Dynamics and Stability of Rolling Viscoelastic Tires

Current steady state rolling tire calculations often do not includetreads because treads destroy the rotational symmetry of the tire.We describe two methodologies to compute time periodic solutions of a two-dimensional viscoelastic tire with treads:solving a minimization problem and solving a system of equations.We also expand on work by Oden and Lin on free spinning rolling elastic tires in which they disovered a hierachy of $N$-peak steady state standing wave solutions.In addition to discovering a two-dimensional hierarchy of standing wave solutions thatincludes their $N$-peak hiearchy,we consider the effects of viscoelasticity on the standing wave solutions.Finally, a commonplace model of viscoelasticity used in our numerical experimentsled to non-physical elastic energy growth for large tire speeds. We show that a viscoelastic model of Govindjee and Reese remedies the problem