## 2 Theoretical Derivation of the Two Dimension Tire Model
### Governing Equations of Hamilton Variation Principle
#### 2.3.1 Mechanical Energy Analysis of the Tire/Wheel Assembly  
The kinetic energy of the flexible belt ring, the tire wheel and the continuous sidewall ring can be referred to as \(T_{c}\), \(T_{w}\), and \(T_{s}\) in Equation (5):  
\[\left\{ \begin{array}{ll}T_{c}=\int\limits_{0}^{2\pi}\frac{1}{2}\rho AR\left| \overrightarrow{R}\right|^{2}d\theta&\text{(a)}\\ T_{w}=\frac{1}{2}m_{w}(\dot{x}^{2}+\dot{y}^{2})+\frac{1}{2}I_{w}(\Omega+\dot{ \gamma})^{2}&\text{(b)}\\ T_{s}=\int\limits_{0}^{2\pi}\frac{1}{2}m_{s}R_{s}\left|\overrightarrow{R}_{s} \right|^{2}d\theta&\text{(c)}\end{array} \right. \tag{5}\]  
where \(\rho A\) is the line density of the flexible belt ring; \(m_{s}\) is the rad density of the continuous sidewall ring; \(m_{w}\) is the wheel mass; \(I_{w}\) is the wheel moment; \(R\) and \(R_{s}\) are, respectively, the belt radii and sidewall ring radii; and \(\gamma\) is the rotating angle of the tire wheel. The whole kinetic energy of the tire/wheel assembly is calculated as Equation (6):  
\[T=T_{c}+T_{w}+T_{s} \tag{6}\]  
The potential energy of the tire/wheel assembly, including strain energy of flexible belt ring and continuous sidewall ring and the spring energy of the sectional sidewall spring respectively refer to \(U_{c}\), \(U_{s}\), and \(U_{sp}\). The pre-tension strain of flexible belt ring resulting from inflation pressure is taken into consideration and the strain energy of flexible belt ring is shown in Equation (7) [22]:  
\[U_{c}=b\int\limits_{0}^{2\pi}\int\limits_{-\frac{\hbar}{2}}^{\frac{\hbar}{2}} \left(\frac{1}{2}\sigma_{r}\varepsilon_{r}+\frac{1}{2}\sigma_{\theta} \varepsilon_{\theta}+\sigma_{\theta}^{0}\varepsilon_{\theta}\right)Rdzd\theta \tag{7}\]  
where the first part of Equation (7) is the radial strain energy of the flexible belt ring and \(\varepsilon_{r}=\frac{\partial u_{r}}{\partial R}=0\); the second part of Equation (7) is the circumferential strain energy of the flexible belt ring and the nonlinear strain \(\varepsilon_{\theta}\):\(\varepsilon_{\theta}=\frac{(u_{\theta}^{\prime}+u_{r})}{R}+\frac{1}{2} \left(\frac{u_{\theta}^{\prime}+u_{r}}{R}\right)^{2}+\frac{1}{2}\left(\frac{ u_{r}^{\prime}-u_{\theta}}{R}\right)^{2}\)[23]; and the third part of Equation (7) is the initial strain energy of flexible belt ring and \(\sigma_{\theta}^{0}\) is the initial strain resulting from the inflation pressure and the centrifugal force caused by rotation and calculated as Equation (8) in Figure 5:  
\[\sigma_{\theta}^{0}A=T_{0}=(\int\limits_{0}^{\pi}\left(\rho A\Omega^{2}R+p_{0 }b\right)Rd\theta)\sin\theta \tag{8}\]  
Figure 5: Pre-tension strain of flexible belt ring due to the inflation pressure and rotation. Thus, \(\sigma_{\theta}^{0}A=\rho A\Omega^{2}R^{2}+p_{0}bR\). The strain energy of continuous sidewall ring \(U_{s}\) is shown in Equation (9):  
\[U_{s}=\int\limits_{0}^{2\pi}\int\limits_{-\frac{h}{2}}^{\frac{h}{2}}(\frac{1}{2 }\sigma_{sr}\epsilon_{sr}+\frac{1}{2}\sigma_{sq}\epsilon_{sq}+\sigma_{sq}^{0} \epsilon_{sq})R_{s}dzd\theta \tag{9}\]  
where the first part of Equation (9) is the radial strain of continuous sidewall ring; the second part of Equation (9) is the circumferential strain energy of continuous sidewall ring and the nonlinear strain \(\epsilon_{sq}\epsilon_{sq}\epsilon_{sq}=\frac{(u_{\theta}{}^{\prime}+u_{gr})}{R _{s}}+\frac{1}{2}\left(\frac{u_{\theta}{}^{\prime}+u_{gr}}{R_{s}}\right)^{2}+ \frac{1}{2}\left(\frac{u_{\theta}{}^{\prime}-u_{gr}}{R_{s}}\right)^{2}\); the third part of Equation (9) is the initial strain energy of continuous sidewall ring; and \(\sigma_{sq}^{0}\) is the initial strain resulting from the centrifugal force caused by rotation, \(\sigma_{sq}^{0}A_{s}=m_{s}\Omega^{2}R_{s}{}^{2}\). The spring energy of the radial and tangential sectional sidewall stiffness:  
\[U_{sp}=\int\limits_{0}^{2\pi} \begin{array}{c}(\frac{1}{2}k_{sr1}(u_{r}-u_{ sr})^{2}+\frac{1}{2}k_{sq1}(u_{\theta}-u_{sq})^{2}+\frac{1}{2}k_{sr2}(u_{sr}-x^{*} \cos\theta-z^{*}\sin\theta)^{2}\\ +\frac{1}{2}k_{sq2}(u_{sq}+x^{*}\sin\theta-z^{*}\cos\theta-r\gamma)^{2})\end{array}