## 2 System Model of the New Wheel Structure
### Preliminary Design of Parameters  
The design of the structural parameters of the new vibration-absorbing wheel includes \(m_{a}\), \(k_{a}\), \(c_{a}\), \(g\) and \(\tau\). These five parameters have a direct impact on the performance of the vehicle vibration reduction. Firstly, the design process of the absorber is borrowed for the design of \(m_{a}\), \(k_{a}\) and \(c_{a}\). The parameters of \(g\) and \(\tau\) are designed in Section 4.2 using PSO for them. The purpose of the new vibration-absorbing wheel is to allow the mass \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(m_{s}\) (kg) & 345 & \(c_{s}\) (N\(\cdot\)s/m) & 1500 \\ \(m_{t}\) (kg) & 40.5 & \(g\) (N\(\cdot\)s/m) & \(-\)30,000–30,000 \\ \(k_{s}\) (N/m) & 17,000 & \(\tau\) (s) & 0–1 \\ \(k_{t}\) (N/m) & 192,000 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vehicle system model parameters. Figure 1: New vibration-absorbing wheel: (**a**) schematic diagram; (**b**) simplified diagram. block to absorb the vibration energy generated by the road excitation. Therefore, the H\({}_{2}\) parameter optimization method is used for parameter design in this chapter. According to the literature [34], the optimal damping and optimal tuning are obtained as follows:  
\[\left\{ \begin{array}{l}\xi_{a}=\sqrt{\frac{\mu(4+3\mu)}{8(1+\mu)(2+\mu)}}\\ f_{a}=\frac{1}{1+\mu}\sqrt{\frac{2+\mu}{2}}\end{array} \right., \tag{2}\]  
where \(\xi_{a}\) is the optimal damping and \(f_{a}\) is the optimal tuning. \(u\) is the mass ratio. After deduction, the following can be obtained:  
\[\left\{ \begin{array}{l}m_{a}=\mu m_{t}\\ k_{a}=f_{a}^{2}\mu(k_{t}+k_{s})\\ c_{a}=2\xi_{a}\sqrt{k_{a}m_{a}}\end{array} \right.. \tag{3}\]  
In the design of vibration absorbers, the mass ratio is generally less than 0.4. By selecting different mass ratios, multiple sets of H\({}_{2}\) optimization parameters with different mass ratios can be obtained according to Equation (3), as shown in Table 2.  
To investigate the effect of different design parameters on the frequency response of the vehicle body in Table 2, a Laplace transform of Equation (1) gives the following:  
\[\left[ \begin{array}{ccc}A_{11}&A_{12}&0\\ A_{21}&A_{22}&A_{23}\\ 0&A_{31}&A_{32}\end{array} \right]\left[ \begin{array}{c}X_{s}(s)\\ X_{t}(s)\\ X_{a}(s)\end{array} \right]=\left[ \begin{array}{c}0\\ k_{t}X_{r}\\ 0\end{array} \right], \tag{4}\]  
where \(s=\omega\)i, and each coefficient matrix item is  
\[\left\{ \begin{array}{l}A_{11}=m_{s}s^{2}+c_{s}s+k_{s}\\ A_{12}=-c_{s}s-k_{s}\\ A_{21}=-c_{s}s-k_{s}\\ A_{22}=m_{t}s^{2}+(c_{a}+c_{s})s-(k_{s}+k_{t}+k_{a})\\ A_{23}=-c_{s}s-k_{a}+ge^{-\tau s}\\ A_{31}=m_{a}s^{2}+c_{a}s+k_{a}-ge^{-\tau s}\\ A_{32}=-c_{a}s-k_{a}\end{array} \right. \tag{5}\]  
From Equation (4), the frequency-response function of the SMA of the vehicle is  
\[|H_{1}(\omega)|=\left|\frac{\ddot{X}_{s}(s)}{X_{r}(s)}\right|=\left|\frac{ \omega^{2}k_{t}A_{12}A_{32}}{A_{12}A_{21}A_{32}-A_{11}A_{22}A_{32}+A_{11}A_{23 }A_{32}}\right|. \tag{6}\]  
The frequency response function curves of different parameters on sprung mass acceleration can be drawn by Equation (6), as shown in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mass Ratio \(\mu\)** & \(m_{a}\) **(kg)** & \(k_{s}\) **(N/m)** & \(c_{a}\) **(N**-\(\mathbf{s}\)/m)** \\ \hline
0.1 & 4.05 & 18,136.36 & 82.62 \\
0.2 & 8.10 & 31,930.56 & 212.28 \\
0.3 & 12.15 & 42,665.68 & 356.97 \\
0.4 & 16.20 & 51,183.67 & 506.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization parameters of H\({}_{2}\) with different mass ratios. It is concluded that the larger the mass ratio, the better the vibration reduction effect on the main system, but the mass ratio should not be too large. In the design of dynamic vibration absorbers, the mass of the absorber is generally selected below 0.4 [35].