A linear complementarity formulation on position level for frictionless impact of planar deformable bodies

Abstract

In this paper, we present an approach for frictionless impact of planar deformable bodies by formulating a linear complementarity problem on position level. This approach is based on the enforcing the Signorini conditions for the impact problem of continua. In doing so, first the bodies are discretized in finite elements and condensed with a finite number of eigenmodes. The equations of motion of the deformable bodies are generated using the well know moving frame of reference approach. Then, the normal gaps between the two impacting bodies are written in terms of the generalized coordinates. In the next step, by following different integration methods the generalized acceleration vector obtained from the equations of motion will be integrated to reach a relationship for the generalized coordinates. At the end, by substituting the generalized coordinates in the relationship of normal gaps, the impact problem is formulated as a linear complementarity problem. Solving this problem leads to the solution of our impact problem considering impact forces and normal gaps. At this point it should be emphasized that in this formulation no coefficient of restitution has to be introduced for obtaining the impact law. Although there exist some approaches for impact modeling of deformable bodies which introduce this coefficient as a measure of energy loss during impact, no coefficient of restitution should be used since the energy loss is taken into account by the damping in the material law of the deformable bodies. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA.