## 2 System Model of the New Wheel Structure
### Dimensionless System Model  
To facilitate the stability analysis of the system, the system parameters are dimensionless. The influence of damping \(c_{a}=0\sim 220\,\mathrm{N}\cdot\mathrm{s}/\mathrm{m}\) of the new vibration-absorbing wheel on system stability is analyzed, and the mathematical model of system control is  
\[\left\{\begin{array}{l}\ddot{x}_{s}{}^{*}+\zeta_{1}(\dot{x}_{s}{}^{*}-\dot{x} _{t}{}^{*})+(x_{s}{}^{*}-x_{t}{}^{*})=0\\ \ddot{x}_{t}{}^{*}+\zeta_{1}\gamma(\dot{x}_{t}{}^{*}-\dot{x}_{s}{}^{*})+\zeta _{2}\gamma(\dot{x}_{t}{}^{*}-\dot{x}_{s}{}^{*})+\gamma(x_{t}{}^{*}-x_{s}{}^{*} )+k_{2}\gamma(x_{t}{}^{*}-x_{s}{}^{*})+k_{1}\gamma(x_{t}{}^{*}-x_{r}{}^{*})+g^{ *}\gamma x_{a}{}^{*}(t{}^{*}-\tau{}^{*})=0\\ \ddot{x}_{a}{}^{*}+\beta\xi_{2}(\dot{x}_{a}{}^{*}-\dot{x}_{t}{}^{*})+k_{2}\beta (x_{a}{}^{*}-x_{t}{}^{*})-g^{*}\beta x_{a}{}^{*}(t{}^{*}-\tau{}^{*})=0\\ \end{array}\right., \tag{7}\]  
where \(\zeta_{1}=\frac{c_{a}}{\sqrt{k_{c}m_{a}}}\), \(\zeta_{2}=\frac{c_{a}}{\sqrt{k_{c}m_{s}}}\), \(k_{1}=\frac{k_{1}}{k_{s}}\), \(k_{2}=\frac{k_{4}}{k_{s}}\), \(\gamma=\frac{m_{s}}{m_{t}}\), \(\beta=\frac{m_{t}}{m_{a}}\), \(\beta=\frac{m_{t}}{m_{a}}\), \(\beta^{*}=\frac{\beta}{k_{s}}\); \(x_{s}{}^{*}\), \(x_{t}{}^{*}\), \(x_{t}{}^{*}\) and \(x_{r}{}^{*}\) are dimensionless body displacement, tire displacement, mass displacement, and external excitation displacement; \(t^{*}\) is the dimensionless time; and \(\tau^{*}\) is the dimensionless time-delay. The body displacement \(x_{s}{}^{*}\), body velocity \(\dot{x}_{s}{}^{*}\), tire displacement \(x_{t}{}^{*}\), tire velocity \(\dot{x}_{t}{}^{*}\), mass block displacement \(x_{a}{}^{*}\), and mass block velocity \(\dot{x}_{a}{}^{*}\) are selected as state variables, and the state space equation of the system can be obtained as  
\[\left\{\begin{array}{l}\dot{\mathbf{X}}=\mathbf{AX}+\mathbf{BU}+\mathbf{Ev}\\ \mathbf{Y}=\mathbf{CX}+\mathbf{Dw}\end{array}\right., \tag{8}\]  
Figure 2: Frequency response function curves of SMA.  
where \(\mathbf{X}=[{x_{s}}^{*},{\dot{x}_{s}}^{*},{x_{t}}^{*},{\dot{x}_{t}}^{*},{x_{a}}^{*},{ \dot{x}_{a}}^{*}]^{\rm T}\) is the state vector; \(\mathbf{Y}=[{\ddot{x}_{s}}^{*},{x_{s}}^{*}-{x_{t}}^{*},{x_{t}}^{*}-{x_{r}}^{*}]^{\rm T}\) is the output vector; \(\mathbf{w}\)= \([{x_{r}}^{*},{\dot{x}_{r}}^{*}]^{\rm T}\) is the excitation vector of the road surface; \(\mathbf{U}\)= \([{x_{s}}^{*}(t^{*}-\tau^{*}),{\dot{x}_{s}}^{*}(t^{*}-\tau^{*}),{\dot{x}_{t}}^{* }(t^{*}-\tau^{*}),{\dot{x}_{t}}^{*}(t^{*}-\tau^{*}),{\dot{x}_{a}}^{*}(t^{*}-\tau ^{*}),{\dot{x}_{a}}^{*}(t^{*}-\tau^{*})]^{\rm T}\) is the state time-delay vector. The matrices in the above equation are  
\[\mathbf{A}=\left[\begin{array}{cccccc}0&1&0&0&0&0\\ -1&-\zeta_{1}&1&\zeta_{1}&0&0\\ 0&0&0&1&0&0\\ \gamma&\gamma\zeta_{1}&-\gamma(1+k_{1}+k_{2})&-\gamma(\zeta_{1}+\zeta_{2})& \gamma k_{2}&\gamma\zeta_{2}\\ 0&0&0&0&0&1\\ 0&0&\beta k_{2}&\beta\xi_{2}&-\beta k_{2}&-\beta\xi_{2}\end{array}\right],\mathbf{B }=\left[\begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-g^{*}\gamma&0\\ 0&0&0&0&0&0\\ 0&0&0&0&g^{*}\beta&0\end{array}\right],\] \[\mathbf{E}=\left[\begin{array}{cccccc}0&0\\ 0&0\\ 0&0\\ k_{1}\gamma&0\\ 0&0\\ 0&0\end{array}\right],\mathbf{C}=\left[\begin{array}{cccccc}-1&-\zeta_{1}&1& \zeta_{1}&0&0\\ 1&0&-1&0&0&0\\ 0&0&1&0&0&0\end{array}\right],\mathbf{D}=\left[\begin{array}{cccccc}0&0\\ 0&0\\ -1&0\end{array}\right],\;\text{respectively}.\]  
The parameters of the vehicle system model in Table 1 can be transformed into dimensionless parameters using Equation (7), as shown in Table 3.