Generalized Coordinate Partitioning for Analysis of Mechanical Systems with Nonholonomic Constraints

1983 ◽  
Vol 105 (3) ◽  
pp. 379-384 ◽  
Author(s):  
P. E. Nikravesh ◽  
E. J. Haug
Keyword(s):  
Differential Equations ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Variable Order ◽  
Step Size ◽  
Euler Parameters ◽  
Integration Algorithm ◽  
Generalized Coordinates

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.

Application of Euler Parameters to the Dynamic Analysis of Three-Dimensional Constrained Mechanical Systems

1982 ◽  
Vol 104 (4) ◽  
pp. 785-791 ◽  
Author(s):  
P. E. Nikravesh ◽  
I. S. Chung
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Dependent Variables

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. Nonlinear holonomic 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 the Euler parameters. Euler parameters, in contrast to Euler angles or any other set of three rotational generalized coordinates, have no critical singular cases. 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, identifies dependent variables, and constructs an influence coefficient matrix relating variations in dependent and indpendent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total 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.

Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems

1982 ◽  
Vol 104 (1) ◽  
pp. 247-255 ◽  
Author(s):  
R. A. Wehage ◽  
E. J. Haug
Keyword(s):  
Differential Equations ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Step Size ◽  
Integration Algorithm ◽  
Independent Variables ◽  
Dependent Variables

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, to facilitate the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matix relating variations in dependent and independent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, is developed that integrates for only the independent variables, yet effectively determines dependent variables. Numerical results are presented for planar motion of two tracked vehicular systems with 13 and 24 degrees of freedom. Computational efficiency of the algorithm is shown to be an order of magnitude better than previously employed algorithms.

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

A geometric approach to the transpositional relations in dynamics of nonholonomic mechanical systems

2018 ◽  
Vol 15 (07) ◽  
pp. 1850112 ◽  
Author(s):  
Mahdi Khajeh Salehani
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Geometric Approach ◽  
Nonholonomic Mechanical Systems ◽  
Nonholonomic Dynamics ◽  
Geometric Objects

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.

Explicit Poincaré equations of motion for general constrained systems. Part I. Analytical results

2007 ◽  
Vol 463 (2082) ◽  
pp. 1421-1434 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Constrained Systems ◽  
Analytical Results ◽  
Multi Body ◽  
D'alembert's Principle

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.

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.

Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

1993 ◽  
Vol 60 (3) ◽  
pp. 662-668 ◽  
Author(s):  
R. E. Kalaba ◽  
F. E. Udwadia
Keyword(s):  
Dynamical Systems ◽  
Closed Form ◽  
Programming Problem ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Closed Form Solution ◽  
Nonholonomic Constraints ◽  
Form Solution ◽  
Constrained Motion ◽  
Constraint Forces

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.

Differential equations of motion of mechanical systems with variable parameters

1974 ◽  
Vol 10 (6) ◽  
pp. 671-674
Author(s):  
V. A. Lazaryan ◽  
L. A. Manashkin ◽  
A. V. Yurchenko
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variable Parameters

Dynamic Analysis of a Spherical Four Bar Mechanism

1992 ◽  
Author(s):  
J. R. Dooley ◽  
J. M. McCarthy
Keyword(s):  
Equations Of Motion ◽  
Joint Angles ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Constraint Equations ◽  
Spherical Mechanisms ◽  
Input And Output ◽  
Closed Chain ◽  
Concentric Spheres ◽  
Spherical Mechanism

Abstract Spherical mechanisms are designed so that the points in each link are constrained to move on concentric spheres This paper develops the equations of motion for spherical four bar mechanisms. While the kinematics of these linkages have been extensively studied, the dynamics equations do not seem to have been previously derived. As design techniques for these mechanisms become more efficient, the equations of motion are required to evaluate their performance. The complete dynamics equations of a four-bar spherical mechanism are derived using the input and output joint angles and the Euler parameters of the coupler as generalized coordinates. These coordinates provide a convenient representation of the constraint equations associated with the closed chain. An example analysis is provided.

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