5. Constrained Rigid Bodies

Abstract

Previous chapter Next chapter Other Titles in Applied Mathematics Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint5. Constrained Rigid Bodiespp.45 - 59Chapter DOI:https://doi.org/10.1137/1.9780898719536.ch5PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt 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 Fm 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 S0 = R1 ×T ( R3 × S3 ) , with the understanding that (t, (u,p), ( u ˙ , p ˙ ) ) and (t, (u,−p), ( u ˙ ,− p ˙ ) ) in R1 ×T ( R3 × S3 ) correspond to the same state. Previous chapter Next chapter RelatedDetails Published:2000ISBN:978-0-89871-446-3eISBN:978-0-89871-953-6 https://doi.org/10.1137/1.9780898719536Book Series Name:Other Titles in Applied MathematicsBook Code:OT68Book Pages:vii + 136Key words:dynamics, rigidity, differentiable algebraic equations, numerical solutions, rigid bodies, Gauss principle, computational methods