A Differential Algebraic Method to Approximate Nonsmooth Mechanical Systems by Ordinary Differential Equations

Nonsmooth mechanical systems, which are mechanical systems involving dry friction and rigid unilateral contact, are usually described as differential inclusions (DIs), that is, differential equations involving discontinuities. Those DIs may be approximated by ordinary differential equations (ODEs) by simply smoothing the discontinuities. Such approximations, however, can produce unrealistic behaviors because the discontinuous natures of the original DIs are lost. This paper presents a new algebraic procedure to approximate DIs describing nonsmooth mechanical systems by ODEs with preserving the discontinuities. The procedure is based on the fact that the DIs can be approximated by differential algebraic inclusions (DAIs), and thus they can be equivalently rewritten as ODEs. The procedure is illustrated by some examples of nonsmooth mechanical systems with simulation results obtained by the fourth-order Runge-Kutta method.