

The mechanical and chemical equations of motion of muscle contraction

NASA Astrophysics Data System (ADS)

Shiner, J. S.; Sieniutycz, Stanislaw

1997-11-01

Up to now no formulation of muscle contraction has provided both the chemical kinetic equations for the reactions responsible for the contraction and the mechanical equation of motion for the muscle. This has most likely been due to the lack of general formalisms for nonlinear systems with chemical-nonchemical coupling valid under the far from equilibrium conditions under which muscle operates physiologically. We have recently developed such formalisms and apply them here to the formulation of muscle contraction to obtain both the chemical and the mechanical equations. The standard formulation up to now has yielded only the dynamic equations for the chemical variables and has considered these to be functions of both time and an appropriate mechanical variable. The macroscopically observable quantities were then obtained by averaging over the mechanical variable. When attempting to derive the dynamics equations for both the chemistry and mechanics this choice of variables leads to conflicting results for the mechanical equation of motion when two different general formalisms are applied. The conflict can be resolved by choosing the variables such that both the chemical variables and the mechanical variables are considered to be functions of time alone. This adds one equation to the set of differential equations to be solved but is actually a simplification of the problem, since these equations are ordinary differential equations, not the partial differential equations of the now standard formulation, and since in this choice of variables the variables themselves are the macroscopic observables the procedure of averaging over the mechanical variable is eliminated. Furthermore, the parameters occurring in the equations at this level of description should be accessible to direct experimental determination.

Earth-moon system: Dynamics and parameter estimation

NASA Technical Reports Server (NTRS)

Breedlove, W. J., Jr.

1975-01-01

A theoretical development of the equations of motion governing the earth-moon system is presented. The earth and moon were treated as finite rigid bodies and a mutual potential was utilized. The sun and remaining planets were treated as particles. Relativistic, non-rigid, and dissipative effects were not included. The translational and rotational motion of the earth and moon were derived in a fully coupled set of equations. Euler parameters were used to model the rotational motions. The mathematical model is intended for use with data analysis software to estimate physical parameters of the earth-moon system using primarily LURE type data. Two program listings are included. Program ANEAMO computes the translational/rotational motion of the earth and moon from analytical solutions. Program RIGEM numerically integrates the fully coupled motions as described above.

The use of the logistic model in space motion sickness prediction

NASA Technical Reports Server (NTRS)

Lin, Karl K.; Reschke, Millard F.

1987-01-01

The one-equation and the two-equation logistic models were used to predict subjects' susceptibility to motion sickness in KC-135 parabolic flights using data from other ground-based motion sickness tests. The results show that the logistic models correctly predicted substantially more cases (an average of 13 percent) in the data subset used for model building. Overall, the logistic models ranged from 53 to 65 percent predictions of the three endpoint parameters, whereas the Bayes linear discriminant procedure ranged from 48 to 65 percent correct for the cross validation sample.

On the synchrotron radiation reaction in external magnetic field

NASA Astrophysics Data System (ADS)

Tursunov, Arman; Kološ, Martin

2017-12-01

We study the dynamics of point electric charges undergoing radiation reaction force due to synchrotron radiation in the presence of external uniform magnetic field. The radiation reaction force cannot be neglected in many physical situations and its presence modifies the equations of motion significantly. The exact form of the equation of motion known as the Lorentz-Dirac equation contains higher order Schott term which leads to the appearance of the runaway solutions. We demonstrate effective computational ways to avoid such unphysical solutions and perform numerical integration of the dynamical equations. We show that in the ultrarelativistic case the Schott term is small and does not have considerable effect to the trajectory of a particle. We compare results with the covariant Landau-Lifshitz equation which is the first iteration of the Lorentz-Dirac equation. Even though the Landau-Lifshitz equation is thought to be approximative solution, we show that in realistic scenarios both approaches lead to identical results.

An Approach for Dynamic Grids

NASA Technical Reports Server (NTRS)

Slater, John W.; Liou, Meng-Sing; Hindman, Richard G.

1994-01-01

An approach is presented for the generation of two-dimensional, structured, dynamic grids. The grid motion may be due to the motion of the boundaries of the computational domain or to the adaptation of the grid to the transient, physical solution. A time-dependent grid is computed through the time integration of the grid speeds which are computed from a system of grid speed equations. The grid speed equations are derived from the time-differentiation of the grid equations so as to ensure that the dynamic grid maintains the desired qualities of the static grid. The grid equations are the Euler-Lagrange equations derived from a variational statement for the grid. The dynamic grid method is demonstrated for a model problem involving boundary motion, an inviscid flow in a converging-diverging nozzle during startup, and a viscous flow over a flat plate with an impinging shock wave. It is shown that the approach is more accurate for transient flows than an approach in which the grid speeds are computed using a finite difference with respect to time of the grid. However, the approach requires significantly more computational effort.

FAST TRACK COMMUNICATION: Semiclassical Klein Kramers and Smoluchowski equations for the Brownian motion of a particle in an external potential

NASA Astrophysics Data System (ADS)

Coffey, W. T.; Kalmykov, Yu P.; Titov, S. V.; Mulligan, B. P.

2007-01-01

The quantum Brownian motion of a particle in an external potential V(x) is treated using the master equation for the Wigner distribution function W(x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(planck4) and in the classical limit, planck → 0, reduces to the Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation for W(x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.

Motion of a pendulum with damping and vibrating axis of suspension at unconventional values of parameters

NASA Astrophysics Data System (ADS)

Demidov, Ivan; Sorokin, Vladislav

2018-05-01

Motion of a pendulum with damping and vibrating axis of suspension is considered at unconventional values of parameters. Case when the frequency of external loading and the natural frequency of the pendulum in the absence of this loading are of the same order is studied. Vibration intensity is assumed to be relatively low. In this case, the corresponding equation of the pendulum's motions doesn't involve an explicit small parameter. To solve the equation a new modification of the method of direct separation of motions is used. As the result, stability conditions of the pendulum inverted position are determined. Effects of damping on these conditions are discussed.

A Study of Shuttlecock's Trajectory in Badminton.

PubMed

Chen, Lung-Ming; Pan, Yi-Hsiang; Chen, Yung-Jen

2009-01-01

The main purpose of this study was to construct and validate a motion equation for the flight of the badminton and to find the relationship between the air resistance force and a shuttlecock's speed. This research method was based on motion laws of aerodynamics. It applied aerodynamic theories to construct motion equation of a shuttlecock's flying trajectory under the effects of gravitational force and air resistance force. The result showed that the motion equation of a shuttlecock's flight trajectory could be constructed by determining the terminal velocity. The predicted shuttlecock trajectory fitted the measured data fairly well. The results also revealed that the drag force was proportional to the square of a shuttlecock velocity. Furthermore, the angle and strength of a stroke could influence trajectory. Finally, this study suggested that we could use a scientific approach to measure a shuttlecock's velocity objectively when testing the quality of shuttlecocks. And could be used to replace the traditional subjective method of the Badminton World Federation based on players' striking shuttlecocks, as well as applying research findings to improve professional knowledge of badminton player training. Key pointsThe motion equation of a shuttlecock's flying trajectory could be constructed by determining the terminal velocity in aerodynamics.Air drag force is proportional to the square of a shuttlecock velocity. Furthermore, the angle and strength of a stroke could influence trajectory.

Nonconservative Lagrangian Mechanics: Purely Causal Equations of Motion

NASA Astrophysics Data System (ADS)

Dreisigmeyer, David W.; Young, Peter M.

2015-06-01

This work builds on the Volterra series formalism presented in Dreisigmeyer and Young (J Phys A 36: 8297, 2003) to model nonconservative systems. Here we treat Lagrangians and actions as `time dependent' Volterra series. We present a new family of kernels to be used in these Volterra series that allow us to derive a single retarded equation of motion using a variational principle.

Nonlinear flight dynamics and stability of hovering model insects

PubMed Central

Liang, Bin; Sun, Mao

2013-01-01

Current analyses on insect dynamic flight stability are based on linear theory and limited to small disturbance motions. However, insects' aerial environment is filled with swirling eddies and wind gusts, and large disturbances are common. Here, we numerically solve the equations of motion coupled with the Navier–Stokes equations to simulate the large disturbance motions and analyse the nonlinear flight dynamics of hovering model insects. We consider two representative model insects, a model hawkmoth (large size, low wingbeat frequency) and a model dronefly (small size, high wingbeat frequency). For small and large initial disturbances, the disturbance motion grows with time, and the insects tumble and never return to the equilibrium state; the hovering flight is inherently (passively) unstable. The instability is caused by a pitch moment produced by forward/backward motion and/or a roll moment produced by side motion of the insect. PMID:23697714

Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s

USGS Publications Warehouse

Boore, D.M.; Atkinson, G.M.

2008-01-01

This paper contains ground-motion prediction equations (GMPEs) for average horizontal-component ground motions as a function of earthquake magnitude, distance from source to site, local average shear-wave velocity, and fault type. Our equations are for peak ground acceleration (PGA), peak ground velocity (PGV), and 5%-damped pseudo-absolute-acceleration spectra (PSA) at periods between 0.01 s and 10 s. They were derived by empirical regression of an extensive strong-motion database compiled by the 'PEER NGA' (Pacific Earthquake Engineering Research Center's Next Generation Attenuation) project. For periods less than 1 s, the analysis used 1,574 records from 58 mainshocks in the distance range from 0 km to 400 km (the number of available data decreased as period increased). The primary predictor variables are moment magnitude (M), closest horizontal distance to the surface projection of the fault plane (RJB), and the time-averaged shear-wave velocity from the surface to 30 m (VS30). The equations are applicable for M=5-8, RJB<200 km, and VS30= 180-1300 m/s. ?? 2008, Earthquake Engineering Research Institute.

Development of a non-linear simulation for generic hypersonic vehicles - ASUHS1

NASA Technical Reports Server (NTRS)

Salas, Juan; Lovell, T. Alan; Schmidt, David K.

1993-01-01

A nonlinear simulation is developed to model the longitudinal motion of a vehicle in hypersonic flight. The equations of motion pertinent to this study are presented. Analytic expressions for the aerodynamic forces acting on a hypersonic vehicle which were obtained from Newtonian Impact Theory are further developed. The control surface forces are further examined to incorporate vehicle elastic motion. The purpose is to establish feasible equations of motion which combine rigid body, elastic, and aeropropulsive dynamics for use in nonlinear simulations. The software package SIMULINK is used to implement the simulation. Also discussed are issues needing additional attention and potential problems associated with the implementation (with proposed solutions).

Schrödinger–Langevin equation with quantum trajectories for photodissociation dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chou, Chia-Chun, E-mail: ccchou@mx.nthu.edu.tw

The Schrödinger–Langevin equation is integrated to study the wave packet dynamics of quantum systems subject to frictional effects by propagating an ensemble of quantum trajectories. The equations of motion for the complex action and quantum trajectories are derived from the Schrödinger–Langevin equation. The moving least squares approach is used to evaluate the spatial derivatives of the complex action required for the integration of the equations of motion. Computational results are presented and analyzed for the evolution of a free Gaussian wave packet, a two-dimensional barrier model, and the photodissociation dynamics of NOCl. The absorption spectrum of NOCl obtained from themore » Schrödinger–Langevin equation displays a redshift when frictional effects increase. This computational result agrees qualitatively with the experimental results in the solution-phase photochemistry of NOCl.« less

Numerical implementation of equations for photon motion in Kerr spacetime

NASA Astrophysics Data System (ADS)

Bursa, Michal

2017-12-01

Raytracing is one of the essential tools for accurate modeling of spectra and variability of various astrophysical objects. It has a major importance in relativistic environments, where light endures to a number of relativistic effects. Because the trajectories of light rays in curved spacetimes, and in Kerr spacetime in particular, are highly non-trivial, we summarize the equations governing the motion of photon (or any other zero rest mass particle) and give analytic solution of the equations that can be further used in practical computer implementations.

Radiation reaction on a classical charged particle: a modified form of the equation of motion.

PubMed

Alcaine, Guillermo García; Llanes-Estrada, Felipe J

2013-09-01

We present and numerically solve a modified form of the equation of motion for a charged particle under the influence of an external force, taking into account the radiation reaction. This covariant equation is integro-differential, as Dirac-Röhrlich's, but has several technical improvements. First, the equation has the form of Newton's second law, with acceleration isolated on the left hand side and the force depending only on positions and velocities: Thus, the equation is linear in the highest derivative. Second, the total four-force is by construction perpendicular to the four-velocity. Third, if the external force vanishes for all future times, the total force and the acceleration automatically vanish at the present time. We show the advantages of this equation by solving it numerically for several examples of external force.

Radiation reaction on a classical charged particle: A modified form of the equation of motion

NASA Astrophysics Data System (ADS)

Alcaine, Guillermo García; Llanes-Estrada, Felipe J.

2013-09-01

We present and numerically solve a modified form of the equation of motion for a charged particle under the influence of an external force, taking into account the radiation reaction. This covariant equation is integro-differential, as Dirac-Röhrlich's, but has several technical improvements. First, the equation has the form of Newton's second law, with acceleration isolated on the left hand side and the force depending only on positions and velocities: Thus, the equation is linear in the highest derivative. Second, the total four-force is by construction perpendicular to the four-velocity. Third, if the external force vanishes for all future times, the total force and the acceleration automatically vanish at the present time. We show the advantages of this equation by solving it numerically for several examples of external force.

Turbulent fluid motion IV-averages, Reynolds decomposition, and the closure problem

NASA Technical Reports Server (NTRS)

Deissler, Robert G.

1992-01-01

Ensemble, time, and space averages as applied to turbulent quantities are discussed, and pertinent properties of the averages are obtained. Those properties, together with Reynolds decomposition, are used to derive the averaged equations of motion and the one- and two-point moment or correlation equations. The terms in the various equations are interpreted. The closure problem of the averaged equations is discussed, and possible closure schemes are considered. Those schemes usually require an input of supplemental information unless the averaged equations are closed by calculating their terms by a numerical solution of the original unaveraged equations. The law of the wall for velocities and temperatures, the velocity- and temperature-defect laws, and the logarithmic laws for velocities and temperatures are derived. Various notions of randomness and their relation to turbulence are considered in light of ergodic theory.

Nonlinear model of a rotating hub-beams structure: Equations of motion

NASA Astrophysics Data System (ADS)

Warminski, Jerzy

2018-01-01

Dynamics of a rotating structure composed of a rigid hub and flexible beams is presented in the paper. A nonlinear model of a beam takes into account bending, extension and nonlinear curvature. The influence of geometric nonlinearity and nonconstant angular velocity on dynamics of the rotating structure is presented. The exact equations of motion and associated boundary conditions are derived on the basis of the Hamilton's principle. The simplification of the exact nonlinear mathematical model is proposed taking into account the second order approximation. The reduced partial differential equations of motion together with associated boundary conditions can be used to study natural or forced vibrations of a rotating structure considering constant or nonconstant angular speed of a rigid hub and an arbitrary number of flexible blades.

Prediction and experimental observation of damage dependent damping in laminated composite beams

NASA Technical Reports Server (NTRS)

Allen, D. H.; Harris, C. E.; Highsmith, A. L.

1987-01-01

The equations of motion are developed for laminated composite beams with load-induced matrix cracking. The damage is accounted for by utilizing internal state variables. The net result of these variables on the field equations is the introduction of both enhanced damping, and degraded stiffness. Both quantities are history dependent and spatially variable, thus resulting in nonlinear equations of motion. It is explained briefly how these equations may be quasi-linearized for laminated polymeric composites under certain types of structural loading. The coupled heat conduction equation is developed, and it is shown that an enhanced Zener damping effect is produced by the introduction of microstructural damage. The resulting equations are utilized to demonstrate how damage dependent material properties may be obtained from dynamic experiments. Finaly, experimental results are compared to model predictions for several composite layups.

Bound Motion of Bodies and Paticles in the Rotating Systems

NASA Astrophysics Data System (ADS)

Pardy, Miroslav

2007-04-01

The Lagrange theory of particle motion in the noninertial systems is applied to the Foucault pendulum, isosceles triangle pendulum and the general triangle pendulum swinging on the rotating Earth. As an analogue, planet orbiting in the rotating galaxy is considered as the giant galactic gyroscope. The Lorentz equation and the Bargmann-Michel-Telegdi equations are generalized for the rotation system. The knowledge of these equations is inevitable for the construction of LHC where each orbital proton “feels” the Coriolis force caused by the rotation of the Earth.

Global Aspects of Charged Particle Motion in Axially Symmetric Multipole Magnetic Fields

NASA Technical Reports Server (NTRS)

Shebalin, John V.

2003-01-01

The motion of a single charged particle in the space outside of a compact region of steady currents is investigated. The charged particle is assumed to produce negligible electromagnetic radiation, so that its energy is conserved. The source of the magnetic field is represented as a point multipole. After a general description, attention is focused on magnetic fields with axial symmetry. Lagrangian dynamical theory is utilized to identify constants of the motion as well as the equations of motion themselves. The qualitative method of Stonner is used to examine charged particle motion in axisymmetric multipole fields of all orders. Although the equations of motion generally have no analytical solutions and must be integrated numerically to produce a specific orbit, a topological examination of dynamics is possible, and can be used, d la Stonner, to completely describe the global aspects of the motion of a single charged particle in a space with an axisymmetric multipole magnetic field.

Bloch equations applied to ion cyclotron resonance spectroscopy: Broadband interconversion between magnetron and cyclotron motion for ion axialization

NASA Astrophysics Data System (ADS)

Guan, Shenheng; Marshall, Alan G.

1993-03-01

Conversion of magnetron motion to cyclotron motion combined with collisional cooling of the cyclotron motion provides an efficient way to reduce the kinetic energy of trapped heavy ions and to reduce their magnetron radii in an ion cyclotron resonance (ICR) ion trap. The coupling of magnetron and cyclotron motion can be realized by azimuthal quadrupolar excitation. Theoretical understanding of the coupling process has until now been based on resonant single-frequency quadrupolar excitation at the combination frequency ωc=ω++ω-, in which ωc is the ion cyclotron orbital frequency in the absence of electrostatic field; and ω+ and ω- are the reduced cyclotron and magnetron frequencies in the presence of an electrostatic trapping potential. In this work, we prove that the magnetron/cyclotron coupling is closely related to a two energy level system whose behavior is described by the well-known Bloch equations. By means of a special transformation, the equations of motion for the coupling may be expressed in Bloch-type equations in spherical coordinates. We show that magnetron-to-cyclotron conversion by single-frequency quadrupolar excitation in ICR is analogous to a 180° pulse in nuclear magnetic resonance (NMR). We go on to show that simultaneous magnetron-to-cyclotron conversion of ions over a finite mass-to-charge ratio range may be produced by quadrupolar frequency-sweep excitation, by analogy to adiabatic rapid passage in magnetic resonance. Axialization by broadband magnetron-to-cyclotron conversion followed by cyclotron cooling is successfully demonstrated experimentally for a crude oil distillate sample.

Modeling the Benchmark Active Control Technology Wind-Tunnel Model for Active Control Design Applications

NASA Technical Reports Server (NTRS)

Waszak, Martin R.

1998-01-01

This report describes the formulation of a model of the dynamic behavior of the Benchmark Active Controls Technology (BACT) wind tunnel model for active control design and analysis applications. The model is formed by combining the equations of motion for the BACT wind tunnel model with actuator models and a model of wind tunnel turbulence. The primary focus of this report is the development of the equations of motion from first principles by using Lagrange's equations and the principle of virtual work. A numerical form of the model is generated by making use of parameters obtained from both experiment and analysis. Comparisons between experimental and analytical data obtained from the numerical model show excellent agreement and suggest that simple coefficient-based aerodynamics are sufficient to accurately characterize the aeroelastic response of the BACT wind tunnel model. The equations of motion developed herein have been used to aid in the design and analysis of a number of flutter suppression controllers that have been successfully implemented.

Flap-lag-torsional dynamic modelling of rotor blades in hover and in forward flight, including the effect of cubic nonlinearities

NASA Technical Reports Server (NTRS)

Crespodasilva, M. R. M.

1981-01-01

The differential equations of motion, and boundary conditions, describing the flap-lead/lag-torsional motion of a flexible rotor blade with a precone angle and a variable pitch angle, which incorporates a pretwist, are derived via Hamilton's principle. The meaning of inextensionality is discussed. The equations are reduced to a set of three integro partial differential equations by elimination of the extension variable. The generalized aerodynamic forces are modelled using Greenberg's extension of Theodorsen's strip theory. The equations of motion are systematically expanded into polynomial nonlinearities with the objective of retaining all terms up to third degree. The blade is modeled as a long, slender, of isotropic Hookean materials. Offsets from the blade's elastic axis through its shear center and the axes for the mass, area and aerodynamic centers, radial nonuniformaties of the blade's stiffnesses and cross section properties are considered and the effect of warp of the cross section is included in the formulation.

Planar dynamics of a uniform beam with rigid bodies affixed to the ends

NASA Technical Reports Server (NTRS)

Storch, J.; Gates, S.

1983-01-01

The planar dynamics of a uniform elastic beam subject to a variety of geometric and natural boundary conditions and external excitations were analyzed. The beams are inextensible and capable of small transverse bending deformations only. Classical beam vibration eigenvalue problems for a cantilever with tip mass, a cantilever with tip body and an unconstrained beam with rigid bodies at each are examined. The characteristic equations, eigenfunctions and orthogonality relations for each are derived. The forced vibration of a cantilever with tip body subject to base acceleration is analyzed. The exact solution of the governing nonhomogeneous partial differential equation with time dependent boundary conditions is presented and compared with a Rayleigh-Ritz approximate solution. The arbitrary planar motion of an elastic beam with rigid bodies at the ends is addressed. Equations of motion are derived for two modal expansions of the beam deflection. The motion equations are cast in a first order form suitable for numerical integration. Selected FORTRAN programs are provided.

Macroscopic damping model for structural dynamics with random polycrystalline configurations

NASA Astrophysics Data System (ADS)

Yang, Yantao; Cui, Junzhi; Yu, Yifan; Xiang, Meizhen

2018-06-01

In this paper the macroscopic damping model for dynamical behavior of the structures with random polycrystalline configurations at micro-nano scales is established. First, the global motion equation of a crystal is decomposed into a set of motion equations with independent single degree of freedom (SDOF) along normal discrete modes, and then damping behavior is introduced into each SDOF motion. Through the interpolation of discrete modes, the continuous representation of damping effects for the crystal is obtained. Second, from energy conservation law the expression of the damping coefficient is derived, and the approximate formula of damping coefficient is given. Next, the continuous damping coefficient for polycrystalline cluster is expressed, the continuous dynamical equation with damping term is obtained, and then the concrete damping coefficients for a polycrystalline Cu sample are shown. Finally, by using statistical two-scale homogenization method, the macroscopic homogenized dynamical equation containing damping term for the structures with random polycrystalline configurations at micro-nano scales is set up.

Equation of Motion of an Interstellar Bussard Ramjet with Radiation and Mass Losses

ERIC Educational Resources Information Center

Semay, Claude; Silvestre-Brac, Bernard

2008-01-01

An interstellar Bussard ramjet is a spaceship using the protons of the interstellar medium in a fusion engine to produce thrust. In recent papers, it was shown that the relativistic equation of motion of an ideal ramjet and that of a ramjet with radiation loss are analytical. When a mass loss appears, the limit speed of the ramjet is more strongly…

Proposed solution methodology for the dynamically coupled nonlinear geared rotor mechanics equations

NASA Technical Reports Server (NTRS)

Mitchell, L. D.; David, J. W.

1983-01-01

The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.

Dynamics of flexible bodies in tree topology - A computer oriented approach

NASA Technical Reports Server (NTRS)

Singh, R. P.; Vandervoort, R. J.; Likins, P. W.

1984-01-01

An approach suited for automatic generation of the equations of motion for large mechanical systems (i.e., large space structures, mechanisms, robots, etc.) is presented. The system topology is restricted to a tree configuration. The tree is defined as an arbitrary set of rigid and flexible bodies connected by hinges characterizing relative translations and rotations of two adjoining bodies. The equations of motion are derived via Kane's method. The resulting equation set is of minimum dimension. Dynamical equations are imbedded in a computer program called TREETOPS. Extensive control simulation capability is built in the TREETOPS program. The simulation is driven by an interactive set-up program resulting in an easy to use analysis tool.

A computer code for three-dimensional incompressible flows using nonorthogonal body-fitted coordinate systems

NASA Technical Reports Server (NTRS)

Chen, Y. S.

1986-01-01

In this report, a numerical method for solving the equations of motion of three-dimensional incompressible flows in nonorthogonal body-fitted coordinate (BFC) systems has been developed. The equations of motion are transformed to a generalized curvilinear coordinate system from which the transformed equations are discretized using finite difference approximations in the transformed domain. The hybrid scheme is used to approximate the convection terms in the governing equations. Solutions of the finite difference equations are obtained iteratively by using a pressure-velocity correction algorithm (SIMPLE-C). Numerical examples of two- and three-dimensional, laminar and turbulent flow problems are employed to evaluate the accuracy and efficiency of the present computer code. The user's guide and computer program listing of the present code are also included.

An Inexpensive Mechanical Model for Projectile Motion

ERIC Educational Resources Information Center

Kagan, David

2011-01-01

As experienced physicists, we see the beauty and simplicity of projectile motion. It is merely the superposition of uniform linear motion along the direction of the initial velocity vector and the downward motion due to the constant acceleration of gravity. We see the kinematic equations as just the mathematical machinery to perform the…

Nonlinear Aeroelastic Equations of Motion of Twisted, Nonuniform, Flexible Horizontal-Axis Wind Turbine Blades

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.

1980-01-01

The second-degree nonlinear equations of motion for a flexible, twisted, nonuniform, horizontal axis wind turbine blade were developed using Hamilton's principle. A mathematical ordering scheme which was consistent with the assumption of a slender beam was used to discard some higher-order elastic and inertial terms in the second-degree nonlinear equations. The blade aerodynamic loading which was employed accounted for both wind shear and tower shadow and was obtained from strip theory based on a quasi-steady approximation of two-dimensional, incompressible, unsteady, airfoil theory. The resulting equations had periodic coefficients and were suitable for determining the aeroelastic stability and response of large horizontal-axis wind turbine blades.

User's manual for interactive LINEAR: A FORTRAN program to derive linear aircraft models

NASA Technical Reports Server (NTRS)

Antoniewicz, Robert F.; Duke, Eugene L.; Patterson, Brian P.

1988-01-01

An interactive FORTRAN program that provides the user with a powerful and flexible tool for the linearization of aircraft aerodynamic models is documented in this report. The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a user-supplied linear or nonlinear aerodynamic model. The nonlinear equations of motion used are six-degree-of-freedom equations with stationary atmosphere and flat, nonrotating earth assumptions. The system model determined by LINEAR consists of matrices for both the state and observation equations. The program has been designed to allow easy selection and definition of the state, control, and observation variables to be used in a particular model.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zubarev, N. M., E-mail: nick@iep.uran.ru; Zubareva, O. V.

The dynamics of a bubble in a dielectric liquid under the influence of a uniform external electric field is considered. It is shown that in the situation where the boundary motion is determined only by electrostatic forces, the special regime of fluid motion can be realized for which the velocity and electric field potentials are linearly related. In the two-dimensional case, the corresponding equations are reduced to an equation similar in structure to the well-known Laplacian growth equation, which, in turn, can be reduced to a finite number of ordinary differential equations. This allows us to obtain exact solutions formore » asymmetric bubble deformations resulting in the formation of a finite-time singularity (cusp)« less

Multiscale turbulence models based on convected fluid microstructure

NASA Astrophysics Data System (ADS)

Holm, Darryl D.; Tronci, Cesare

2012-11-01

The Euler-Poincaré approach to complex fluids is used to derive multiscale equations for computationally modeling Euler flows as a basis for modeling turbulence. The model is based on a kinematic sweeping ansatz (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zero-mean) motion of fluid parcels at the unresolved scales. Even in the simplest two-scale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modeling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods.

Identification of aerodynamic models for maneuvering aircraft

NASA Technical Reports Server (NTRS)

Chin, Suei; Lan, C. Edward

1990-01-01

Due to the requirement of increased performance and maneuverability, the flight envelope of a modern fighter is frequently extended to the high angle-of-attack regime. Vehicles maneuvering in this regime are subjected to nonlinear aerodynamic loads. The nonlinearities are due mainly to three-dimensional separated flow and concentrated vortex flow that occur at large angles of attack. Accurate prediction of these nonlinear airloads is of great importance in the analysis of a vehicle's flight motion and in the design of its flight control system. A satisfactory evaluation of the performance envelope of the aircraft may require a large number of coupled computations, one for each change in initial conditions. To avoid the disadvantage of solving the coupled flow-field equations and aircraft's motion equations, an alternate approach is to use a mathematical modeling to describe the steady and unsteady aerodynamics for the aircraft equations of motion. Aerodynamic forces and moments acting on a rapidly maneuvering aircraft are, in general, nonlinear functions of motion variables, their time rate of change, and the history of maneuvering. A numerical method was developed to analyze the nonlinear and time-dependent aerodynamic response to establish the generalized indicial function in terms of motion variables and their time rates of change.

A theoretical analysis of airplane longitudinal stability and control as affected by wind shear

NASA Technical Reports Server (NTRS)

Sherman, W. L.

1977-01-01

The longitudinal equations of motion with wind shear terms were used to analyze the stability and motions of a jet transport. A positive wind shear gives a decreasing head wind or changes a head wind into a tail wind. A negative wind shear gives a decreasing tail wind or changes a tail wind into a head wind. It was found that wind shear had very little effect on the short period mode and that negative wind shear, although it affected the phugoid, did not cause stability problems. On the other hand, it was found that positive wind shear can cause the phugoid to become aperiodic and unstable. In this case, a stability boundary for the phugoid was found that is valid for most aircraft at all flight speeds. Calculations of aircraft motions confirmed the results of the stability analysis. It was found that a flight path control automatic pilot and an airspeed control system provide good control in all types of wind shear. Appendixes give equations of motion that include the effects of downdrafts and updrafts and extend the longitudinal equations of motion for shear to six degrees of freedom.

Analysis of swimming motions.

NASA Technical Reports Server (NTRS)

Gallenstein, J.; Huston, R. L.

1973-01-01

This paper presents an analysis of swimming motion with specific attention given to the flutter kick, the breast-stroke kick, and the breast stroke. The analysis is completely theoretical. It employs a mathematical model of the human body consisting of frustrums of elliptical cones. Dynamical equations are written for this model including both viscous and inertia forces. These equations are then applied with approximated swimming strokes and solved numerically using a digital computer. The procedure is to specify the input of the swimming motion. The computer solution then provides the output displacement, velocity, and rotation or body roll of the swimmer.

Research on modeling and motion simulation of a spherical space robot with telescopic manipulator based on virtual prototype technology

NASA Astrophysics Data System (ADS)

Shi, Chengkun; Sun, Hanxu; Jia, Qingxuan; Zhao, Kailiang

2009-05-01

For realizing omni-directional movement and operating task of spherical space robot system, this paper describes an innovated prototype and analyzes dynamic characteristics of a spherical rolling robot with telescopic manipulator. Based on the Newton-Euler equations, the kinematics and dynamic equations of the spherical robot's motion are instructed detailedly. Then the motion simulations of the robot in different environments are developed with ADAMS. The simulation results validate the mathematics model of the system. And the dynamic model establishes theoretical basis for the latter job.

Slackline dynamics and the Helmholtz-Duffing oscillator

NASA Astrophysics Data System (ADS)

Athanasiadis, Panos J.

2018-01-01

Slacklining is a new, rapidly expanding sport, and understanding its physics is paramount for maximizing fun and safety. Yet, compared to other sports, very little has been published so far on slackline dynamics. The equations of motion describing a slackline are fundamentally nonlinear, and assuming linear elasticity, they lead to a form of the Duffing equation. Following this approach, characteristic examples of slackline motion are simulated, including trickline bouncing, leash falls and longline surfing. The time-dependent solutions of the differential equations describing the system are acquired by numerical integration. A simple form of energy dissipation (linear drag) is added in some cases. It is recognized in this study that geometric nonlinearity is a fundamental aspect characterizing the dynamics of slacklines. Sports, and particularly slackline, is an excellent way of engaging young people with physics. A slackline is a simple yet insightful example of a nonlinear oscillator. It is very easy to model in the laboratory, as well as to rig and try on a university campus. For instructive purposes, its behaviour can be explored by numerically integrating the respective equations of motion. A form of the Duffing equation emerges naturally in the analysis and provides a powerful introduction to nonlinear dynamics. The material is suitable for graduate students and undergraduates with a background in classical mechanics and differential equations.

A simple method for calculating the characteristics of the Dutch roll motion of an airplane

NASA Technical Reports Server (NTRS)

Klawans, Bernard B

1956-01-01

A simple method for calculating the characteristics of the Dutch roll motion of an airplane is obtained by arranging the lateral equations of motion in such form and order that an iterative process is quickly convergent.

Atwood and Poggendorff: an insightful analogy

NASA Astrophysics Data System (ADS)

Coelho, R. L.; Borges, P. F.; Karam, R.

2016-11-01

Atwood’s treatise, in which the Atwood machine appears, was published in 1784. About 70 years later, Poggendorff showed experimentally that the weight of an Atwood machine is reduced when it is brought to motion. In the present paper, a twofold connection between this experiment and the Atwood machine is established. Firstly, if the Poggendorff apparatus is taken as an ideal one, the equations of motion of the apparatus coincide with the equations of motion of the compound Atwood machine. Secondly, if the Poggendorff apparatus, which works as a lever, is rebalanced, the equations of this equilibrium provide us with the solution for a compound Atwood machine with the same bodies. This analogy is pedagogically useful because it illustrates a common strategy to transform a dynamic in a static situation improving students’ comprehension of Newton’s laws and equilibrium.

User's Guide for ECAP2D: an Euler Unsteady Aerodynamic and Aeroelastic Analysis Program for Two Dimensional Oscillating Cascades, Version 1.0

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.

1995-01-01

This guide describes the input data required for using ECAP2D (Euler Cascade Aeroelastic Program-Two Dimensional). ECAP2D can be used for steady or unsteady aerodynamic and aeroelastic analysis of two dimensional cascades. Euler equations are used to obtain aerodynamic forces. The structural dynamic equations are written for a rigid typical section undergoing pitching (torsion) and plunging (bending) motion. The solution methods include harmonic oscillation method, influence coefficient method, pulse response method, and time integration method. For harmonic oscillation method, example inputs and outputs are provided for pitching motion and plunging motion. For the rest of the methods, input and output for pitching motion only are given.

Asymptotic analysis of quasilinear parabolic-hyperbolic equations describing the large longitudinal motion of a light viscoelastic bar with a heavy attachment

NASA Astrophysics Data System (ADS)

Yip, Shui Cheung

We study the longitudinal motion of a nonlinearly viscoelastic bar with one end fixed and the other end attached to a heavy tip mass. This problem is a precise continuum mechanical analog of the basic discrete mechanical problem of the motion of a mass point on a (massless) spring. This motion is governed by an initial-boundary-value problem for a class of third-order quasilinear parabolic-hyperbolic partial differential equations subject to a nonstandard boundary condition, which is the equation of motion of the tip mass. The ratio of the mass of the bar to that of the tip mass is taken to be a small parameter varepsilon. We prove that this problem has a unique regular solution that admits a valid asymptotic expansion, including an initial-layer expansion, in powers of varepsilon for varepsilon near 0. The fundamental constitutive hypothesis that the tension be a uniformly monotone function of the strain rate plays a critical role in a delicate proof that each term of the initial layer expansion decays exponentially in time. These results depend on new decay estimates for the solution of quasilinear parabolic equations. The constitutive hypothesis that the viscosity become large where the bar nears total compression leads to important uniform bounds for the strain and the strain rate. Higher-order energy estimates support the proof by the Schauder Fixed-Point Theorem of the existence of solutions having a level of regularity appropriate for the asymptotics.

Dynamics of omnidirectional unmanned rescue vehicle with mecanum wheels

NASA Astrophysics Data System (ADS)

Typiak, Andrzej; Łopatka, Marian Janusz; Rykała, Łukasz; Kijek, Magdalena

2018-01-01

The work presents the dynamic equations of motion of a unmanned six-wheeled vehicle with mecanum wheels for rescue applications derived with the of Lagrange equations of the second kind with multipliers. Analysed vehicle through using mecanum wheels has three degrees of freedom and can move on a flat ground in any direction with any configuration of platform's frame. In order to derive dynamic equations of motion of mentioned object, kinetic potential of the system and generalized forces affecting the system are determined. The results of a solution of inverse dynamics problem are also published.

Modelling of charged satellite motion in Earth's gravitational and magnetic fields

NASA Astrophysics Data System (ADS)

Abd El-Bar, S. E.; Abd El-Salam, F. A.

2018-05-01

In this work Lagrange's planetary equations for a charged satellite subjected to the Earth's gravitational and magnetic force fields are solved. The Earth's gravity, and magnetic and electric force components are obtained and expressed in terms of orbital elements. The variational equations of orbit with the considered model in Keplerian elements are derived. The solution of the problem in a fully analytical way is obtained. The temporal rate of changes of the orbital elements of the spacecraft are integrated via Lagrange's planetary equations and integrals of the normalized Keplerian motion obtained by Ahmed (Astron. J. 107(5):1900, 1994).

The Pendulum and the Calculus.

ERIC Educational Resources Information Center

Sworder, Steven C.

A pair of experiments, appropriate for the lower division fourth semester calculus or differential equations course, are presented. The second order differential equation representing the equation of motion of a simple pendulum is derived. The period of oscillation for a particular pendulum can be predicted from the solution to this equation. As a…

Conformational statistics of stiff macromolecules as solutions to partial differential equations on the rotation and motion groups

PubMed

Chirikjian; Wang

2000-07-01

Partial differential equations (PDE's) for the probability density function (PDF) of the position and orientation of the distal end of a stiff macromolecule relative to its proximal end are derived and solved. The Kratky-Porod wormlike chain, the Yamakawa helical wormlike chain, and the original and revised Marko-Siggia models are examples of stiffness models to which the present formulation is applied. The solution technique uses harmonic analysis on the rotation and motion groups to convert PDE's governing the PDF's of interest into linear algebraic equations which have mathematically elegant solutions.

Recursive linearization of multibody dynamics equations of motion

NASA Technical Reports Server (NTRS)

Lin, Tsung-Chieh; Yae, K. Harold

1989-01-01

The equations of motion of a multibody system are nonlinear in nature, and thus pose a difficult problem in linear control design. One approach is to have a first-order approximation through the numerical perturbations at a given configuration, and to design a control law based on the linearized model. Here, a linearized model is generated analytically by following the footsteps of the recursive derivation of the equations of motion. The equations of motion are first written in a Newton-Euler form, which is systematic and easy to construct; then, they are transformed into a relative coordinate representation, which is more efficient in computation. A new computational method for linearization is obtained by applying a series of first-order analytical approximations to the recursive kinematic relationships. The method has proved to be computationally more efficient because of its recursive nature. It has also turned out to be more accurate because of the fact that analytical perturbation circumvents numerical differentiation and other associated numerical operations that may accumulate computational error, thus requiring only analytical operations of matrices and vectors. The power of the proposed linearization algorithm is demonstrated, in comparison to a numerical perturbation method, with a two-link manipulator and a seven degrees of freedom robotic manipulator. Its application to control design is also demonstrated.

Derivation of equations of motion for multi-blade rotors employing coupled modes and including high twist capability

NASA Technical Reports Server (NTRS)

Sopher, R.

1975-01-01

The equations of motion are derived for a multiblade rotor. A high twist capability and coupled flatwise-edgewise assumed normal modes are employed instead of uncoupled flatwise - edgewise assumed normal models. The torsion mode is uncoupled. Support system models, consisting of complete helicopters in free flight, or grounded flexible supports, arbitrary rotor-induced inflow, and arbitrary vertical gust models are also used.

Differential equations in airplane mechanics

NASA Technical Reports Server (NTRS)

Carleman, M T

1922-01-01

In the following report, we will first draw some conclusions of purely theoretical interest, from the general equations of motion. At the end, we will consider the motion of an airplane, with the engine dead and with the assumption that the angle of attack remains constant. Thus we arrive at a simple result, which can be rendered practically utilizable for determining the trajectory of an airplane descending at a constant steering angle.

Contraction of high eccentricity satellite orbits using uniformly regular KS canonical elements with oblate diurnally varying atmosphere.

NASA Astrophysics Data System (ADS)

Raj, Xavier James

2016-07-01

Accurate orbit prediction of an artificial satellite under the influence of air drag is one of the most difficult and untraceable problem in orbital dynamics. The orbital decay of these satellites is mainly controlled by the atmospheric drag effects. The effects of the atmosphere are difficult to determine, since the atmospheric density undergoes large fluctuations. The classical Newtonian equations of motion, which is non linear is not suitable for long-term integration. Many transformations have emerged in the literature to stabilize the equations of motion either to reduce the accumulation of local numerical errors or allowing the use of large integration step sizes, or both in the transformed space. One such transformation is known as KS transformation by Kustaanheimo and Stiefel, who regularized the nonlinear Kepler equations of motion and reduced it into linear differential equations of a harmonic oscillator of constant frequency. The method of KS total energy element equations has been found to be a very powerful method for obtaining numerical as well as analytical solution with respect to any type of perturbing forces, as the equations are less sensitive to round off and truncation errors. The uniformly regular KS canonical equations are a particular canonical form of the KS differential equations, where all the ten KS Canonical elements αi and βi are constant for unperturbed motion. These equations permit the uniform formulation of the basic laws of elliptic, parabolic and hyperbolic motion. Using these equations, developed analytical solution for short term orbit predictions with respect to Earth's zonal harmonic terms J2, J3, J4. Further, these equations were utilized to include the canonical forces and analytical theories with air drag were developed for low eccentricity orbits (e < 0.2) with different atmospheric models. Using uniformly regular KS canonical elements developed analytical theory for high eccentricity (e > 0.2) orbits by assuming the atmosphere to be oblate only. In this paper a new non-singular analytical theory is developed for the motion of high eccentricity satellite orbits with oblate diurnally varying atmosphere in terms of the uniformly regular KS canonical elements. The analytical solutions are generated up to fourth-order terms using a new independent variable and c (a small parameter dependent on the flattening of the atmosphere). Due to symmetry, only two of the nine equations need to be solved analytically to compute the state vector and change in energy at the end of each revolution. The theory is developed on the assumption that density is constant on the surfaces of spheroids of fixed ellipticity ɛ (equal to the Earth's ellipticity, 0.00335) whose axes coincide with the Earth's axis. Numerical experimentation with the analytical solution for a wide range of perigee height, eccentricity, and orbital inclination has been carried out up to 100 revolutions. Comparisons are made with numerically integrated values and found that they match quite well. Effectiveness of the present analytical solutions will be demonstrated by comparing the results with other analytical solutions in the literature.

Applying Boundary Conditions Using a Time-Dependent Lagrangian for Modeling Laser-Plasma Interactions

NASA Astrophysics Data System (ADS)

Reyes, Jonathan; Shadwick, B. A.

2016-10-01

Modeling the evolution of a short, intense laser pulse propagating through an underdense plasma is of particular interest in the physics of laser-plasma interactions. Numerical models are typically created by first discretizing the equations of motion and then imposing boundary conditions. Using the variational principle of Chen and Sudan, we spatially discretize the Lagrangian density to obtain discrete equations of motion and a discrete energy conservation law which is exactly satisfied regardless of the spatial grid resolution. Modifying the derived equations of motion (e.g., enforcing boundary conditions) generally ruins energy conservation. However, time-dependent terms can be added to the Lagrangian which force the equations of motion to have the desired boundary conditions. Although some foresight is needed to choose these time-dependent terms, this approach provides a mechanism for energy to exit the closed system while allowing the conservation law to account for the loss. An appropriate time discretization scheme is selected based on stability analysis and resolution requirements. We present results using this variational approach in a co-moving coordinate system and compare such results to those using traditional second-order methods. This work was supported by the U. S. Department of Energy under Contract No. DE-SC0008382 and by the National Science Foundation under Contract No. PHY- 1104683.

Equations of Motion for the g-LIMIT Microgravity Vibration Isolation System

NASA Technical Reports Server (NTRS)

Kim, Y. K.; Whorton, M. S.

2001-01-01

A desirable microgravity environment for experimental science payloads may require an active vibration isolation control system. A vibration isolation system named g-LIMIT (GLovebox Integrated Microgravity Isolation Technology) is being developed by NASA Marshall Space Flight Center to support microgravity science experiments using the microgravity science glovebox. In this technical memorandum, the full six-degree-of-freedom nonlinear equations of motion for g-LIMIT are derived. Although the motivation for this model development is control design and analysis of g-LIMIT, the equations are derived for a general configuration and may be used for other isolation systems as well.

Comparisons of ground motions from the 1999 Chi-Chi, earthquake with empirical predictions largely based on data from California

USGS Publications Warehouse

Boore, D.M.

2001-01-01

This article has the modest goal of comparing the ground motions recorded during the 1999 Chi-Chi, Taiwan, mainshock with predictions from four empirical-based equations commonly used for western North America; these empirical predictions are largely based on data from California. Comparisons are made for peak acceleration and 5%-damped response spectra at periods between 0.1 and 4 sec. The general finding is that the Chi-Chi ground motions are smaller than those predicted from the empirically based equations for periods less than about 1 sec by factors averaging about 0.4 but as small as 0.26 (depending on period, on which equation is used, and on whether the sites are assumed to be rock or soil). There is a trend for the observed motions to approach or even exceed the predicted motions for longer periods. Motions at similar distances (30-60 km) to the east and to the west of the fault differ dramatically at periods between about 2 and 20 sec: Long-duration wave trains are present on the motions to the west, and when normalized to similar amplitudes at short periods, the response spectra of the motions at the western stations are as much as five times larger than those of motions from eastern stations. The explanation for the difference is probably related to site and propagation effects; the western stations are on the Coastal Plain, whereas the eastern stations are at the foot of young and steep mountains, either in the relatively narrow Longitudinal Valley or along the eastern coast-the sediments underlying the eastern stations are probably shallower and have higher velocity than those under the western stations.

Effect of Configuration Pitching Motion on Twin Tail Buffet Response

NASA Technical Reports Server (NTRS)

Sheta, Essam F.; Kandil, Osama A.

1998-01-01

The effect of dynamic pitch-up motion of delta wing on twin-tail buffet response is investigated. The computational model consists of a delta wing-twin tail configuration. The computations are carried out on a dynamic multi-block grid structure. This multidisciplinary problem is solved using three sets of equations which consists of the unsteady Navier-Stokes equations, the aeroelastic equations, and the grid displacement equations. The configuration is pitched-up from zero up to 60 deg. angle of attack, and the freestream Mach number and Reynolds number are 0.3 and 1.25 million, respectively. With the twin tail fixed as rigid surfaces and with no-forced pitch-up motion, the problem is solved for the initial flow conditions. Next, the problem is solved for the twin-tail response for uncoupled bending and torsional vibrations due to the unsteady loads on the twin tail and due to the forced pitch-up motion. The dynamic pitch-up problem is also solved for the flow response with the twin tail kept rigid. The configuration is investigated for inboard position of the twin tail which corresponds to a separation distance between the twin tail of 33% wing chord. The computed results are compared with the available experimental data.

Yang-Baxter maps, discrete integrable equations and quantum groups

NASA Astrophysics Data System (ADS)

Bazhanov, Vladimir V.; Sergeev, Sergey M.

2018-01-01

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq (sl (2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.

Steady state conductance in a double quantum dot array: the nonequilibrium equation-of-motion Green function approach.

PubMed

Levy, Tal J; Rabani, Eran

2013-04-28

We study steady state transport through a double quantum dot array using the equation-of-motion approach to the nonequilibrium Green functions formalism. This popular technique relies on uncontrolled approximations to obtain a closure for a hierarchy of equations; however, its accuracy is questioned. We focus on 4 different closures, 2 of which were previously proposed in the context of the single quantum dot system (Anderson impurity model) and were extended to the double quantum dot array, and develop 2 new closures. Results for the differential conductance are compared to those attained by a master equation approach known to be accurate for weak system-leads couplings and high temperatures. While all 4 closures provide an accurate description of the Coulomb blockade and other transport properties in the single quantum dot case, they differ in the case of the double quantum dot array, where only one of the developed closures provides satisfactory results. This is rationalized by comparing the poles of the Green functions to the exact many-particle energy differences for the isolate system. Our analysis provides means to extend the equation-of-motion technique to more elaborate models of large bridge systems with strong electronic interactions.

Optimal trajectories for the aeroassisted flight experiment, 1988-89

NASA Technical Reports Server (NTRS)

Miele, A.

1989-01-01

Research is summarized on optimal trajectories for the aeroassisted flight experiment, performed by the Aero-Astronautics Group of Rice University during the period 1988 through 1989. This research includes the following topics: (1) equations of motion in an Earth-fixed system; (2) equations of motion in an inertial system; (3) formultion of the optimal trajectory problem; (4) results on the optimal trajectory problem; and (5) guidance implications.

Analytical solutions for the motion of a charged particle in electric and magnetic fields via non-singular fractional derivatives

NASA Astrophysics Data System (ADS)

Morales-Delgado, V. F.; Gómez-Aguilar, J. F.; Taneco-Hernandez, M. A.

2017-12-01

In this work we propose fractional differential equations for the motion of a charged particle in electric, magnetic and electromagnetic fields. Exact solutions are obtained for the fractional differential equations by employing the Laplace transform method. The temporal fractional differential equations are considered in the Caputo-Fabrizio-Caputo and Atangana-Baleanu-Caputo sense. Application examples consider constant, ramp and harmonic fields. In addition, we present numerical results for different values of the fractional order. In all cases, when α = 1, we recover the standard electrodynamics.

Enhanced Modeling of First-Order Plant Equations of Motion for Aeroelastic and Aeroservoelastic Applications

NASA Technical Reports Server (NTRS)

Pototzky, Anthony S.

2010-01-01

A methodology is described for generating first-order plant equations of motion for aeroelastic and aeroservoelastic applications. The description begins with the process of generating data files representing specialized mode-shapes, such as rigid-body and control surface modes, using both PATRAN and NASTRAN analysis. NASTRAN executes the 146 solution sequence using numerous Direct Matrix Abstraction Program (DMAP) calls to import the mode-shape files and to perform the aeroelastic response analysis. The aeroelastic response analysis calculates and extracts structural frequencies, generalized masses, frequency-dependent generalized aerodynamic force (GAF) coefficients, sensor deflections and load coefficients data as text-formatted data files. The data files are then re-sequenced and re-formatted using a custom written FORTRAN program. The text-formatted data files are stored and coefficients for s-plane equations are fitted to the frequency-dependent GAF coefficients using two Interactions of Structures, Aerodynamics and Controls (ISAC) programs. With tabular files from stored data created by ISAC, MATLAB generates the first-order aeroservoelastic plant equations of motion. These equations include control-surface actuator, turbulence, sensor and load modeling. Altitude varying root-locus plot and PSD plot results for a model of the F-18 aircraft are presented to demonstrate the capability.

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

NASA Astrophysics Data System (ADS)

Dakel, Mzaki; Baguet, Sébastien; Dufour, Régis

2014-05-01

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.

Solar wind and the motion of dust grains

NASA Astrophysics Data System (ADS)

Klačka, J.; Petržala, J.; Pástor, P.; Kómar, L.

2012-04-01

In this paper, we investigate the action of solar wind on an arbitrarily shaped interplanetary dust particle. The final relativistically covariant equation of motion of the particle also contains the change of the particle's mass. The non-radial solar wind velocity vector is also included. The covariant equation of motion reduces to the Poynting-Robertson effect in the limiting case when a spherical particle is treated, when the speed of the incident solar wind corpuscles tends to the speed of light and when the corpuscles spread radially from the Sun. The results of quantum mechanics have to be incorporated into the physical considerations, in order to obtain the limiting case. If the solar wind affects the motion of a spherical interplanetary dust particle, then ?. Here, p'in and p'out are the incoming and outgoing radiation momenta (per unit time), respectively, measured in the proper frame of reference of the particle, and ? and ? are the solar wind pressure and the total scattering cross-sections, respectively. An analytical solution of the derived equation of motion yields a qualitative behaviour consistent with numerical calculations. This also holds if we consider a decrease of the particle's mass. Using numerical integration of the derived equation of motion, we confirm our analytical result that the non-radial solar wind (with a constant value of angle between the radial direction and the direction of the solar wind velocity) causes outspiralling of the dust particle from the Sun for large values of the particle's semimajor axis. The non-radial solar wind also increases the time the particle spirals towards the Sun. If we consider the periodical variability of the solar wind with the solar cycle, then there are resonances between the particle's orbital period and the period of the solar cycle.

A Review of System Identification Methods Applied to Aircraft

NASA Technical Reports Server (NTRS)

Klein, V.

1983-01-01

Airplane identification, equation error method, maximum likelihood method, parameter estimation in frequency domain, extended Kalman filter, aircraft equations of motion, aerodynamic model equations, criteria for the selection of a parsimonious model, and online aircraft identification are addressed.

Adding In-Plane Flexibility to the Equations of Motion of a Single Rotor Helicopter

NASA Technical Reports Server (NTRS)

Curtiss, H. C., Jr.

2000-01-01

This report describes a way to add the effects of main rotor blade flexibility in the in- plane or lead-lag direction to a large set of non-linear equations of motion for a single rotor helicopter with rigid blades(l). Differences between the frequency of the regressing lag mode predicted by the equations of (1) and that measured in flight (2) for a UH-60 helicopter indicate that some element is missing from the analytical model of (1) which assumes rigid blades. A previous study (3) noted a similar discrepancy for the CH-53 helicopter. Using a relatively simple analytical model in (3), compared to (1), it was shown that a mechanical lag damper increases significantly the coupling between the rigid lag mode and the first flexible mode. This increased coupling due to a powerful lag damper produces an increase in the lowest lag frequency when viewed in a frame rotating with the blade. Flight test measurements normally indicate the frequency of this mode in a non-rotating or fixed frame. This report presents the additions necessary to the full equations of motion, to include main rotor blade lag flexibility. Since these additions are made to a very complex nonlinear dynamic model, in order to provide physical insight, a discussion of the results obtained from a simplified set of equations of motion is included. The reduced model illustrates the physics involved in the coupling and should indicate trends in the full model.

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

DOE Office of Scientific and Technical Information (OSTI.GOV)

Boubendir, Yassine; Mendez, Vicenc; Rotstein, Horacio G.

2010-09-15

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delaymore » time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.« less

Modeling Attitude Dynamics in Simulink: A Study of the Rotational and Translational Motion of a Spacecraft Given Torques and Impulses Generated by RMS Hand Controllers

NASA Technical Reports Server (NTRS)

Mauldin, Rebecca H.

2010-01-01

In order to study and control the attitude of a spacecraft, it is necessary to understand the natural motion of a body in orbit. Assuming a spacecraft to be a rigid body, dynamics describes the complete motion of the vehicle by the translational and rotational motion of the body. The Simulink Attitude Analysis Model applies the equations of rigid body motion to the study of a spacecraft?s attitude in orbit. Using a TCP/IP connection, Matlab reads the values of the Remote Manipulator System (RMS) hand controllers and passes them to Simulink as specified torque and impulse profiles. Simulink then uses the governing kinematic and dynamic equations of a rigid body in low earth orbit (LE0) to plot the attitude response of a spacecraft for five seconds given known applied torques and impulses, and constant principal moments of inertia.

Advanced control concepts. [trim solution for space shuttle

NASA Technical Reports Server (NTRS)

Hutton, M. F.; Friedland, B.

1973-01-01

The selection of a trim solution that provides the space shuttle with the highest level of performance and dynamic control in the presense of wind disturbances and bias torques due to misalignment of rocket engines is described. It was determined that engine gimballing is insufficient to provide control to trim the vehicle for headwind and sidewind disturbances, and that it is necessary to use aerodynamic surfaces in conjunction with engine gimballing to achieve trim. The algebraic equations for computing the trim solution were derived from the differential equations describing the motion of the vehicle by substituting the desired trim conditions. The general problem of showing how the trim equations are derived from the equations of motion and the mathematical forms of the performance criterion is discussed in detail, along with the general equations for studying the dynamic response of the trim solution.

Primer vector theory applied to the linear relative-motion equations. [for N-impulse space trajectory optimization

NASA Technical Reports Server (NTRS)

Jezewski, D.

1980-01-01

Prime vector theory is used in analyzing a set of linear relative-motion equations - the Clohessy-Wiltshire (C/W) equations - to determine the criteria and necessary conditions for an optimal N-impulse trajectory. The analysis develops the analytical criteria for improving a solution by: (1) moving any dependent or independent variable in the initial and/or final orbit, and (2) adding intermediate impulses. If these criteria are violated, the theory establishes a sufficient number of analytical equations. The subsequent satisfaction of these equations will result in the optimal position vectors and times of an N-impulse trajectory. The solution is examined for the specific boundary conditions of: (1) fixed-end conditions, two impulse, and time-open transfer; (2) an orbit-to-orbit transfer; and (3) a generalized renezvous problem.

Minimizing Secular J2 Perturbation Effects on Satellite Formations

DTIC Science & Technology

2008-03-01

linear set of differential equations describing the relative motion was established by Hill as well as Clohessy and Wiltshire , with a slightly... Wiltshire (CW) equations, and Hill- Clohessy - Wiltshire (HCW) equations. In the simplest form these differential equations can be expressed as: 2 2 2 3 2...different orientation. Because these equations are much alike, the differential equations established are referred to as Hill’s equations, Clohessy

High-precision numerical integration of equations in dynamics

NASA Astrophysics Data System (ADS)

Alesova, I. M.; Babadzanjanz, L. K.; Pototskaya, I. Yu.; Pupysheva, Yu. Yu.; Saakyan, A. T.

2018-05-01

An important requirement for the process of solving differential equations in Dynamics, such as the equations of the motion of celestial bodies and, in particular, the motion of cosmic robotic systems is high accuracy at large time intervals. One of effective tools for obtaining such solutions is the Taylor series method. In this connection, we note that it is very advantageous to reduce the given equations of Dynamics to systems with polynomial (in unknowns) right-hand sides. This allows us to obtain effective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. In the paper, these questions are discussed and appropriate algorithms are considered.

Evaluation of MOSTAS computer code for predicting dynamic loads in two bladed wind turbines

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.; Janetzke, D. C.; Sullivan, T. L.

1979-01-01

Calculated dynamic blade loads were compared with measured loads over a range of yaw stiffnesses of the DOE/NASA Mod-O wind turbine to evaluate the performance of two versions of the MOSTAS computer code. The first version uses a time-averaged coefficient approximation in conjunction with a multi-blade coordinate transformation for two bladed rotors to solve the equations of motion by standard eigenanalysis. The second version accounts for periodic coefficients while solving the equations by a time history integration. A hypothetical three-degree of freedom dynamic model was investigated. The exact equations of motion of this model were solved using the Floquet-Lipunov method. The equations with time-averaged coefficients were solved by standard eigenanalysis.

Flap-lag-torsional dynamics of helicopter rotor blades in forward flight

NASA Technical Reports Server (NTRS)

Crespodasilva, M. R. M.

1986-01-01

A perturbation/numerical methodology to analyze the flap-lead/lag motion of a centrally hinged spring restrained rotor blade that is valid for both hover and for forward flight was developed. The derivation of the nonlinear differential equations of motion and the analysis of the stability of the steady state response of the blade were conducted entirely in a Symbolics 3670 Machine using MACSYMA to perform all the lengthy symbolic manipulations. It also includes generation of the fortran codes and plots of the results. The Floquet theory was also applied to the differential equations of motion in order to compare results with those obtained from the perturbation analysis. The results obtained from the perturbation methodology and from Floquet theory were found to be very close to each other, which demonstrates the usefullness of the perturbation methodology. Another problem under study consisted in the analysis of the influence of higher order terms in the response and stability of a flexible rotor blade in forward flight using Computerized Symbolic Manipulation and a perturbation technique to bypass the Floquet theory. The derivation of the partial differential equations of motion is presented.

Dynamic behaviors of cavitation bubble for the steady cavitating flow

NASA Astrophysics Data System (ADS)

Cai, Jun; Huai, Xiulan; Li, Xunfeng

2009-12-01

In this paper, by introducing the flow velocity item into the classical Rayleigh-Plesset dynamic equation, a new equation, which does not involve the time term and can describe the motion of cavitation bubble in the steady cavitating flow, has been obtained. By solving the new motion equation using Runge-Kutta fourth order method with adaptive step size control, the dynamic behaviors of cavitation bubble driven by the varying pressure field downstream of a venturi cavitation reactor are numerically simulated. The effects of liquid temperature (corresponding to the saturated vapor pressure of liquid), cavitation number and inlet pressure of venturi on radial motion of bubble and pressure pulse due to the radial motion are analyzed and discussed in detail. Some dynamic behaviors of bubble different from those in previous papers are displayed. In addition, the internal relationship between bubble dynamics and process intensification is also discussed. The simulation results reported in this work reveal the variation laws of cavitation intensity with the flow conditions of liquid, and will lay a foundation for the practical application of hydrodynamic cavitation technology.

Self-similar motion of a Nambu-Goto string

NASA Astrophysics Data System (ADS)

Igata, Takahisa; Houri, Tsuyoshi; Harada, Tomohiro

2016-09-01

We study the self-similar motion of a string in a self-similar spacetime by introducing the concept of a self-similar string, which is defined as the world sheet to which a homothetic vector field is tangent. It is shown that in Nambu-Goto theory, the equations of motion for a self-similar string reduce to those for a particle. Moreover, under certain conditions such as the hypersurface orthogonality of the homothetic vector field, the equations of motion for a self-similar string simplify to the geodesic equations on a (pseudo)Riemannian space. As a concrete example, we investigate a self-similar Nambu-Goto string in a spatially flat Friedmann-Lemaître-Robertson-Walker expanding universe with self-similarity and obtain solutions of open and closed strings, which have various nontrivial configurations depending on the rate of the cosmic expansion. For instance, we obtain a circular solution that evolves linearly in the cosmic time while keeping its configuration by the balance between the effects of the cosmic expansion and string tension. We also show the instability for linear radial perturbation of the circular solutions.

Large-Amplitude, High-Rate Roll Oscillations of a 65 deg Delta Wing at High Incidence

NASA Technical Reports Server (NTRS)

Chaderjian, Neal M.; Schiff, Lewis B.

2000-01-01

The IAR/WL 65 deg delta wing experimental results provide both detail pressure measurements and a wide range of flow conditions covering from simple attached flow, through fully developed vortex and vortex burst flow, up to fully-stalled flow at very high incidence. Thus, the Computational Unsteady Aerodynamics researchers can use it at different level of validating the corresponding code. In this section a range of CFD results are provided for the 65 deg delta wing at selected flow conditions. The time-dependent, three-dimensional, Reynolds-averaged, Navier-Stokes (RANS) equations are used to numerically simulate the unsteady vertical flow. Two sting angles and two large- amplitude, high-rate, forced-roll motions and a damped free-to-roll motion are presented. The free-to-roll motion is computed by coupling the time-dependent RANS equations to the flight dynamic equation of motion. The computed results are compared with experimental pressures, forces, moments and roll angle time history. In addition, surface and off-surface flow particle streaks are also presented.

A New Twisting Somersault: 513XD

NASA Astrophysics Data System (ADS)

Tong, William; Dullin, Holger R.

2017-12-01

We present the mathematical framework of an athlete modelled as a system of coupled rigid bodies to simulate platform and springboard diving. Euler's equations of motion are generalised to non-rigid bodies and are then used to innovate a new dive sequence that in principle can be performed by real-world athletes. We begin by assuming that shape changes are instantaneous so that the equations of motion simplify enough to be solved analytically, and then use this insight to present a new dive (513XD) consisting of 1.5 somersaults and five twists using realistic shape changes. Finally, we demonstrate the phenomenon of converting pure somersaulting motion into pure twisting motion by using a sequence of impulsive shape changes, which may have applications in other fields such as space aeronautics.

Translation of time-reversal violation in the neutral K-meson system into a table-top mechanical system

NASA Astrophysics Data System (ADS)

Reiser, Andreas; Schubert, Klaus R.; Stiewe, Jürgen

2012-08-01

Weak interactions break time-reversal (T) symmetry in the two-state system of neutral K-mesons. We present and discuss a two-state mechanical system, i.e. a Foucault-type pendulum on a rotating table, for a full representation of {K^0}{{\\overlineK}{}^0} transitions by the pendulum motions including T violation. The pendulum moves with two different oscillation frequencies and two different magnetic dampings. Its equation of motion is identical to the differential equation for the real part of the CPT-symmetric K-meson wavefunction. The pendulum is able to represent microscopic CP and T violation with CPT symmetry owing to the macroscopic Coriolis force, which breaks the symmetry under reversal-of-motion. Video clips of the pendulum motions are given as supplementary material.

Testing for a cosmological influence on local physics using atomic and gravitational clocks

NASA Technical Reports Server (NTRS)

Adams, P. J.; Hellings, R. W.; Canuto, V. M.; Goldman, I.

1983-01-01

The existence of a possible influence of the large-scale structure of the universe on local physics is discussed. A particular realization of such an influence is discussed in terms of the behavior in time of atomic and gravitational clocks. Two natural categories of metric theories embodying a cosmic infuence exist. The first category has geodesic equations of motion in atomic units, while the second category has geodesic equations of motion in gravitational units. Equations of motion for test bodies are derived for both categories of theories in the appropriate parametrized post-Newtonian limit and are applied to the Solar System. Ranging data to the Viking lander on Mars are of sufficient precision to reveal (1) if such a cosmological influence exists at the level of Hubble's constant, and (2) which category of theories is appropriate for a descripton of the phenomenon.

Stochastic analysis of pitch angle scattering of charged particles by transverse magnetic waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lemons, Don S.; Liu Kaijun; Winske, Dan

2009-11-15

This paper describes a theory of the velocity space scattering of charged particles in a static magnetic field composed of a uniform background field and a sum of transverse, circularly polarized, magnetic waves. When that sum has many terms the autocorrelation time required for particle orbits to become effectively randomized is small compared with the time required for the particle velocity distribution to change significantly. In this regime the deterministic equations of motion can be transformed into stochastic differential equations of motion. The resulting stochastic velocity space scattering is described, in part, by a pitch angle diffusion rate that ismore » a function of initial pitch angle and properties of the wave spectrum. Numerical solutions of the deterministic equations of motion agree with the theory at all pitch angles, for wave energy densities up to and above the energy density of the uniform field, and for different wave spectral shapes.« less

Derivation of charts for determining the horizontal tail load variation with any elevator motion

NASA Technical Reports Server (NTRS)

Pearson, Henry A

1943-01-01

The equations relating the wing and tail loads are derived for a unit elevator displacement. These equations are then converted into a nondimensional form and charts are given by which the wing- and tail-load-increment variation may be determined under dynamic conditions for any type of elevator motion and for various degrees of airplane stability. In order to illustrate the use of the charts, several examples are included in which the wing and tail loads are evaluated for a number of types of elevator motion. Methods are given for determining the necessary derivatives from results of wind-tunnel tests when such tests are available.

Collisional redistribution of radiation. II - The effects of degeneracy on the equations of motion for the density matrix. III - The equation of motion for the correlation function and the scattered spectrum

NASA Technical Reports Server (NTRS)

Burnett, K.; Cooper, J.

1980-01-01

The effect of correlations between an absorber atom and perturbers in the binary-collision approximation are applied to degenerate atomic systems. A generalized absorption profile which specifies the final state of the atom after an absorption event is related to the total intensities of Rayleigh scattering and fluorescence from the atom. It is suggested that additional dynamical information to that obtainable from ordinary absorption experiments is required in order to describe redistributed atomic radiation. The scattering of monochromatic radiation by a degenerate atom is computed in a binary-collision approximation; an equation of motion is derived for the correlation function which is valid outside the quantum-regression regime. Solutions are given for the weak-field conditions in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles.

Visualization in mechanics: the dynamics of an unbalanced roller

NASA Astrophysics Data System (ADS)

Cumber, Peter S.

2017-04-01

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical user interface for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the system dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration the simulation tool has been applied to a range of roller-pivot arm configurations. In addition, approximations to the equations of motion are explored and a second-order model is shown to be accurate for a limited range of parameters.

An extension of stochastic hierarchy equations of motion for the equilibrium correlation functions

NASA Astrophysics Data System (ADS)

Ke, Yaling; Zhao, Yi

2017-06-01

A traditional stochastic hierarchy equations of motion method is extended into the correlated real-time and imaginary-time propagations, in this paper, for its applications in calculating the equilibrium correlation functions. The central idea is based on a combined employment of stochastic unravelling and hierarchical techniques for the temperature-dependent and temperature-free parts of the influence functional, respectively, in the path integral formalism of the open quantum systems coupled to a harmonic bath. The feasibility and validity of the proposed method are justified in the emission spectra of homodimer compared to those obtained through the deterministic hierarchy equations of motion. Besides, it is interesting to find that the complex noises generated from a small portion of real-time and imaginary-time cross terms can be safely dropped to produce the stable and accurate position and flux correlation functions in a broad parameter regime.

Equations of motion for coupled n-body systems

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1980-01-01

Computer program, developed to analyze spacecraft attitude dynamics, can be applied to large class of problems involving objects that can be simplified into component parts. Systems of coupled rigid bodies, point masses, symmetric wheels, and elastically flexible bodies can be analyzed. Program derives complete set of non-linear equations of motion in vectordyadic format. Numerical solutions may be printed out. Program is in FORTRAN IV for batch execution and has been implemented on IBM 360.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Unseren, M.A.

A general framework for solving the dynamic load distribution when two manipulators hold a rigid object is proposed. The underspecified problem of solving for the contact forces and torques based on the object`s equations of motion is transformed into a well specified problem. This is accomplished by augmenting the object`s equations of motion with additional equations which relate a new vector variable quantifying the internal contact force and torque degrees of freedom (DOF) as a linear function of the contact forces and torques. The resulting augmented system yields a well specified solution for the contact forces and torques in whichmore » they are separated into their motion inducing and internal components. A particular solution is suggested which enables the designer to conveniently specify what portion of the payload`s mass each manipulator is to bear. It is also shown that the results of the previous work are just a special case of the general load distribution framework described here.« less

A new method for solving the quantum hydrodynamic equations of motion: application to two-dimensional reactive scattering.

PubMed

Pauler, Denise K; Kendrick, Brian K

2004-01-08

The de Broglie-Bohm hydrodynamic equations of motion are solved using a meshless method based on a moving least squares approach and an arbitrary Lagrangian-Eulerian frame of reference. A regridding algorithm adds and deletes computational points as needed in order to maintain a uniform interparticle spacing, and unitary time evolution is obtained by propagating the wave packet using averaged fields. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. The methodology is applied to a two-dimensional model collinear reaction with an activation barrier. Reaction probabilities are computed as a function of both time and energy, and are in excellent agreement with those based on the quantum trajectory method. (c) 2004 American Institute of Physics

Lagrangian equations of motion of particles and photons in a Schwarzschild field

NASA Astrophysics Data System (ADS)

Ritus, V. I.

2015-11-01

The equations of motion of a particle in the gravitational field of a black hole are considered in a formulation that uses generalized coordinates, velocities, and accelerations and is convenient for finding the integrals of motion. The equations are rewritten in terms of the physical velocities and accelerations measured in the Schwarzschild frame by a stationary observer using proper local length and time standards. The attractive force due to the field and the centripetal acceleration of a particle is proportional to the particle kinetic energy m/\\sqrt{1 - v^2}, consistently with the fact that the particle kinetic energy and the photon energy \\hbarω in the field increase by the same factor compared with their values without a field. The attraction exerted on particles and photons by a gravitational field source is proportional to their kinetic energies. The particle trajectory in the ultrarelativistic limit v \\to 1 coincides with the photon trajectory.

The primer vector in linear, relative-motion equations. [spacecraft trajectory optimization

NASA Technical Reports Server (NTRS)

1980-01-01

Primer vector theory is used in analyzing a set of linear, relative-motion equations - the Clohessy-Wiltshire equations - to determine the criteria and necessary conditions for an optimal, N-impulse trajectory. Since the state vector for these equations is defined in terms of a linear system of ordinary differential equations, all fundamental relations defining the solution of the state and costate equations, and the necessary conditions for optimality, can be expressed in terms of elementary functions. The analysis develops the analytical criteria for improving a solution by (1) moving any dependent or independent variable in the initial and/or final orbit, and (2) adding intermediate impulses. If these criteria are violated, the theory establishes a sufficient number of analytical equations. The subsequent satisfaction of these equations will result in the optimal position vectors and times of an N-impulse trajectory. The solution is examined for the specific boundary conditions of (1) fixed-end conditions, two-impulse, and time-open transfer; (2) an orbit-to-orbit transfer; and (3) a generalized rendezvous problem. A sequence of rendezvous problems is solved to illustrate the analysis and the computational procedure.

Numerical simulation of fluid flow through simplified blade cascade with prescribed harmonic motion using discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Vimmr, Jan; Bublík, Ondřej; Prausová, Helena; Hála, Jindřich; Pešek, Luděk

2018-06-01

This paper deals with a numerical simulation of compressible viscous fluid flow around three flat plates with prescribed harmonic motion. This arrangement presents a simplified blade cascade with forward wave motion. The aim of this simulation is to determine the aerodynamic forces acting on the flat plates. The mathematical model describing this problem is formed by Favre-averaged system of Navier-Stokes equations in arbitrary Lagrangian-Eulerian (ALE) formulation completed by one-equation Spalart-Allmaras turbulence model. The simulation was performed using the developed in-house CFD software based on discontinuous Galerkin method, which offers high order of accuracy.

Reference equations of motion for automatic rendezvous and capture

NASA Technical Reports Server (NTRS)

Henderson, David M.

1992-01-01

The analysis presented in this paper defines the reference coordinate frames, equations of motion, and control parameters necessary to model the relative motion and attitude of spacecraft in close proximity with another space system during the Automatic Rendezvous and Capture phase of an on-orbit operation. The relative docking port target position vector and the attitude control matrix are defined based upon an arbitrary spacecraft design. These translation and rotation control parameters could be used to drive the error signal input to the vehicle flight control system. Measurements for these control parameters would become the bases for an autopilot or feedback control system (FCS) design for a specific spacecraft.

Estimating unknown input parameters when implementing the NGA ground-motion prediction equations in engineering practice

USGS Publications Warehouse

Kaklamanos, James; Baise, Laurie G.; Boore, David M.

2011-01-01

The ground-motion prediction equations (GMPEs) developed as part of the Next Generation Attenuation of Ground Motions (NGA-West) project in 2008 are becoming widely used in seismic hazard analyses. However, these new models are considerably more complicated than previous GMPEs, and they require several more input parameters. When employing the NGA models, users routinely face situations in which some of the required input parameters are unknown. In this paper, we present a framework for estimating the unknown source, path, and site parameters when implementing the NGA models in engineering practice, and we derive geometrically-based equations relating the three distance measures found in the NGA models. Our intent is for the content of this paper not only to make the NGA models more accessible, but also to help with the implementation of other present or future GMPEs.

Multi-component ground motion response spectra for coupled horizontal, vertical, angular accelerations, and tilt

USGS Publications Warehouse

Kalkan, E.; Graizer, V.

2007-01-01

Rotational and vertical components of ground motion are almost always ignored in design or in the assessment of structures despite the fact that vertical motion can be twice as much as the horizontal motion and may exceed 2g level, and rotational excitation may reach few degrees in the proximity of fault rupture. Coupling of different components of ground excitation may significantly amplify the seismic demand by introducing additional lateral forces and enhanced P-?? effects. In this paper, a governing equation of motion is postulated to compute the response of a SDOF oscillator under a multi-component excitation. The expanded equation includes secondary P-?? components associated with the combined impacts of tilt and vertical excitations in addition to the inertial forcing terms due to the angular and translational accelerations. The elastic and inelastic spectral ordinates traditionally generated considering the uniaxial input motion are compared at the end with the multi-component response spectra of coupled horizontal, vertical and tilting motions. The proposed multi-component response spectrum reflects kinematic characteristics of the ground motion that are not identifiable by the conventional spectrum itself, at least for the near-fault region where high intensity vertical shaking and rotational excitation are likely to occur.

Analysis of Formation Flying in Eccentric Orbits Using Linearized Equations of Relative Motion

NASA Technical Reports Server (NTRS)

Lane, Christopher; Axelrad, Penina

2004-01-01

Geometrical methods for formation flying design based on the analytical solution to Hill's equations have been previously developed and used to specify desired relative motions in near circular orbits. By generating relationships between the vehicles that are intuitive, these approaches offer valuable insight into the relative motion and allow for the rapid design of satellite configurations to achieve mission specific requirements, such as vehicle separation at perigee or apogee, minimum separation, or a specific geometrical shape. Furthermore, the results obtained using geometrical approaches can be used to better constrain numerical optimization methods; allowing those methods to converge to optimal satellite configurations faster. This paper presents a set of geometrical relationships for formations in eccentric orbits, where Hill.s equations are not valid, and shows how these relationships can be used to investigate formation designs and how they evolve with time.

Revisiting Newtonian and Non-Newtonian Fluid Mechanics Using Computer Algebra

ERIC Educational Resources Information Center

Knight, D. G.

2006-01-01

This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward,…

Brans-Dicke Galileon and the variational principle

NASA Astrophysics Data System (ADS)

Quiros, Israel; García-Salcedo, Ricardo; Gonzalez, Tame; Horta-Rangel, F. Antonio; Saavedra, Joel

2016-09-01

This paper is aimed at a (mostly) pedagogical exposition of the derivation of the motion equations of certain modifications of general relativity. Here we derive in all detail the motion equations in the Brans-Dicke theory with cubic self-interaction. This is a modification of the Brans-Dicke theory by the addition of a term in the Lagrangian which is non-linear in the derivatives of the scalar field: it contains second-order derivatives. This is the basis of the so-called Brans-Dicke Galileon. We pay special attention to the variational principle and to the algebraic details of the derivation. It is shown how higher order derivatives of the fields appearing in the intermediate computations cancel out leading to second order motion equations. The reader will find useful tips for the derivation of the field equations of modifications of general relativity such as the scalar-tensor theories and f(R) theories, by means of the (stationary action) variational principle. The content of this paper is particularly recommended to those graduate and postgraduate students who are interested in the study of the mentioned modifications of general relativity.

Modeling the Benchmark Active Control Technology Wind-Tunnel Model for Application to Flutter Suppression

NASA Technical Reports Server (NTRS)

Waszak, Martin R.

1996-01-01

This paper describes the formulation of a model of the dynamic behavior of the Benchmark Active Controls Technology (BACT) wind-tunnel model for application to design and analysis of flutter suppression controllers. The model is formed by combining the equations of motion for the BACT wind-tunnel model with actuator models and a model of wind-tunnel turbulence. The primary focus of this paper is the development of the equations of motion from first principles using Lagrange's equations and the principle of virtual work. A numerical form of the model is generated using values for parameters obtained from both experiment and analysis. A unique aspect of the BACT wind-tunnel model is that it has upper- and lower-surface spoilers for active control. Comparisons with experimental frequency responses and other data show excellent agreement and suggest that simple coefficient-based aerodynamics are sufficient to accurately characterize the aeroelastic response of the BACT wind-tunnel model. The equations of motion developed herein have been used to assist the design and analysis of a number of flutter suppression controllers that have been successfully implemented.

Analytical and experimental study of the vibration of bonded beams with a lap joint

NASA Technical Reports Server (NTRS)

Rao, M. D.; Crocker, M. J.

1990-01-01

A theoretical model to study the flexural vibration of a bonded lap joint system is described in this paper. First, equations of motion at the joint region are derived using a differential element approach. The transverse displacements of the upper and lower beam are considered to be different. The adhesive is assumed to be linearly viscoelastic and the widely used Kelvin-Voight model is used to represent the viscoelastic behavior of the adhesive. The shear force at the interface between the adhesive and the beam is obtained from the simple bending motion equations of the two beams. The resulting equations of motion are combined with the equations of transverse vibration of the beams in the unjointed regions. These are later solved as a boundary value problem to obtain the eigenvalues and eigenvectors of the system. The model can be used to predict the natural frequencies, modal damping ratios, and mode shapes of the system for free vibration. Good agreement between numerical and experimental results was obtained for a system of graphite epoxy beams lap-jointed by an epoxy adhesive.

Dynamic Theory of Relativistic Electrons Stochastic Heating by Whistler Mode Waves with Application to the Earth Magnetosphere

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Tel'nikhin, A. A.; Kronberg, T. K.

2007-01-01

In the Hamiltonian approach an electron motion in a coherent packet of the whistler mode waves propagating along the direction of an ambient magnetic field is studied. The physical processes by which these particles are accelerated to high energy are established. Equations governing a particle motion were transformed in to a closed pair of nonlinear difference equations. The solutions of these equations have shown there exists the energetic threshold below that the electron motion is regular, and when the initial energy is above the threshold an electron moves stochastically. Particle energy spectra and pitch angle electron scattering are described by the Fokker-Planck-Kolmogorov equations. Calculating the stochastic diffusion of electrons due to a spectrum of whistler modes is presented. The parametric dependence of the diffusion coefficients on the plasma particle density, magnitude of wave field, and the strength of magnetic field is studies. It is shown that significant pitch angle diffusion occurs for the Earth radiation belt electrons with energies from a few keV up to a few MeV.

Exact Descriptions of General Relativity Derived from Newtonian Mechanics within Curved Geometries

NASA Astrophysics Data System (ADS)

Savickas, David

2015-04-01

General relativity and Newtonian mechanics are shown to be exactly related when Newton's second law is written in a curved geometry by using the physical components of a vector as is defined in tensor calculus. By replacing length within the momentum's velocity by the vector metric in a curved geometry the second law can then be shown to be exactly identical to the geodesic equation of motion occurring in general relativity. When time's vector direction is constant, as similarly occurs in Newtonian mechanics, this equation can be reduced to a curved three-dimensional equation of motion that yields the the Schwarzschild equations of motion for an isolated particle. They can be used to describe gravitational behavior for any array of masses for which the Newtonian gravitational potential is known, and is shown to describe a mass particle's behavior in the gravitational field of a thin mass-rod. This use of Newton's laws allows relativistic behavior to be described in a physically comprehensible manner. D. Savickas, Int. J. Mod. Phys. D 23 1430018, (2014).

The one-dimensional asymmetric persistent random walk

NASA Astrophysics Data System (ADS)

Rossetto, Vincent

2018-04-01

Persistent random walks are intermediate transport processes between a uniform rectilinear motion and a Brownian motion. They are formed by successive steps of random finite lengths and directions travelled at a fixed speed. The isotropic and symmetric 1D persistent random walk is governed by the telegrapher’s equation, also called the hyperbolic heat conduction equation. These equations have been designed to resolve the paradox of the infinite speed in the heat and diffusion equations. The finiteness of both the speed and the correlation length leads to several classes of random walks: Persistent random walk in one dimension can display anomalies that cannot arise for Brownian motion such as anisotropy and asymmetries. In this work we focus on the case where the mean free path is anisotropic, the only anomaly leading to a physics that is different from the telegrapher’s case. We derive exact expression of its Green’s function, for its scattering statistics and distribution of first-passage time at the origin. The phenomenology of the latter shows a transition for quantities like the escape probability and the residence time.

On the stability of dust orbits in mean-motion resonances perturbed by from an interstellar wind

NASA Astrophysics Data System (ADS)

Pástor, Pavol

2014-09-01

Circumstellar dust particles can be captured in a mean-motion resonance (MMR) with a planet and simultaneously be affected by non-gravitational effects. It is possible to describe the secular variations of a particle orbit in the MMR analytically using averaged resonant equations. We derive the averaged resonant equations from the equations of motion in near-canonical form. The secular variations of the particle orbit depending on the orientation of the orbit in space are taken into account. The averaged resonant equations can be derived/confirmed also from Lagrange's planetary equations. We apply the derived theory to the case when the non-gravitational effects are the Poynting-Robertson effect, the radial stellar wind, and an interstellar wind. The analytical and numerical results obtained are in excellent agreement. We found that the types of orbits correspond to libration centers of the conservative problem. The averaged resonant equations can lead to a system of equations which holds for stationary points in a subset of resonant variables. Using this system we show analytically that for the considered non-gravitational effects, all stationary points should correspond to orbits which are stationary in interplanetary space after an averaging over a synodic period. In an exact resonance, the stationary orbits are stable. The stability is achieved by a periodic repetition of the evolution during the synodic period. Numerical solutions of this system show that there are no stationary orbits for either the exact or non-exact resonances.

NBOD2- PROGRAM TO DERIVE AND SOLVE EQUATIONS OF MOTION FOR COUPLED N-BODY SYSTEMS

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1994-01-01

The analysis of the dynamic characteristics of a complex system, such as a spacecraft or a robot, is usually best accomplished through the study of a simulation model. The simulation model must have the same dynamic characteristics as the complex system, while lending itself to mathematical quantification. The NBOD2 computer program was developed to aid in the analysis of spacecraft attitude dynamics. NBOD2 is a very general program that may be applied to a large class of problems involving coupled N-body systems. NBOD2 provides the dynamics analyst with the capability to automatically derive and numerically solve the equations of motion for any system that can be modeled as a topological tree of coupled rigid bodies, flexible bodies, point masses, and symmetrical momentum wheels. NBOD2 uses a topological tree model of the dynamic system to derive the vector-dyadic equations of motion for the system. The user builds this topological tree model by using rigid and flexible bodies, point masses, and symmetrical momentum wheels with appropriate connections. To insure that the relative motion between contiguous bodies is kinematically constrained, NBOD2 assumes that contiguous rigid and flexible bodies are connected by physically reliable 0, 1, 2, and 3-degrees-of-freedom gimbals. These gimbals prohibit relative translational motion, while permitting up to 3 degrees of relative rotational freedom at hinge points. Point masses may have 0, 1, 2, or 3-degrees of relative translational freedom, and symmetric momentum wheels may have a single degree of rotational freedom relative to the body in which they are imbedded. Flexible bodies may possess several degrees of vibrational freedom in addition to the degrees of freedom associated with the connection gimbals. Data concerning the natural modes and vibrations of the flexible bodies must be supplied by the user. NBOD2 combines the best features of the discrete-body approach and the nested body approach to reduce the topological tree to a complete set of nonlinear equations of motion in vector-dyadic form for the system being analyzed. NBOD2 can then numerically solve the equations of motion. Input to NBOD2 consists of a user-supplied description of the system to be modeled. The NBOD2 system includes an interactive, tutorial, input support program to aid the NBOD2 user in preparing input data. Output from NBOD2 consists of a listing of the complete set of nonlinear equations of motion in vector-dyadic form and any userspecified set of system state variables. The NBOD2 program is written in FORTRAN 77 for batch execution and has been implemented on a DEC VAX-11/780 computer. The NBOD2 program was developed in 1978 and last updated in 1982.

Report on progress at the Center for Engineering Strong Motion Data (CESMD)

USGS Publications Warehouse

Haddadi, H.; Shakal, A.; Huang, M.; Parrish, J.; Stephens, C.; Savage, William U.; Leith, William S.

2012-01-01

The CESMD now provides strong-motion records from lower magnitude (

Comparison between collective coordinate models for domain wall motion in PMA nanostrips in the presence of the Dzyaloshinskii-Moriya interaction

NASA Astrophysics Data System (ADS)

Vandermeulen, J.; Nasseri, S. A.; Van de Wiele, B.; Durin, G.; Van Waeyenberge, B.; Dupré, L.

2018-03-01

Lagrangian-based collective coordinate models for magnetic domain wall (DW) motion rely on an ansatz for the DW profile and a Lagrangian approach to describe the DW motion in terms of a set of time-dependent collective coordinates: the DW position, the DW magnetization angle, the DW width and the DW tilting angle. Another approach was recently used to derive similar equations of motion by averaging the Landau-Lifshitz-Gilbert equation without any ansatz, and identifying the relevant collective coordinates afterwards. In this paper, we use an updated version of the semi-analytical equations to compare the Lagrangian-based collective coordinate models with micromagnetic simulations for field- and STT-driven (spin-transfer torque-driven) DW motion in Pt/CoFe/MgO and Pt/Co/AlOx nanostrips. Through this comparison, we assess the accuracy of the different models, and provide insight into the deviations of the models from simulations. It is found that the lack of terms related to DW asymmetry in the Lagrangian-based collective coordinate models significantly contributes to the discrepancy between the predictions of the most accurate Lagrangian-based model and the micromagnetic simulations in the field-driven case. This is in contrast to the STT-driven case where the DW remains symmetric.

Dynamic analysis of propulsion mechanism directly driven by wave energy for marine mobile buoy

NASA Astrophysics Data System (ADS)

Yu, Zhenjiang; Zheng, Zhongqiang; Yang, Xiaoguang; Chang, Zongyu

2016-07-01

Marine mobile buoy(MMB) have many potential applications in the maritime industry and ocean science. Great progress has been made, however the technology in this area is far from maturity in theory and faced with many difficulties in application. A dynamic model of the propulsion mechanism is very necessary for optimizing the parameters of the MMB, especially with consideration of hydrodynamic force. The principle of wave-driven propulsion mechanism is briefly introduced. To set a theory foundation for study on the MMB, a dynamic model of the propulsion mechanism of the MMB is obtained. The responses of the motion of the platform and the hydrofoil are obtained by using a numerical integration method to solve the ordinary differential equations. A simplified form of the motion equations is reached by omitting terms with high order small values. The relationship among the heave motion of the buoy, stiffness of the elastic components, and the forward speed can be obtained by using these simplified equations. The dynamic analysis show the following: The angle of displacement of foil is fairly small with the biggest value around 0.3 rad; The speed of mobile buoy and the angle of hydrofoil increased gradually with the increase of heave motion of buoy; The relationship among heaven motion, stiffness and attack angle is that heave motion leads to the angle change of foil whereas the item of speed or push function is determined by vertical velocity and angle, therefore, the heave motion and stiffness can affect the motion of buoy significantly if the size of hydrofoil is kept constant. The proposed model is provided to optimize the parameters of the MMB and a foundation is laid for improving the performance of the MMB.

Motions in Taub-NUT-de Sitter spinning spacetime

NASA Astrophysics Data System (ADS)

Banu, Akhtara

2012-09-01

We investigate the geodesic motion of pseudo-classical spinning particles in the Taub-NUT-de Sitter spacetime. We obtain the conserved quantities from the solutions of the generalized Killing equations for spinning spaces. Applying the formalism the motion of a pseudo-classical Dirac fermion is analyzed on a cone and plane.

Perturbed soliton excitations of Rao-dust Alfvén waves in magnetized dusty plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kavitha, L., E-mail: louiskavitha@yahoo.co.in; The Abdus Salam International Centre for Theoretical Physics, Trieste; Lavanya, C.

We investigate the propagation dynamics of the perturbed soliton excitations in a three component fully ionized dusty magnetoplasma consisting of electrons, ions, and heavy charged dust particulates. We derive the governing equation of motion for the two dimensional Rao-dust magnetohydrodynamic (R-D-MHD) wave by employing the inertialess electron equation of motion, inertial ion equation of motion, the continuity equations in a plasma with immobile charged dust grains, together with the Maxwell's equations, by assuming quasi neutrality and neglecting the displacement current in Ampere's law. Furthermore, we assume the massive dust particles are practically immobile since we are interested in timescales muchmore » shorter than the dusty plasma period, thereby neglecting any damping of the modes due to the grain charge fluctuations. We invoke the reductive perturbation method to represent the governing dynamics by a perturbed cubic nonlinear Schrödinger (pCNLS) equation. We solve the pCNLS, along the lines of Kodama-Ablowitz multiple scale nonlinear perturbation technique and explored the R-D-MHD waves as solitary wave excitations in a magnetized dusty plasma. Since Alfvén waves play an important role in energy transport in driving field-aligned currents, particle acceleration and heating, solar flares, and the solar wind, this representation of R-D-MHD waves as soliton excitations may have extensive applications to study the lower part of the earth's ionosphere.« less

A study of numerical methods of solution of the equations of motion of a controlled satellite under the influence of gravity gradient torque

NASA Technical Reports Server (NTRS)

Thompson, J. F.; Mcwhorter, J. C.; Siddiqi, S. A.; Shanks, S. P.

1973-01-01

Numerical methods of integration of the equations of motion of a controlled satellite under the influence of gravity-gradient torque are considered. The results of computer experimentation using a number of Runge-Kutta, multi-step, and extrapolation methods for the numerical integration of this differential system are presented, and particularly efficient methods are noted. A large bibliography of numerical methods for initial value problems for ordinary differential equations is presented, and a compilation of Runge-Kutta and multistep formulas is given. Less common numerical integration techniques from the literature are noted for further consideration.

Optimization of cascade blade mistuning. I - Equations of motion and basic inherent properties

NASA Technical Reports Server (NTRS)

Nissim, E.

1985-01-01

Attention is given to the derivation of the equations of motion of mistuned compressor blades, interpolating aerodynamic coefficients by means of quadratic expressions in the reduced frequency. If the coefficients of the quadratic expressions are permitted to assume complex values, excellent accuracy is obtained and Pade rational expressions are obviated. On the basis of the resulting equations, it is shown analytically that the sum of all the real parts of the eigenvalues is independent of the mistuning introduced into the system. Blade mistuning is further treated through the aerodynamic energy approach, and the limiting vibration modes associated with alternative mistunings are identified.

Perturbed Equations of Motion for Formation Flight Near the Sun-Earth L2 Point

NASA Technical Reports Server (NTRS)

Luquette, Richard; Segerman, A. M.; Zedd, M. F.

2005-01-01

NASA is planning missions to the vicinity of the Sun-Earth L(sub 2) point, some involving a distributed system of telescope spacecraft, configured in a plane about a hub. Several sets of differential equations are written for the formation flight of such telescopes relative to the hub, with varying levels of fidelity. Effects are cast as additive perturbations to the circular restricted three-body problem, expanded in terms of the system distanced, to an accuracy of 10-20 m. These include Earth's orbital eccentricity, lunar motion, solar radiation pressure, and small thrusting forces. Simulations validating the expanded differential equations are presented.

Comparisons of ground motions from five aftershocks of the 1999 Chi-Chi, Taiwan, earthquake with empirical predictions largely based on data from California

USGS Publications Warehouse

Wang, G.-Q.; Boore, D.M.; Igel, H.; Zhou, X.-Y.

2004-01-01

The observed ground motions from five large aftershocks of the 1999 Chi-Chi, Taiwan, earthquake are compared with predictions from four equations based primarily on data from California. The four equations for active tectonic regions are those developed by Abrahamson and Silva (1997), Boore et al. (1997), Campbell (1997, 2001), and Sadigh et al. (1997). Comparisons are made for horizontal-component peak ground accelerations and 5%-damped pseudoacceleration response spectra at periods between 0.02 sec and 5 sec. The observed motions are in reasonable agreement with the predictions, particularly for distances from 10 to 30 km. This is in marked contrast to the motions from the Chi-Chi mainshock, which are much lower than the predicted motions for periods less than about 1 sec. The results indicate that the low motions in the mainshock are not due to unusual, localized absorption of seismic energy, because waves from the mainshock and the aftershocks generally traverse the same section of the crust and are recorded at the same stations. The aftershock motions at distances of 30-60 km are somewhat lower than the predictions (but not nearly by as small a factor as those for the mainshock), suggesting that the ground motion attenuates more rapidly in this region of Taiwan than it does in the areas we compare with it. We provide equations for the regional attenuation of response spectra, which show increasing decay of motion with distance for decreasing oscillator periods. This observational study also demonstrates that ground motions have large earthquake-location-dependent variability for a specific site. This variability reduces the accuracy with which an earthquake-specific prediction of site response can be predicted. Online Material: PGAs and PSAs from the 1999 Chi-Chi earthquake and five aftershocks.

Equation-of-motion coupled cluster method for high spin double electron attachment calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Musiał, Monika, E-mail: musial@ich.us.edu.pl; Lupa, Łukasz; Kucharski, Stanisław A.

The new formulation of the equation-of-motion (EOM) coupled cluster (CC) approach applicable to the calculations of the double electron attachment (DEA) states for the high spin components is proposed. The new EOM equations are derived for the high spin triplet and quintet states. In both cases the new equations are easier to solve but the substantial simplification is observed in the case of quintets. Out of 21 diagrammatic terms contributing to the standard DEA-EOM-CCSDT equations for the R{sub 2} and R{sub 3} amplitudes only four terms survive contributing to the R{sub 3} part. The implemented method has been applied tomore » the calculations of the excited states (singlets, triplets, and quintets) energies of the carbon and silicon atoms and potential energy curves for selected states of the Na{sub 2} (triplets) and B{sub 2} (quintets) molecules.« less

Microscopic Interpretation and Generalization of the Bloch-Torrey Equation for Diffusion Magnetic Resonance

PubMed Central

Seroussi, Inbar; Grebenkov, Denis S.; Pasternak, Ofer; Sochen, Nir

2017-01-01

In order to bridge microscopic molecular motion with macroscopic diffusion MR signal in complex structures, we propose a general stochastic model for molecular motion in a magnetic field. The Fokker-Planck equation of this model governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad-hoc selection of a propagator. In particular, the boundary conditions that implicitly incorporate the microstructure into the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal. PMID:28242566

Trajectory And Heating Of A Hypervelocity Projectile

NASA Technical Reports Server (NTRS)

Tauber, Michael E.

1992-01-01

Technical paper presents derivation of approximate, closed-form equation for relationship between velocity of projectile and density of atmosphere. Results of calculations based on approximate equation agree well with results from numerical integrations of exact equations of motion. Comparisons of results presented in series of graphs.

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

NASA Astrophysics Data System (ADS)

Chelnokov, Yu. N.

2017-11-01

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo-Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper. A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues-Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given. Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo-Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass. The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.

Modelling of Dynamics of a Wheeled Mobile Robot with Mecanum Wheels with the use of Lagrange Equations of the Second Kind

NASA Astrophysics Data System (ADS)

Hendzel, Z.; Rykała, Ł.

2017-02-01

The work presents the dynamic equations of motion of a wheeled mobile robot with mecanum wheels derived with the use of Lagrange equations of the second kind. Mecanum wheels are a new type of wheels used in wheeled mobile robots and they consist of freely rotating rollers attached to the circumference of the wheels. In order to derive dynamic equations of motion of a wheeled mobile robot, the kinetic energy of the system is determined, as well as the generalised forces affecting the system. The resulting mathematical model of a wheeled mobile robot was generated with the use of Maple V software. The results of a solution of inverse and forward problems of dynamics of the discussed object are also published.

Equation-of-motion coupled cluster method for the description of the high spin excited states

DOE Office of Scientific and Technical Information (OSTI.GOV)

Musiał, Monika, E-mail: musial@ich.us.edu.pl; Lupa, Łukasz; Kucharski, Stanisław A.

2016-04-21

The equation-of-motion (EOM) coupled cluster (CC) approach in the version applicable for the excitation energy (EE) calculations has been formulated for high spin components. The EE-EOM-CC scheme based on the restricted Hartree-Fock reference and standard amplitude equations as used in the Davidson diagonalization procedure yields the singlet states. The triplet and higher spin components require separate amplitude equations. In the case of quintets, the relevant equations are much simpler and easier to solve. Out of 26 diagrammatic terms contributing to the R{sub 1} and R{sub 2} singlet equations in the case of quintets, only R{sub 2} operator survives with 5more » diagrammatic terms present. In addition all terms engaging three body elements of the similarity transformed Hamiltonian disappear. This indicates a substantial simplification of the theory. The implemented method has been applied to the pilot study of the excited states of the C{sub 2} molecule and quintet states of C and Si atoms.« less

On the orbital stability of pendulum-like vibrations of a rigid body carrying a rotor

NASA Astrophysics Data System (ADS)

Yehia, Hamad M.; El-Hadidy, E. G.

2013-09-01

One of the most notable effects in mechanics is the stabilization of the unstable upper equilibrium position of a symmetric body fixed from one point on its axis of symmetry, either by giving the body a suitable angular velocity or by adding a suitably spinned rotor along its axis. This effect is widely used in technology and in space dynamics. The aim of the present article is to explore the effect of the presence of a rotor on a simple periodic motion of the rigid body and its motion as a physical pendulum. The equation in the variation for pendulum vibrations takes the form in which α depends on the moments of inertia, ρ on the gyrostatic momentum of the rotor and ν (the modulus of the elliptic function) depends on the total energy of the motion. This equation, which reduces to Lame's equation when ρ = 0, has not been studied to any extent in the literature. The determination of the zones of stability and instability of plane motion reduces to finding conditions for the existence of primitive periodic solutions (with periods 4 K( ν), 8 K( ν)) with those parameters. Complete analysis of primitive periodic solutions of this equation is performed analogously to that of Ince for Lame's equation. Zones of stability and instability are determined analytically and illustrated in a graphical form by plotting surfaces separating them in the three-dimensional space of parameters. The problem is also solved numerically in certain regions of the parameter space, and results are compared to analytical ones.

Celestial dynamics and astrometry in expanding universe

NASA Astrophysics Data System (ADS)

Kopeikin, Sergei

2012-08-01

Post - Newtonian theory of motion of celestial bodies and propagation of light was instrumental in conducting the critical experimental tests of general relativity and in building the astronomical ephemerides of celestial bodies in the solar system with an unparalleled precision. The cornerstone of the theory is the postulate that the solar system is gravitationally isolated from the rest of the universe and the background spacetime is asymptotically flat. The present talk abolishes this postulate and lays down the principles of celestial dynamics of particles and light moving in gravitational field of a localized astronomical system embedded to the expanding universe. We formulate the precise mathematical concept of the Newtonian limit of Einstein ’s field equations in the conformally - flat spacetime and analyse the geodesic equations of motion o f particles and light in this limit. We demonstrate that the equations of motion of particles and light can be reduced to their Newtonian counterparts by doing conformal transformations of time and space coordinates. However, the Newtonian equations for particles and light differ by terms of the first order in the Hubble constant. This leads to the important conclusion that the equations of motion used currently by Space Navigation Centres and Astronomical Observatories for calculating orbits of celestial bodies, are incomplete and missing some terms of cosmological origin. We explicitly identify the missing terms and demonstrate that they bring about a noticeable discrepancy between the observed and calculated astronomical ephemerides. We argue that a number of observed celestial anomalies in the solar system can be explained as caused by the Hubble expansion of the universe.

Vibration of a hydrostatic gas bearing due to supply pressure oscillations

NASA Technical Reports Server (NTRS)

Branch, H. D.; Watkins, C. B.; Eronini, I. E.

1984-01-01

The vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating supply pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and the sleeve is solved numerically together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The results presented include Bode plots of bearing oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency. The results are compared with the results of an earlier study involving the response of a similar bearing to oscillating exhaust pressure.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sergyeyev, Artur; Krtous, Pavel; Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holesovickach 2, Prague

We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in [J. High Energy Phys. 02 (2007) 004] and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordon equation obtained in [J. High Energy Phys. 02 (2007) 005] are joint eigenfunctions for all of thesemore » operators. We also present an explicit form of the zero mode for the Klein-Gordon equation with zero mass. In the semiclassical approximation we find that the separated solutions of the Hamilton-Jacobi equation for geodesic motion are also solutions for a set of Hamilton-Jacobi-type equations which correspond to the quadratic conserved quantities arising from the above Killing tensors.« less

A general method to determine the stability of compressible flows

NASA Technical Reports Server (NTRS)

Guenther, R. A.; Chang, I. D.

1982-01-01

Several problems were studied using two completely different approaches. The initial method was to use the standard linearized perturbation theory by finding the value of the individual small disturbance quantities based on the equations of motion. These were serially eliminated from the equations of motion to derive a single equation that governs the stability of fluid dynamic system. These equations could not be reduced unless the steady state variable depends only on one coordinate. The stability equation based on one dependent variable was found and was examined to determine the stability of a compressible swirling jet. The second method applied a Lagrangian approach to the problem. Since the equations developed were based on different assumptions, the condition of stability was compared only for the Rayleigh problem of a swirling flow, both examples reduce to the Rayleigh criterion. This technique allows including the viscous shear terms which is not possible in the first method. The same problem was then examined to see what effect shear has on stability.

An exact sum-rule for the Hubbard model: an historical/pedagogical approach

NASA Astrophysics Data System (ADS)

Di Matteo, S.; Claveau, Y.

2017-07-01

The aim of the present article is to derive an exact integral equation for the Green function of the Hubbard model through an equation-of-motion procedure, like in the original Hubbard papers. Though our exact integral equation does not allow to solve the Hubbard model, it represents a strong constraint on its approximate solutions. An analogous sum rule has been already obtained in the literature, through the use of a spectral moment technique. We think however that our equation-of-motion procedure can be more easily related to the historical procedure of the original Hubbard papers. We also discuss examples of possible applications of the sum rule and propose and analyse a solution, fulfilling it, that can be used for a pedagogical introduction to the Mott-Hubbard metal-insulator transition.

Gyrokinetic equations and full f solution method based on Dirac's constrained Hamiltonian and inverse Kruskal iteration

DOE Office of Scientific and Technical Information (OSTI.GOV)

Heikkinen, J. A.; Nora, M.

2011-02-15

Gyrokinetic equations of motion, Poisson equation, and energy and momentum conservation laws are derived based on the reduced-phase-space Lagrangian and inverse Kruskal iteration introduced by Pfirsch and Correa-Restrepo [J. Plasma Phys. 70, 719 (2004)]. This formalism, together with the choice of the adiabatic invariant J= as one of the averaging coordinates in phase space, provides an alternative to the standard gyrokinetics. Within second order in gyrokinetic parameter, the new equations do not show explicit ponderomotivelike or polarizationlike terms. Pullback of particle information with an iterated gyrophase and field dependent gyroradius function from the gyrocenter position defined by gyroaveraged coordinates allowsmore » direct numerical integration of the gyrokinetic equations in particle simulation of the field and particles with full distribution function. As an example, gyrokinetic systems with polarization drift either present or absent in the equations of motion are considered.« less

Self-propulsion of a planar electric or magnetic microbot immersed in a polar viscous fluid

NASA Astrophysics Data System (ADS)

Felderhof, B. U.

2011-05-01

A planar sheet immersed in an electrically polar liquid like water can propel itself by means of a plane wave charge density propagating in the sheet. The corresponding running electric wave polarizes the fluid and causes an electrical torque density to act on the fluid. The sheet is convected by the fluid motion resulting from the conversion of rotational particle motion, generated by the torque density, into translational fluid motion by the mechanism of friction and spin diffusion. Similarly, a planar sheet immersed in a magnetic ferrofluid can propel itself by means of a plane wave current density in the sheet and the torque density acting on the fluid corresponding to the running wave magnetic field and magnetization. The effect is studied on the basis of the micropolar fluid equations of motion and Maxwell’s equations of electrostatics or magnetostatics, respectively. An analytic expression is derived for the velocity of the sheet by perturbation theory to second order in powers of the amplitude of the driving charge or current density. Under the assumption that the equilibrium magnetic equation of state may be used in linearized form and that higher harmonics than the first may be neglected, a set of self-consistent integral equations is derived which can be solved numerically by iteration. In typical situations the second-order perturbation theory turns out to be quite accurate.

Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades

NASA Technical Reports Server (NTRS)

Hodges, D. H.; Dowell, E. H.

1974-01-01

The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.

Small-Caliber Projectile Target Impact Angle Determined From Close Proximity Radiographs

DTIC Science & Technology

2006-10-01

discrete motion data that can be numerically modeled using linear aerodynamic theory or 6-degrees-of- freedom equations of motion. The values of Fφ...Prediction Excel® Spreadsheet shown in figure 9. The Gamma at Impact Spreadsheet uses the linear aerodynamics model , equations 5 and 6, to calculate αT...trajectory angle error via consideration of the RMS fit errors of the actual firings. However, the linear aerodynamics model does not include this effect

A simple method to design non-collision relative orbits for close spacecraft formation flying

NASA Astrophysics Data System (ADS)

Jiang, Wei; Li, JunFeng; Jiang, FangHua; Bernelli-Zazzera, Franco

2018-05-01

A set of linearized relative motion equations of spacecraft flying on unperturbed elliptical orbits are specialized for particular cases, where the leader orbit is circular or equatorial. Based on these extended equations, we are able to analyze the relative motion regulation between a pair of spacecraft flying on arbitrary unperturbed orbits with the same semi-major axis in close formation. Given the initial orbital elements of the leader, this paper presents a simple way to design initial relative orbital elements of close spacecraft with the same semi-major axis, thus preventing collision under non-perturbed conditions. Considering the mean influence of J 2 perturbation, namely secular J 2 perturbation, we derive the mean derivatives of orbital element differences, and then expand them to first order. Thus the first order expansion of orbital element differences can be added to the relative motion equations for further analysis. For a pair of spacecraft that will never collide under non-perturbed situations, we present a simple method to determine whether a collision will occur when J 2 perturbation is considered. Examples are given to prove the validity of the extended relative motion equations and to illustrate how the methods presented can be used. The simple method for designing initial relative orbital elements proposed here could be helpful to the preliminary design of the relative orbital elements between spacecraft in a close formation, when collision avoidance is necessary.

Recursion Operators and Bi-Hamiltonian Structures in Multidimensions II,

DTIC Science & Technology

1986-07-01

a Symmifetry (1.2). For example the Kadomtsev - Petviashvili (KP) equation and the Davey-Stewartson (DS) equation admit two such hierarchies of...Degasperis, Nuovo Cimento, 398, 1 (1977). [16] P. Caudrey, Discrete and Periodic Spectral Transforms Related to the Kadomtsev - Petviashvili Equation ...these equations possess infinitely many time dependent symmetries and constants of motion. The master symmetries T for these equations are simply derived

Documenting the NASA Armstrong Flight Research Center Oblate Earth Simulation Equations of Motion and Integration Algorithm

NASA Technical Reports Server (NTRS)

Clarke, R.; Lintereur, L.; Bahm, C.

2016-01-01

A desire for more complete documentation of the National Aeronautics and Space Administration (NASA) Armstrong Flight Research Center (AFRC), Edwards, California legacy code used in the core simulation has led to this e ort to fully document the oblate Earth six-degree-of-freedom equations of motion and integration algorithm. The authors of this report have taken much of the earlier work of the simulation engineering group and used it as a jumping-o point for this report. The largest addition this report makes is that each element of the equations of motion is traced back to first principles and at no point is the reader forced to take an equation on faith alone. There are no discoveries of previously unknown principles contained in this report; this report is a collection and presentation of textbook principles. The value of this report is that those textbook principles are herein documented in standard nomenclature that matches the form of the computer code DERIVC. Previous handwritten notes are much of the backbone of this work, however, in almost every area, derivations are explicitly shown to assure the reader that the equations which make up the oblate Earth version of the computer routine, DERIVC, are correct.

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

PubMed

Kataoka; Tsutahara; Akuzawa

2000-02-14

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.

Orbital motions of astronomical bodies and their centre of mass from different reference frames: a conceptual step between the geocentric and heliocentric models

NASA Astrophysics Data System (ADS)

Guerra, André G. C.; Simeão Carvalho, Paulo

2016-09-01

The motion of astronomical bodies and the centre of mass of the system is not always well perceived by students. One of the struggles is the conceptual change of reference frame, which is the same that held back the acceptance of the Heliocentric model over the Geocentric one. To address the question, the notion of centre of mass, motion equations (and their numerical solution for a system of multiple bodies), and change of frame of reference is introduced. The discussion is done based on conceptual and real world examples, using the solar system. Consequently, through the use of simple ‘do it yourself’ methods and basic equations, students can debate complex motions, and have a wider and potentially effective understanding of physics.

Dynamic field theory and equations of motion in cosmology

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kopeikin, Sergei M., E-mail: kopeikins@missouri.edu; Petrov, Alexander N., E-mail: alex.petrov55@gmail.com

2014-11-15

We discuss a field-theoretical approach based on general-relativistic variational principle to derive the covariant field equations and hydrodynamic equations of motion of baryonic matter governed by cosmological perturbations of dark matter and dark energy. The action depends on the gravitational and matter Lagrangian. The gravitational Lagrangian depends on the metric tensor and its first and second derivatives. The matter Lagrangian includes dark matter, dark energy and the ordinary baryonic matter which plays the role of a bare perturbation. The total Lagrangian is expanded in an asymptotic Taylor series around the background cosmological manifold defined as a solution of Einstein’s equationsmore » in the form of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric tensor. The small parameter of the decomposition is the magnitude of the metric tensor perturbation. Each term of the series expansion is gauge-invariant and all of them together form a basis for the successive post-Friedmannian approximations around the background metric. The approximation scheme is covariant and the asymptotic nature of the Lagrangian decomposition does not require the post-Friedmannian perturbations to be small though computationally it works the most effectively when the perturbed metric is close enough to the background FLRW metric. The temporal evolution of the background metric is governed by dark matter and dark energy and we associate the large scale inhomogeneities in these two components as those generated by the primordial cosmological perturbations with an effective matter density contrast δρ/ρ≤1. The small scale inhomogeneities are generated by the condensations of baryonic matter considered as the bare perturbations of the background manifold that admits δρ/ρ≫1. Mathematically, the large scale perturbations are given by the homogeneous solution of the linearized field equations while the small scale perturbations are described by a particular solution of these equations with the bare stress–energy tensor of the baryonic matter. We explicitly work out the covariant field equations of the successive post-Friedmannian approximations of Einstein’s equations in cosmology and derive equations of motion of large and small scale inhomogeneities of dark matter and dark energy. We apply these equations to derive the post-Friedmannian equations of motion of baryonic matter comprising stars, galaxies and their clusters.« less

Thermal coupling effect on the vortex dynamics of superconducting thin films: time-dependent Ginzburg–Landau simulations

NASA Astrophysics Data System (ADS)

Jing, Ze; Yong, Huadong; Zhou, Youhe

2018-05-01

In this paper, vortex dynamics of superconducting thin films are numerically investigated by the generalized time-dependent Ginzburg–Landau (TDGL) theory. Interactions between vortex motion and the motion induced energy dissipation is considered by solving the coupled TDGL equation and the heat diffusion equation. It is found that thermal coupling has significant effects on the vortex dynamics of superconducting thin films. Branching in the vortex penetration path originates from the coupling between vortex motion and the motion induced energy dissipation. In addition, the environment temperature, the magnetic field ramp rate and the geometry of the superconducting film also greatly influence the vortex dynamic behaviors. Our results provide new insights into the dynamics of superconducting vortices, and give a mesoscopic understanding on the channeling and branching of vortex penetration paths during flux avalanches.

The Modelling of Axially Translating Flexible Beams

NASA Astrophysics Data System (ADS)

Theodore, R. J.; Arakeri, J. H.; Ghosal, A.

1996-04-01

The axially translating flexible beam with a prismatic joint can be modelled by using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, a non-dimensional form of the Euler Bernoulli beam equation is presented, obtained by using the concept of group velocity, and also the conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions leads to a time-dependent frequency equation for the translating flexible beam. A novel method is presented for solving this time dependent frequency equation by using a differential form of the frequency equation. The assume mode/Lagrangian formulation of dynamics is employed to derive closed form equations of motion. It is shown by using Lyapunov's first method that the dynamic responses of flexural modal variables become unstable during retraction of the flexible beam, which the dynamic response during extension of the beam is stable. Numerical simulation results are presented for the uniform axial motion induced transverse vibration for a typical flexible beam.

Response of Pendulums to Translational and Rotational Components of Ground Motion

NASA Astrophysics Data System (ADS)

Graizer, V.; Kalkan, E.

2008-12-01

Dynamic response of most seismological instruments and many engineering structures to ground shaking can be represented via response of a pendulum (single-degree-of-freedom oscillator). Pendulum response is usually simplified by considering the input from uni-axial translational motion only. Complete ground motion however, includes not only translational components but also rotations (tilt and torsion). We consider complete equations of motion for three following types of pendulum: (i) conventional mass-on-rod, (ii) mass- on-spring type, and (iii) inverted (astatic), then their response sensitivities to each component of complex ground motion are examined. Inverted pendulums are used in seismology for more than 100 years, for example, classical Wiechert's horizontal seismograph built around 1905 and still used at some seismological observatories, and recent Guralp's horizontal seismometers CMG-40T and CMG-3T. Inverted pendulums also have significant importance for engineering applications where they are often used to simulate the dynamic response of various structural systems. The results of this study show that a horizontal pendulum similar to a modern accelerometer used in strong motion measurements is practically sensitive to translational motion and tilt only, while inverted pendulum is sensitive not only to translational components, but also to angular accelerations and tilt. For better understanding of the inverted pendulum's dynamic behavior under complex ground excitation, relative contribution of each component of motion on response variants is carefully isolated. The responses of pendulums are calculated in time-domain using close-form solution Duhamel's integral with complex input forcing functions. As compared to a common horizontal pendulum, response of an inverted pendulum is sensitive to acceleration of gravity and vertical acceleration when it reaches the level close to 1.0 g. Gravity effect introduces nonlinearity into the differential equation of motion, and results in shift of the frequency response to lower frequencies. The equations of inverted pendulum represent elastic response of pendulums (as material behavior), with nonlinearity created by time and amplitude dependence of equation coefficients. Sensitivity of inverted pendulum to angular acceleration of tilt is proportional to the length of a pendulum, and should be taken into consideration since it can produce significant effect especially for long pendulums, idealizing for instance, bridge piers, bents, elevated water tanks, telecommunication towers, etc.

Nonlinear Drift-Kinetic Equation in the Presence of a Circularly Polarized Wave

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Krivorutsky, E. N.; Whitaker, Ann F. (Technical Monitor)

2001-01-01

Equations of the single particle motion and nonlinear kinetic equation for plasma in the presence of a circularly polarized wave of arbitrary frequency in the drift approximation are presented. The nonstationarity and inhomogeneity of the plasma-wave system are taken into account.

Two-Dimensional Motions of Rockets

ERIC Educational Resources Information Center

Kang, Yoonhwan; Bae, Saebyok

2007-01-01

We analyse the two-dimensional motions of the rockets for various types of rocket thrusts, the air friction and the gravitation by using a suitable representation of the rocket equation and the numerical calculation. The slope shapes of the rocket trajectories are discussed for the three types of rocket engines. Unlike the projectile motions, the…

Air motions inside dome room of Big Telescope Alt-azimuth at Special Astrophysical Observatory RAS. Numerical solutions of Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Nosov, V. V.; Lukin, V. P.; Nosov, E. V.; Torgaev, A. V.

2017-11-01

The structure of air turbulent motion inside the closed dome room of Big Telescope Alt-azimuth at Special Astrophysical Observatory of the Russian Academy of Sciences (RAS) has been experimentally and theoretically studied. Theoretical results have been reached by numerical solving of boundary value problem for Navier-Stokes equations. Solitary large vortices (coherent structures, topological solitons) are observed indoors. Coherent breakdown of these vortices leads to the coherent turbulence. In the case of identical boundary conditions the pattern of air motions as a result of the simulation and the pattern, registered experimentally using the compact portable ultrasonic weather station, are practically the same.

Celestial ephemerides in an expanding universe

NASA Astrophysics Data System (ADS)

Kopeikin, Sergei M.

2012-09-01

The post-Newtonian theory of motion of celestial bodies and propagation of light was instrumental in conducting the critical experimental tests of general relativity and in building the astronomical ephemerides of celestial bodies in the Solar System with unparalleled precision. The cornerstone of the theory is the postulate that the Solar System is gravitationally isolated from the rest of the Universe and the background spacetime is asymptotically flat. The present article extends this theoretical concept and formulates the principles of celestial dynamics of particles and light moving in the gravitational field of a localized astronomical system embedded to the expanding Friedmann-Lemaître-Robertson-Walker universe. We formulate the precise mathematical concept of the Newtonian limit of Einstein’s field equations in the conformally flat Friedmann-Lemaître-Robertson-Walker spacetime and analyze the geodesic motion of massive particles and light in this limit. We prove that by doing conformal spacetime transformations, one can reduce the equations of motion of particles and light to the classical form of the Newtonian theory. However, the time arguments in the equations of motion of particles and light differ from each other in terms being proportional to the Hubble constant H. This leads to the important conclusion that the equations of light propagation used currently by space navigation centers for fitting range and Doppler-tracking observations of celestial bodies are missing some terms of the cosmological origin that are proportional to the Hubble constant H. We also analyze the effect of the cosmological expansion on motion of electrons in atoms. We prove that the Hubble expansion does not affect the atomic frequencies and hence does not affect the atomic time scale used in the creation of astronomical ephemerides. We derive the cosmological correction to the light travel time equation and argue that its measurement opens an exciting opportunity to determine the local value of the Hubble constant H in the Solar System independently of cosmological observations.

Sul moto del baricentro del sistema Terra-Luna intorno al Sole in assenza di perturbazioni planetarie

NASA Astrophysics Data System (ADS)

Sambo, Alberto

2003-09-01

The Sun, the Earth and the Moon are considered from the point of view of a dynamical problem of three point masses. In this setting, we are interested in investigating the motion of the barycentre C of the Earth/Moon system with respect to the Sun. The differential equation of the motion considered is obtained in vectorial form from the first principles. Its investigation allows to conclude that the motion of the barycentre C of the Earth/Moon system around the Sun is not keplerian, even in absence of planetary perturbations. The equation is derived without specific assumptions, and can thus be applied to any other "three body" system.

Understanding reversals of a rattleback

NASA Astrophysics Data System (ADS)

Rauch-Wojciechowski, Stefan; Przybylska, Maria

2017-07-01

A counterintuitive unidirectional (say counterclockwise) motion of a toy rattleback takes place when it is started by tapping it at a long side or by spinning it slowly in the clockwise sense of rotation. We study the motion of a toy rattleback having an ellipsoidal-shaped bottom by using frictionless Newton equations of motion of a rigid body rolling without sliding in a plane. We simulate these equations for tapping and spinning initial conditions to see the contact trajectory, the force arm and the reaction force responsible for torque turning the rattleback in the counterclockwise sense of rotation. Long time behavior of such a rattleback is, however, quasi-periodic and a rattleback starting with small transversal oscillations turns in the clockwise direction.

A two-dimensional solution of the FW-H equation for rectilinear motion of sources

NASA Astrophysics Data System (ADS)

Bozorgi, Alireza; Siozos-Rousoulis, Leonidas; Nourbakhsh, Seyyed Ahmad; Ghorbaniasl, Ghader

2017-02-01

In this paper, a subsonic solution of the two-dimensional Ffowcs Williams and Hawkings (FW-H) equation is presented for calculation of noise generated by sources moving with constant velocity in a medium at rest or in a moving medium. The solution is represented in the frequency domain and is valid for observers located far from the noise sources. In order to verify the validity of the derived formula, three test cases are considered, namely a monopole, a dipole, and a quadrupole source in a medium at rest or in motion. The calculated results well coincide with the analytical solutions, validating the applicability of the formula to rectilinear subsonic motion problems.

Covariant Uniform Acceleration

NASA Astrophysics Data System (ADS)

Friedman, Yaakov; Scarr, Tzvi

2013-04-01

We derive a 4D covariant Relativistic Dynamics Equation. This equation canonically extends the 3D relativistic dynamics equation , where F is the 3D force and p = m0γv is the 3D relativistic momentum. The standard 4D equation is only partially covariant. To achieve full Lorentz covariance, we replace the four-force F by a rank 2 antisymmetric tensor acting on the four-velocity. By taking this tensor to be constant, we obtain a covariant definition of uniformly accelerated motion. This solves a problem of Einstein and Planck. We compute explicit solutions for uniformly accelerated motion. The solutions are divided into four Lorentz-invariant types: null, linear, rotational, and general. For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends 1D hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion. We use Generalized Fermi-Walker transport to construct a uniformly accelerated family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the Weak Hypothesis of Locality, we obtain local spacetime transformations from a uniformly accelerated frame K' to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame. We obtain velocity and acceleration transformations from a uniformly accelerated system K' to an inertial frame K. We introduce the 4D velocity, an adaptation of Horwitz and Piron s notion of "off-shell." We derive the general formula for the time dilation between accelerated clocks. We obtain a formula for the angular velocity of a uniformly accelerated object. Every rest point of K' is uniformly accelerated, and its acceleration is a function of the observer's acceleration and its position. We obtain an interpretation of the Lorentz-Abraham-Dirac equation as an acceleration transformation from K' to K.

Miscellaneous: Various Low-Mach-Number Fluid Problems and Motions

NASA Astrophysics Data System (ADS)

Zeytounian, Radyadour Kh.

In this last chapter, we consider, first, in Sect. 7.1, mainly the asymptotic derivation of the KZK equation of nonlinear acoustics, which generalizes the well-known Burgers' unsteady one-dimensional dissipative model equation (Burgers 1948) to an equation with a diffraction and parabolic effect.

Acoustic resonances in cylinder bundles oscillating in a compressibile fluid

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lin, W.H.; Raptis, A.C.

1984-12-01

This paper deals with an analytical study on acoustic resonances of elastic oscillations of a group of parallel, circular, thin cylinders in an unbounded volume of barotropic, compressible, inviscid fluid. The perturbed motion of the fluid is assumed due entirely to the flexural oscillations of the cylinders. The motion of the fluid disturbances is first formulated in a three-dimensional wave form and then casted into a two-dimensional Helmholtz equation for the harmonic motion in time and in axial space. The acoustic motion in the fluid and the elastic motion in the cylinders are solved simultaneously. Acoustic resonances were approximately determinedmore » from the secular (eigenvalue) equation by the method of successive iteration with the use of digital computers for a given set of the fluid properties and the cylinders' geometry and properties. Effects of the flexural wavenumber and the configuration of and the spacing between the cylinders on the acoustic resonances were thoroughly investigated.« less

Simple Harmonics Motion experiment based on LabVIEW interface for Arduino

NASA Astrophysics Data System (ADS)

Tong-on, Anusorn; Saphet, Parinya; Thepnurat, Meechai

2017-09-01

In this work, we developed an affordable modern innovative physics lab apparatus. The ultrasonic sensor is used to measure the position of a mass attached on a spring as a function of time. The data acquisition system and control device were developed based on LabVIEW interface for Arduino UNO R3. The experiment was designed to explain wave propagation which is modeled by simple harmonic motion. The simple harmonic system (mass and spring) was observed and the motion can be realized using curve fitting to the wave equation in Mathematica. We found that the spring constants provided by Hooke’s law and the wave equation fit are 9.9402 and 9.1706 N/m, respectively.

Advancements of In-Flight Mass Moment of Inertia and Structural Deflection Algorithms for Satellite Attitude Simulators

DTIC Science & Technology

2015-03-26

pendulum [15] to estimate the MOI. The benefit to this methodology is that instead of a direct comparison to Euler’s equations when using an on-board ACS...the equations of motion of pendulum motion are evaluated to estimate the resistance to angular acceleration. Instead of attempting to compare noisy...sensor data instantaneously when using on-board ACS data, the pendulum oscillation frequency is estimated, which can be globally smoothed for highly

The integration of the motion equations of low-orbiting earth satellites using Taylor's method

NASA Astrophysics Data System (ADS)

Krivov, A. V.; Chernysheva, N. A.

1990-04-01

A method for the numerical integration of the equations of motion for a satellite is proposed, taking the earth's oblateness and atmospheric drag into account. The method is based on Taylor's representation of the solution to the corresponding polynomial system. The algorithm for choosing the integration step and error estimation is constructed. The method is realized as a subrouting package. The method is applied to a low-orbiting earth satellite and the results are compared with those obtained using Everhart's method.

Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates

DOE PAGES

Goodman, Roy H.; Kevrekidis, P. G.; Carretero-González, R.

2015-04-14

We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. In this study, we uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals inmore » the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system, we are able to construct complex periodic orbits in the original, PDE, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.« less

Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates

DOE Office of Scientific and Technical Information (OSTI.GOV)

Goodman, Roy H.; Kevrekidis, P. G.; Carretero-González, R.

We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. In this study, we uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals inmore » the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system, we are able to construct complex periodic orbits in the original, PDE, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.« less

Coupled fluid-structure interaction. Part 1: Theory. Part 2: Application

NASA Technical Reports Server (NTRS)

Felippa, Carlos A.; Ohayon, Roger

1991-01-01

A general three dimensional variational principle is obtained for the motion of an acoustic field enclosed in a rigid or flexible container by the method of canonical decomposition applied to a modified form of the wave equation in the displacement potential. The general principle is specialized to a mixed two-field principle that contains the fluid displacement potential and pressure as independent fields. Semidiscrete finite element equations of motion based on this principle are derived and sample cases are given.

Investigation of Liquid Sloshing in Spin-Stabilized Satellites.

DTIC Science & Technology

1993-01-31

deformation of the spinning structure in addition to the rigid body motion . A Lagrangian approach was used to develop the equations of motion which include...nonlinear relationships for the unknown rigid body motions and linear terms for the relatively small elastic deformations of the members. Appendix F...the rigid body motion of the test assembly. A pendulum analogy was used to model the sloshing liquid in that early program. Several numerical

Spontaneous mode-selection in the self-propelled motion of a solid/liquid composite driven by interfacial instability

NASA Astrophysics Data System (ADS)

Takabatake, Fumi; Magome, Nobuyuki; Ichikawa, Masatoshi; Yoshikawa, Kenichi

2011-03-01

Spontaneous motion of a solid/liquid composite induced by a chemical Marangoni effect, where an oil droplet attached to a solid soap is placed on a water phase, was investigated. The composite exhibits various characteristic motions, such as revolution (orbital motion) and translational motion. The results showed that the mode of this spontaneous motion switches with a change in the size of the solid scrap. The essential features of this mode-switching were reproduced by ordinary differential equations by considering nonlinear friction with proper symmetry.

Modeling moving systems with RELAP5-3D

DOE PAGES

Mesina, G. L.; Aumiller, David L.; Buschman, Francis X.; ...

2015-12-04

RELAP5-3D is typically used to model stationary, land-based reactors. However, it can also model reactors in other inertial and accelerating frames of reference. By changing the magnitude of the gravitational vector through user input, RELAP5-3D can model reactors on a space station or the moon. The field equations have also been modified to model reactors in a non-inertial frame, such as occur in land-based reactors during earthquakes or onboard spacecraft. Transient body forces affect fluid flow in thermal-fluid machinery aboard accelerating crafts during rotational and translational accelerations. It is useful to express the equations of fluid motion in the acceleratingmore » frame of reference attached to the moving craft. However, careful treatment of the rotational and translational kinematics is required to accurately capture the physics of the fluid motion. Correlations for flow at angles between horizontal and vertical are generated via interpolation where no experimental studies or data exist. The equations for three-dimensional fluid motion in a non-inertial frame of reference are developed. As a result, two different systems for describing rotational motion are presented, user input is discussed, and an example is given.« less

Boundary layer flow of MHD tangent hyperbolic nanofluid over a stretching sheet: A numerical investigation

NASA Astrophysics Data System (ADS)

Khan, Mair; Hussain, Arif; Malik, M. Y.; Salahuddin, T.; Khan, Farzana

This article presents the two-dimensional flow of MHD hyperbolic tangent fluid with nanoparticles towards a stretching surface. The mathematical modelling of current flow analysis yields the nonlinear set of partial differential equations which then are reduce to ordinary differential equations by using suitable scaling transforms. Then resulting equations are solved by using shooting technique. The behaviour of the involved physical parameters (Weissenberg number We , Hartmann number M , Prandtl number Pr , Brownian motion parameter Nb , Lewis number Le and thermophoresis number Nt) on velocity, temperature and concentration are interpreted in detail. Additionally, local skin friction, local Nusselt number and local Sherwood number are computed and analyzed. It has been explored that Weissenberg number and Hartmann number are decelerate fluid motion. Brownian motion and thermophoresis both enhance the fluid temperature. Local Sherwood number is increasing function whereas Nusselt number is reducing function for increasing values of Brownian motion parameter Nb , Prandtl number Pr , thermophoresis parameter Nt and Lewis number Le . Additionally, computed results are compared with existing literature to validate the accuracy of solution, one can see that present results have quite resemblance with reported data.

Equivalent equations of motion for gravity and entropy

DOE Office of Scientific and Technical Information (OSTI.GOV)

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel

We demonstrate an equivalence between the wave equation obeyed by the entanglement entropy of CFT subregions and the linearized bulk Einstein equation in Anti-de Sitter space. In doing so, we make use of the formalism of kinematic space and fields on this space. We show that the gravitational dynamics are equivalent to a gauge invariant wave-equation on kinematic space and that this equation arises in natural correspondence to the conformal Casimir equation in the CFT.

Equivalent equations of motion for gravity and entropy

DOE PAGES

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; ...

2017-02-01

We demonstrate an equivalence between the wave equation obeyed by the entanglement entropy of CFT subregions and the linearized bulk Einstein equation in Anti-de Sitter space. In doing so, we make use of the formalism of kinematic space and fields on this space. We show that the gravitational dynamics are equivalent to a gauge invariant wave-equation on kinematic space and that this equation arises in natural correspondence to the conformal Casimir equation in the CFT.

MEGNO-analysis of the orbital evolution of GEO objects. (Russian Title: MEGNO-анализ орбитальной эволюции объектов зоны ГЕО )

NASA Astrophysics Data System (ADS)

Aleksandrova, A. G.; Chuvashov, I. N.; Bordovitsyna, T. V.

2011-07-01

The results of investigations of the instability of orbits in the GEO are presentеd. Average parameter MEGNO as main indicator of chaotic state has been used. The parameter is computed by combined numerical integration of equations of the motion, equations in variation and equations of MEGNO parameters. The results have been obtained using software package "Numerical model of the systems artificial satellite motion", implemented on the cluster "Skiff Cyberia".

Optimal Control of Stochastic Systems Driven by Fractional Brownian Motions

DTIC Science & Technology

2014-10-09

problems for stochastic partial differential equations driven by fractional Brownian motions are explicitly solved. For the control of a continuous time...linear systems with Brownian motion or a discrete time linear system with a white Gaussian noise and costs 1. REPORT DATE (DD-MM-YYYY) 4. TITLE AND...Army Research Office P.O. Box 12211 Research Triangle Park, NC 27709-2211 stochastic optimal control, fractional Brownian motion , stochastic

Smooth particle hydrodynamics: theory and application to the origin of the moon

DOE Office of Scientific and Technical Information (OSTI.GOV)

Benz, W.

1986-01-01

The origin of the moon is modeled by the so-called smooth particle hydrodynamics (SPH) method (Lucy, 1977, Monaghan 1985) which substitutes to the fluid a finite set of extended particles, the hydrodynamics equations reduce to the equation of motion of individual particles. These equations of motion differ only from the standard gravitational N-body problem insofar that pressure gradients and viscosity terms have to be added to the gradient of the potential to derive the forces between the particles. The numerical tools developed for ''classical'' N-body problems can therefore be readily applied to solve 3 dimensional hydroynamical problems. 12 refs., 1more » fig.« less

Technique to eliminate computational instability in multibody simulations employing the Lagrange multiplier

NASA Technical Reports Server (NTRS)

Watts, G.

1992-01-01

A programming technique to eliminate computational instability in multibody simulations that use the Lagrange multiplier is presented. The computational instability occurs when the attached bodies drift apart and violate the constraints. The programming technique uses the constraint equation, instead of integration, to determine the coordinates that are not independent. Although the equations of motion are unchanged, a complete derivation of the incorporation of the Lagrange multiplier into the equation of motion for two bodies is presented. A listing of a digital computer program which uses the programming technique to eliminate computational instability is also presented. The computer program simulates a solid rocket booster and parachute connected by a frictionless swivel.

Study of stability of the difference scheme for the model problem of the gaslift process

NASA Astrophysics Data System (ADS)

Temirbekov, Nurlan; Turarov, Amankeldy

2017-09-01

The paper studies a model of the gaslift process where the motion in a gas-lift well is described by partial differential equations. The system describing the studied process consists of equations of motion, continuity, equations of thermodynamic state, and hydraulic resistance. A two-layer finite-difference Lax-Vendroff scheme is constructed for the numerical solution of the problem. The stability of the difference scheme for the model problem is investigated using the method of a priori estimates, the order of approximation is investigated, the algorithm for numerical implementation of the gaslift process model is given, and the graphs are presented. The development and investigation of difference schemes for the numerical solution of systems of equations of gas dynamics makes it possible to obtain simultaneously exact and monotonic solutions.

Equations of Interdoublet Separation during Flagella Motion Reveal Mechanisms of Wave Propagation and Instability

PubMed Central

Bayly, Philip V.; Wilson, Kate S.

2014-01-01

The motion of flagella and cilia arises from the coordinated activity of dynein motor protein molecules arrayed along microtubule doublets that span the length of axoneme (the flagellar cytoskeleton). Dynein activity causes relative sliding between the doublets, which generates propulsive bending of the flagellum. The mechanism of dynein coordination remains incompletely understood, although it has been the focus of many studies, both theoretical and experimental. In one leading hypothesis, known as the geometric clutch (GC) model, local dynein activity is thought to be controlled by interdoublet separation. The GC model has been implemented as a numerical simulation in which the behavior of a discrete set of rigid links in viscous fluid, driven by active elements, was approximated using a simplified time-marching scheme. A continuum mechanical model and associated partial differential equations of the GC model have remained lacking. Such equations would provide insight into the underlying biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this article, the equations of motion for the flagellum and its doublets are derived from mechanical equilibrium principles and simple constitutive models. These equations are analyzed to reveal mechanisms of wave propagation and instability in the GC model. With parameter values in the range expected for Chlamydomonas flagella, solutions to the fully nonlinear equations closely resemble observed waveforms. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other leading models. PMID:25296329

The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies

NASA Astrophysics Data System (ADS)

Routh, Edward John

2013-03-01

Preface; 1. Moving axes and relative motion; 2. Oscillations about equilibrium; 3. Oscillations about a state of motion; 4. Motion of a body under no forces; 5. Motion of a body under any forces; 6. Nature of the motion given by linear equations and the conditions of stability; 7. Free and forced oscillations; 8. Determination of the constants of integration in terms of the initial conditions; 9. Calculus of finite differences; 10. Calculus of variations; 11. Precession and nutation; 12. Motion of the moon about its centre; 13. Motion of a string or chain; 14. Motion of a membrane; Notes.

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

NASA Astrophysics Data System (ADS)

Majima, H.; Suzuki, A.

2011-12-01

We consider the one-dimensional motion of a particle immersed in a potential field U(x) under the influence of a frictional (dissipative) force linear in velocity ( -γẋ) and a time-dependent external force ( K(t)). The dissipative system subject to these forces is discussed by introducing the extended Bateman's system, which is described by the Lagrangian: ℒ=mẋẏ-U(x+{1}/{2}y)+U(x-{1}/{2}y)+{γ}/{2}(xẏ-yẋ)-xK(t)+yK(t), which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting U(x±y/2)={1}/{2}k( specifically for a dual extended damped-amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman's Hamiltonian ℋ. The Heisenberg equations of motion utilizing the quantized Hamiltonian ℋ̂ surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K(t) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped-amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.

Motion of small bodies in classical field theory

NASA Astrophysics Data System (ADS)

Gralla, Samuel E.

2010-04-01

I show how prior work with R. Wald on geodesic motion in general relativity can be generalized to classical field theories of a metric and other tensor fields on four-dimensional spacetime that (1) are second-order and (2) follow from a diffeomorphism-covariant Lagrangian. The approach is to consider a one-parameter-family of solutions to the field equations satisfying certain assumptions designed to reflect the existence of a body whose size, mass, and various charges are simultaneously scaled to zero. (That such solutions exist places a further restriction on the class of theories to which our results apply.) Assumptions are made only on the spacetime region outside of the body, so that the results apply independent of the body’s composition (and, e.g., black holes are allowed). The worldline “left behind” by the shrinking, disappearing body is interpreted as its lowest-order motion. An equation for this worldline follows from the “Bianchi identity” for the theory, without use of any properties of the field equations beyond their being second-order. The form of the force law for a theory therefore depends only on the ranks of its various tensor fields; the detailed properties of the field equations are relevant only for determining the charges for a particular body (which are the “monopoles” of its exterior fields in a suitable limiting sense). I explicitly derive the force law (and mass-evolution law) in the case of scalar and vector fields, and give the recipe in the higher-rank case. Note that the vector force law is quite complicated, simplifying to the Lorentz force law only in the presence of the Maxwell gauge symmetry. Example applications of the results are the motion of “chameleon” bodies beyond the Newtonian limit, and the motion of bodies in (classical) non-Abelian gauge theory. I also make some comments on the role that scaling plays in the appearance of universality in the motion of bodies.

Nonlinear flap-lag axial equations of a rotating beam

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.; Kvaternik, R. G.

1977-01-01

It is possible to identify essentially four approaches by which analysts have established either the linear or nonlinear governing equations of motion for a particular problem related to the dynamics of rotating elastic bodies. The approaches include the effective applied load artifice in combination with a variational principle and the use of Newton's second law, written as D'Alembert's principle, applied to the deformed configuration. A third approach is a variational method in which nonlinear strain-displacement relations and a first-degree displacement field are used. The method introduced by Vigneron (1975) for deriving the linear flap-lag equations of a rotating beam constitutes the fourth approach. The reported investigation shows that all four approaches make use of the geometric nonlinear theory of elasticity. An alternative method for deriving the nonlinear coupled flap-lag-axial equations of motion is also discussed.

SU-G-JeP3-09: Tumor Location Prediction Using Natural Respiratory Volume for Respiratory Gated Radiation Therapy (RGRT): System Verification Study

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kim, M; Jung, J; Yoon, D

Purpose: Respiratory gated radiation therapy (RGRT) gives accurate results when a patient’s breathing is stable and regular. Thus, the patient should be fully aware during respiratory pattern training before undergoing the RGRT treatment. In order to bypass the process of respiratory pattern training, we propose a target location prediction system for RGRT that uses only natural respiratory volume, and confirm its application. Methods: In order to verify the proposed target location prediction system, an in-house phantom set was used. This set involves a chest phantom including target, external markers, and motion generator. Natural respiratory volume signals were generated using themore » random function in MATLAB code. In the chest phantom, the target takes a linear motion based on the respiratory signal. After a four-dimensional computed tomography (4DCT) scan of the in-house phantom, the motion trajectory was derived as a linear equation. The accuracy of the linear equation was compared with that of the motion algorithm used by the operating motion generator. In addition, we attempted target location prediction using random respiratory volume values. Results: The correspondence rate of the linear equation derived from the 4DCT images with the motion algorithm of the motion generator was 99.41%. In addition, the average error rate of target location prediction was 1.23% for 26 cases. Conclusion: We confirmed the applicability of our proposed target location prediction system for RGRT using natural respiratory volume. If additional clinical studies can be conducted, a more accurate prediction system can be realized without requiring respiratory pattern training.« less

Mixed quantum/classical theory of rotationally and vibrationally inelastic scattering in space-fixed and body-fixed reference frames

NASA Astrophysics Data System (ADS)

Semenov, Alexander; Babikov, Dmitri

2013-11-01

We formulated the mixed quantum/classical theory for rotationally and vibrationally inelastic scattering process in the diatomic molecule + atom system. Two versions of theory are presented, first in the space-fixed and second in the body-fixed reference frame. First version is easy to derive and the resultant equations of motion are transparent, but the state-to-state transition matrix is complex-valued and dense. Such calculations may be computationally demanding for heavier molecules and/or higher temperatures, when the number of accessible channels becomes large. In contrast, the second version of theory requires some tedious derivations and the final equations of motion are rather complicated (not particularly intuitive). However, the state-to-state transitions are driven by real-valued sparse matrixes of much smaller size. Thus, this formulation is the method of choice from the computational point of view, while the space-fixed formulation can serve as a test of the body-fixed equations of motion, and the code. Rigorous numerical tests were carried out for a model system to ensure that all equations, matrixes, and computer codes in both formulations are correct.

Research of small bodies motion prognosis effectivity on cluster "Skif Cyberia". (Russian Title: Исследование эффективности прогнозирования движения малых тел Солнечной системы на кластере "Skif Cyberia")

NASA Astrophysics Data System (ADS)

Baturin, A. P.

2010-12-01

The results of the experimental estimations on cluster "Skif Cyberia" of Everhart's numerical integration accuracy and rapidness are presented. The integration has been carried out for celestial bodies' equations of motion such as N-body problem equations and perturbed two-body problem equations. In the last case the perturbing bodies' coordinates are being taked during calculations from the ephemeris DE406. The accuracy and rapidness estimations have been made by means of forward and backward integrations with various values of Everhart method parameters of motion equations of the short-periodic comet Herschel-Rigollet. The optimal combinations of these parameters have been obtained. The research has been made both for 16-digit decimal accuracy and for 34-digit one.

Quantum asymmetry between time and space

PubMed Central

2016-01-01

An asymmetry exists between time and space in the sense that physical systems inevitably evolve over time, whereas there is no corresponding ubiquitous translation over space. The asymmetry, which is presumed to be elemental, is represented by equations of motion and conservation laws that operate differently over time and space. If, however, the asymmetry was found to be due to deeper causes, this conventional view of time evolution would need reworking. Here we show, using a sum-over-paths formalism, that a violation of time reversal (T) symmetry might be such a cause. If T symmetry is obeyed, then the formalism treats time and space symmetrically such that states of matter are localized both in space and in time. In this case, equations of motion and conservation laws are undefined or inapplicable. However, if T symmetry is violated, then the same sum over paths formalism yields states that are localized in space and distributed without bound over time, creating an asymmetry between time and space. Moreover, the states satisfy an equation of motion (the Schrödinger equation) and conservation laws apply. This suggests that the time–space asymmetry is not elemental as currently presumed, and that T violation may have a deep connection with time evolution. PMID:26997899

Theory and procedure for determining loads and motions in chine-immersed hydrodynamic impacts of prismatic bodies

NASA Technical Reports Server (NTRS)

Schnitzer, Emanuel

1953-01-01

A theoretical method is derived for the determination of the motions and loads during chine-immersed water landings of prismatic bodies. This method makes use of a variation of two-dimensional deflected water mass over the complete range of immersion, modified by a correction for three-dimensional flow. Equations are simplified through omission of the term proportional to the acceleration of the deflected mass for use in calculation of loads on hulls having moderate and heavy beam loading. The effects of water rise at the keel are included in these equations. In order to make a direct comparison of theory with experiment, a modification of the equations was made to include the effect of finite test-carriage mass. A simple method of computation which can be applied without reading the body of this report is presented as an appendix along with the required theoretical plots for determination of loads and motions in chine-immersed landings.

Aeroelastic effects in multi-rotor vehicles with application to a hybrid heavy lift system. Part 1: Formulation of equations of motion

NASA Technical Reports Server (NTRS)

Venkatesan, C.; Friedman, P.

1984-01-01

This report presents a set of governing coupled differential equations for a model of a hybrid aircraft. The model consists of multiple rotor systems connected by an elastic interconnecting structure, with options to add any combination of or all of the following components; i.e., thrusters, a buoyant hull, and an underslung weight. The dynamic equations are written for the individual blade with hub motions, for the rigid body motions of the whole model, and also for the flexible modes of the interconnecting structure. One of the purposes of this study is to serve as the basis of a numerical study aimed at determining the aeroelastic stability and structural response characteristics of a Hybrid Heavy Lift Airship (HHLA). It is also expected that the formulation may be applicable to analyzing stability and responses of dual rotor helicopters such as a Heavy Lift Helicopter (HLH). Futhermore, the model is capable of representing coupled rotor/body aeromechanical problems of single rotor helicopters.

Analytic study of a rolling sphere on a rough surface

NASA Astrophysics Data System (ADS)

Florea, Olivia A.; Rosca, Ileana C.

2016-11-01

In this paper it is realized an analytic study of the rolling's sphere on a rough horizontal plane under the action of its own gravity. The necessities of integration of the system of dynamical equations of motion lead us to find a reference system where the motion equations should be transformed into simpler expressions and which, in the presence of some significant hypothesis to permit the application of some original methods of analytical integration. In technical applications, the bodies may have a free rolling motion or a motion constrained by geometrical relations in assemblies of parts and machine parts. This study involves a lot of investigations in the field of tribology and of applied dynamics accompanied by experiments. Multiple recordings of several trajectories of the sphere, as well as their treatment of images, also followed by statistical processing experimental data allowed highlighting a very good agreement between the theoretical findings and experimental results.

On the Milankovitch orbital elements for perturbed Keplerian motion

NASA Astrophysics Data System (ADS)

Rosengren, Aaron J.; Scheeres, Daniel J.

2014-03-01

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace-Runge-Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.

Post-Newtonian Circular Restricted 3-Body Problem: Schwarzschild primaries

NASA Astrophysics Data System (ADS)

Dubeibe, F. L.; Lora-Clavijo, F. D.; González, G. A.

2017-07-01

The restricted three-body problem (RTBP) has been extensively studied to investigate the stability of the solar system, extra-solar subsystems, asteroid capture, and the dynamics of two massive black holes orbited by a sun. In the present work, we study the stability of the planar circular restricted three-body problem in the context of post-Newtonian approximations. First of all, we review the results obtained from the post-Newtonian equations of motion calculated in the framework of the Einstein-Infeld-Hoffmann formalism (EIH). Therefore, using the Fodor-Hoenselers-Perjes formalism (FHP), we have performed an expansion of the gravitational potential for two primaries, deriving a new system of equations of motion, which unlike the EIH-approach, preserves the Jacobian integral of motion. Additionally, we have obtained approximate expressions for the Lagrange points in terms of a mass parameter μ, where it is found that the deviations from the classical regime are larger for the FHP than for the EIH equations.

The Versatile Elastohydrodynamics of a Free Particle near a Thin Soft Wall

NASA Astrophysics Data System (ADS)

Salez, Thomas; Saintyves, Baudouin; Mahadevan, L.

2015-03-01

We address the free motion of a buoyant particle inside a viscous fluid, in the vicinity of a thin compressible elastic wall. After discussing the main scalings, we obtain analytically the dominant drag forces within the soft lubrication approximation. By including those into the equations of motion of the particle, we establish a general governing system of three coupled nonlinear and singular differential equations, that describe the three essential motions: sedimentation, hydroplaning, and hydrospinning, through four dimensionless control parameters. Numerical integration allows us to predict a wide zoology of exotic solutions - despite the low-Reynolds feature of the flow - including: spontaneous oscillation, Magnus-like effect, enhanced sedimentation, and boomerang-like effect. We compare these predictions to experiments. The presented elementary approach could be of interest in the description of a broad variety of elastohydrodynamical phenomena, including: landslides, ageing of cartilaginous joints, and motion of a cell in a microfluidic channel or in a blood vessel.

A space-time tensor formulation for continuum mechanics in general curvilinear, moving, and deforming coordinate systems

NASA Technical Reports Server (NTRS)

Avis, L. M.

1976-01-01

Tensor methods are used to express the continuum equations of motion in general curvilinear, moving, and deforming coordinate systems. The space-time tensor formulation is applicable to situations in which, for example, the boundaries move and deform. Placing a coordinate surface on such a boundary simplifies the boundary condition treatment. The space-time tensor formulation is also applicable to coordinate systems with coordinate surfaces defined as surfaces of constant pressure, density, temperature, or any other scalar continuum field function. The vanishing of the function gradient components along the coordinate surfaces may simplify the set of governing equations. In numerical integration of the equations of motion, the freedom of motion of the coordinate surfaces provides a potential for enhanced resolution of the continuum field function. An example problem of an incompressible, inviscid fluid with a top free surface is considered, where the surfaces of constant pressure (including the top free surface) are coordinate surfaces.

Entropy, extremality, euclidean variations, and the equations of motion

NASA Astrophysics Data System (ADS)

Dong, Xi; Lewkowycz, Aitor

2018-01-01

We study the Euclidean gravitational path integral computing the Rényi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton's constant G N , providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in G N . We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.

Dynamic Analysis and Control of Lightweight Manipulators with Flexible Parallel Link Mechanisms. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Lee, Jeh Won

1990-01-01

The objective is the theoretical analysis and the experimental verification of dynamics and control of a two link flexible manipulator with a flexible parallel link mechanism. Nonlinear equations of motion of the lightweight manipulator are derived by the Lagrangian method in symbolic form to better understand the structure of the dynamic model. The resulting equation of motion have a structure which is useful to reduce the number of terms calculated, to check correctness, or to extend the model to higher order. A manipulator with a flexible parallel link mechanism is a constrained dynamic system whose equations are sensitive to numerical integration error. This constrained system is solved using singular value decomposition of the constraint Jacobian matrix. Elastic motion is expressed by the assumed mode method. Mode shape functions of each link are chosen using the load interfaced component mode synthesis. The discrepancies between the analytical model and the experiment are explained using a simplified and a detailed finite element model.

Dynamic fluid sloshing in a one-dimensional array of coupled vessels

NASA Astrophysics Data System (ADS)

Huang, Y. H.; Turner, M. R.

2017-12-01

This paper investigates the coupled motion between the dynamics of N vessels coupled together in a one-dimensional array by springs and the motion of the inviscid fluid sloshing within each vessel. We develop a fully nonlinear model for the system relative to a moving frame such that the fluid in each vessel is governed by the Euler equations and the motion of each vessel is modeled by a forced spring equation. By considering a linearization of the model, the characteristic equation for the natural frequencies of the system is derived and analyzed for a variety of nondimensional parameter regimes. It is found that the problem can exhibit a variety of resonance situations from the 1 :1 resonance to (N +1 ) -fold 1 :⋯:1 resonance, as well as more general r :s :⋯:t resonances for natural numbers r ,s ,t . This paper focuses in particular on determining the existence of regions of parameter space where the (N +1 ) -fold 1 :⋯:1 resonance can be found.

Constants of the motion, universal time and the Hamilton-Jacobi function in general relativity

NASA Astrophysics Data System (ADS)

O'Hara, Paul

2013-04-01

In most text books of mechanics, Newton's laws or Hamilton's equations of motion are first written down and then solved based on initial conditions to determine the constants of the motions and to describe the trajectories of the particles. In this essay, we take a different starting point. We begin with the metrics of general relativity and show how they can be used to construct by inspection constants of motion, which can then be used to write down the equations of the trajectories. This will be achieved by deriving a Hamiltonian-Jacobi function from the metric and showing that its existence requires all of the above mentioned properties. The article concludes by showing that a consistent theory of such functions also requires the need for a universal measure of time which can be identified with the "worldtime" parameter, first introduced by Steuckelberg and later developed by Horwitz and Piron.

Modification of equation of motion of fluid-conveying pipe for laminar and turbulent flow profiles

NASA Astrophysics Data System (ADS)

Guo, C. Q.; Zhang, C. H.; Païdoussis, M. P.

2010-07-01

Considering the non-uniformity of the flow velocity distribution in fluid-conveying pipes caused by the viscosity of real fluids, the centrifugal force term in the equation of motion of the pipe is modified for laminar and turbulent flow profiles. The flow-profile-modification factors are found to be 1.333, 1.015-1.040 and 1.035-1.055 for laminar flow in circular pipes, turbulent flow in smooth-wall circular pipes and turbulent flow in rough-wall circular pipes, respectively. The critical flow velocities for divergence in the above-mentioned three cases are found to be 13.4%, 0.74-1.9% and 1.7-2.6%, respectively, lower than that with plug flow, while those for flutter are even lower, which could reach 36% for the laminar flow profile. By introducing two new concepts of equivalent flow velocity and equivalent mass, fluid-conveying pipe problems with different flow profiles can be solved with the equation of motion for plug flow.

Trajectory of Charged Particle in Combined Electric and Magnetic Fields Using Interactive Spreadsheets

ERIC Educational Resources Information Center

Tambade, Popat S.

2011-01-01

The objective of this article is to graphically illustrate to the students the physical phenomenon of motion of charged particle under the action of simultaneous electric and magnetic fields by simulating particle motion on a computer. Differential equations of motions are solved analytically and path of particle in three-dimensional space are…

Manipulator interactive design with interconnected flexible elements

NASA Technical Reports Server (NTRS)

Singh, R. P.; Likins, P. W.

1983-01-01

This paper describes the development of an analysis tool for the interactive design of control systems for manipulators and similar electro-mechanical systems amenable to representation as structures in a topological chain. The chain consists of a series of elastic bodies subject to small deformations and arbitrary displacements. The bodies are connected by hinges which permit kinematic constraints, control, or relative motion with six degrees of freedom. The equations of motion for the chain configuration are derived via Kane's method, extended for application to interconnected flexible bodies with time-varying boundary conditions. A corresponding set of modal coordinates has been selected. The motion equations are imbedded within a simulation that transforms the vector-dyadic equations into scalar form for numerical integration. The simulation also includes a linear, time-invariant controler specified in transfer function format and a set of sensors and actuators that interface between the structure and controller. The simulation is driven by an interactive set-up program resulting in an easy-to-use analysis tool.

A relativistic generalisation of rigid motions

NASA Astrophysics Data System (ADS)

Llosa, J.; Molina, A.; Soler, D.

2012-07-01

A weaker substitute for the too restrictive class of Born-rigid motions is proposed, which we call radar-holonomic motions. The definition is expressed as a set of differential equations. Integrability conditions and Cauchy problem are studied. We finally obtain an example of a radar-holonomic congruence containing a given worldline with a given value of the rotation on this line.

Investigation on Stability in Roll of Square Section Missile at High Angle of Attack

NASA Astrophysics Data System (ADS)

Tao, Yang; Fan, Zhaolin; Wu, Jifei; Wu, Wenhua

An experimental investigation of the stability in roll of a square section missile at high incidence was conducted in FL-23 wind tunnel. Dynamic motions were obtained on a square section missile that is free to rotate about its longitudinal axis. Different dynamic rolling motions were observed depending on the incidence of the model sting. These dynamic regimes include damped oscillations, quasi-limit-cycle wing-rock motion, and constant rolling. A coupling numerical method was established by solving the fluid dynamics equations and the rigid-body dynamics equations synchronously in order to predict the onset and the development of uncommented motions and then explore the unsteady movement characteristics of the aircraft. The study indicates that the aircraft loss stability at high incidence is caused by the asymmetric vertex on the level fin tip liftoff and attach alternately. The computation results are in line with the experiment results.

Acceleration constraints in modeling and control of nonholonomic systems

NASA Astrophysics Data System (ADS)

Bajodah, Abdulrahman H.

2003-10-01

Acceleration constraints are used to enhance modeling techniques for dynamical systems. In particular, Kane's equations of motion subjected to bilateral constraints, unilateral constraints, and servo-constraints are modified by utilizing acceleration constraints for the purpose of simplifying the equations and increasing their applicability. The tangential properties of Kane's method provide relationships between the holonomic and the nonholonomic partial velocities, and hence allow one to describe nonholonomic generalized active and inertia forces in terms of their holonomic counterparts, i.e., those which correspond to the system without constraints. Therefore, based on the modeling process objectives, the holonomic and the nonholonomic vector entities in Kane's approach are used interchangeably to model holonomic and nonholonomic systems. When the holonomic partial velocities are used to model nonholonomic systems, the resulting models are full-order (also called nonminimal or unreduced) and separated in accelerations. As a consequence, they are readily integrable and can be used for generic system analysis. Other related topics are constraint forces, numerical stability of the nonminimal equations of motion, and numerical constraint stabilization. Two types of unilateral constraints considered are impulsive and friction constraints. Impulsive constraints are modeled by means of a continuous-in-velocities and impulse-momentum approaches. In controlled motion, the acceleration form of constraints is utilized with the Moore-Penrose generalized inverse of the corresponding constraint matrix to solve for the inverse dynamics of servo-constraints, and for the redundancy resolution of overactuated manipulators. If control variables are involved in the algebraic constraint equations, then these tools are used to modify the controlled equations of motion in order to facilitate control system design. An illustrative example of spacecraft stabilization is presented.

Boundary layers and resistance on liquid motion with only slight friction

NASA Technical Reports Server (NTRS)

1980-01-01

The laws of fluid motion are examined systematically for the case where friction is assumed to be very slight. Calculations are carried out with the appropriate differential equation and practical investigations are illustrated.

Asymptotic of the Solutions of Hyperbolic Equations with a Skew-Symmetric Perturbation

NASA Astrophysics Data System (ADS)

Gallagher, Isabelle

1998-12-01

Using methods introduced by S. Schochet inJ. Differential Equations114(1994), 476-512, we compute the first term of an asymptotic expansion of the solutions of hyperbolic equations perturbated by a skew-symmetric linear operator. That result is first applied to two systems describing the motion of geophysic fluids: the rotating Euler equations and the primitive system of the quasigeostrophic equations. Finally in the last part, we study the slightly compressible Euler equations by application of that same result.

Mathematical Development and Computational Analysis of Harmonic Phase-Magnetic Resonance Imaging (HARP-MRI) Based on Bloch Nuclear Magnetic Resonance (NMR) Diffusion Model for Myocardial Motion.

PubMed

Dada, Michael O; Jayeoba, Babatunde; Awojoyogbe, Bamidele O; Uno, Uno E; Awe, Oluseyi E

2017-09-13

Harmonic Phase-Magnetic Resonance Imaging (HARP-MRI) is a tagged image analysis method that can measure myocardial motion and strain in near real-time and is considered a potential candidate to make magnetic resonance tagging clinically viable. However, analytical expressions of radially tagged transverse magnetization in polar coordinates (which is required to appropriately describe the shape of the heart) have not been explored because the physics required to directly connect myocardial deformation of tagged Nuclear Magnetic Resonance (NMR) transverse magnetization in polar geometry and the appropriate harmonic phase parameters are not yet available. The analytical solution of Bloch NMR diffusion equation in spherical geometry with appropriate spherical wave tagging function is important for proper analysis and monitoring of heart systolic and diastolic deformation with relevant boundary conditions. In this study, we applied Harmonic Phase MRI method to compute the difference between tagged and untagged NMR transverse magnetization based on the Bloch NMR diffusion equation and obtained radial wave tagging function for analysis of myocardial motion. The analytical solution of the Bloch NMR equations and the computational simulation of myocardial motion as developed in this study are intended to significantly improve healthcare for accurate diagnosis, prognosis and treatment of cardiovascular related deceases at the lowest cost because MRI scan is still one of the most expensive anywhere. The analysis is fundamental and significant because all Magnetic Resonance Imaging techniques are based on the Bloch NMR flow equations.

The dynamics and control of large flexible asymmetric spacecraft

NASA Astrophysics Data System (ADS)

Humphries, T. T.

1991-02-01

This thesis develops the equations of motion for a large flexible asymmetric Earth observation satellite and finds the characteristics of its motion under the influence of control forces. The mathematical model of the structure is produced using analytical methods. The equations of motion are formed using an expanded momentum technique which accounts for translational motion of the spacecraft hub and employs orthogonality relations between appendage and vehicle modes. The controllability and observability conditions of the full spacecraft motions using force and torque actuators are defined. A three axis reaction wheel control system is implemented for both slewing the spacecraft and controlling its resulting motions. From minor slew results it is shown that the lowest frequency elastic mode of the spacecraft is more important than higher frequency modes, when considering the effects of elastic motion on instrument pointing from the hub. Minor slews of the spacecraft configurations considered produce elastic deflections resulting in rotational attitude motions large enough to contravene pointing accuracy requirements of instruments aboard the spacecraft hub. Active vibration damping is required to reduce these hub motions to acceptable bounds in sufficiently small time. A comparison between hub mounted collocated and hub/appendage mounted non-collocated control systems verifies that provided the non-collocated system is stable, it can more effectively damp elastic modes whilst maintaining adequate damping of rigid modes. Analysis undertaken shows that the reaction wheel controller could be replaced by a thruster control system which decouples the modes of the spacecraft motion, enabling them to be individually damped.

Radially Symmetric Motions of Nonlinearly Viscoelastic Bodies Under Live Loads

NASA Astrophysics Data System (ADS)

Stepanov, Alexey B.; Antman, Stuart S.

2017-12-01

This paper treats radially symmetric motions of nonlinearly viscoelastic circular-cylindrical and spherical shells subjected to the live loads of centrifugal force and (time-dependent) hydrostatic pressures. The governing equations are exact versions of those for 3-dimensional continuum mechanics (so shell does not connote an approximate via some shell theory). These motions are governed by quasilinear third-order parabolic-hyperbolic equations having but one independent spatial variable. The principal part of such a partial differential equation is determined by a general family of nonlinear constitutive equations. The presence of strains in two orthogonal directions requires a careful treatment of constitutive restrictions that are physically natural and support the analysis. The interaction of geometrically exact formulations, the compatible use of general constitutive equations for material response, and the presence of live loads show how these factors play crucial roles in the behavior of solutions. In particular, for different kinds of live loads there are thresholds separating materials that produce qualitatively different dynamical behavior. The analysis (using classical methods) covers infinite-time blowup for cylindrical shells subject to centrifugal forces, infinite-time blowup for cylindrical shells subject to steady and time-dependent hydrostatic pressures, finite-time blowup for spherical shells subject to steady and time-dependent hydrostatic pressures, and the preclusion of total compression. This paper concludes with a sketch (using some modern methods) of the existence of regular solutions until the time of blowup.

Modeling and simulation of dynamics of a planar-motion rigid body with friction and surface contact

NASA Astrophysics Data System (ADS)

Wang, Xiaojun; Lv, Jing

2017-07-01

The modeling and numerical method for the dynamics of a planar-motion rigid body with frictional contact between plane surfaces were presented based on the theory of contact mechanics and the algorithm of linear complementarity problem (LCP). The Coulomb’s dry friction model is adopted as the friction law, and the normal contact forces are expressed as functions of the local deformations and their speeds in contact bodies. The dynamic equations of the rigid body are obtained by the Lagrange equation. The transition problem of stick-slip motions between contact surfaces is formulated and solved as LCP through establishing the complementary conditions of the friction law. Finally, a numerical example is presented as an example to show the application.

An intrinsic approach in the curved n-body problem: The negative curvature case

NASA Astrophysics Data System (ADS)

Diacu, Florin; Pérez-Chavela, Ernesto; Reyes Victoria, J. Guadalupe

We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincaré disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincaré upper half plane in order to analyze the hyperbolic and parabolic relative equilibria. Using techniques of Riemannian geometry, we characterize each of the above classes of periodic orbits. For n=2 and n=3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting.

Equations for estimating horizontal response spectra and peak acceleration from western North American earthquakes: A summary of recent work

USGS Publications Warehouse

Boore, D.M.; Joyner, W.B.; Fumal, T.E.

1997-01-01

In this paper we summarize our recently-published work on estimating horizontal response spectra and peak acceleration for shallow earthquakes in western North America. Although none of the sets of coefficients given here for the equations are new, for the convenience of the reader and in keeping with the style of this special issue, we provide tables for estimating random horizontal-component peak acceleration and 5 percent damped pseudo-acceleration response spectra in terms of the natural, rather than common, logarithm of the ground-motion parameter. The equations give ground motion in terms of moment magnitude, distance, and site conditions for strike-slip, reverse-slip, or unspecified faulting mechanisms. Site conditions are represented by the shear velocity averaged over the upper 30 m, and recommended values of average shear velocity are given for typical rock and soil sites and for site categories used in the National Earthquake Hazards Reduction Program's recommended seismic code provisions. In addition, we stipulate more restrictive ranges of magnitude and distance for the use of our equations than in our previous publications. Finally, we provide tables of input parameters that include a few corrections to site classifications and earthquake magnitude (the corrections made a small enough difference in the ground-motion predictions that we chose not to change the coefficients of the prediction equations).

Equations of motion of test particles for solving the spin-dependent Boltzmann–Vlasov equation

DOE PAGES

Xia, Yin; Xu, Jun; Li, Bao-An; ...

2016-06-16

A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann–Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. Themore » resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin–orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.« less

Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions.

PubMed

Banik, Suman Kumar; Bag, Bidhan Chandra; Ray, Deb Shankar

2002-05-01

Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).

Forced Convection Heat Transfer in Circular Pipes

ERIC Educational Resources Information Center

Tosun, Ismail

2007-01-01

One of the pitfalls of engineering education is to lose the physical insight of the problem while tackling the mathematical part. Forced convection heat transfer (the Graetz-Nusselt problem) certainly falls into this category. The equation of energy together with the equation of motion leads to a partial differential equation subject to various…

MHD Jeffrey nanofluid past a stretching sheet with viscous dissipation effect

NASA Astrophysics Data System (ADS)

Zokri, S. M.; Arifin, N. S.; Salleh, M. Z.; Kasim, A. R. M.; Mohammad, N. F.; Yusoff, W. N. S. W.

2017-09-01

This study investigates the influence of viscous dissipation on magnetohydrodynamic (MHD) flow of Jeffrey nanofluid over a stretching sheet with convective boundary conditions. The nonlinear partial differential equations are reduced into the nonlinear ordinary differential equations by utilizing the similarity transformation variables. The Runge-Kutta Fehlberg method is used to solve the problem numerically. The numerical solutions obtained are presented graphically for several dimensionless parameters such as Brownian motion, Lewis number and Eckert number on the specified temperature and concentration profiles. It is noted that the temperature profile is accelerated due to increasing values of Brownian motion parameter and Eckert number. In contrast, both the Brownian motion parameter and Lewis number have caused the deceleration in the concentration profiles.

The thermal-vortex equations

NASA Technical Reports Server (NTRS)

Shebalin, John V.

1987-01-01

The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.

What can we learn from the Wells, NV earthquake sequence about seismic hazard in the intermountain west?

USGS Publications Warehouse

Petersen, M.D.; Pankow, K.L.; Biasi, G.P.; Meremonte, M.

2008-01-01

The February 21, 2008 Wells, NV earthquake (M 6) was felt throughout eastern Nevada, southern Idaho, and western Utah. The town of Wells sustained significant damage to unreinforced masonry buildings. The earthquake occurred in a region of low seismic hazard with little seismicity, low geodetic strain rates, and few mapped faults. The peak horizontal ground acceleration predicted by the USGS National Seismic Hazard Maps is about 0.2 g at 2% probability of exceedance in 50 years, with the contributions coming mostly from the Ruby Mountain fault and background seismicity (M5-7.0). The hazard model predicts that the probability of occurrence of an M>6 event within 50 km of Wells is about 15% in 100 years. Although the earthquake was inside the USArray Transportable Array network, the nearest on-scale recordings of ground motions from the mainshock were too distant to estimate accelerations in town. The University of Nevada Reno, the University of Utah, and the U.S. Geological Survey deployed portable instruments to capture the ground motions from aftershocks of this rare normal-faulting event. Shaking from a M 4.7 aftershock recorded on portable instruments at distances less than 10 km exceeded 0.3 g, and sustained accelerations above 0.1 g lasted for about 5 seconds. For a magnitude 5 earthquake at 10 km distance the NGA equations predict median peak ground accelerations about 0.1 g. Ground motions from normal faulting earthquakes are poorly represented in the ground motion prediction equations. We compare portable and Transportable Array ground-motion recordings with prediction equations. Advanced National Seismic System stations in Utah recorded ground motions 250 km from the mainshock of about 2% g. The maximum ground motion recorded in Salt Lake City was in the center of the basin. We analyze the spatial variability of ground motions (rock vs. soil) and the influence of the Salt Lake Basin in modifying the ground motions. We then compare this data with the September 28, 2004 Parkfield aftershocks to contrast the differences between strike-slip and normal ground motions.

Satellite Formation Design for Space Based Radar Applications

DTIC Science & Technology

2007-07-30

communications. While the Clohessy - Wiltshire Hills (CWH) equations have been in existence for sometime, it is more recently that they have been... Clohessy - Wiltshire equations. To get the state transition matrix for relative position and velocity, these differential equations are integrated to...Practical Guidance Methodology for Relative Motion of LEO Spacecraft Based on the Clohessy - Wiltshire Equations,” AAS Paper 04-252, AAS/AIAA Space

Covariant Formulation of Hooke's Law.

ERIC Educational Resources Information Center

Gron, O.

1981-01-01

Introducing a four-vector strain and a four-force stress, Hooke's law is written as a four-vector equation. This formulation is shown to clarify seemingly paradoxical results in connection with uniformly accelerated motion, and rotational motion with angular acceleration. (Author/JN)

An analytical model of a curved beam with a T shaped cross section

NASA Astrophysics Data System (ADS)

Hull, Andrew J.; Perez, Daniel; Cox, Donald L.

2018-03-01

This paper derives a comprehensive analytical dynamic model of a closed circular beam that has a T shaped cross section. The new model includes in-plane and out-of-plane vibrations derived using continuous media expressions which produces results that have a valid frequency range above those available from traditional lumped parameter models. The web is modeled using two-dimensional elasticity equations for in-plane motion and the classical flexural plate equation for out-of-plane motion. The flange is modeled using two sets of Donnell shell equations: one for the left side of the flange and one for the right side of the flange. The governing differential equations are solved with unknown wave propagation coefficients multiplied by spatial domain and time domain functions which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to results from finite element analysis.

Preliminary Planar Formation: Flight Dynamics Near Sun-Earth L2 Point

NASA Technical Reports Server (NTRS)

Segerman, Alan M.; Zedd, Michael F.

2003-01-01

NASA's Goddard Space Flight Center is planning a series of missions in the vicinity of the Sun-Earth L2 libration point. Some of these projects will involve a distributed space system of telescope spacecraft acting together as a single telescope for high-resolution. The individual telescopes will be configured in a plane, surrounding a hub, where the telescope plane can be aimed toward various astronomical targets of interest. In preparation for these missions, it is necessary to develop an improved understanding of the dynamical behavior of objects in a planar configuration near L2. The classical circular restricted three body problem is taken as the basis for the analysis. At first order, the motion of such a telescope relative to the hub is described by a system of linear second order differential equations. These equations are identical to the circular restricted problem's linear equations describing the hub motion about L2. Therefore, the fundamental frequencies, both parallel to and normal to the ecliptic plane, are the same for the relative telescope motion as for the hub motion. To maintain the telescope plane for the duration necessary for the planned observations, a halo-type orbit of the telescopes about the hub is investigated. By using a halo orbit, the individual telescopes remain in approximately the same plane over the observation duration. For such an orbit, the fundamental periods parallel to and normal to the ecliptic plane are forced to be the same by careful selection of the initial conditions in order to adjust the higher order forces. The relative amplitudes of the resulting oscillations are associated with the orientation of the telescope plane relative to the ecliptic. As in the circular restricted problem, initial conditions for the linearized equations must be selected so as not to excite the convergent or divergent linear modes. In a higher order analysis, the telescope relative motion equations include the effects of the position of the hub relative to L2. In this paper, the differential equations are developed through second order in the distance of the hub from the libration point. A modified Lindstedt-Poincad perturbation method is employed to construct the solution of these differential equations through that same order of magnitude. In the course of the solution process, relationships are determined between the initial conditions of the telescopes, selected in order to avoid resonance excitation. As the differential equations include the hub position, it is necessary to simultaneously develop the solution for the hub. As has been done in past analyses of the circular restricted problem, the hub position is written in a power series formulation in terms of its distance from L2. Then, in order to be included in the telescope equations, the hub solution is cast in terms of the nonlinear frequency of the relative telescope motion. In the course of the analysis, it is determined that the hub should also maintain a halo orbit - about L2. Additionally, relationships are formed between the initial conditions of the telescopes and the hub. These relationships may be used to associate sets of initial conditions with particular orientations of the telescope plane. The accuracy of the analytical solution is verified through various simulations and comparison to numerical integration of the differential equations. The results of the simulations are presented, along with a graphical representation of the relationships between the initial conditions of the telescopes and hub.

Elimination of secular terms from the differential equations for the elements of perturbed two-body motion

NASA Technical Reports Server (NTRS)

Bond, Victor R.; Fraietta, Michael F.

1991-01-01

In 1961, Sperling linearized and regularized the differential equations of motion of the two-body problem by changing the independent variable from time to fictitious time by Sundman's transformation (r = dt/ds) and by embedding the two-body energy integral and the Laplace vector. In 1968, Burdet developed a perturbation theory which was uniformly valid for all types of orbits using a variation of parameters approach on the elements which appeared in Sperling's equations for the two-body solution. In 1973, Bond and Hanssen improved Burdet's set of differential equations by embedding the total energy (which is a constant when the potential function is explicitly dependent upon time.) The Jacobian constant was used as an element to replace the total energy in a reformulation of the differential equations of motion. In the process, another element which is proportional to a component of the angular momentum was introduced. Recently trajectories computed during numerical studies of atmospheric entry from circular orbits and low thrust beginning in near-circular orbits exhibited numerical instability when solved by the method of Bond and Gottlieb (1989) for long time intervals. It was found that this instability was due to secular terms which appear on the righthand sides of the differential equations of some of the elements. In this paper, this instability is removed by the introduction of another vector integral called the delta integral (which replaces the Laplace Vector) and another scalar integral which removes the secular terms. The introduction of these integrals requires a new derivation of the differential equations for most of the elements. For this rederivation, the Lagrange method of variation of parameters is used, making the development more concise. Numerical examples of this improvement are presented.

Large eddy simulation of incompressible turbulent channel flow

NASA Technical Reports Server (NTRS)

Moin, P.; Reynolds, W. C.; Ferziger, J. H.

1978-01-01

The three-dimensional, time-dependent primitive equations of motion were numerically integrated for the case of turbulent channel flow. A partially implicit numerical method was developed. An important feature of this scheme is that the equation of continuity is solved directly. The residual field motions were simulated through an eddy viscosity model, while the large-scale field was obtained directly from the solution of the governing equations. An important portion of the initial velocity field was obtained from the solution of the linearized Navier-Stokes equations. The pseudospectral method was used for numerical differentiation in the horizontal directions, and second-order finite-difference schemes were used in the direction normal to the walls. The large eddy simulation technique is capable of reproducing some of the important features of wall-bounded turbulent flows. The resolvable portions of the root-mean square wall pressure fluctuations, pressure velocity-gradient correlations, and velocity pressure-gradient correlations are documented.

The Motion of a Leaking Oscillator: A Study for the Physics Class

ERIC Educational Resources Information Center

Rodrigues, Hilário; Panza, Nelson; Portes, Dirceu; Soares, Alexandre

2014-01-01

This paper is essentially about the general form of Newton's second law for variable mass problems. We develop a model for describing the motion of the one-dimensional oscillator with a variable mass within the framework of classroom physics. We present a simple numerical procedure for the solution of the equation of motion of the system to…

The Effect of Solar Radiation Pressure on the Motion of an Artificial Satellite

NASA Technical Reports Server (NTRS)

Bryant, Robert W.

1961-01-01

The effects of solar radiation pressure on the motion of an artificial satellite are obtained, including the effects of the intermittent acceleration which results from the eclipsing of the satellite by the earth. Vectorial methods have been utilized to obtain the nonlinear equations describing the motion, and the method of Kryloff-Bogoliuboff has been applied in their solution.

Turbulent solutions of equations of fluid motion

NASA Technical Reports Server (NTRS)

Deissler, R. G.

1985-01-01

Some turbulent solutions of the unaveraged Navier-Stokes equations (equations of fluid motion) are reviewed. Those equations are solved numerically in order to study the nonlinear physics of incompressible turbulent flow. The three components of the mean-square velocity fluctuations are initially equal for the conditions chosen. The resulting solutions show characteristics of turbulence, such as the linear and nonlinear excitation of small-scale fluctuations. For the stronger fluctuations the initially nonrandom flow develops into an apparently random turbulence. The cases considered include turbulence that is statistically homogeneous or inhomogeneous and isotropic or anisotropic. A statistically steady-state turbulence is obtained by using a spatially periodic body force. Various turbulence processes, including the transfer of energy between eddy sizes and between directional components and the production, dissipation, and spatial diffusion of turbulence, are considered. It is concluded that the physical processes occurring in turbulence can be profitably studied numerically.

Characteristics of manipulator for industrial robot with three rotational pairs having parallel axes

NASA Astrophysics Data System (ADS)

Poteyev, M. I.

1986-01-01

The dynamics of a manipulator with three rotatinal kinematic pairs having parallel axes are analyzed, for application in an industrial robot. The system of Lagrange equations of the second kind, describing the motion of such a mechanism in terms of kinetic energy in generalized coordinates, is reduced to equations of motion in terms of Newton's laws. These are useful not only for either determining the moments of force couples which will produce a prescribed motion or, conversely determining the motion which given force couples will produce but also for solving optimization problems under constraints in both cases and for estimating dynamic errors. As a specific example, a manipulator with all three axes of vertical rotation is considered. The performance of this manipulator, namely the parameters of its motion as functions of time, is compared with that of a manipulator having one rotational and two translational kinematic pairs. Computer aided simulation of their motion on the basis of ideal models, with all three links represented by identical homogeneous bars, has yielded velocity time diagrams which indicate that the manipulator with three rotational pairs is 4.5 times faster.

Theoretical model of chirality-induced helical self-propulsion

NASA Astrophysics Data System (ADS)

Yamamoto, Takaki; Sano, Masaki

2018-01-01

We recently reported the experimental realization of a chiral artificial microswimmer exhibiting helical self-propulsion [T. Yamamoto and M. Sano, Soft Matter 13, 3328 (2017), 10.1039/C7SM00337D]. In the experiment, cholesteric liquid crystal (CLC) droplets dispersed in surfactant solutions swam spontaneously, driven by the Marangoni flow, in helical paths whose handedness is determined by the chirality of the component molecules of CLC. To study the mechanism of the emergence of the helical self-propelled motion, we propose a phenomenological model of the self-propelled helical motion of the CLC droplets. Our model is constructed by symmetry argument in chiral systems, and it describes the dynamics of CLC droplets with coupled time-evolution equations in terms of a velocity, an angular velocity, and a tensor variable representing the symmetry of the helical director field of the droplet. We found that helical motions as well as other chiral motions appear in our model. By investigating bifurcation behaviors between each chiral motion, we found that the chiral coupling terms between the velocity and the angular velocity, the structural anisotropy of the CLC droplet, and the nonlinearity of model equations play a crucial role in the emergence of the helical motion of the CLC droplet.

Validation of the Passenger Ride Quality Apparatus (PRQA) for simulation of aircraft motions for ride-quality research

NASA Technical Reports Server (NTRS)

Bigler, W. B., II

1977-01-01

The NASA passenger ride quality apparatus (PRQA), a ground based motion simulator, was compared to the total in flight simulator (TIFS). Tests were made on PRQA with varying stimuli: motions only; motions and noise; motions, noise, and visual; and motions and visual. Regression equations for the tests were obtained and subsequent t-testing of the slopes indicated that ground based simulator tests produced comfort change rates similar to actual flight data. It was recommended that PRQA be used in the ride quality program for aircraft and that it be validated for other transportation modes.

The effect of memory in the stochastic master equation analyzed using the stochastic Liouville equation of motion. Electronic energy migration transfer between reorienting donor-donor, donor-acceptor chromophores

NASA Astrophysics Data System (ADS)

Håkansson, Pär; Westlund, Per-Olof

2005-01-01

This paper discusses the process of energy migration transfer within reorientating chromophores using the stochastic master equation (SME) and the stochastic Liouville equation (SLE) of motion. We have found that the SME over-estimates the rate of the energy migration compared to the SLE solution for a case of weakly interacting chromophores. This discrepancy between SME and SLE is caused by a memory effect occurring when fluctuations in the dipole-dipole Hamiltonian ( H( t)) are on the same timescale as the intrinsic fast transverse relaxation rate characterized by (1/ T2). Thus the timescale critical for energy-transfer experiments is T2≈10 -13 s. An extended SME is constructed, accounting for the memory effect of the dipole-dipole Hamiltonian dynamics. The influence of memory on the interpretation of experiments is discussed.

String junction as a baryonic constituent

NASA Astrophysics Data System (ADS)

Kalashnikova, Yu. S.; Nefediev, A. V.

1996-02-01

We extend the model for QCD string with quarks to consider the Mercedes Benz string configuration describing the three-quark baryon. Under the assumption of adiabatic separation of quark and string junction motion we formulate and solve the classical equation of motion for the junction. We dare to quantize the motion of the junction, and discuss the impact of these modes on the baryon spectra.

Solution of the exact equations for three-dimensional atmospheric entry using directly matched asymptotic expansions

NASA Technical Reports Server (NTRS)

Busemann, A.; Vinh, N. X.; Culp, R. D.

1976-01-01

The problem of determining the trajectories, partially or wholly contained in the atmosphere of a spherical, nonrotating planet, is considered. The exact equations of motion for three-dimensional, aerodynamically affected flight are derived. Modified Chapman variables are introduced and the equations are transformed into a set suitable for analytic integration using asymptotic expansions. The trajectory is solved in two regions: the outer region, where the force may be considered a gravitational field with aerodynamic perturbations, and the inner region, where the force is predominantly aerodynamic, with gravity as a perturbation. The two solutions are matched directly. A composite solution, valid everywhere, is constructed by additive composition. This approach of directly matched asymptotic expansions applied to the exact equations of motion couched in terms of modified Chapman variables yields an analytical solution which should prove to be a powerful tool for aerodynamic orbit calculations.

Motion of the angular momentum vector in body coordinates for torque-free dual-spin spacecraft

NASA Technical Reports Server (NTRS)

Fedor, J. V.

1981-01-01

The motion of the angular momentum vector in body coordinates for torque free, asymmetric dual spin spacecraft without and, for a special case, with energy dissipation on the main spacecraft is investigated. Without energy dissipation, two integrals can be obtained from the Euler equations of motion. Using the classical method of elimination of variable, the motion about the equilibrium points (six for the general case) are derived with these integrals. For small nutation angle, theta, the trajectories about the theta = 0 deg and theta = 180 deg points readily show the requirements for stable motion about these points. Also the conditions needed to eliminate stable motion about the theta = 180 deg point as well as the other undesireable equilibrium points follow directly from these equations. For the special case where the angular momentum vector moves about the principal axis which contains the momentum wheel, the notion of 'free variable' azimuth angle is used. Physically this angle must vary from 0 to 2 pi in a circular periodic fashion. Expressions are thus obtained for the nutation angle in terms of the free variable and other spacecraft parameters. Results show that in general there are two separate trajectory expressions that govern the motion of the angular momentum vector in body coordinates.

A new performance index for the repetitive motion of mobile manipulators.

PubMed

Xiao, Lin; Zhang, Yunong

2014-02-01

A mobile manipulator is a robotic device composed of a mobile platform and a stationary manipulator fixed to the platform. To achieve the repetitive motion control of mobile manipulators, the mobile platform and the manipulator have to realize the repetitive motion simultaneously. To do so, a novel quadratic performance index is, for the first time, designed and presented in this paper, of which the effectiveness is analyzed by following a neural dynamics method. Then, a repetitive motion scheme is proposed by combining the criterion, physical constraints, and integrated kinematical equations of mobile manipulators, which is further reformulated as a quadratic programming (QP) subject to equality and bound constraints. In addition, two important Bridge theorems are established to prove that such a QP can be converted equivalently into a linear variational inequality, and then equivalently into a piecewise-linear projection equation (PLPE). A real-time numerical algorithm based on PLPE is thus developed and applied for the online solution of the resultant QP. Two tracking-path tasks demonstrate the effectiveness and accuracy of the repetitive motion scheme. In addition, comparisons between the nonrepetitive and repetitive motion further validate the superiority and novelty of the proposed scheme.

Nonlinear motion of cantilevered SWNT and Its Meaning to Phonon Dynamics

NASA Astrophysics Data System (ADS)

Koh, Heeyuen; Cannon, James; Chiashi, Shohei; Shiomi, Junichiro; Maruyama, Shigeo

2013-03-01

Based on the finding that the lowest frequency mode of cantilevered SWNT is described by the continuum beam theory in frequency domain, we considered its effect of the symmetric structure for the coupling of orthogonal transverse modes to explain the nonlinear motion of free thermal vibration. This nonlinear motion calculated by our molecular dynamics simulation, once regarded as noise, is observed to have the periodic order with duffing and beating, which is dependent on aspect ratio and temperature. It could be dictated by the governing equation from the Green Lagrangian strain tensor. The nonlinear beam equation from strain tensor described the motion well for various models which has different aspect ratio in molecular dynamics simulation. Since this motion is nothing but the interaction between 2nd mode of radial, tangential mode and 1st longitudinal mode, it was found that Green Lagrangian strain tensor is capable to deal such coupling. The free thermal motion of suspended SWNT is also considered without temperature gradient. The Q factor measured by this theoretical analysis will be discussed. Part of this work was financially supported by Grant-in-Aid for Scientific Research (19054003 and 22226006), and Global COE Program 'Global Center for Excellence for Mechanical Systems Innovation'

Multi-Hamiltonian structure of equations of hydrodynamic type

NASA Astrophysics Data System (ADS)

Gümral, H.; Nutku, Y.

1990-11-01

The discussion of the Hamiltonian structure of two-component equations of hydrodynamic type is completed by presenting the Hamiltonian operators for Euler's equation governing the motion of plane sound waves of finite amplitude and another quasilinear second-order wave equation. There exists a doubly infinite family of conserved Hamiltonians for the equations of gas dynamics that degenerate into one, namely, the Benney sequence, for shallow-water waves. Infinite sequences of conserved quantities for these equations are also presented. In the case of multicomponent equations of hydrodynamic type, it is shown, that Kodama's generalization of the shallow-water equations admits bi-Hamiltonian structure.

Analytical solution of the time-dependent Bloch NMR flow equations: a translational mechanical analysis

NASA Astrophysics Data System (ADS)

Awojoyogbe, O. B.

2004-08-01

Various biological and physiological properties of living tissue can be studied by means of nuclear magnetic resonance techniques. Unfortunately, the basic physics of extracting the relevant information from the solution of Bloch nuclear magnetic resource (NMR) equations to accurately monitor the clinical state of biological systems is still not yet fully understood. Presently, there are no simple closed solutions known to the Bloch equations for a general RF excitation. Therefore the translational mechanical analysis of the Bloch NMR equations presented in this study, which can be taken as definitions of new functions to be studied in detail may reveal very important information from which various NMR flow parameters can be derived. Fortunately, many of the most important but hidden applications of blood flow parameters can be revealed without too much difficulty if appropriate mathematical techniques are used to solve the equations. In this study we are concerned with a mathematical study of the laws of NMR physics from the point of view of translational mechanical theory. The important contribution of this study is that solutions to the Bloch NMR flow equations do always exist and can be found as accurately as desired. We shall restrict our attention to cases where the radio frequency field can be treated by simple analytical methods. First we shall derive a time dependant second-order non-homogeneous linear differential equation from the Bloch NMR equation in term of the equilibrium magnetization M0, RF B1( t) field, T1 and T2 relaxation times. Then, we would develop a general method of solving the differential equation for the cases when RF B1( t)=0, and when RF B1( t)≠0. This allows us to obtain the intrinsic or natural behavior of the NMR system as well as the response of the system under investigation to a specific influence of external force to the system. Specifically, we consider the case where the RF B1 varies harmonically with time. Here the complete motion of the system consists of two parts. The first part describes the motion of the transverse magnetization My in the absence of RF B( t) field. The second part of the motion described by the particular integral of the derived differential equation does not decay with time but continues its periodic behavior indefinitely. The complete motion of the NMR flow system is then quantitatively and qualitatively described.

Measurements of Aerodynamic Damping in the MIT Transonic Rotor

NASA Technical Reports Server (NTRS)

Crawley, E. F.

1981-01-01

A method was developed and demonstrated for the direct measurement of aerodynamic forcing and aerodynamic damping of a transonic compressor. The method is based on the inverse solution of the structural dynamic equations of motion of the blade disk system in order to determine the forces acting on the system. The disturbing and damping forces acting on a given blade are determined if the equations of motion are expressed in individual blade coordinates. If the structural dynamic equations are transformed to multiblade coordinates, the damping can be measured for blade disk modes, and related to a reduced frequency and interblade phase angle. In order to measure the aerodynamic damping in this way, the free response to a known excitation is studied.

Formulation of the relativistic moment implicit particle-in-cell method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Noguchi, Koichi; Tronci, Cesare; Zuccaro, Gianluca

2007-04-15

A new formulation is presented for the implicit moment method applied to the time-dependent relativistic Vlasov-Maxwell system. The new approach is based on a specific formulation of the implicit moment method that allows us to retain the same formalism that is valid in the classical case despite the formidable complication introduced by the nonlinear nature of the relativistic equations of motion. To demonstrate the validity of the new formulation, an implicit finite difference algorithm is developed to solve the Maxwell's equations and equations of motion. A number of benchmark problems are run: two stream instability, ion acoustic wave damping, Weibelmore » instability, and Poynting flux acceleration. The numerical results are all in agreement with analytical solutions.« less

A Simplified Treatment of Brownian Motion and Stochastic Differential Equations Arising in Financial Mathematics

ERIC Educational Resources Information Center

Parlar, Mahmut

2004-01-01

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper--which led to the awarding of the 1997 Nobel prize in Economic Sciences--Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its…

Stability of sequences generated by nonlinear differential systems. [for analysis of glider jet aircraft motion

NASA Technical Reports Server (NTRS)

Brown, R. L.

1979-01-01

A local stability analysis is presented for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. It is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and the one-leg k-step numerical solution can be approximated. Attention is given to the example of the two-dimensional problem involving the stability of the longitudinal equations of motion of a gliding jet aircraft.

Classical integrable many-body systems disconnected with semi-simple Lie algebras

NASA Astrophysics Data System (ADS)

Inozemtsev, V. I.

2017-05-01

The review of the results in the theory of integrable many-body systems disconnected with semisimple Lie algebras is done. The one-dimensional systems of light Calogero-Sutherland-Moser particles interacting with one particle of infinite mass located at the origin are described in detail. In some cases the exact solutions of the equations of motion are obtained. The general theory of integration of the equations of motion needs the methods of algebraic geometry. The Lax pairs with spectral parameter are constructed for this purpose. The theory still contains many unsolved problems.

The mechanics of solids in the plastically-deformable state

NASA Technical Reports Server (NTRS)

Mises, R. V.

1986-01-01

The mechanics of continua, which is based on the general stress model of Cauchy, up to the present has almost exclusively been applied to liquid and solid elastic bodies. Saint-Venant has developed a theory for the plastic or remaining form changes of solids, but it does not give the required number of equations for determining motion. A complete set of equations of motion for plastic deformable bodies is derived. This is done within the framework of Cauch mechanics. And it is supported by certain experimental facts which characterize the range of applications.

A microscopic derivation of nuclear collective rotation-vibration model and its application to nuclei

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gulshani, P., E-mail: matlap@bell.net

We derive a microscopic version of the successful phenomenological hydrodynamic model of Bohr-Davydov-Faessler-Greiner for collective rotation-vibration motion of an axially symmetric deformed nucleus. The derivation is not limited to small oscillation amplitude. The nuclear Schrodinger equation is canonically transformed to collective co-ordinates, which is then linearized using a constrained variational method. The associated constraints are imposed on the wavefunction rather than on the particle co-ordinates. The approach yields three self-consistent, time-reversal invariant, cranking-type Schrodinger equations for the rotation-vibration and intrinsic motions, and a self-consistency equation. For harmonic oscillator mean-field potentials, these equations are solved in closed forms for excitation energy,more » cut-off angular momentum, and other nuclear properties for the ground-state rotational band in some deformed nuclei. The results are compared with measured data.« less

Premixed flames in closed cylindrical tubes

NASA Astrophysics Data System (ADS)

Metzener, Philippe; Matalon, Moshe

2001-09-01

We consider the propagation of a premixed flame, as a two-dimensional sheet separating unburned gas from burned products, in a closed cylindrical tube. A nonlinear evolution equation, that describes the motion of the flame front as a function of its mean position, is derived. The equation contains a destabilizing term that results from the gas motion induced by thermal expansion and has a memory term associated with vorticity generation. Numerical solutions of this equation indicate that, when diffusion is stabilizing, the flame evolves into a non-planar form whose shape, and its associated symmetry properties, are determined by the Markstein parameter, and by the initial data. In particular, we observe the development of convex axisymmetric or non-axisymmetric flames, tulip flames and cellular flames.

Nonclassical acoustics

NASA Technical Reports Server (NTRS)

Kentzer, C. P.

1976-01-01

A statistical approach to sound propagation is considered in situations where, due to the presence of large gradients of properties of the medium, the classical (deterministic) treatment of wave motion is inadequate. Mathematical methods for wave motions not restricted to small wavelengths (analogous to known methods of quantum mechanics) are used to formulate a wave theory of sound in nonuniform flows. Nonlinear transport equations for field probabilities are derived for the limiting case of noninteracting sound waves and it is postulated that such transport equations, appropriately generalized, may be used to predict the statistical behavior of sound in arbitrary flows.

The equations of motion of an artificial satellite in nonsingular variables

NASA Technical Reports Server (NTRS)

Giacaglia, G. E. O.

1975-01-01

The equations of motion of an artificial satellite are given in nonsingular variables. Any term in the geopotential is considered as well as luni-solar perturbations up to an arbitrary power of r/r prime; r prime being the geocentric distance of the disturbing body. Resonances with tesseral harmonics and with the moon or sun are also considered. By neglecting the shadow effect, the disturbing function for solar radiation is also developed in nonsingular variables for the long periodic perturbations. Formulas are developed for implementation of the theory in actual computations.

Application of parametric equations of motion to study the laser induced multiphoton dissociation of H2+ in intense laser field.

PubMed

Kalita, Dhruba J; Rao, Akshay; Rajvanshi, Ishir; Gupta, Ashish K

2011-06-14

We have applied parametric equations of motion (PEM) to study photodissociation dynamics of H(2)(+). The resonances are extracted using smooth exterior scaling method. This is the first application of PEM to non-Hermitian Hamiltonian that includes resonances and the continuum. Here, we have studied how the different resonance states behave with respect to the change in field amplitude. The advantage of this method is that one can easily trace the different states that are changing as the field parameter changes.

The Kardar-Parisi-Zhang Equation as Scaling Limit of Weakly Asymmetric Interacting Brownian Motions

NASA Astrophysics Data System (ADS)

Diehl, Joscha; Gubinelli, Massimiliano; Perkowski, Nicolas

2017-09-01

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara (Arch Ration Mech Anal 212(2):597-644, 2014) and the corresponding uniqueness result of Gubinelli and Perkowski (Energy solutions of KPZ are unique, 2015).

Second order nonlinear equations of motion for spinning highly flexible line-elements. [for spacecraft solar sail

NASA Technical Reports Server (NTRS)

Salama, M.; Trubert, M.

1979-01-01

A formulation is given for the second order nonlinear equations of motion for spinning line-elements having little or no intrinsic structural stiffness. Such elements have been employed in recent studies of structural concepts for future large space structures such as the Heliogyro solar sailer. The derivation is based on Hamilton's variational principle and includes the effect of initial geometric imperfections (axial, curvature, and twist) on the line-element dynamics. For comparison with previous work, the nonlinear equations are reduced to a linearized form frequently found in the literature. The comparison has revealed several new spin-stiffening terms that have not been previously identified and/or retained. They combine geometric imperfections, rotary inertia, Coriolis, and gyroscopic terms.

Generating Complaint Motion of Objects with an Articulated Hand

DTIC Science & Technology

1985-06-01

we consider the motions of rigid objects as the solhtions to a con- straint problem. We will examine the task of manipulation in the context of...describe the motion of a rigid object is equivalent to specifying sufficient constraint equations on these unknowns such that they are uniquely...assumption of rigidity . When a rigid object is constrained by a set of contacts, its motion must be consistent with those of the contacts, i.e. its

Multibody dynamics model building using graphical interfaces

NASA Technical Reports Server (NTRS)

Macala, Glenn A.

1989-01-01

In recent years, the extremely laborious task of manually deriving equations of motion for the simulation of multibody spacecraft dynamics has largely been eliminated. Instead, the dynamicist now works with commonly available general purpose dynamics simulation programs which generate the equations of motion either explicitly or implicitly via computer codes. The user interface to these programs has predominantly been via input data files, each with its own required format and peculiarities, causing errors and frustrations during program setup. Recent progress in a more natural method of data input for dynamics programs: the graphical interface, is described.

Numerical simulation of fluid flow around a scramaccelerator projectile

NASA Technical Reports Server (NTRS)

Pepper, Darrell W.; Humphrey, Joseph W.; Sobota, Thomas H.

1991-01-01

Numerical simulations of the fluid motion and temperature distribution around a 'scramaccelerator' projectile are obtained for Mach numbers in the 5-10 range. A finite element method is used to solve the equations of motion for inviscid and viscous two-dimensional or axisymmetric compressible flow. The time-dependent equations are solved explicitly, using bilinear isoparametric quadrilateral elements, mass lumping, and a shock-capturing Petrov-Galerkin formulation. Computed results indicate that maintaining on-design performance for controlling and stabilizing oblique detonation waves is critically dependent on projectile shape and Mach number.

The Shock and Vibration Digest. Volume 16, Number 11

DTIC Science & Technology

1984-11-01

wave [19], a secular equation for Rayleigh waves on ing, seismic risk, and related problems are discussed. the surface of an anisotropic half-space...waves in an !so- tive equation of an elastic-plastic rack medium was....... tropic linear elastic half-space with plane material used; the coefficient...pair of semi-linear hyperbolic partial differential -- " Conditions under which the equations of motion equations governing slow variations in amplitude

Active Brownian particles with velocity-alignment and active fluctuations

NASA Astrophysics Data System (ADS)

Großmann, R.; Schimansky-Geier, L.; Romanczuk, P.

2012-07-01

We consider a model of active Brownian particles (ABPs) with velocity alignment in two spatial dimensions with passive and active fluctuations. Here, active fluctuations refers to purely non-equilibrium stochastic forces correlated with the heading of an individual active particle. In the simplest case studied here, they are assumed to be independent stochastic forces parallel (speed noise) and perpendicular (angular noise) to the velocity of the particle. On the other hand, passive fluctuations are defined by a noise vector independent of the direction of motion of a particle, and may account, for example, for thermal fluctuations. We derive a macroscopic description of the ABP gas with velocity-alignment interaction. Here, we start from the individual-based description in terms of stochastic differential equations (Langevin equations) and derive equations of motion for the coarse-grained kinetic variables (density, velocity and temperature) via a moment expansion of the corresponding probability density function. We focus here on the different impact of active and passive fluctuations on onset of collective motion and show how active fluctuations in the active Brownian dynamics can change the phase-transition behaviour of the system. In particular, we show that active angular fluctuations lead to an earlier breakdown of collective motion and to the emergence of a new bistable regime in the mean-field case.

Similarity-transformed equation-of-motion vibrational coupled-cluster theory.

PubMed

Faucheaux, Jacob A; Nooijen, Marcel; Hirata, So

2018-02-07

A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. The first transformation uses an exponential excitation operator, defining the equation-of-motion vibrational coupled-cluster (EOM-XVCC) method, and the second uses an exponential excitation-deexcitation operator. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. We furthermore elucidate the similarities and differences between electronic and vibrational STEOM methods and between STEOM-XVCC and vibrational many-body Green's function theory based on the Dyson equation, which is also an anharmonic one-mode theory. The latter comparison inspires three approximate STEOM-XVCC methods utilizing the common approximations made in the Dyson equation: the diagonal approximation, a perturbative expansion of the Dyson self-energy, and the frequency-independent approximation. The STEOM-XVCC method including up to the simultaneous four-mode excitation operator in a quartic force field and its three approximate variants are formulated and implemented in computer codes with the aid of computer algebra, and they are applied to small test cases with varied degrees of anharmonicity.

Similarity-transformed equation-of-motion vibrational coupled-cluster theory

NASA Astrophysics Data System (ADS)

Faucheaux, Jacob A.; Nooijen, Marcel; Hirata, So

2018-02-01

A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. The first transformation uses an exponential excitation operator, defining the equation-of-motion vibrational coupled-cluster (EOM-XVCC) method, and the second uses an exponential excitation-deexcitation operator. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. We furthermore elucidate the similarities and differences between electronic and vibrational STEOM methods and between STEOM-XVCC and vibrational many-body Green's function theory based on the Dyson equation, which is also an anharmonic one-mode theory. The latter comparison inspires three approximate STEOM-XVCC methods utilizing the common approximations made in the Dyson equation: the diagonal approximation, a perturbative expansion of the Dyson self-energy, and the frequency-independent approximation. The STEOM-XVCC method including up to the simultaneous four-mode excitation operator in a quartic force field and its three approximate variants are formulated and implemented in computer codes with the aid of computer algebra, and they are applied to small test cases with varied degrees of anharmonicity.

Theory of relativistic Brownian motion in the presence of electromagnetic field in (1+1) dimension

NASA Astrophysics Data System (ADS)

Mukhopadhyay, Annesh; Bandyopadhyay, M.; Bhamidipati, C.

2018-04-01

In this work, we consider the relativistic generalization of the theory of Brownian motion for the (1+1) dimensional case, which is again consistent with Einstein's special theory of relativity and reduces to standard Brownian motion in the Newtonian limit. All the generalizations are made considering Special theory of relativity into account. The particle under consideration has a velocity close to the speed of light and is a free Brownian particle suspended in a heat bath. With this generalization the velocity probability density functions are also obtained using Ito, Stratonovich and Hanggi-Klimontovich approach of pre-point, mid-point and post-point discretization rule. Subsequently, in our work, we have obtained the relativistic Langevin equations in the presence of an electromagnetic field. Finally, taking a special case of a constant vector potential and a constant electric field into account the Langevin equations are solved for the momentum and subsequently the velocity of the particle. Using a similar approach to the Fokker-planck equations of motion, the velocity distributions are also obtained in the presence of a constant vector potential and are plotted, which shows essential deviations from the one obtained without a potential. Our constant potential model can be realized in an optical potential.

Radiation from Accelerating Electric Charges: The Third Derivative of Position

NASA Astrophysics Data System (ADS)

Butterworth, Edward

2010-03-01

While some textbooks appear to suggest that acceleration of an electric charge is both a necessary and sufficient cause for the generation of electromagnetic radiation, the question has in fact had an intricate and involved history. In particular, the acceleration of a charge in hyperbolic motion, the behavior of a charge supported against a gravitational force (and its implications for the Equivalence Principle), and a charge accelerated by a workless constraint have been the subject of repeated investigation. The present paper examines specifically the manner in which the third derivative of position enters into the equations of motion, and the implications this has for the emission of radiation. Plass opens his review article with the statement that ``A fundamental property of all charged particles is that electromagnetic energy is radiated whenever they are accelerated'' (Plass 1961; emphasis mine). His treatment of the equations of motion, however, emphasizes the importance of the occurrence of the third derivative of position therein, present in linear motion only when the rate of acceleration is increasing or decreasing. There appears to be general agreement that the presence of a nonzero third derivative indicates that this charge is radiating; but does its absence preclude radiation? This question leads back to the issues of charges accelerated by a uniform gravitational field. We will examine the equations of motion as presented in Fulton & Rohrlich (1960), Plass (1961), Barut (1964), Teitelboim (1970) and Mo & Papas (1971) in the light of more recent literature in an attempt to clarify this question.

An Application of the A* Search to Trajectory Optimization

DTIC Science & Technology

1990-05-11

linearized model of orbital motion called the Clohessy - Wiltshire Equations and a node search technique called A*. The planner discussed in this thesis starts...states while transfer time is left unspecified. 13 Chapter 2. Background HILL’S ( CLOHESSY - WILTSHIRE ) EQUATIONS The Euler-Hill equations describe... Clohessy - Wiltshire equations. The coordinate system used in this thesis is commonly referred to as Local Vertical, Local Horizontal or LVLH reference frame

Nonlinear Drift-Kinetic Equation in the Presence of a Circularly Polarized Wave

NASA Technical Reports Server (NTRS)

Khazanov, G. V.; Krivorutsky, E. N.; Six, N. Frank (Technical Monitor)

2002-01-01

Equations of the single particle motion and nonlinear kinetic equation for plasma in the presence of a circularly polarized wave of arbitrary frequency in the drift approximation are presented. The nonstationarity and inhomogeneity of the plasma-wave system are taken into account. The time dependent part of the ponderomotive force is discussed.

HAM2D: 2D Shearing Box Model

NASA Astrophysics Data System (ADS)

Gammie, Charles F.; Guan, Xiaoyue

2012-10-01

HAM solves non-relativistic hyperbolic partial differential equations in conservative form using high-resolution shock-capturing techniques. This version of HAM has been configured to solve the magnetohydrodynamic equations of motion in axisymmetry to evolve a shearing box model.

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Testing different concepts of the equations of motion, describing runout time and distance of slow-moving gravitational slides and flows.

NASA Astrophysics Data System (ADS)

van Asch, Th. W. J.; Daehne, A.; Spickermann, A.; Travelletti, J.; Bégueria-Portuguès, S.

2010-05-01

The kinematics of rapid and slow moving landslides is commonly described by equations of motion, which in case of a viscous component are based on the Navier-Stokes equation. They consist of inertial terms related to the change in velocity in time (local acceleration) and space (convective acceleration) and terms related to respectively the gravity, pressure and viscous forces. These viscous resistance forces in the mass balance can be accompanied or replaced by other rheological (frictional and cohesive) terms depending on the liquid/solid ratio of the moving mass. We designed a 1D and a GIS based 2.5 D model with a numerical implementation for these equations which gave a reasonable simple compromise solution that achieved a desired level of stability, accuracy and controlled diffusion. An explicit finite difference (Eulerian) mesh, i.e. the moving mass was described by variation in the conservative variables at point fixed coordinates (i,j) as a function of time (n). A central difference forward scheme is used for the numerical solutions of the mass and momentum balance equations. A number of case studies of fast debris flows ranging in velocity between 1 and 10 m s-1, carried out in the Faucon torrent French Alps, the Wartschenbach torrent in Austria, near the Turnoff Creek in British Columbia, the Peringalam catchment in SW-India and the Jagüeyes landslide in the Guantánamo province Cuba, showed that the models were able to describe velocity, deposition and run-out reasonable well using different rheological characteristics. Despite the fact that many authors include an inertial term in the equation of motion for slow moving mass movements it appeared that our 1D and GIS based 2.5 D models were not able to simulate properly the velocity of slower moving debris flows or landslides with velocities ranging from 1 to 2 m min-1 until 30 mm y-1.Deletion of the inertial term related to the local acceleration in the equation of motion, thus assuming that there is a permanent equilibrium between gravity, pressure and Coulomb-viscous forces, produced a more flexible tool, able to describe the velocity, deposition and run-out of mass movements with a wide range of values. Examples of successful simulations in 1-D and 2.5-D exist already. In this contribution we will compare 1D simulations with and without a local acceleration term and analyze the results. A slow moving debris flow which developed on the Super-Sauze mudslide and a slow moving landslide in varved clays near Monestier-du-Percy in the French Alps were selected to test the calibration performances of these two options in the equation of motion.

A method for the determination of the coefficient of rolling friction using cycloidal pendulum

NASA Astrophysics Data System (ADS)

Ciornei, M. C.; Alaci, S.; Ciornei, F. C.; Romanu, I. C.

2017-08-01

The paper presents a method for experimental finding of coefficient of rolling friction appropriate for biomedical applications based on the theory of cycloidal pendulum. When a mobile circle rolls over a fixed straight line, the points from the circle describe trajectories called normal cycloids. To materialize this model, it is sufficient that a small region from boundary surfaces of a moving rigid body is spherical. Assuming pure rolling motion, the equation of motion of the cycloidal pendulum is obtained - an ordinary nonlinear differential equation. The experimental device is composed by two interconnected balls rolling over the material to be studied. The inertial characteristics of the pendulum can be adjusted via weights placed on a rod. A laser spot oscillates together to the pendulum and provides the amplitude of oscillations. After finding the experimental parameters necessary in differential equation of motion, it can be integrated using the Runge-Kutta of fourth order method. The equation was integrated for several materials and found values of rolling friction coefficients. Two main conclusions are drawn: the coefficient of rolling friction influenced significantly the amplitude of oscillation but the effect upon the period of oscillation is practically imperceptible. A methodology is proposed for finding the rolling friction coefficient and the pure rolling condition is verified.

Brownian motion of massive skyrmions in magnetic thin films

DOE Office of Scientific and Technical Information (OSTI.GOV)

Troncoso, Roberto E., E-mail: r.troncoso.c@gmail.com; Núñez, Álvaro S., E-mail: alnunez@dfi.uchile.cl

2014-12-15

We report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced equation for the motion of skyrmion has the form of a stochastic generalized Thiele’s equation. We propose an ansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By using this ansatz it is found that the skyrmion mass tensor is closely related to intrinsic skyrmion parameters, such as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the average drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal andmore » transverse mobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of the skyrmion mass.« less

Hydraulic modeling of unsteady debris-flow surges with solid-fluid interactions

USGS Publications Warehouse

Iverson, Richard M.

1997-01-01

Interactions of solid and fluid constituents produce the unique style of motion that typifies debris flows. To simulate this motion, a new hydraulic model represents debris flows as deforming masses of granular solids variably liquefied by viscous pore fluid. The momentum equation of the model describes how internal and boundary forces change as coarse-grained surge heads dominated by grain-contact friction grade into muddy debris-flow bodies more strongly influenced by fluid viscosity and pressure. Scaling analysis reveals that pore-pressure variations can cause flow resistance in surge heads to surpass that in debris-flow bodies by orders of magnitude. Numerical solutions of the coupled momentum and continuity equations provide good predictions of unsteady, nonuniform motion of experimental debris flows from initiation through deposition.

Rigid rotators. [deriving the time-independent energy states associated with rotational motions of the molecule

NASA Technical Reports Server (NTRS)

1976-01-01

The two-particle, steady-state Schroedinger equation is transformed to center of mass and internuclear distance vector coordinates, leading to the free particle wave equation for the kinetic energy motion of the molecule and a decoupled wave equation for a single particle of reduced mass moving in a spherical potential field. The latter describes the vibrational and rotational energy modes of the diatomic molecule. For fixed internuclear distance, this becomes the equation of rigid rotator motion. The classical partition function for the rotator is derived and compared with the quantum expression. Molecular symmetry effects are developed from the generalized Pauli principle that the steady-state wave function of any system of fundamental particles must be antisymmetric. Nuclear spin and spin quantum functions are introduced and ortho- and para-states of rotators, along with their degeneracies, are defined. Effects of nuclear spin on entropy are deduced. Next, rigid polyatomic rotators are considered and the partition function for this case is derived. The patterns of rotational energy levels for nonlinear molecules are discussed for the spherical symmetric top, for the prolate symmetric top, for the oblate symmetric top, and for the asymmetric top. Finally, the equilibrium energy and specific heat of rigid rotators are derived.

Automatic generation of the non-holonomic equations of motion for vehicle stability analysis

NASA Astrophysics Data System (ADS)

Minaker, B. P.; Rieveley, R. J.

2010-09-01

The mathematical analysis of vehicle stability has been utilised as an important tool in the design, development, and evaluation of vehicle architectures and stability controls. This paper presents a novel method for automatic generation of the linearised equations of motion for mechanical systems that is well suited to vehicle stability analysis. Unlike conventional methods for generating linearised equations of motion in standard linear second order form, the proposed method allows for the analysis of systems with non-holonomic constraints. In the proposed method, the algebraic constraint equations are eliminated after linearisation and reduction to first order. The described method has been successfully applied to an assortment of classic dynamic problems of varying complexity including the classic rolling coin, the planar truck-trailer, and the bicycle, as well as in more recent problems such as a rotor-stator and a benchmark road vehicle with suspension. This method has also been applied in the design and analysis of a novel three-wheeled narrow tilting vehicle with zero roll-stiffness. An application for determining passively stable configurations using the proposed method together with a genetic search algorithm is detailed. The proposed method and software implementation has been shown to be robust and provides invaluable conceptual insight into the stability of vehicles and mechanical systems.

Numerical solution of the exact cavity equations of motion for an unstable optical resonator.

PubMed

Bowers, M S; Moody, S E

1990-09-20

We solve numerically, we believe for the first time, the exact cavity equations of motion for a realistic unstable resonator with a simple gain saturation model. The cavity equations of motion, first formulated by Siegman ["Exact Cavity Equations for Lasers with Large Output Coupling," Appl. Phys. Lett. 36, 412-414 (1980)], and which we term the dynamic coupled modes (DCM) method of solution, solve for the full 3-D time dependent electric field inside the optical cavity by expanding the field in terms of the actual diffractive transverse eigenmodes of the bare (gain free) cavity with time varying coefficients. The spatially varying gain serves to couple the bare cavity transverse modes and to scatter power from mode to mode. We show that the DCM method numerically converges with respect to the number of eigenmodes in the basis set. The intracavity intensity in the numerical example shown reaches a steady state, and this steady state distribution is compared with that computed from the traditional Fox and Li approach using a fast Fourier transform propagation algorithm. The output wavefronts from both methods are quite similar, and the computed output powers agree to within 10%. The usefulness and advantages of using this method for predicting the output of a laser, especially pulsed lasers used for coherent detection, are discussed.

Using Statcast to lift the discussion of projectile motion

NASA Astrophysics Data System (ADS)

Siegel, P. B.

2017-04-01

Home run data from Major League Baseball's Statcast can be described by adding a lift force to the equations of projectile motion commonly used in undergraduate computational physics courses. We discuss how the Statcast data can be implemented in the classroom.

Conformal collineations and anisotropic fluids in general relativity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Duggal, K.L.; Sharma, R.

1986-10-01

Recently, Herrera et al. (L. Herrera, J. Jimenez, L. Leal, J. Ponce de Leon, M. Esculpi, and V. Galino, J. Math. Phys. 25, 3274 (1984)) studied the consequences of the existence of a one-parameter group of conformal motions for anisotropic matter. They concluded that for special conformal motions, the stiff equation of state (p = ..mu..) is singled out in a unique way, provided the generating conformal vector field is orthogonal to the four-velocity. In this paper, the same problem is studied by using conformal collineations (which include conformal motions as subgroups). It is shown that, for a special conformalmore » collineation, the stiff equation of state is not singled out. Non-Einstein Ricci-recurrent spaces are considered as physical models for the fluid matter.« less

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

NASA Technical Reports Server (NTRS)

Van Dommelen, Leon L.; Cowley, Stephen J.

1990-01-01

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

On the Lagrangian description of unsteady boundary layer separation. Part 1: General theory

NASA Technical Reports Server (NTRS)

Vandommelen, Leon L.; Cowley, Stephen J.

1989-01-01

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

On the Analysis of Multistep-Out-of-Grid Method for Celestial Mechanics Tasks

NASA Astrophysics Data System (ADS)

Olifer, L.; Choliy, V.

2016-09-01

Occasionally, there is a necessity in high-accurate prediction of celestial body trajectory. The most common way to do that is to solve Kepler's equation analytically or to use Runge-Kutta or Adams integrators to solve equation of motion numerically. For low-orbit satellites, there is a critical need in accounting geopotential and another forces which influence motion. As the result, the right side of equation of motion becomes much bigger, and classical integrators will not be quite effective. On the other hand, there is a multistep-out-of-grid (MOG) method which combines Runge-Kutta and Adams methods. The MOG method is based on using m on-grid values of the solution and n × m off-grid derivative estimations. Such method could provide stable integrators of maximum possible order, O (hm+mn+n-1). The main subject of this research was to implement and analyze the MOG method for solving satellite equation of motion with taking into account Earth geopotential model (ex. EGM2008 (Pavlis at al., 2008)) and with possibility to add other perturbations such as atmospheric drag or solar radiation pressure. Simulations were made for satellites on low orbit and with various eccentricities (from 0.1 to 0.9). Results of the MOG integrator were compared with results of Runge-Kutta and Adams integrators. It was shown that the MOG method has better accuracy than classical ones of the same order and less right-hand value estimations when is working on high orders. That gives it some advantage over "classical" methods.

Effects of spacecraft motions on fluids experiments

NASA Technical Reports Server (NTRS)

Gans, R. F.

1981-01-01

The equations of motion governing an incompressible fluid contained in an orbiting laboratory were examined to isolate various fictitious forces and their relative influence on the fluid. The forces are divided into those arising from the orbital motions and those arising from small local motions of the spacecraft about its center of mass. The latter dominate the nonrotating experiments. Both are important for rotating experiments. A brief discussion of the onset of time-dependence and violent instability in earth-based rotating and processing systems is given.

Implications of next generation attenuation ground motion prediction equations for site coefficients used in earthquake resistant design

USGS Publications Warehouse

Borcherdt, Roger D.

2014-01-01

Proposals are developed to update Tables 11.4-1 and 11.4-2 of Minimum Design Loads for Buildings and Other Structures published as American Society of Civil Engineers Structural Engineering Institute standard 7-10 (ASCE/SEI 7–10). The updates are mean next generation attenuation (NGA) site coefficients inferred directly from the four NGA ground motion prediction equations used to derive the maximum considered earthquake response maps adopted in ASCE/SEI 7–10. Proposals include the recommendation to use straight-line interpolation to infer site coefficients at intermediate values of (average shear velocity to 30-m depth). The NGA coefficients are shown to agree well with adopted site coefficients at low levels of input motion (0.1 g) and those observed from the Loma Prieta earthquake. For higher levels of input motion, the majority of the adopted values are within the 95% epistemic-uncertainty limits implied by the NGA estimates with the exceptions being the mid-period site coefficient, Fv, for site class D and the short-period coefficient, Fa, for site class C, both of which are slightly less than the corresponding 95% limit. The NGA data base shows that the median value  of 913 m/s for site class B is more typical than 760 m/s as a value to characterize firm to hard rock sites as the uniform ground condition for future maximum considered earthquake response ground motion estimates. Future updates of NGA ground motion prediction equations can be incorporated easily into future adjustments of adopted site coefficients using procedures presented herein. 

A practical application of the geometrical theory on fibered manifolds to an autonomous bicycle motion in mechanical system with nonholonomic constraints

NASA Astrophysics Data System (ADS)

Haddout, Soufiane

2018-01-01

The equations of motion of a bicycle are highly nonlinear and rolling of wheels without slipping can only be expressed by nonholonomic constraint equations. A geometrical theory of general nonholonomic constrained systems on fibered manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by O. Krupková (Rossi) in 1990's. Her approach is suitable for study of all kinds of mechanical systems-without restricting to Lagrangian, time-independent, or regular ones, and is applicable to arbitrary constraints (holonomic, semiholonomic, linear, nonlinear or general nonholonomic). The goal of this paper is to apply Krupková's geometric theory of nonholonomic mechanical systems to study a concrete problem in nonlinear nonholonomic dynamics, i.e., autonomous bicycle. The dynamical model is preserved in simulations in its original nonlinear form without any simplifying. The results of numerical solutions of constrained equations of motion, derived within the theory, are in good agreement with measurements and thus they open the possibility of direct application of the theory to practical situations.

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

NASA Astrophysics Data System (ADS)

Mingari Scarpello, Giovanni; Ritelli, Daniele

2018-06-01

The present study highlights the dynamics of a body moving about a fixed point and provides analytical closed form solutions. Firstly, for the symmetrical heavy body, that is the Lagrange-Poisson case, we compute the second (precession, ψ ) and third (spin, φ) Euler angles in explicit and real form by means of multiple hypergeometric (Lauricella) functions. Secondly, releasing the weight assumption but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle ψ completing the treatment of the Euler-Poinsot case. Thirdly, by integrating the relevant differential equation, we reach the finite polar equation of a special motion trajectory named the herpolhode. Finally, we keep the symmetry of the first problem, but without weight, and take into account a viscous dissipation. The use of motion first integrals—adopted for the first two problems—is no longer practicable in this situation; therefore, the Euler equations, faced directly, are driving to particular occurrences of Bessel functions of order - 1/2.

Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique

NASA Technical Reports Server (NTRS)

Nordmann, R.; Weiser, P.

1989-01-01

The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.

Application of the multireference equation of motion coupled cluster method, including spin-orbit coupling, to the atomic spectra of Cr, Mn, Fe and Co

NASA Astrophysics Data System (ADS)

Liu, Zhebing; Huntington, Lee M. J.; Nooijen, Marcel

2015-10-01

The recently introduced multireference equation of motion (MR-EOM) approach is combined with a simple treatment of spin-orbit coupling, as implemented in the ORCA program. The resulting multireference equation of motion spin-orbit coupling (MR-EOM-SOC) approach is applied to the first-row transition metal atoms Cr, Mn, Fe and Co, for which experimental data are readily available. Using the MR-EOM-SOC approach, the splittings in each L-S multiplet can be accurately assessed (root mean square (RMS) errors of about 70 cm-1). The RMS errors for J-specific excitation energies range from 414 to 783 cm-1 and are comparable to previously reported J-averaged MR-EOM results using the ACESII program. The MR-EOM approach is highly efficient. A typical MR-EOM calculation of a full spin-orbit spectrum takes about 2 CPU hours on a single processor of a 12-core node, consisting of Intel XEON 2.93 GHz CPUs with 12.3 MB of shared cache memory.

Exact finite difference schemes for the non-linear unidirectional wave equation

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1985-01-01

Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.

Relativistic equation-of-motion coupled-cluster method using open-shell reference wavefunction: Application to ionization potential

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pathak, Himadri, E-mail: hmdrpthk@gmail.com; Sasmal, Sudip, E-mail: sudipsasmal.chem@gmail.com; Vaval, Nayana

2016-08-21

The open-shell reference relativistic equation-of-motion coupled-cluster method within its four-component description is successfully implemented with the consideration of single- and double- excitation approximations using the Dirac-Coulomb Hamiltonian. At the first attempt, the implemented method is employed to calculate ionization potential value of heavy atomic (Ag, Cs, Au, Fr, and Lr) and molecular (HgH and PbF) systems, where the effect of relativity does really matter to obtain highly accurate results. Not only the relativistic effect but also the effect of electron correlation is crucial in these heavy atomic and molecular systems. To justify the fact, we have taken two further approximationsmore » in the four-component relativistic equation-of-motion framework to quantify how the effect of electron correlation plays a role in the calculated values at different levels of theory. All these calculated results are compared with the available experimental data as well as with other theoretically calculated values to judge the extent of accuracy obtained in our calculations.« less

Aeroelastic Stability of Rotor Blades Using Finite Element Analysis

NASA Technical Reports Server (NTRS)

Chopra, I.; Sivaneri, N.

1982-01-01

The flutter stability of flap bending, lead-lag bending, and torsion of helicopter rotor blades in hover is investigated using a finite element formulation based on Hamilton's principle. The blade is divided into a number of finite elements. Quasi-steady strip theory is used to evaluate the aerodynamic loads. The nonlinear equations of motion are solved for steady-state blade deflections through an iterative procedure. The equations of motion are linearized assuming blade motion to be a small perturbation about the steady deflected shape. The normal mode method based on the coupled rotating natural modes is used to reduce the number of equations in the flutter analysis. First the formulation is applied to single-load-path blades (articulated and hingeless blades). Numerical results show very good agreement with existing results obtained using the modal approach. The second part of the application concerns multiple-load-path blades, i.e. bearingless blades. Numerical results are presented for several analytical models of the bearingless blade. Results are also obtained using an equivalent beam approach wherein a bearingless blade is modelled as a single beam with equivalent properties. Results show the equivalent beam model.

Stochastic ground motion simulation

USGS Publications Warehouse

Rezaeian, Sanaz; Xiaodan, Sun; Beer, Michael; Kougioumtzoglou, Ioannis A.; Patelli, Edoardo; Siu-Kui Au, Ivan

2014-01-01

Strong earthquake ground motion records are fundamental in engineering applications. Ground motion time series are used in response-history dynamic analysis of structural or geotechnical systems. In such analysis, the validity of predicted responses depends on the validity of the input excitations. Ground motion records are also used to develop ground motion prediction equations(GMPEs) for intensity measures such as spectral accelerations that are used in response-spectrum dynamic analysis. Despite the thousands of available strong ground motion records, there remains a shortage of records for large-magnitude earthquakes at short distances or in specific regions, as well as records that sample specific combinations of source, path, and site characteristics.

Coulomb Thrusting Application Study

DTIC Science & Technology

2006-01-20

Acceleration Magnitudes To study the relative motion of spacecraft in nearly circular orbits, the Clohessy - Wiltshire - Hill equations are commonly...Modeling The Clohessy - Wiltshire -Hill’s equations12–14 for one of the spacecraft in the 2-craft Coulomb tether formation is given by ẍ1 − 2nẏ1...equa- tion was obtained using the Clohessy - Wiltshire - Hill equations, while the linearized differential equations of ψ and θ were derived from the full

A theoretical investigation of the rolling oscillations of an airplane with ailerons free

NASA Technical Reports Server (NTRS)

Cohen, Doris

1944-01-01

An analysis is made of the stability of an airplane with ailerons free, with particular attention to the motions when the ailerons have a tendency to float against the wind. The present analysis supersedes the aileron investigation contained in NACA Technical Report no. 709. The equations of motion are first written to include yawing and sideslipping, and it is demonstrated that the principal effects of freeing the ailerons can be determined without regard to these motions. If the ailerons tend to float against the wind and have a high degree of aerodynamic balance, rolling oscillations, in addition to the normal lateral oscillations, are likely to occur. On the basis of the equations including only the rolling motion and the aileron deflection, formulas derived for the stability and damping of the rolling oscillations in terms of the hinge-moment derivatives are also presented showing the oscillatory regions and stability boundaries for a fictitious airplane of conventional proportion. The effects of friction in the control system are investigated and discussed.

Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders.

PubMed

Kanazawa, Kiyoshi; Sueshige, Takumi; Takayasu, Hideki; Takayasu, Misako

2018-03-30

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders' trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.

Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders

NASA Astrophysics Data System (ADS)

Kanazawa, Kiyoshi; Sueshige, Takumi; Takayasu, Hideki; Takayasu, Misako

2018-03-01

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders' trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.

Dynamic modeling and motion simulation for a winged hybrid-driven underwater glider

NASA Astrophysics Data System (ADS)

Wang, Shu-Xin; Sun, Xiu-Jun; Wang, Yan-Hui; Wu, Jian-Guo; Wang, Xiao-Ming

2011-03-01

PETREL, a winged hybrid-driven underwater glider is a novel and practical marine survey platform which combines the features of legacy underwater glider and conventional AUV (autonomous underwater vehicle). It can be treated as a multi-rigid-body system with a floating base and a particular hydrodynamic profile. In this paper, theorems on linear and angular momentum are used to establish the dynamic equations of motion of each rigid body and the effect of translational and rotational motion of internal masses on the attitude control are taken into consideration. In addition, due to the unique external shape with fixed wings and deflectable rudders and the dual-drive operation in thrust and glide modes, the approaches of building dynamic model of conventional AUV and hydrodynamic model of submarine are introduced, and the tailored dynamic equations of the hybrid glider are formulated. Moreover, the behaviors of motion in glide and thrust operation are analyzed based on the simulation and the feasibility of the dynamic model is validated by data from lake field trials.

Tunnelling from non-localised initial states

NASA Technical Reports Server (NTRS)

Bowcock, Peter; Gregory, Ruth

1991-01-01

An approach for calculating tunneling amplitudes from a nonlocalized initial state is presented. Generalizing the matching conditions and equations of motion to allow for complex momentum permits a description of tunneling in the presence of so-called classical motion. Possible applications of the method are presented.

Brownian motion of a particle with arbitrary shape.

PubMed

Cichocki, Bogdan; Ekiel-Jeżewska, Maria L; Wajnryb, Eligiusz

2015-06-07

Brownian motion of a particle with an arbitrary shape is investigated theoretically. Analytical expressions for the time-dependent cross-correlations of the Brownian translational and rotational displacements are derived from the Smoluchowski equation. The role of the particle mobility center is determined and discussed.

Rotating Wavepackets

ERIC Educational Resources Information Center

Lekner, John

2008-01-01

Any free-particle wavepacket solution of Schrodinger's equation can be converted by differentiations to wavepackets rotating about the original direction of motion. The angular momentum component along the motion associated with this rotation is an integral multiple of [h-bar]. It is an "intrinsic" angular momentum: independent of origin and…

Image defects from surface and alignment errors in grazing incidence telescopes

NASA Technical Reports Server (NTRS)

Saha, Timo T.

1989-01-01

The rigid body motions and low frequency surface errors of grazing incidence Wolter telescopes are studied. The analysis is based on surface error descriptors proposed by Paul Glenn. In his analysis, the alignment and surface errors are expressed in terms of Legendre-Fourier polynomials. Individual terms in the expression correspond to rigid body motions (decenter and tilt) and low spatial frequency surface errors of mirrors. With the help of the Legendre-Fourier polynomials and the geometry of grazing incidence telescopes, exact and approximated first order equations are derived in this paper for the components of the ray intercepts at the image plane. These equations are then used to calculate the sensitivities of Wolter type I and II telescopes for the rigid body motions and surface deformations. The rms spot diameters calculated from this theory and OSAC ray tracing code agree very well. This theory also provides a tool to predict how rigid body motions and surface errors of the mirrors compensate each other.

Universal current-velocity relation of skyrmion motion in chiral magnets

NASA Astrophysics Data System (ADS)

Iwasaki, Junichi; Mochizuki, Masahito; Nagaosa, Naoto

2013-02-01

Current-driven motion of the magnetic domain wall in ferromagnets is attracting intense attention because of potential applications such as racetrack memory. There, the critical current density to drive the motion is ~109-1012 A m-2. The skyrmions recently discovered in chiral magnets have much smaller critical current density of ~105-106 A m-2, but the microscopic mechanism is not yet explored. Here we present a numerical simulation of Landau-Lifshitz-Gilbert equation, which reveals a remarkably robust and universal current-velocity relation of the skyrmion motion driven by the spin-transfer-torque unaffected by either impurities or nonadiabatic effect in sharp contrast to the case of domain wall or spin helix. Simulation results are analysed using a theory based on Thiele’s equation, and it is concluded that this behaviour is due to the Magnus force and flexible shape-deformation of individual skyrmions and skyrmion crystal, which enable them to avoid pinning centres.

Langevin dynamics for ramified structures

NASA Astrophysics Data System (ADS)

Méndez, Vicenç; Iomin, Alexander; Horsthemke, Werner; Campos, Daniel

2017-06-01

We propose a generalized Langevin formalism to describe transport in combs and similar ramified structures. Our approach consists of a Langevin equation without drift for the motion along the backbone. The motion along the secondary branches may be described either by a Langevin equation or by other types of random processes. The mean square displacement (MSD) along the backbone characterizes the transport through the ramified structure. We derive a general analytical expression for this observable in terms of the probability distribution function of the motion along the secondary branches. We apply our result to various types of motion along the secondary branches of finite or infinite length, such as subdiffusion, superdiffusion, and Langevin dynamics with colored Gaussian noise and with non-Gaussian white noise. Monte Carlo simulations show excellent agreement with the analytical results. The MSD for the case of Gaussian noise is shown to be independent of the noise color. We conclude by generalizing our analytical expression for the MSD to the case where each secondary branch is n dimensional.

Every Heavenly Body When Created Will Have No Motion, Linear, Rotational and/or Vibratory Motion, Singly or in Some Combination

NASA Astrophysics Data System (ADS)

Brekke, Stewart

2010-02-01

Each galaxy, star and planet is in a state of no motion, linear, rotational and/or vibratory motion. Orbital motion is linear motion in a force field such as gravity. These motions were created in the formation of the galaxy, star or planet unless modified by external events such as colliding galaxies or impacts such as meteors. Some motions, such as rotations and vibrations may be differential such as in the cases of our sun and the Milky Way galaxy. The basic equation for each heavenly body is as follows. E = mc^2 + 1/2mv^2 + 1/2I2̂+ 1/2Kx^2 + WG+ WE+ WM. )

Full three-body problem in effective-field-theory models of gravity

NASA Astrophysics Data System (ADS)

Battista, Emmanuele; Esposito, Giampiero

2014-10-01

Recent work in the literature has studied the restricted three-body problem within the framework of effective-field-theory models of gravity. This paper extends such a program by considering the full three-body problem, when the Newtonian potential is replaced by a more general central potential which depends on the mutual separations of the three bodies. The general form of the equations of motion is written down, and they are studied when the interaction potential reduces to the quantum-corrected central potential considered recently in the literature. A recursive algorithm is found for solving the associated variational equations, which describe small departures from given periodic solutions of the equations of motion. Our scheme involves repeated application of a 2×2 matrix of first-order linear differential operators.

Relative motion of orbiting particles under the influence of perturbing forces. Volume 2: Analytical results. [equations of motion and mathematical solutions

NASA Technical Reports Server (NTRS)

Eades, J. B., Jr.

1974-01-01

The mathematical developments carried out for this investigation are reported. In addition to describing and discussing the solutions which were acquired, there are compendia of data presented herein which summarize the equations and describe them as representative trace geometries. In this analysis the relative motion problems have been referred to two particular frames of reference; one which is inertially aligned, and one which is (local) horizon oriented. In addition to obtaining the classical initial values solutions, there are results which describe cases having applied specific forces serving as forcing functions. Also, in order to provide a complete state representation the speed components, as well as the displacements, have been described. These coordinates are traced on representative planes analogous to the displacement geometries. By this procedure a complete description of a relative motion is developed; and, as a consequence range rate as well as range information is obtained.

Phase dynamics of oscillating magnetizations coupled via spin pumping

NASA Astrophysics Data System (ADS)

Taniguchi, Tomohiro

2018-05-01

A theoretical formalism is developed to simultaneously solve equation of motion of the magnetizations in two ferromagnets and the spin-pumping induced spin transport equation. Based on the formalism, a coupled motion of the magnetizations in a self-oscillation state is studied. The spin pumping is found to induce an in-phase synchronization of the magnetizations for the oscillation around the easy axis. For an out-of-plane self-oscillation around the hard axis, on the other hand, the spin pumping leads to an in-phase synchronization in a small current region, whereas an antiphase synchronization is excited in a large current region. An analytical theory based on the phase equation reveals that the phase difference between the magnetizations in a steady state depends on the oscillation direction, clockwise or counterclockwise, of the magnetizations.

One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ishii, M.

1977-10-01

In view of the practical importance of the drift-flux model for two-phase flow analysis in general and in the analysis of nuclear-reactor transients and accidents in particular, the kinematic constitutive equation for the drift velocity has been studied for various two-phase flow regimes. The constitutive equation that specifies the relative motion between phases in the drift-flux model has been derived by taking into account the interfacial geometry, the body-force field, shear stresses, and the interfacial momentum transfer, since these macroscopic effects govern the relative velocity between phases. A comparison of the model with various experimental data over various flow regimesmore » and a wide range of flow parameters shows a satisfactory agreement.« less

Nonlinear flap-lag-axial equations of a rotating beam with arbitrary precone angle

NASA Technical Reports Server (NTRS)

Kvaternik, R. G.; White, W. F., Jr.; Kaza, K. R. V.

1978-01-01

In an attempt both to unify and extend the analytical basis of several aspects of the dynamic behavior of flexible rotating beams, the second-degree nonlinear equations of motion for the coupled flapwise bending, lagwise bending, and axial extension of an untwisted, torsionally rigid, nonuniform, rotating beam having an arbitrary angle of precone with the plane perpendicular to the axis of rotation are derived using Hamilton's principle. The derivation of the equations is based on the geometric nonlinear theory of elasticity and the resulting equations are consistent with the assumption that the strains are negligible compared to unity. No restrictions are imposed on the relative displacements or angular rotations of the cross sections of the beam other than those implied by the assumption of small strains. Illustrative numerical results, obtained by using an integrating matrix as the basis for the method of solution, are presented both for the purpose of validating the present method of solution and indicating the range of applicability of the equations of motion and the method of solution.

Quantum Hamilton equations of motion for bound states of one-dimensional quantum systems

NASA Astrophysics Data System (ADS)

Köppe, J.; Patzold, M.; Grecksch, W.; Paul, W.

2018-06-01

On the basis of Nelson's stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton's principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson's diffusion paths in the ground state.

Aeromechanical stability of helicopters with a bearingless main rotor. Part 1: Equations of motion

NASA Technical Reports Server (NTRS)

Hodges, D. H.

1978-01-01

Equations of motion for a coupled rotor-body system were derived for the purpose of studying air and ground resonance characteristics of helicopters that have bearingless main rotors. For the fuselage, only four rigid body degrees of freedom are considered; longitudinal and lateral translations, pitch, and roll. The rotor is assumed to consist of three or more rigid blades. Each blade is joined to the hub by means of a flexible beam segment (flexbeam or strap). Pitch change is accomplished by twisting the flexbeam with the pitch-control system, the characteristics of which are variable. Thus, the analysis is capable of implicitly treating aeroelastic couplings generated by the flexbeam elastic deflections, the pitch-control system, and the angular offsets of the blade and flexbeam. The linearized equations are written in the nonrotating system retaining only the cyclic rotor modes; thus, they comprise a system of homogeneous ordinary differential equations with constant coefficients. All contributions to the linearized perturbation equations from inertia, gravity, quasi-steady aerodynamics, and the flexbeam equilibrium deflections are retained exactly.

Dynamic Nonlinear Elastic Stability of Helicopter Rotor Blades in Hover and in Forward Flight

NASA Technical Reports Server (NTRS)

Friedmann, P.; Tong, P.

1972-01-01

Equations for large coupled flap-lag motion of hingeless elastic helicopter blades are consistently derived. Only torsionally-rigid blades excited by quasi-steady aerodynamic loads are considered. The nonlinear equations of motion in the time and space variables are reduced to a system of coupled nonlinear ordinary differential equations with periodic coefficients, using Galerkin's method for the space variables. The nonlinearities present in the equations are those arising from the inclusion of moderately large deflections in the inertia and aerodynamic loading terms. The resulting system of nonlinear equations has been solved, using an asymptotic expansion procedure in multiple time scales. The stability boundaries, amplitudes of nonlinear response, and conditions for existence of limit cycles are obtained analytically. Thus, the different roles played by the forcing function, parametric excitation, and nonlinear coupling in affecting the solution can be easily identified, and the basic physical mechanism of coupled flap-lag response becomes clear. The effect of forward flight is obtained with the requirement of trimmed flight at fixed values of the thrust coefficient.

Forces Associated with Nonlinear Nonholonomic Constraint Equations

NASA Technical Reports Server (NTRS)

Roithmayr, Carlos M.; Hodges, Dewey H.

2010-01-01

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.

The dissolution or growth of a sphere

NASA Technical Reports Server (NTRS)

Shankar, N.; Wiltshire, Timothy J.; Subramanian, R. Shankar

1984-01-01

The problem of the dissolution or growth of an isolated stationary sphere in a large fluid body is analyzed. The motion of the boundary as well as the the resulting motion in the liquid are properly taken into account. The governing equations are solved using a recently developed technique (Subramanian and Weinberg, 1981) which employs an asymptotic expansion in time. Results for the radius of the sphere as a function of time are calculated. The range of utility of the present solution is established by comparison with a numerical solution of the governing equations obtained by the method of finite differences.

Regulation of landslide motion by dilatancy and pore pressure feedback

USGS Publications Warehouse

Iverson, R.M.

2005-01-01

A new mathematical model clarifies how diverse styles and rates of landslide motion can result from regulation of Coulomb friction by dilation or contraction of water-saturated basal shear zones. Normalization of the model equations shows that feedback due to coupling between landslide motion, shear zone volume change, and pore pressure change depends on a single dimensionless parameter ??, which, in turn, depends on the dilatancy angle ?? and the intrinsic timescales for pore pressure generation and dissipation. If shear zone soil contracts during slope failure, then ?? 0, and negative feedback permits slow, steady landslide motion to occur while positive pore pressure is supplied by rain infiltration. Steady state slip velocities v0 obey v0 = -(K/??) p*e, where K is the hydraulic conductivity and p*e is the normalized (dimensionless) negative pore pressure generated by dilation. If rain infiltration and attendant pore pressure growth continue unabated, however, their influence ultimately overwhelms the stabilizing influence of negative p*e. Then, unbounded landslide acceleration occurs, accentuated by an instability that develops if ?? diminishes as landslide motion proceeds. Nonetheless, numerical solutions of the model equations show that slow, nearly steady motion of a clay-rich landslide may persist for many months as a result of negative pore pressure feedback that regulates basal Coulomb friction. Similarly stabilized motion is less likely to occur in sand-rich landslides that are characterized by weaker negative feedback.

Nonlinear equations of motion for Landau resonance interactions with a whistler mode wave

NASA Technical Reports Server (NTRS)

Inan, U. S.; Tkalcevic, S.

1982-01-01

A simple set of equations is presented for the description of the cyclotron averaged motion of Landau resonant particles in a whistler mode wave propagating at an angle to the static magnetic field. A comparison is conducted of the wave magnetic field and electric field effects for the parameters of the magnetosphere, and the parameter ranges for which the wave magnetic field effects would be negligible are determined. It is shown that the effect of the wave magnetic field can be neglected for low pitch angles, high normal wave angles, and/or high normalized wave frequencies.

On the dynamics of chain systems. [applications in manipulator and human body models

NASA Technical Reports Server (NTRS)

Huston, R. L.; Passerello, C. E.

1974-01-01

A computer-oriented method for obtaining dynamical equations of motion for chain systems is presented. A chain system is defined as an arbitrarily assembled set of rigid bodies such that adjoining bodies have at least one common point and such that closed loops are not formed. The equations of motion are developed through the use of Lagrange's form of d'Alembert's principle. The method and procedure is illustrated with an elementary study of a tripod space manipulator. The method is designed for application with systems such as human body models, chains and cables, and dynamic finite-segment models.

Euler flow predictions for an oscillating cascade using a high resolution wave-split scheme

NASA Technical Reports Server (NTRS)

Huff, Dennis L.; Swafford, Timothy W.; Reddy, T. S. R.

1991-01-01

A compressible flow code that can predict the nonlinear unsteady aerodynamics associated with transonic flows over oscillating cascades is developed and validated. The code solves the two dimensional, unsteady Euler equations using a time-marching, flux-difference splitting scheme. The unsteady pressures and forces can be determined for arbitrary input motions, although only harmonic pitching and plunging motions are addressed. The code solves the flow equations on a H-grid which is allowed to deform with the airfoil motion. Predictions are presented for both flat plate cascades and loaded airfoil cascades. Results are compared to flat plate theory and experimental data. Predictions are also presented for several oscillating cascades with strong normal shocks where the pitching amplitudes, cascade geometry and interblade phase angles are varied to investigate nonlinear behavior.

Generalized extended Lagrangian Born-Oppenheimer molecular dynamics

DOE PAGES

Niklasson, Anders M. N.; Cawkwell, Marc J.

2014-10-29

Extended Lagrangian Born-Oppenheimer molecular dynamics based on Kohn-Sham density functional theory is generalized in the limit of vanishing self-consistent field optimization prior to the force evaluations. The equations of motion are derived directly from the extended Lagrangian under the condition of an adiabatic separation between the nuclear and the electronic degrees of freedom. We show how this separation is automatically fulfilled and system independent. The generalized equations of motion require only one diagonalization per time step and are applicable to a broader range of materials with improved accuracy and stability compared to previous formulations.