Cyclic steady states of treaded rolling bodies

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