Variational Equations Of Motion Research Articles

Basis Set Generation for Quantum Dynamics Simulations Using Simple Trajectory-Based Methods

Methods for solving the time-dependent Schrödinger equation generally employ either a global static basis set, which is fixed at the outset, or a dynamic basis set, which evolves according to classical-like or variational equations of motion; the former approach results in the well-known exponential scaling with system size, while the latter can suffer from challenging numerical problems, such as singular matrices, as well as violation of energy conservation. Here, we suggest a middle road: building a basis set using trajectories to place time-independent basis functions in the regions of phase space relevant to wave function propagation. This simple approach, which potentially circumvents many of the problems traditionally associated with global or dynamic basis sets, is successfully demonstrated for two challenging benchmark problems in quantum dynamics, namely, relaxation dynamics following photoexcitation in pyrazine, and the spin Boson model.

Bending Vibration Analysis of Thick Honeycomb Sandwich Panel

A simple approach to reduce the governing equations for thick honeycomb sandwich panels to a single equation containing only one displacement function is proposed in this article. The variational equations of motion are derived using Hamilton’s principle and by introducing differential operators. The exact solution of the natural frequencies for rectangular honeycomb sandwich panels with all edges simply-supported is obtained and leading to a satisfied agreement with theoretical results. The influence of the ratio of core thickness to face sheet thickness and aspect ratio on the natural frequencies is studied. The analysis results demonstrate that the presence of inertia of rotation will decrease natural frequencies, and the aspect ration changes not only the value of natural frequencies of the sandwich panel but also the mode order.

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J2). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.

Fundamental frequency of an orthotropic rectangular plate with an internal centre point support

A method of calculating the fundamental frequency of an orthotropic rectangular plate with a centrally located point support and free edges is presented. The variational equation of motion is derived by applying Hamilton’s principle. The analytical approach determining the fundamental frequency of the plate is developed using the generalised Galerkin method and verified by comparison with the results of the finite element modal analysis. The comparisons of the computational results indicate that the fundamental frequency of the centre-supported plates can be calculated with sufficient accuracy using the analytical technique developed in this work. The approach presented in the paper can be efficiently employed when designing composite rectangular plates for a specified value of the fundamental frequency.

Fundamental frequency of the CCCF composite sandwich plate

Abstract The analytical closed-form solution providing the value of fundamental frequency of the CCCF composite sandwich plate is proposed in this article. The variational equations of motion are derived using Hamilton’s principle in conjunction with the first-order shear deformation theory. The solution approach involves reduction of the governing equations of motion to a system of ordinary equations using Kantorovich procedure and subsequent application of the generalised Galerkin method. The analytical solution is verified using the finite element modal analysis. The comparisons of computational results have shown that the fundamental frequency of the CCCF sandwich plates can be calculated with sufficient accuracy using the analytical technique developed in this work.

Appropriate SCF basis sets for orbital studies of galaxies and a ‘quantum-mechanical’ method to compute them

We address the question of an appropriate choice of basis functions for the self-consistent field (SCF) method of simulation of the N-body problem. Our criterion is based on a comparison of the orbits found in N-body realizations of analytical potential-density models of triaxial galaxies, in which the potential is fitted by the SCF method using a variety of basis sets, with those of the original models. Our tests refer to maximally triaxial Dehnen gamma-models for values of $\gamma$ in the range 0<=gamma<=1. When an N-body realization of a model is fitted by the SCF method, the choice of radial basis functions affects significantly the way the potential, forces, or derivatives of the forces are reproduced, especially in the central regions of the system. We find that this results in serious discrepancies in the relative amounts of chaotic versus regular orbits, or in the distributions of the Lyapunov characteristic exponents, as found by different basis sets. Numerical tests include the Clutton-Brock and the Hernquist-Ostriker (HO) basis sets, as well as a family of numerical basis sets which are `close' to the HO basis set. The family of numerical basis sets is parametrized in terms of a quantity $\epsilon$ which appears in the kernel functions of the Sturm-Liouville (SL) equation defining each basis set. The HO basis set is the $\epsilon=0$ member of the family. We demonstrate that grid solutions of the SL equation yielding numerical basis sets introduce large errors in the variational equations of motion. We propose a quantum-mechanical method of solution of the SL equation which overcomes these errors. We finally give criteria for a choice of optimal value of $\epsilon$ and calculate the latter as a function of the value of gamma.

The numerical analysis of the trapezoidal thread rolling process

AbstractThe thread rolling is difficult technological process. Improve quality and contemporary reduce manufacture cost of the trapezoidal thread requires acquaintances of physical phenomena observed in the contact zone between rolls and deform work‐pieces. Therefore, in this paper the physical and mathematical models of deformations (displacements and strains) and stress in the cold process of trapezoidal thread rolling, were developed. The process is considered as a geometrical and physical non‐linear, initial as well as boundary value problem. The phenomena on a typical incremental step were described using a step‐by‐step incremental procedure, with an updated Lagrangian formulation. The state of strains was described by Green‐Lagrange's tensor, while the state of stress by the second symmetrical Pioli‐Kirchhoff's tensor. The object was treated as an elastic (in the reversible zone) and visco‐plastic body (in non‐reversible zone) with mixed hardening. The variational equation of motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. The application developed for in the ANSYS programme, which provides a complex time analysis for displacement, strains and stresses occurring in the object. The recommendations concern modeling the trapezoidal thread rolling process, where reduce degrees of freedom in numerical model is very important and provide convergence calculated results for maximum stress and strain values in the thread surface layer, were elaborated. The influence a various process conditions on the states deformation and stress for examples calculations, were presented. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

구속된 다물체시스템의 선형화에 관한 연구

This research proposes an implementation method of linearized equations of motion for multibody systems with closed loops. The null space of the constraint Jacobian is first pre-multiplied to the equations of motion to eliminate the Lagrange multiplier and the equations of motion are reduced down to a minimum set of ordinary differential equations. The resulting differential equations are functions of ail relative coordinates, velocities, and accelerations. Since the coordinates, velocities, and accelerations are tightly coupled by the position, velocity, and acceleration level constraints, direct substitution of the relationships among these variables yields very complicated equations to be implemented. As a consequence, the reduced equations of motion are perturbed with respect to the variations of all coordinates, velocities, and accelerations, which are coupled by the constraints. The position, velocity and acceleration level constraints are also perturbed to obtain the relationships between the variations of all relative coordinates, velocities, and accelerations and variations of the independent ones. The perturbed constraint equations are then simultaneously solved for variations of all coordinates, velocities, and accelerations only in terms of the variations of the independent coordinates, velocities, and accelerations. Finally, the relationships between the variations of all coordinates, velocities, accelerations and these of the independent ones are substituted into the variational equations of motion to obtain the linearized equations of motion only in terms of the independent coordinate, velocity, and acceleration variations.

Numerical Analysis Of Thermal Phenomena And DeformationsIn Processing Zone In The Centerless Continuous Grinding Process

The paper presents the physical and mathematical models of deformations and heat transfer in the continuous centerless grinding process for axial-symmetric workpieces. The process is considered as a geometrical, physical and thermal initial and boundary value problem. The phenomena on a typical incremental step were described using a step-by-step incremental procedure, with updated Lagrangian formulation. The variational equation of motion and heat transfer in three dimensions for th~s case has been proposed. Then, we have used the finite elements methods (FEM) and dynamic explicit method (DEM) to obtain the solution. Examples of calculations of the displacement, strain and stress field in the surface layer zones for two methods of calculation are presented.

1305 任意周期関数励振を受けるばね-質量系の衝突振動 : 調和および分数調波振動の解析

This paper deals with impact vibration in spring-mass system excited by periodic force with arbitrary function. The analytical model is steady impact vibration of a mass with spring and damping in the case where the system is subjected to excitation by displacement. The restoring force, which has an unsymmetric piecewise-linear characteristics, is elastic collision to unsymmetric faces. In order to analyze the main and the 1/2nd order subharmonic resonances, the Fourier series method is applied to this system. Following the theoretical analysis, numerical calculation is performed and the resonance curves are made from the wave of resulting vibration. Effects of the excitation wave form and the excitation amplitude ratio on the resonance curves are shown by numerical results. Also, the stability problems of these vibrations are discussed utilizing Hill's infinite determinant which is obtained from a variational equation of motion. As a result, the stable branches of resonance curves are distinguished from unstable ones. For verification of the numerical method, the experiments are performed. The analytical results are in a fairly good agreement with the experimental ones.

Impulsive Feedback Control to Establish Specific Mean Orbit Elements of Spacecraft Formations

An impulsive feedback control is developed to establish specie c relative orbits for spacecraft formation e ying. The relative orbit tracking errors are expressed in terms of mean orbit elements. The feedback control, based on Gauss’ s variational equations of motion, allows specie c orbit elements or orbit element sets to be controlled with minimal impact on the remaining osculating orbit elements. This is advantageous when J2-invariant orbits are to be controlled, where only the argument of perigee and mean anomaly will drift apart at equal and opposite rates. The advantage of this impulsive feedback control, compared to optimal control solutions, is that it can operate with little computational effort and in a near-optimal manner, whilerequiring only a marginal penalty in fuel cost. When applied to the spacecraft formation e ying problem, this control could also be used to perform general orbit corrections. Formulas are developed providing accurate estimates of the sensitivities of the mean semimajor axis and mean eccentricity with respect to the osculating inclination angle. With these sensitivities, the tracking error in semimajor axis, eccentricity, and inclination angle can be canceled within one orbit.

Numerical modeling of nonlinear deformation and buckling of composite plate-shell structures under pulsed loading

Nonlinear three-dimensional problems of dynamic deformation, buckling, and posteritical behavior of composite shell structures under pulsed loads are analyzed. The structure is assumed to be made of rigidly joined plates and shells of revolution along the lines coinciding with the coordinate directions of the joined elements. Individual structural elements can be made of both composite and conventional isotropic materials. The kinematic model of deformation of the structural elements is based on Timoshenko-type hypotheses. This approach is oriented to the calculation of nonstationary deformation processes in composite structures under small deformations but large displacements and rotation angles, and is implemented in the context of a simplified version of the geometrically nonlinear theory of shells. The physical relations in the composite structural elements are based on the theory of effective moduli for individual layers or for the package as a whole, whereas in the metallic elements this is done in the framework of the theory of plastic flow. The equations of motion of a composite shell structure are derived based on the principle of virtual displacements with some additional conditions allowing for the joint operation of structural elements. To solve the initial boundary-value problem formulated, an efficient numerical method is developed based on the finite-difference discretization of variational equations of motion in space variables and an explicit second-order time-integration scheme. The permissible time-integration step is determined using Neumann's spectral criterion. The above method is especially efficient in calculating thin-walled shells, as well as in the case of local loads acting on the structural element, when the discretization grid has to be condensed in the zones of rapidly changing solutions in space variables. The results of analyzing the nonstationary deformation processes and critical loads are presented for composite and isotropic cylindrical shells reinforced with a set of discrete ribs in the case of pulsed axial compression and external pressure.

On the Ordinary, Generalized, and Pseudo-Variational Equations of Motion in Nonlinear Elasticity, Piezoelectricity, and Classical Plate Theories

Ordinary, generalized, and pseudo-variational equations of motion in three-dimensional theories of nonlinear elasticity and piezoelectricity are presented systematically. These are applied to the derivations of plate equations of the classical type. In contrast to the derivations of plate equations that include thickness and higher-order effects, it is shown that the volume and surface integrals in a three-dimensional ordinary variational equation of motion must now be used jointly in a coupled manner. Details are demonstrated by first treating a classical linear plate. Equations of the classical type for large deflections of laminated composite and piezoelectric plates are then derived, with the famous von Ka´rma´n equations of an isotropic homogeneous plate deducible as a special case. Interrelationship among various plate equations is emphasized.

Some Recent Advances in Linear and Nonlinear Dynamical Modeling of Elastic and Piezoelectric Plates

A previous review has covered our early efforts in dynamics and vibration of homogeneous and layered elastic plates and shells, including sandwiches. As an update, the present review summarizes our recent efforts in the same general area, now further including laminated composite and piezoelectric plates. These reflect the increasing importance of laminates, the growing interest in distributed active vibration control through the use of piezoelectric layers and often a combination of both. The main purpose of this paper is to again provide a systematic overview of linear and nonlinear dynamical modeling of both homogeneous and layered plates. As many of our contributions have been made by means of the variational equations of motion in three-dimensional elasticity and piezoelectricity, an extended discussion of these equations is included as an integral part of this review. Recent applications of these to the linear and nonlinear dynamical modeling of elastic laminates and piezoelectric layers are described.

Characterization of Chaotic Behavior in Nonlinear Vibration Systems. (Applications of the Methods of Statistical Mechanics without System Equations).

To characterize chaotic oscillation arising in nonlinear systems, a method of statistical mechanics is applied. The chaotic oscillation is characterized in terms of the dynamic structure functions : the fluctuation spectrum, the q-weighted average and the q-weighted variance. In the previous paper, in order to obtain the local expansion rates that lead to the dynamic structure functions, the variational equations of motion were used. However the expansion rates cannot be obtained from the variational equations if the equations of motion are not known. Therefore, in the present paper, under the assumption that the equations of motion are not known, it is shown that the method of estimating matrices based on Poincare maps of ocsillation is applicable. As examples, the parametrically forced pendulum system and the Duffing system are taken up. The results of the method of estimating matrices are similar to those of the variational method.

Dynamic analysis of delamination growth

T dynamic problem of the growth of a thin homogeneous strip delamination in a thick base plate under axial loading has been studied with a Griffith-type fracture criterion adopted. The variational equation of motion and the dynamic local growth condition at the crack tip are derived. The resulting nonlinear free boundary problem is solved by a numerical technique that combines both the finite element method and the Newton-Raphson iterative method/The local growth condition at the delamination front is examined at each time step of the integration process. It is found that the ratio of the applied compressive strain to the bifurcation strain of the delaminated layer is an important parameter in determining the delamination growth behavior. The results of the present analysis are compared with those of the quasidynamic analysis.

Elastodynamic Formulation of the Eulerian-Lagrangian Kinematic Description

An extension of the Eulerian-Lagrangian kinematic description (Haber, 1984) to elastodynamic problems is presented. Expressions are derived for field variables and material time derivatives using the new kinematic description. The variational equation of motion is written in a weak form suitable for use with isoparametric finite elements. The new kinematic model allows a finite element mesh to continuously adjust for changes in the structural geometry, material interfaces, or the domain of the boundary conditions without a discrete remeshing process. Applications of the new model to mode I dynamic crack propagation demonstrates its advantages over moving mesh methods based on conventional Lagrangian kinematic models. Numerical results show excellent agreement with analytic predictions.

Motion near the triangular points in the elliptic restricted problem of three bodies

In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given.

Application of Rayleigh-Ritz and Galerkin methods to non-linear vibration of plates

The difference between the three variational equations of motion and the dynamic analogue of the von Karman equations governing the non-linear vibration of plates is considered in the context of the Rayleigh-Ritz and Galerkin methods. The non-linear free vibrations of two types of plates with in-plane constraints are considered as examples. In the analysis a single mode expansion is assumed for the transverse displacement and the in-plane inertia terms are neglected. The convergence of the non-linear time period is studied by increasing the number of terms in the modal expansions for the in-plane displacements. It is observed that the Rayleigh-Ritz and Galerkin approximations converge from opposite directions, thus suggesting a possible approach to bound the solutions on either side. The general significance of this observation is discussed.

Nutational Stability of a Dual-Spin Satellite under the Influence of Applied Reaction Torques

It is assumed that since the solar paddle attachments to the hub of the spacecraft are not rigidly locked, the effect of the solar panels can be replaced by a constant reaction torque acting on the hub of the spacecraft. This could result in the satellite having an equilibrium motion about an axis displaced from the nominal axis of symmetry. The variational equations of motion are developed about such an equilibrium position using the SAS-A spacecraft as a model. Energy dissipation on the rotor as well as the main body is included. This nonautonomous set of differential equations are linearized and transformed to an autonomous set using the Liapunov reducibility theorem. The stability of the kinematically similar system is examined numerically using representative SAS-A parameters for the case when either pair of solar panels is assumed to be loosely attached. Stability is verified for small system nutation angles (0.1 degree).