A new approximate model of tyre accounting for both deformed state and dry friction forces in the contact spot on the background of the coupled model

A new approximate model of the tire rolling accounting for coupled longitudinal and lateral sliding as well as the spinning and the deformed state resulting in elastic forces is proposed. The main goal of this investigation consists in the construction of simple models with a few of degrees of freedom allowing one to implement them analytically in the engineering practice, primarily for the estimation of the rolling stability and for the prognosis of the instable rolling so-called ”shimmy phenomenon” and to numerical simulation of the transient dynamics of rolling wheels with sliding effects. Such a model could become useful at the earlier stages of the engineering design instead of complex numerical models that are usually resource consuming. The known models of the shimmy phenomenon are usually based on the nonholonomic condition of the steady rolling and consider as the main cause of instability only the tire deformation while the sliding and spinning are assumed to vanish, i. e. the dry friction effects are neglected. Another type of models consists in the ”rigid wheel” assumption, in other words only the dry friction effects are accounted on the background of the coupled dry friction theory whereas the deformed state effects are neglected. Such a theory is based on the complete accounting of the combined kinematics due to simultaneous sliding and spin and shows its efficiency for slightly deformed wheels and the unsteady rolling regimes. Our goal consists in the formulation of the combined model that takes into account as well as the deformed state as the dry friction effects. This model is based on the solution of some model problems for the tire using solid finite element simulation or various shell theories, the computing of a set of specific generalized rigidity factors for the tire model, and on the accounting for the shape of the contact spot and the contact pressure distribution after the numerical simulation of the tire-road contact interaction in quasi-statics. As a result, we obtain a model with only a few degrees of freedom but more general that the Keldysh’s or Klimov’s ones