I intend to put some notes here about rigid body dynamics, and its connections with other issues in mathematics (pure and applied) and engineering. The things that makes these problems diffficult are Coulomb friction and impacts (or, more generally, impulsive forces).
Rigid body dynamics is the study of the mechanics of bodies that cannot be deformed, unlike elasticity. The bodies can move and rotate, but cannot be stretched or compressed. Mathematically, the big difference is that rigid bodies only need a finite number of parameters to describe their state or configuration, while elastic bodies need functions to describe their state - which is usually the deformation function. That's because different parts of an elastic body can be deformed differently.
No physical bodies are actually completely rigid. They deform to some extent. But for a large number of materials (metals, stone, hard plastics) bodies made of these materials behave very much as if they were rigid under normal conditions (low speed, low to moderate forces, small to medium size bodies). The rigid body model is just a model, just as elasticity is also a model. After all, real bodies are not continua, but are made up of atoms. But for many purposes, the rigid body model is an excellent model.
Applications of rigid body dynamics include robotics, analysis of machines (gearing, clutches), and motion (walking, grasping).
Contact occurs when two bodies touch. Rigid bodies cannot overlap, of course, but they can touch. When they touch, they can impart a contact force or impulse to the other body. Mathematically, contact can be described in terms of a function giving the distance between the bodies: f(q) >= 0. The normal contact force N has a complementary relationship to f :
N >= 0, f(q) >= 0, N.f(q) = 0.
Rigid bodies cannot interpenetrate, so f(q) >= 0 for all feasible configurations q.
Last modified: Thu Mar 27 13:01:57 EST