Current steady state rolling tire calculations often do not include treads 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 that includes 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 experiments led to non-physical elastic energy growth for large tire speeds. We show that a viscoelastic model of Govindjee and Reese remedies the problem.

The study of spinning axisymmetric cylinders undergoing finite deformation is a classic problem in several industrial settings - the tire industry in particular. We present a stability analysis of spinning elastic and viscoelastic cylinders using ARPACK to compute eigenvalues and eigenfunctions of finite element discretizations of the linearized evolution operator. We show that the eigenmodes correspond to N-peak standing or traveling waves for the linearized problem with an additional index describing the number of oscillations in the radial direction. We find a second hierarchy of bifurcations to standing waves where these eigenvalues cross zero, and confirm numerically the existence of finite-amplitude standing waves for the nonlinear problem on one of the new branches. In the viscoelastic case, this analysis permits us to study the validity of two popular models of finite viscoelasticity. We show that a commonly used finite deformation linear convolution model results in non-physical energy growth and finite-time blow-up when the system is perturbed in a linearly unstable direction and followed nonlinearly in time. On the other hand, Sidoroff-style viscoelastic models are seen to be linearly and nonlinearly stable, as is physically required. © 2014 Elsevier Ltd. All rights reserved.

The analysis of spinning axisymmetric bodies undergoing finite deformation is useful for understanding the behavior of a wide variety of engineering systems - for example, rolling tires. However, progress beyond the axisymmetric case has been lacking. In this work, we present a methodology for the treatment of treaded rolling bodies by devising a Newton-Krylov shooting scheme for the computation of cyclic steady states of motion. The scheme advocated permits one to determine the entire transient (cyclically steady) motion of a spinning treaded body with consideration of viscoelasticity as well as contact. We demonstrate the viability of the method through three examples, (1) a viscoelastic cylinder with eight sinusoidal tread blocks, (2) a viscoelastic cylinder with 16 square tread blocks, and (3) a viscoelastic oval in which repeated contact and separation from a rigid plane excites distinct transient vibrational modes in the body during each cycle. For practical values of the viscoelastic relaxation time, the new approach is found to converge faster than the naïve approach of evolving an initial guess over many cycles until a cyclic steady state is reached.