Computational Dynamics, 3rd Edition

Chapter 4. FORMS OF THE DYNAMIC EQUATIONS

The focus of Chapter 3 was on the analysis of kinematically driven systems in which all the degrees of freedom are specified. Since the system configuration can be completely determined when the degrees of freedom are known, the analysis of kinematically driven systems leads to a system of algebraic equations that can be solved for the coordinates, velocities, and accelerations without the need for a force analysis. However, if one or more of the system degrees of freedom are not known a priori, the force analysis becomes necessary and the system equations of motion must be formulated to obtain a number of equations equal to the number of the unknown variables. In the case of unconstrained motion, the equations of motion of the system take a simple known form defined by Newton-Euler equations, and therefore, the selection of the system coordinates is not the subject of much argument. In the case of constrained multibody dynamics, on the other hand, different numbers of coordinates can be selected, leading to different forms of the dynamic equations. Some formulations that employ redundant coordinates lead to a relatively large system of equations expressed in terms of the constraint forces, while some other formulations lead to a minimum set of differential equations of motion expressed in terms of the degrees of freedom. Since the degrees of freedom, by definition, are independent and are not related by kinematic relationships, it is expected ...