Bahram Ravani

Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems

1. Concepts and definitions 2. Topology and kinematic architecture 3. Transformation matrices in kinematics 4. Modeling mechanisms and multibody systems with transformation matrices 5. Position analysis by kinematic equations 6. Differential kinematics and numeric solution of posture equations 7. Velocity analysis 8. Acceleration analysis 9. Modeling dynamic aspects of mechanisms and multibody systems 10. Dynamic equations of motion 11. Linearized equations of motion 12. Equilibrium position analysis 13. Frequency response of mechanisms and multibody systems 14. Time response of mechanisms and multibody systems 15. Collision detection 16. Impact analysis 17. Constraint force analysis.

Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems: Dynamic Equations of Motion

Introduction Throughout earlier chapters we have carefully formulated our equations in a very general, multi-degree of freedom form. In fact, our only two limiting assumptions so far have been: (1) that all bodies of our system are totally rigid, allowing no deformation or deflection, and (2) that all joints act precisely as described by their mathematical models shown in section 4.6, exhibiting no effects such as backlash or clearances. Indeed, our efforts have produced a kinematic model of our system that is extremely general and powerful. Even though its solution may be tedious for hand calculation, we recognize that evaluation is intended by digital computation and we hope to continue this generality and precision throughout our work in dynamics. Lagranges Equation Although it may be possible to formulate the equations of motion for a general dynamic system by sketching free-body diagrams, assigning sign conventions and notation, and applying Newtons laws, such an approach is not used here because we are interested in complex and diversified three-dimensional mechanisms and multibody systems and our focus is on developing methods that can be coded for computation in a general setting. An approach based on energy and Lagranges equation is adopted here, which results in a very general form and minimizes the potential for errors in formulation. Before we discuss the method, however, let us review a very brief history of energy methods in mechanics.

Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems: Modeling Dynamic Aspects of Mechanisms and Multibody Systems

Introduction In the very beginning of this text, section 1.1, we observed that the science of mechanics is composed of two parts called statics and dynamics, first distinguished by Euler in 1765. His advice is, perhaps, worth repeating here [1]:n The investigation of the motion of a rigid body may be conveniently separated into two parts, the one geometrical, the other mechanical. In the first part, the transference of the body from a given position to any other position must be investigated without respect to the causes of the motion, and must be represented by analytical formulae which will define the position of each point of the body after the transference with respect to its initial placement. This investigation will therefore be referable solely to geometry, or rather to stereomety [the art of stone-cutting]. It is clear that by the separation of this part of the question from the other, which belongs properly to Mechanics, the determination of the motion from dynamic principles will be made much easier than if the two parts were undertaken conjointly. We also noted that dynamics is made up of two major disciplines, later recognized as the distinct sciences of kinematics and kinetics, which treat the motion and the forces producing it, respectively.

Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems: Transformation Matrices in Kinematics

Introduction Before formulating a numeric method for design analysis of mechanisms and multibody systems, let us first consider the essential characteristics of the problem being addressed. What are the chief difficulties encountered in the design analysis of a mechanism or multibody system? It is not the laws of mechanics as such that cause difficulty. It is the fact that, once a problem has been formulated, it is often too formidable algebraically to be easily solved. This complexity does not arise from static and dynamic force relationships, but from the kinematics – the changing geometry. The basic constraint equations that govern the motions within a machine or multibody system come from the fact that rigid bodies hold their respective joint elements in constant spatial relationships to one another. This type of constraint invariably leads to a set of highly nonlinear simultaneous algebraic equations. Because the difficulties in an analytic approach to mechanism and multibody system analysis stem from the geometry, it is wise to choose a mathematical formulation suited to this type of problem. One such formulation is based on the use of matrices to represent the transformation equations between strategically located coordinate systems fixed in successive bodies. This approach has been developed into an extremely general and powerful technique for mechanism and multibody system analysis, and the next several chapters are devoted to its presentation. Before this can be presented effectively, however, we must become familiar with a number of basic operations that render matrix algebra so useful in performing coordinate transformations. The purpose of this chapter, therefore, is to develop this foundation.