On Energy Rate Theorems for Linear First-Order Nonholonomic Systems

1991 ◽  
Vol 58 (2) ◽  
pp. 536-544 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Recent Work ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Systems ◽  
General Linear ◽  
Energy Rate ◽  
First Order ◽  
Power Equation ◽  
Boltzmann Hamel Equations ◽  
Power Reaction

Starting with the Boltzmann/Hamel equations of motion of mechanical systems subject to general linear and first-order nonholonomic and rheonomic constraints, and in nonholonomic coordinates, this paper derives the general (reactionless) energy rate, or power, equation for such systems, in a straightforward and physically clear fashion. A power (reaction-containing) equation for these systems, but in holonomic coordinates, is also derived for completeness. An example of a sphere rolling on a uniformly spinning plane serves to illustrate these power theorems. The paper extends and corrects the recent work by Kane and Levinson (1988) on the same topic.

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

On the Boltzmann-Hamel Equations of Motion: A Vectorial Treatment

1994 ◽  
Vol 61 (2) ◽  
pp. 453-459 ◽  
Author(s):  
J. G. Papastavridis
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Discrete Mechanical Systems ◽  
The Common ◽  
Boltzmann Hamel Equations ◽  
Nonlinear Velocity

This paper presents a direct vectorial derivation of the famous Boltzmann-Hamel equations of motion of discrete mechanical systems, in general nonlinear nonholonomic coordinates and under general nonlinear (velocity) nonholonomic constraints. The connection between particle and system vectors is stressed throughout, in all relevant kinematic and kinetic quantities/principles/theorems. The specialization of these results to the common case of linear nonholonomic coordinates and linear nonholonomic (i.e., Pfaffian) constraints is carried out in the paper’s Appendix.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

1993 ◽  
Author(s):  
Timothy A. Loduha ◽  
Bahram Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Relaxation Method ◽  
Nonholonomic Systems ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Constraint Relaxation ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

Abstract In this paper we present a method for obtaining first-order decoupled equations of motion for multi-rigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or more complex dynamical systems, where the appropriate congruency transformation may be difficult to obtain, we present a constraint relaxation method based on the use of orthogonal complements. The results are illustrated using several examples. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

On the Transformation Properties of the Nonlinear Hamel Equations

1995 ◽  
Vol 62 (4) ◽  
pp. 924-929 ◽  
Author(s):  
J. G. Papastavridis
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Inertia Force ◽  
Discrete Mechanical Systems ◽  
Inertial Terms ◽  
The Individual ◽  
Boltzmann Hamel Equations ◽  
System Inertia

This paper discusses the transformation properties of the famous Johnsen-Hamel equations of motion of discrete mechanical systems in general nonlinear nonholonomic coordinates and constraints (i.e., the nonlinear extension of the well-known Boltzmann-Hamel equations), under general nonlinear (local) quasi-velocity transformations. It is shown that the individual kinematico-inertial terms making up the system inertia force, or system acceleration, such as the nonlinear nonholonomic Euler-Lagrange operator and nonholonomic correction (or deviation) terms, in general, do not transform as nonholonomic covariant vectors; although taken as a whole they do, as expected. This work extends and completes the work of Papastavridis (1994), and it is strongly recommended that it be read after that paper.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

1995 ◽  
Vol 62 (1) ◽  
pp. 216-222 ◽  
Author(s):  
T. A. Loduha ◽  
B. Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Numerical Implementation ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Holonomic Systems ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or holonomic systems with unreduced configuration coordinates, we incorporate an orthogonal complement in conjunction with the congruency transformation. A pair of examples illustrate the results. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

Numerical Integration of a New Set of Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Nikolaos Potosakis ◽  
Elias Paraskevopoulos ◽  
Sotirios Natsiavas
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Weak Form ◽  
Integration Scheme ◽  
Weak Formulation ◽  
First Order ◽  
Motion Constraints ◽  
Time Integration Scheme

Some new theoretical and numerical results are presented on the dynamic response of a class of mechanical systems with equality motion constraints. At the beginning, the equations of motion of the corresponding unconstrained system are presented, first in strong and then in a weak form. Next, the formulation is extended to systems with holonomic and/or nonholonomic constraints. The formulation is based on a new set of equations of motion, represented by a system of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the motion constraints. Moreover, the position, velocity and momentum type quantities are assumed to be independent, forming a three field set of equations. The weak formulation developed was first used to cast the equations of motion as a set of first order ODEs in the coordinates and the corresponding momenta. Then, the same formulation was also employed as a basis for producing a suitable time integration scheme for the systems examined. The validity and efficiency of this scheme was tested and illustrated by applying it to a number of characteristic example systems.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

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