## 4 Analytical Coupling Modal of the Flexible Belt and the Continuous Sidewall  
### Simplification Forms of the Kinematic Equation  
The in-extensibility assumption [23] (the circumferential length of the middle surface of the flexible ring, such as the tire belt and the tire sidewall, is constant during tire deformation) is utilized to simplify the dynamic equation of flexible belt:  
\[{u_{\theta}}^{\prime}+u_{r}=0 \tag{30}\]  
Substituting Equation (30) into Equation (17a,b), then differentiating Equation (17a) with respect to \(\theta\) and adding it to Equation (17b), the kinematic equation of the flexible belt ring with the in-extensibility belt ring assumption is transformed as:  
\[\begin{array}{l}-\frac{EI}{R^{4}}({u_{\theta}}^{(6)}+2{u_{\theta}}^{(4)}+{u_ {\theta}}^{\prime})+\frac{{\sigma_{\theta}^{2}A}}{{\dot{R}}^{2}}({u_{\theta} }^{(4)}+2{u_{\theta}}^{\prime}+  
The harmonic vibration characteristic of the heavy load radial tire is influenced by the material and geometrical parameters of the tire, moreover is independent with the wheel motion. The wheel is assumed to be fixed and non-rotating, so:  
\[x^{*}=0,\;y^{*}=0\;\text{and}\;\gamma=0 \tag{34}\]  
The kinematic equations of the proposed flexible ring on an elastic continuous foundation tire model are transformed as:  
\[\left\{ \begin{array}{ll}-\frac{EI}{R^{4}}(u_{\theta}{}^{(6)}+2u_{\theta}{}^{ (4)}+u_{\theta}{}^{\prime\prime})+\frac{\sigma_{\theta}^{0}A}{R^{2}}(u_{ \theta}{}^{(4)}+2u_{\theta}{}^{\prime\prime}+u_{\theta})-k_{\text{s}\tau 1}(u_{ \theta}{}^{\prime\prime}-u^{\prime\prime}{}_{\delta\theta})\\ +k_{\text{s}\theta 1}(u_{\theta}-u_{\delta\theta})-\frac{p_{0}p_{0}}{R}(u_{ \theta}+u_{\theta}{}^{\prime\prime})+\rho A(\ddot{u}_{\theta}-\ddot{u}{}^{ \prime\prime}{}_{\theta})=q^{\prime}{}_{w}+q_{v}\\ -\frac{E_{s}L_{s}}{R^{4}}(u_{s\theta}{}^{(6)}+2u_{s\theta}{}^{(4)}+u_{s\theta }{}^{\prime\prime})-k_{s\tau 2}u_{s\theta}{}^{\prime\prime}+k_{s\theta}2u_{s \theta}-k_{s\tau 1}(-u_{\theta}{}^{\prime\prime}+u_{s\theta}{}^{\prime\prime})\\ -k_{s\theta 1}(u_{\theta}-u_{s\theta})+m_{s}(\ddot{u}_{s\theta}-\ddot{u}_{s\theta }{}^{\prime\prime})=0\end{array} \right. \tag{35}\]  
Modal Solution of the Planar Coupling Vibration between the Flexible Belt and the Continuous Sidewall  
The rotating feature and the movement of wheel is taken into the completing kinematic equations (Equation (17)) and the simplified forms of the proposed tire model with the In-extensibility assumption are derived as Equation (35). Planar coupling vibration characteristic between the flexible belt ring and the continuous sidewall ring would be further investigated and solved with modal expansion method. The free vibration mode of the flexible belt and the continuous sidewall is assumed in the sinusoidal series illustrated in Equation (36), in which, Equation (36a,b) respectively refer to the circumference vibration modal of the flexible belt ring and the continuous sidewall ring:  
\[\left\{ \begin{array}{ll}u_{\theta}(\theta,t)=\sum\limits_{1}^{\infty}V_{ \theta,n}\cos(n(\theta-\varphi))\sin\omega_{n}t&\text{(a)}\\ u_{s\theta}(\theta,t)=\sum\limits_{1}^{\infty}V_{s\theta,n}\cos(n(\theta- \varphi))\sin\omega_{n}t&\text{(b)}\end{array} \right. \tag{36}\]  
Substituting Equation (36) into Equation (35), the planar dynamics of the heavy load radial tire with a large flat ratio are given as:  
\[\left[ \begin{array}{cc}-\rho A(1+n^{2})\omega^{2}+K_{11}&K_{12}\\ K_{21}&-m_{s}(1+n^{2})\omega^{2}+K_{22}\end{array} \right]\left[ \begin{array}[] {c}V_{\theta,n}\\ V_{s\theta,n}\end{array} \right]=0 \tag{37}\]  
where \(K_{11}=\frac{EI}{R^{4}}(n^{6}-2n^{4}+n^{2})+\frac{p_{0}p_{0}^{6}}{R}(n^{4}-n^{2 })+k_{s\tau 1}n^{2}+k_{s\theta 1}\), \(K_{12}=K_{21}=-k_{s\tau 1}n^{2}-k_{s\theta 1}\), \(K_{22}=\frac{E_{s}L_{s}}{R^{4}}(n^{6}-2n^{4}+n^{2})+k_{s\tau 1}n^{2}+k_{s\theta 1}+k_{s \tau 2}n^{2}+k_{s\theta 2}\).