Finite-dimensional contact mechanics

In this paper, the continuous and numerical formulations of rigid–body dynamics based on measure differential inclusions and time–stepping methods recently developed are described and extended to include a finite number of elastic modes of vibration. The time-stepping methods already incorporate Coulomb friction, and are able to handle situations such as PainlevÉ's famous problem where impulsive forces occur without a collision. The elastic modes of vibration can be incorporated directly into the continuous formulation, but due to the stiffness typical of elastic vibrations, the numerical methods used need to be modified to incorporate them directly. The resulting numerical methods are dissipative in the limit, but only dissipate energy while there is contact.