This article proposes (1) a two-dimensional tire model that extends the deformable ring on elastic foundation (REF) model by treating the ring as a laminated beam and (2) a feedback compensation approach to solve the tire-road contact problem as facilitated by the laminated REF model. The internal layer of the laminated ring is formulated using the Timoshenko beam theory that can also be easily regressed to an Euler beam. The external layer of the laminated ring is modeled as a circular beam that primarily takes into account the strain energy contributed by the tire tread in the transverse or radial direction.

The elastic foundations are assumed to have a predeformation in the radial direction, which can model tire inflation pressure in pneumatic tires or foundation precompression/pretension for nonpneumatic tires. The analytical solution for the static deformation response of this laminated REF model due to an arbitrary external force is detailed first. Then, a feedback p-controller algorithm that penalizes geometry errors in the contact region is outlined as a unified approach that can be used to solve frictionless contact problems between a tire and arbitrary road profiles. The performances of the proposed model and algorithm are compared against those obtained from a detailed finite element analysis. Both flat surface and cleat contact responses are shown to illustrate the utility of this laminated REF model and the contact algorithm.

Wang, C., Ayalew, B., Rhyne, T., Cron, S., and Adcox, J., Tire Science and Technology, TSTCA, Vol. 45, No. 4, October–December 2017,
pp. 307–334.

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