Convergence of viscoelastic constraints to nonholonomic idealization
Abstract Rolling motion, which is usually described by means of nonholonomic constraints, can occur in many technical systems e.g. roller bearings or gear wheels in gearboxes. The idealized modeling of mechanical multibody systems with rolling elements leads to differential algebraic equations (DAEs) The kinematical condition of a vanishing relative velocity is enforced by constraint forces. However contact areas are not ideally rigid, but compliant due to local deformations of asperities and elasticity of the contacting bodies. For this reason a sensible physical description should take these effects into account. Thus the contact forces are modeled in the present paper using a tangential viscoelastic force element in the contact. The rolling motion is then enforced by applied forces, instead of constraint forces in the ideally rigid case. The objective of this work is to show that under certain conditions, solutions of the general multibody system with viscoelastic contact model converge to the solutions of the multibody system containing idealized nonholonomic constraint equations, if the viscoelastic constants approach infinity. An ansatz in form of an asymptotic series expansion with initial layer terms is introduced to prove the convergence under appropriate assumptions on the viscoelastic parameters. In order to suppress high frequency oscillations in the contact, the choice of the damping parameter is inspired by the critical damping, known in linear systems theory.