Numerical simulation of nonsmooth multibody systems with switching bilateral constraints

This paper addresses a special class of multibody systems where switching functions trigger instantaneous changes in the bilateral constraints, thus leading to a time-discontinuous response. These switching functions define switching surfaces, that partition the system dynamics into different domains with nonsmooth transitions. Switching surfaces are thus instrumental in the proposed modelling framework, as they orchestrate the geometry of the constraint space. The equations of motion can either be expressed as hybrid differential-algebraic equations or as an equality of differential measures with constraints. At switching, an impact law based on an intermediate gradient is introduced and we suggest to determine this intermediate gradient by interpolation between the pre- and post-switch gradients. Theoretical arguments and numerical results show that the choice of this intermediate gradient drives the energy behavior at switching. The numerical integration demands special attention, as classical DAE solvers fail to handle discontinuities at switching surfaces. Event-driven and time-stepping schemes from nonsmooth dynamics are well-suited for this class of problems. We use a benchmark test to compare the solutions obtained from classical and nonsmooth versions of the generalized-α method. Further, using the nonsmooth generalized-α method, three examples of multibody systems with switching bilateral constraints are successfully simulated.