## 4 Analytical Coupling Modal of the Flexible Belt and the Continuous Sidewall
### Simplification Forms of the Kinematic Equation
The kinematic equation (Equation (37)) is reorganized as:  
\[\rho Am_{s}(1+n^{2})^{2}\omega^{4}-\left[\rho A(1+n^{2})K_{22}+m_{s}(1+n^{2}) K_{11}\right]\omega^{2}+K_{11}K_{22}-K_{12}K_{21}=0 \tag{38}\]  
The roots of Equation (38) are obtained as Equation (39). Meanwhile, the deformation relationship between the flexible belt ring and the continuous sidewall ring refers to Equation (40):  
\[\omega_{n}{}^{2}=\frac{\left[\rho AK_{22}+m_{s}K_{11}\right]\pm\sqrt{\left[ \rho AK_{22}+m_{s}K_{11}\right]^{2}-4\rho Am_{s}(K_{11}K_{22}-K_{12}K_{21})}}{ 2\rho Am_{s}(1+n^{2})} \tag{39}\]  
\[\frac{V_{\theta,n}}{V_{s\theta,n}}=\frac{K_{12}}{\rho A(1+n^{2})\omega^{2}-K_{1 1}}=\frac{m_{s}(1+n^{2})\omega^{2}-K_{22}}{K_{21}} \tag{40}\]Analytical modal resonance frequency of the flexible belt ring and the continuous sidewall ring is derived as:  
\[f_{n1,n2}=\frac{\omega_{n1,n2}}{2\pi} \tag{41}\]  
The deformation relationship between the flexible belt ring and the continuous sidewall ring is concluded as:  
\[f=f_{n1}=\frac{\sqrt{\frac{\left[\rho AK_{22}+m_{s}K_{11}\right]-\sqrt{\left[ \rho AK_{22}+m_{s}K_{11}\right]^{2}-4\rho Am_{s}(K_{11}K_{22}-K_{12}K_{21})}}{2 \rho Am_{s}(1+n^{2})}}}{2\pi},\]  
\[\frac{V_{\theta,n}}{V_{s\theta,n}}=\frac{2m_{s}K_{12}}{\left[\rho AK_{22}-m_{s} K_{11}\right]+\sqrt{\left[\rho AK_{22}+m_{s}K_{11}\right]^{2}-4\rho Am_{s}(K_{11} K_{22}-K_{12}K_{21})}} \tag{42}\]  
\[f=f_{n2}=\frac{\sqrt{\frac{\left[\rho AK_{22}+m_{s}K_{11}\right]+\sqrt{\left[ \rho AK_{22}+m_{s}K_{11}\right]^{2}-4\rho Am_{s}(K_{11}K_{22}-K_{12}K_{21})}}{ 2\rho Am_{s}(1+n^{2})}}}{2\pi},\]  
\[\frac{V_{\theta,n}}{V_{s\theta,n}}=\frac{2m_{s}K_{12}}{\left[\rho AK_{22}-m_{s} K_{11}\right]-\sqrt{\left[\rho AK_{22}+m_{s}K_{11}\right]^{2}-4\rho Am_{s}(K_{11} K_{22}-K_{12}K_{21})}} \tag{43}\]  
The coupling relationship of the flexible belt and the continuous sidewall is discussed by comparing the two roots of Equation (29). The Equation (32) is the identical equation and substituted into Equations (42) and (43) respectively. \[\sqrt{\left[\rho AK_{22}+m_{s}K_{11}\right]^{2}-4\rho Am_{s}(K_{11}K_{22}-K_{ 12}K_{21})}>\left|\rho AK_{22}-m_{s}K_{11}\right| \tag{44}\]  
Substituting Equation (44) into Equations (42) and (43), the conclusion is made that:  
1. \(f=f_{n1}\), substituting Equation (44) into Equation (42) and \(\frac{V_{\theta,n}}{V_{\theta,n}}>0\);
2. \(f=f_{n2}\), substituting Equation (44) into Equation (43) and \(\frac{V_{\theta,n}}{V_{s\theta,n}}<0\);
3. The positive ratio of \(V_{\theta,n}\) and \(V_{s\theta,n}\) with \(f=f_{n1}\) implies the same vibration direction of flexible belt ring and continuous sidewall ring, which is consistent with Figure 10c,d and the conclusion given by the flexible beam on an elastic continuous sidewall beam tire model [(23)]; and
4. The negative ratio of \(V_{\theta,n}\) and \(V_{s\theta,n}\) with \(f=f_{n2}\) implies the opposite vibration direction of flexible belt ring and the continuous sidewall ring, which is consistent with Figure 10c,d, and the conclusion given by the flexible beam on an elastic continuous sidewall beam tire model [(23)].