Numerical Integration of Systems with Unilateral Constraints

Abstract

If a mechanical system with coordinates q = (q₁...q_n)^t is subject to the unilateral constraint

$$f\left( {\underline q ,t} \right) \geqslant 0,\quad f \in {{C}^{2}}\left[ {U \subset {{\mathbb{R}}^{{n + 1}}}} \right],\quad t - physical\,time$$

(1)

then discontinuities can occur in the velocities (in contrast to systems with the classical bilateral constraint f (q,t) =0 which does not cause such discontinuities). For the velocity jumps algebraic equations are known (see refs.[1, 2]). When a system of differential equations of motion subject to the constraint (1) is solved numerically the usual procedure is as follows. First the impact time t_i is determined iteratively. The numerical integration is then interrupted at t=t_i and the velocity jumps are computed. With new initial conditions the numerical integration is then continued. The present paper describes a technique for computing highly accurate solutions without having to interrupt the numerical integration and without having to solve algebraic equations for velocity jumps. Any standard integration routine with stepsize control can be used.