5. Constrained Rigid Bodies
Society for Industrial and Applied Mathematics eBooks · 2000
In this chapter, we derive the equations of motion of a constrained rigid body, or system of rigid bodies, in direct continuation of what was done in Chapter 4 in the unconstrained case. Our goal is to obtain a primary DAE formulation, namely, the system (5.49) below or (5.18) if only one body is involved. In addition, several other versions of these systems are given for cases involving certain special features. Specifically, while the main system (5.49) is a DAE on the configuration space, and hence a DAE on a manifold, the system (5.51) is a simpler DAE in Euclidian space in the case when extensions of the constraint function and of the force and moment fields are readily available. Alternately, geometric constraints lead to a special form of (5.49) given by the system (5.56). Although we set up the equations of motion, we do not address here the existence of solutions for the corresponding systems. This will be done later in Chapters 6 and 7, where, among other things, existence theorems for initial value problems are proved for DAEs in Euclidian space (Chapter 6) and manifolds (Chapter 7). 5.1 Kinematic Constraints We will retain here the same notation developed in Chapter 4 for the unconstrained motion of a rigid body ℬ. But, for the sake of brevity, it is useful to introduce the simplification Fm (t,u,p, u ˙ , p ˙ ) ≔2H (p)⊤ N (t,u,p, u ˙ , p ˙ ) +8H (p)⊤ H (p) H ( p ˙ )⊤ JH ( p ˙ ) p, 5.1 where the subscript “m” is a reminder that represents a moment force. With this the equations of unconstrained motion (4.22) assume the more concise form Mü= Fr (t,u,p, u ˙ , p ˙ ), 4H (p) ⊤ JH (p) p ¨ = Fm (t,u,p, u ˙ , p ˙ ). 5.2 The unconstrained state space is , with the understanding that and in correspond to the same state.