Multibody mechanics and the residual flexibility method

Abstract

This thesis provides first an introduction to an area of continuum mechanics the author terms multibody mechanics. Therein, a continuum is separated into a finite number of bodies. By merging ideas from classical and multibody dynamics with contemporary rational mechanics, approximate methods such as the finite element method and substructuring techniques are presented as natural approximation schemes of the differential equations generated. When such approximations are done for each body, which are in turn connected to form a system, differential-algebraic equations with index three result. The difficulty in solving index three differential-algebraic equations is demonstrated with a simple example and alternative strategies are discussed. In most situations, the alternatives either destroy the natural sparse structure of the matrices or employ artificial techniques to control constraint drift. For flexible bodies, the benefits of using the residual flexibility method are demonstrated. The method naturally retains the sparse (mostly diagonal) matrix structures while also resulting in differential-algebraic equations with index one. As is well-known, the numerical solution of index one equations is more easily accomplished than that of equations with a higher index. Therefore, the residual flexibility method represents a remarkable approach for not only modeling the flexibility but also reducing the index of the governing differential-algebraic equations.