APPLICATION OF THE VIBRATION ANALYSIS OF LINEAR SYSTEMS WITH TIME-PERIODIC COEFFICIENTS TO THE DYNAMICS OF A ROLLING TYRE

In this article, theories of rolling tyre vibrations are presented. In previous publications, tread pattern was neglected and authors have studied vibrations in smooth tyres. When heterogeneity caused by a tread pattern on the tyre belt is introduced, it is shown that vibrations can be described by linear equations with time periodic coefficients. Firstly, the perturbation method is applied for a nearly smooth tyre, and the “self-excitation” phenomenon, a general feature in time periodic linear systems, is illustrated with the semi-analytic expressions obtained. Then, the generalization to a strong heterogeneity is achieved using the Bloch wave theory. This theoretical background suggests the decomposition of experimental data of noise in time signals for a given phase as compared to the wheel rotation. Finally, an effective method for numerical computations of vibrations is proposed; it uses the Floquet theory, a consequence of the Bloch theory. Finite element formulation and algorithm are derived for the heterogeneous “circular ring model”.