Dynamics of mechanical systems with nonlinear nonholonomic constraints - II Differential equations of motion

Author(s):  
D.N. Zeković
1983 ◽  
Vol 105 (3) ◽  
pp. 379-384 ◽  
Author(s):  
P. E. Nikravesh ◽  
E. J. Haug

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems with holonomic and nonholonomic constraints. Holonomic and nonholonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are Euler parameters. The maximal set of generalized coordinates facilitates the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix and identifies independent coordinates and velocities. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.


2018 ◽  
Vol 15 (07) ◽  
pp. 1850112 ◽  
Author(s):  
Mahdi Khajeh Salehani

Exploring the geometry of mechanical systems subject to nonholonomic constraints and using various bundle and variational structures intrinsically present in the nonholonomic setting, we study the structure of the equations of motion in a way that aids the analysis and helps to isolate the important geometric objects that govern the motion of such systems. Furthermore, we show that considering different sets of transpositional relations corresponding to different transitivity choices provides different variational structures associated with nonholonomic dynamics, but the derived equations (being referred to as the generalized Hamel–Voronets equations) are equivalent to the Lagrange–d’Alembert equations. To illustrate results of this work and as some applications of the generalized Hamel–Voronets formalisms discussed in this paper, we conclude with considering the balanced Tennessee racer, as well as its modification being referred to as the generalized nonholonomic cart, and an [Formula: see text]-snake with three wheeled planar platforms whose snake-like motion is induced by shape variations of the system.


Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri

This paper gives the general constrained Poincaré equations of motion for mechanical systems subjected to holonomic and/or nonholonomic constraints that may or may not satisfy d'Alembert's principle at each instant of time. It also extends Gauss's principle of least constraint to include quasi-accelerations when the constraints are ideal, thereby expanding the compass of this principle considerably. The new equations provide deeper insights into the dynamics of multi-body systems and point to new ways for controlling them.


1994 ◽  
Vol 61 (2) ◽  
pp. 453-459 ◽  
Author(s):  
J. G. Papastavridis

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.


1993 ◽  
Vol 60 (3) ◽  
pp. 662-668 ◽  
Author(s):  
R. E. Kalaba ◽  
F. E. Udwadia

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.


1974 ◽  
Vol 10 (6) ◽  
pp. 671-674
Author(s):  
V. A. Lazaryan ◽  
L. A. Manashkin ◽  
A. V. Yurchenko

2014 ◽  
Vol 11 (03) ◽  
pp. 1450017
Author(s):  
G. F. Torres del Castillo ◽  
O. Sosa-Rodríguez

It is shown that for a mechanical system with a finite number of degrees of freedom, subject to nonholonomic constraints, there exists an infinite number of Hamiltonians and symplectic structures such that the equations of motion can be written as the Hamilton equations, with the original constraints incorporated in the Hamiltonian structure.


1962 ◽  
Vol 66 (619) ◽  
pp. 457-460 ◽  
Author(s):  
S. P. Chan ◽  
H. L. Cox ◽  
W. A. Benfield

This paper presents a numerical method, derived directly from the basic differential equations of motion and expressed in the form of recurrence-matrix of finite differences, that can be generally applied to all multi-degree-of-freedom structures subjected to dynamic forces or forced displacements on any masses at any instants of time. The movements of the system may be described by any form of generalised co-ordinates.


Author(s):  
Hiroaki Yoshimura

The paper illustrates the Lagrangian formalism of mechanical systems with nonholonomic constraints using the ideas of geometric mechanics. We first review a Lagrangian system for a conservative mechanical system in the context of variational principle of Hamilton, and we investigate the case that a given Lagrangian is hyperregular, which can be illustrated in the context of the symplectic structure on the tangent bundle of a configuration space by using the Legendre transformation. The Lagrangian system is denoted by the second order vector field and the Lagrangian one- and two-forms associated with a given hyperregular Lagrangian. Then, we demonstrate that a mechanical system with nonholonomic constraints can be formulated on the tangent bundle of a configuration manifold by using Lagrange multipliers. To do this, we investigate the Lagrange-d’Alembert principle from geometric points of view and we also show the intrinsic expression of the Lagrange-d’Alembert equations of motion for nonholonomic mechanical systems with nonconservative force fields.


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