On the rigid body displacements and rotations in unilateral contact problems and applications

Abstract

This paper deals with the treatment of rigid body displacements and rotations in unilateral contact problems. In the presence of rigid body modes the equivalent formulations of the problem, i.e. the variational inequality, the quadratic programming and the linear complementarity formulation involve positive semidefinite matrices. The basic differences from the classical bilateral problems is that in unilateral problems the rigid body modes must be compatible with the inequality constraints. This is a fairly difficult problem to solve and in its generality was open until now. In this paper a systematic analysis including convergence results is presented with respect to some methods already in use, namely the method of coupling with additional elastic bodies or ‘soft’ springs and the method of consideration of certain nodes as fixed. Moreover a new linear complementarity formulation of the problem which explicitly includes the rigid body ‘displacements’ is proposed and is numerically treated by a complementary pivoting technique. Necessary and sufficient conditions for the solution of the problem are derived and the theory is illustrated by examples from structural analysis and from robotics.