A mathematical treatment of the radial tire modeled as a rotating cylindrical shell on an elastic foundation

A mathematical model has been developed which adequately represents the static and dynamic behavior of a radial tire. The tire has been treated as a rotating cylindrical shell on an elastic foundation. The motion has been restricted to the plane of the tire only. The principle of minimum energy is employed to derive the equations of motion. Both free and forced vibration cases are discussed. An expression for the natural frequencies has been obtained. For the forced vibration case, the applied force and the solutions for the radial and tangential deflections have been written in Fourier series expansion form. Three different footprint pressure distribution profiles are assumed. The Laplace transforms and convolution theorem are used for the analysis. Both steady-state and transient solutions for the deflections have been obtained. As a limiting case of the steady state analysis, the static deflection case has been derived. The condition of instability has been examined to determine the limiting speed at or above which standing wave phenomena occur.

The shell equations have been applied to determine the stresses in the tire belt in terms of the deflections for both static and dynamic cases. Variations of static stresses and cord forces around the periphery of the tire have been evaluated. The analysis predicts the natural frequencies, various mode shapes and contour shapes under static loadings with reasonable accuracy. The influences of different constructional and design parameters, including the effect of speed of rotation, on the natural frequencies are shown. The variations of cord forces with load have also been presented. The influence of the speed of rolling on the vertical deflection has been determined. The results are in close agreement with the experimental observations.