## 2 Theoretical Derivation of the Two Dimension Tire Model
### Governing Equations of Hamilton Variation Principle
#### 2.3.3. Kinematic Dynamics of Flexible Ring on an Elastic Continuous Sidewall Ring Tire Model  
Equation (5) (kinetic energy of the tire/wheel assembly), Equation (11) (potential energy of the tire/wheel assembly) and Equation (12) are substituted into Equation (4) and the independent variables, including \(u_{r}\), \(u_{\theta}\), \(u_{sr}\), \(u_{s\theta}\), \(x^{*}\), \(y^{*}\), and \(\gamma\), are manipulated. With the means of the variation principle, the planar coupling kinematic equations between the flexible belt and the continuous sidewall are obtained as:  
\[\left\{\begin{array}{l}\frac{EI}{R^{4}}({u_{r}}^{(4)}-{u_{\theta}}^{(3)})+ \frac{EA}{R^{2}}({u_{r}}+{u_{\theta}}^{\prime})+\frac{{\sigma_{\theta}^{0}A}}{R ^{2}}({u_{\theta}}^{\prime}-{u_{r}}^{*})+k_{sr1}({u_{r}}-{u_{sr}})-\frac{{P_{ \theta}^{0}}b}{R}({u_{r}}+{u_{\theta}}^{\prime})\\ +\rho A({\tilde{u}_{r}}-2\Omega{\dot{u}_{\theta}}-\Omega^{2}u_{r})=q_{r}\\ \frac{EI}{R^{4}}({u_{r}}^{(3)}-{u_{\theta}}^{\prime})-\frac{EA}{R^{2}}({u_{r} }^{\prime}+{u_{\theta}}^{\prime\prime})+\frac{{\sigma_{\theta}^{0}A}}{R^{2}}( {u_{\theta}}-{u_{r}}^{\prime})+k_{s\theta 1}({u_{\theta}}-{u_{s\theta}})+\frac{{P_{ \theta}^{0}}b}{R}({u_{r}}^{\prime}-{u_{\theta}})\\ +\rho A({\tilde{u}_{\theta}}+2\Omega{\dot{u}_{r}}-\Omega^{2}u_{\theta})=q_{v} \\ \frac{E_{\frac{I}{R^{4}}}}{R_{\theta}^{4}}({u_{sr}}^{(4)}-{u_{\theta}}^{(3)})+ \frac{E_{\frac{A}{R^{2}}}A_{\theta}}{R_{\theta}^{2}}({u_{sr}}+{u_{s\theta}}^{ \prime})+\frac{{\sigma_{\theta}^{0}A}}{R_{\theta}^{2}}({u_{s\theta}}^{\prime} -{u_{sr}}^{\prime})+{m_{s}}({\tilde{u}_{sr}}-2\Omega{\dot{u}_{s\theta}}-\Omega ^{2}u_{sr})\\ +k_{s\pi 2}({u_{sr}}-{y^{*}}\sin\theta-{x^{*}}\cos\theta)-k_{s\pi 1}({u_{r}}-{u_{sr}})=0 \\ \frac{E_{\frac{I}{R^{4}}}}{R_{\theta}^{4}}({u_{sr}}^{(3)}-{u_{\theta}}^{(n)}- \frac{E_{\pi}A_{\theta}}{R_{\theta}^{2}}({u_{sr}}^{\prime}+{u_{s\theta}}^{ \prime\prime})+\frac{{\sigma_{\theta}^{0}A}}{R_{\theta}^{2}}({u_{s\theta}}-{u _{sr}}^{\prime})+{m_{s}}({\tilde{u}_{s\theta}}+2\Omega{\dot{u}_{s\theta}}- \Omega^{2}u_{s\theta})\\ +k_{s\theta 2}({u_{s\theta}}-{y^{*}}\cos\theta+{x^{*}}\sin\theta-\gamma r)-k_{s \theta 1}({u_{\theta}}-{u_{s\theta}})=0\\ m_{w}({\tilde{x}}^{*}-2\Omega{\dot{y}}^{*}-\Omega^{2}{x^{*}})+\pi R_{s}({k_{ s\pi 2}}+{k_{s\theta 2}}){x^{*}}-{R_{s}}\int\limits_{0}^{2\pi}({k_{s\pi 2}}{u_{sr}}\cos \theta-{k_{s\theta 2}}{u_{s\theta}}\sin\theta)d\theta=F_{wx}&(\text{e})\\ m_{w}({\tilde{y}}^{*}+2\Omega{\dot{x}}^{*}-\Omega^{2}{y^{*}})+\pi R_{s}({k_{ s\pi 2}}+{k_{s\theta 2}}){y^{*}}-{R_{s}}\int\limits_{0}^{2\pi}({k_{s\pi 2}}{u_{sr}}\sin \theta+{k_{s\theta 2}}{u_{s\theta}}\cos\theta)d\theta=F_{wy}&(\text{f})\\ l_{w}{\tilde{\gamma}}+2\pi{k_{s\theta 1}}{R_{s}}^{3}\gamma-{R_{s}}^{2}\int\limits_{0}^{2\pi}{k_{ s\theta 2}}{u_{s\theta}}{d\theta}=T_{w}&(\text{g})\end{array}\right. \tag{17}\]  
where Equation (17a) gives the radial vibration equation of the flexible belt ring; Equation (17b) shows the tangential vibration equation of the flexible belt ring; Equation (17c) gives the radial vibration equation of the continuous sidewall ring; Equation (17d) presents the tangential vibration equation of the continuous sidewall ring; Equation (17e,f,g) respectively illustrate the horizontal, vertical, and rotating movement of the wheel.