Explicit Equation of Motion of Constrained Systems

Abstract

This chapter develops a new, simple, general, and explicit form of the equations of motion for general nonlinear constrained mechanical systems that can have holonomic and/or nonholonomic constraints that may or may not be ideal, and that may contain either positive semi-definite or positive definite mass matrices. This is done through the replacement of the actual unconstrained mechanical system, which may have a positive semi-definite mass matrix, with an unconstrained auxiliary system that is then subjected to the same holonomic and/or nonholonomic constraints as those applied to the actual unconstrained mechanical system. The unconstrained auxiliary system is subjected to the same “given” force as the actual mechanical system, and its mass matrix is appropriately augmented to make it positive definite so that the so-called fundamental equation can then be directly and simply applied to obtain the closed-form acceleration of the actual constrained mechanical system. Furthermore, it is shown that by appropriately augmenting the “given” force that acts on the actual unconstrained mechanical system, the auxiliary system directly provides the constraint force that needs to be imposed on the actual unconstrained mechanical system so that it satisfies the given holonomic and/or nonholonomic constraints. Thus, irrespective of whether the mass matrix of the actual unconstrained mechanical system is positive definite or positive semi-definite, a simple, unified fundamental equation results that give a closed-form representation of both the acceleration of the constrained mechanical system and the constraint force. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, constrained mechanical systems, such as those encountered in multi-body dynamics. Several examples are provided.