Self-excited Lateral Vibrations of Rolling Tires

In this study, a low degree-of-freedom mechanical model of a rolling tire is constructed, in which the lateral deformation of the contact patch and tire carcass is considered. The so-called delayed contact patch model is implemented and combined with a simple tire carcass model. The interaction between the contact patch and the carcass together with the lateral mode of the attached suspension system is modeled by means of minimum number of relevant parameters in a simplified way in order to construct analytical results. Critical parameter ranges of selfexcited vibrations are determined against the longitudinal speed of the tire. The intricate shapes of the corresponding tire deformations are presented by means of numerical simulations. NOMENCLATURE a Half length of the contact patch. m Mass of the wheel. k Specific stiffness of the tire. ks Lateral stiffness of the suspension. q Lateral deformation of the tire in the contact patch. R Radius of the undeformed tire. t Time. v Longitudinal speed of the center point of the wheel. w Lateral deformation outside the contact patch. YC Lateral displacement of the center point of the wheel. ωc Natural angular frequency of a tire brush element. ρA Distributed mass of the tire. τ1 Time delay in the contact patch. τ2 Time delay outside the contact patch. INTRODUCTION The shimmy of motorcycles [1, 2], steered wheels of cars and the airplane gears [3–5] is a well known phenomenon in vehicle dynamics. Although this phenomenon dates back to the appearance of the first vehicles [6], it requires special attention of engineers in the design stage even nowadays. The safety hazard of shimmy induced many publications about the lateral vibration of towed wheels. Moreover, the investigation of shimmy leads to improved and detailed models of tires and tire/road interactions. It was recognized very early that the tire/ground contact patch is responsible for a kind of memory effect [7], but the available mathematical theories and methods did not allow engineers at that time to analyze delay differential equations, while an engineering approximation of the tire lateral deformation in the contact patch helped to explore some properties of the shimmying tire [8, 9]. Later, the accurate modeling of the contact patch lateral deformation and the analysis of the corresponding delay differential equations provided new explanations for some quasi-periodic vibrations [10] and also uncovered new parameter domains where so-called micro-shimmy exists [11]. The application of this time delayed contact patch description in the single track car model [12] and in the single track car-trailer model [13] also identified parameter ranges where small amplitude lateral vibrations may appear. These vibrations can be neglected in view of the lateral stability of the vehicles but they are relevant components in the noise generation. 1 Copyright c © 2015 by ASME Noise generation of tires has become an important aspect of tire design in recent decades. There are several studies (see, for example, [14–17]) in this field, which include detailed tire/ground contact patch and tire carcass models but do not consider the dynamics of the attached vehicle system. This is due to the fact that the major part of the noise of tires is originated in the so-called horn effect, which does not have any relation to the tire attached suspension system. Here, we present a simple mechanical model of a brush tire model that is supported by a laterally elastic suspension. This mechanical mode is a special case of classical shimmy models in case of infinite caster length, that is, the wheel cannot rotate about the vertical axis. Namely, a lateral mode of the attached suspension-wheel system is considered, in which the delayed contact patch model is combined with a tire carcass model. Critical parameter ranges of self-excited vibrations are determined versus the longitudinal speed of the tire with special attention to the frequencies of these vibrations. Some numerical simulation are presented to illustrate the traveling deformation waves along the tire circumference both in the contact patch and outside the contact patch. MECHANICAL MODEL The mechanical model in question is shown in Fig. 1. The wheel of elastic tire is rolling in the ground-fixed (X ,Y,Z) coordinate system; its longitudinal speed v is kept constant. The rotational axis of the wheel of tire is supported in lateral direction by the spring of stiffness ks. The lateral position of the wheel center point is described by YC(t) as a function of the time t. The mass of the wheel is denoted by m. The (x,y,z) coordinate system is fixed to the wheel center point, its axes are parallel to the axes of the (X ,Y,Z) coordinate system. The radius of the undeformed tire is R. The tire is in contact with the ground along the contact patch of length 2a. The deformation in the longitudinal dimension of the tire is neglected in this study, while the lateral tire deformation is described by q(x, t) in the contact patch and by w(χ , t) outside the contact patch. As it is shown in Fig. 1, the angle χ ∈ [0,β ] (where β = 2(π −α) and α = arcsin(a/R)) sweeps along the circumference of the tire starting from the trailing edge R end ending at the leading edge L of the contact patch. In this study, we use the so-called brush tire model, which considers separated tire particles along the circumference of the tire. These tire particles are viewed like the thread elements of the tire, namely, their lateral deformations are independent. The distributed mass of the tire particles is characterized by ρA [kg/m], and the specific lateral stiffness k [N/m2] relates to the distributed elastic support of the tire particles. X L