Standing Waves in Tires

Abstract

The phenomenon of standing waves in tires occurs in the sidewall and tire tread area when the vehicle speed exceeds a critical speed. Two approaches can be adopted in the study of standing waves: the analytical approach and FEA. The analytical approach, such as the adoption of a membrane model or elastic ring model, is further classified into the wave propagation approach and resonance approach. This chapter discusses the analytical approach and FEA for standing waves.

Notes

Note 15.1 Eq. (15.10)

Equation (15.8) uses the fixed coordinate system, while Eq. (15.10) uses the rotating coordinate system. The relations

$$\begin{aligned} \frac{\partial }{\partial t}_{\text{rotation}} & = \frac{\partial }{\partial t}_{\text{fixed}} + \frac{\partial }{\partial \xi }\frac{\partial \xi }{\partial t}_{\text{fixed}} = \frac{\partial }{\partial t}_{\text{fixed}} + V\frac{\partial }{\partial \xi } \\ \frac{{\partial^{2} }}{{\partial t^{2} }}_{\text{rotation}} & = \frac{{\partial^{2} }}{{\partial t^{2} }}_{\text{fixed}} + 2V\frac{{\partial^{2} }}{\partial \xi \partial t}_{\text{fixed}} + V^{2} \frac{{\partial^{2} }}{{\partial \xi^{2} }} \\ \end{aligned}$$

(15.73)

are thus satisfied for the time derivatives in each coordinate system. The second terms in the above equations are related to the Coriolis force, and the contributions of the second terms are smaller than those of the first and third terms.

Note 15.2 Eqs. (15.29) and (15.30)

Figure 15.19 gives

$$\begin{array}{*{20}l} {d = R_{\text{D}} \left( {1 - \cos \psi_{0} } \right) + a\left( {1 - \cos \theta_{0} } \right) \approx \frac{1}{2}\left( {R_{\text{D}} \psi_{0}^{2} + a\theta_{0}^{2} } \right)} \hfill \\ {R_{\text{D}} \psi_{0} = a\theta_{0} } \hfill \\ {d = \frac{1}{2}\left\{ {R_{\text{D}} \left( {\frac{{a\theta_{0} }}{{R_{\text{D}} }}} \right)^{2} + a\theta_{0}^{2} } \right\} = \frac{{\theta_{0}^{2} }}{2}a\left( {1 + \frac{a}{{R_{\text{D}} }}} \right).} \hfill \\ \end{array}$$

(15.74)

When the shape of the drum is expressed in the coordinate system of the tire, the distance between the tire surface and the drum surface, w₀, which corresponds to the displacement from the tire surface to the drum surface, is given by

$$w_{0} = \frac{{a^{2} \theta^{2} }}{2}\left( {\frac{1}{a} + \frac{1}{{R_{\text{D}} }}} \right) - d.$$

(15.75)

Fig. 15.19

Tire deformation on a drum

Using the relations ξ₀ = aθ₀ and ξ = aθ, w₀ is rewritten as

$$w_{0} = \frac{{\xi^{2} - \xi_{0}^{2} }}{2}\left( {\frac{1}{a} + \frac{1}{{R_{\text{D}} }}} \right).$$

(15.76)