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.

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.

A unified tensor approach to the analysis of mechanical systems

1997 ◽  
Vol 211 (4) ◽  
pp. 313-322 ◽  
Author(s):  
A A Fogarasy ◽  
M R Smith
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Equation Of Motion ◽  
Newtonian Mechanics ◽  
Kinematic Constraint ◽  
Lagrangian Equation ◽  
Generalized Coordinates ◽  
Lagrangian Equations

It is shown in this paper that all methods of dynamic analysis of mechanisms used in practice can be derived from an invariant formed from the Lagrangian equation of motion. For the dynamic analysis of mechanisms subjected to kinematic constraint conditions, the Lagrangian equations of motion are far more suitable than the Newtonian approach. Since the Lagrangian equations are tensor equations, they are valid irrespective of what kind of generalized coordinates are used. This is not so, however, when the Newtonian approach is used. It is demonstrated by a simple example that a careless use of Newtonian mechanics can lead to erroneous results.

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.

Analysis and Optimal Design of Spatial Mechanical Systems

1987 ◽  
Author(s):  
H. Ashrafeiuon ◽  
N. K. Mani
Keyword(s):  
Sensitivity Analysis ◽  
Optimal Design ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Design Sensitivity ◽  
General Purpose ◽  
Design Sensitivity Analysis ◽  
Analysis And Design ◽  
Spatial Systems

Abstract This paper presents a new approach to optimal design of large multibody spatial mechanical systems. This approach uses symbolic computing to generate the necessary equations for dynamic analysis and design sensitivity analysis. Identification of system topology is carried out using graph theory. The equations of motion are formulated in terms of relative joint coordinates through the use of velocity transformation matrix. Design sensitivity analysis is carried out using the Direct Differentiation method applied to the relative joint coordinate formulation for spatial systems. Symbolic manipulation programs are used to develop subroutines which provide information for dynamic and design sensitivity analysis. These subroutines are linked to a general purpose computer program which performs dynamic analysis, design sensitivity analysis, and optimization. An example is presented to demonstrate the efficiency of the approach.

On the Dynamics of a Weighted Bowling Ball

1979 ◽  
Vol 46 (4) ◽  
pp. 937-943 ◽  
Author(s):  
R. L. Huston ◽  
C. Passerello ◽  
J. M. Winget ◽  
J. Sears
Keyword(s):  
Angular Velocity ◽  
Velocity Component ◽  
Equations Of Motion ◽  
Mass Center ◽  
Governing Equations ◽  
Euler Parameters ◽  
Factors Affecting ◽  
Dependent Variables ◽  
And Performance ◽  
Component Parallel

An analysis of the dynamics and performance of a weighted, slipping/rolling bowling ball is presented. The analysis uses Euler parameters and angular velocity components as dependent variables. The governing equations of motion are integrated using standard digital/numerical procedures. Particular attention is given to factors affecting ball performance (“hook”) and the lane oil tracing on the ball. It is found that factors most affecting hook are the mass-center location, the lane conditions (friction), and the initial angular velocity component parallel to the lane.

SYMBOLIC TREATMENT FOR THE EQUATIONS OF MOTION FOR RIGID MULTIBODY SYSTEMS

2010 ◽  
Vol 34 (1) ◽  
pp. 37-55 ◽  
Author(s):  
Bukoko C. Ikoki ◽  
Marc J. Richard ◽  
Mohamed Bouazara ◽  
Sélim Datoussaïd
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
The World ◽  
Rigid Multibody Systems ◽  
Symbolic Approach ◽  
Rigid Multibody

The library of symbolic C++ routines is broadly used throughout the world. In this article, we consider its application in the symbolic treatment of rigid multibody systems through a new software KINDA (KINematic & Dynamic Analysis). Besides the attraction which represents the symbolic approach and the effectiveness of this algorithm, the capacities of algebraical manipulations of symbolic routines are exploited to produce concise and legible differential equations of motion for reduced size mechanisms. These equations also constitute a powerful tool for the validation of symbolic generation algorithms other than by comparing results provided by numerical methods. The appeal in the software KINDA resides in the capability to generate the differential equations of motion from the choice of the multibody formalism adopted by the analyst.

scholarly journals New Aspects Concerning Dynamics of Rigid Solid Body in Plane-Parallel Motion Subjected to Real Constraints. Part One

2019 ◽  
Vol 24 (2) ◽  
pp. 175-180
Author(s):  
Vladimir Dragoş Tătaru ◽  
Mircea Bogdan Tătaru
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Analysis ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Original Method ◽  
Horizontal Force ◽  
Constraint Forces ◽  
Inclined Plane ◽  
Parallel Motion

Abstract The present paper approaches in an original manner the dynamic analysis of a wheel which climbs on an inclined plane under the action of a horizontal force. The wheel rolls and slides in the same time. The two movements, rolling and sliding are considered to be independent of each other. Therefore we are dealing with a solid rigid body with two degrees of freedom. The difficulty of approaching the problem lies in the fact that in the differential equations describing the motion of the solid rigid body are also present the constraint forces and these are unknown. For this reason they must be eliminated from the differential equations of motion. The paper presents as well an original method of the constraint forces elimination.

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
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