Dynamics of planar and spatial rigid-body systems

Abstract

In this chapter the equations of spatial and planar motion of unconstrained rigid bodies will be derived based on the laws of Newton and Euler (such as in standard textbooks like [44], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65]). The equations of motion will be written with respect to a general body-fixed reference point P i (P i ≠ C i , C i is the center of mass) of a body i. In Section 4.1 the notions of linear momentum and angular momentum of a rigid body will be introduced and rewritten in a form suitable for representing the Newton—Euler equations in a desired form. Here the notions of the center of mass and the inertia matrix of a rigid body will also be introduced, and some properties of this matrix will be briefly discussed. In Section 4.2 the Newton—Euler equations will be derived for planar and spatial motion of a rigid body represented with respect to the reference points P i and C i . (In Appendix A.2 the Lagrange formalism, applied to a rigid body under spatial motion, will be briefly discussed). The equations of motion of unconstrained and constrained planar and spatial rigid-body mechanisms will be collected in Section 4.3, combining the kinematic constraint equations of Section 3.1 and the rigid-body equations of Section 4.2.4. This provides model equations in DE form for unconstrained rigid bodies and in DAE form for constrained rigid bodies and rigid-body mechanisms. A few aspects concerning the numerical solution of DAEs will be briefly discussed in Section 4.4.