# A Hydraulic Semi-Active Suspension Based on Road Statistical Properties and Its Road Identification
## 3 Road Identification Method  
### Vehicle Vertical Acceleration Response  
To create the road identification method for vehicles equipped with the proposed semi-active suspension system, the functional relationship among the vertical vibration acceleration of the body \(\ddot{z}_{2}\), the running speed \(v\), the PSD \(G_{q}\), the damping ratio \(\xi\), and other related parameters of vehicles needs to be deduced.  
In Figure 1, the motion equation of the vehicle with the proposed semi-active suspension system can be expressed as:  
\[\left\{\begin{array}{l}m_{2}\ddot{z}_{2}+C(\ddot{z}_{2}-\dot{z}_{1})+K_{ \mathrm{s}}(z_{2}-z_{1})=0\\ m_{1}\ddot{z}_{1}+C(\ddot{z}_{1}-\dot{z}_{2})+K_{\mathrm{s}}(z_{1}-z_{2})+K_{ \mathrm{t}}(z_{1}-q)=0\end{array}\right. \tag{1}\]  
The Laplace transformation of Equation (1) can be expressed as:  
\[\left\{\begin{array}{l}m_{2}Z_{2}s^{2}+C(Z_{2}-Z_{1})s+K(Z_{2}-Z_{1})=0\\ m_{1}Z_{1}s^{2}+C(Z_{1}-Z_{2})s+K(Z_{1}-Z_{2})+K_{\mathrm{t}}(Z_{1}-Q)=0\end{array}\right. \tag{2}\]  
The following auxiliary variables are introduced:  
\[r_{\mathrm{k}}=\frac{K_{\mathrm{t}}}{K},r_{\mathrm{m}}=\frac{m_{2}}{m_{1}}, \ \omega_{0}=\sqrt{\frac{K}{m_{2}}},\ \xi=\frac{C}{2\sqrt{m_{2}K}},\ \lambda=\frac{\omega}{\omega_{0}}\]  
where \(r_{\mathrm{k}}\) is the stiffness ratio, \(r_{\mathrm{m}}\) is the mass ratio, \(\omega_{0}\) is the natural frequency of the car body, \(\lambda\) is the frequency ratio, and \(\xi\) is the damping ratio of the semi-active suspension system.  
Substituting \(s=\mathrm{j}\omega\) into Equation (2), the frequency response function \(H(\mathrm{j}\omega)_{z_{2}\sim q}\) between the vertical displacement \(z_{2}\) of the car body and the road input \(q\) can be obtained:  
\[H(\mathrm{j}\omega)_{z_{2}\sim q}=\frac{r_{\mathrm{k}}+2r_{\mathrm{k}}\xi \lambda\mathrm{j}}{(1-\lambda^{2})\Big{(}1+r_{\mathrm{k}}-\frac{\lambda^{2}}{r _{\mathrm{m}}}\Big{)}-1+\Big{[}r_{\mathrm{k}}-\Big{(}\frac{1+r_{\mathrm{m}}}{ r_{\mathrm{m}}}\Big{)}\lambda^{2}\Big{]}2^{\xi}\lambda\mathrm{j}} \tag{3}\]  
The frequency response function \(H(\mathrm{j}\omega)_{\ddot{z}_{2}\sim q}\) of the vertical acceleration \(\ddot{z}_{2}\) of the body to the road excitation velocity \(\dot{q}\) can be expressed as:  
\[H(\mathrm{j}\omega)_{\ddot{z}_{2}\sim\dot{q}}=\mathrm{j}\omega H(\mathrm{j} \omega)_{z_{2}\sim q} \tag{4}\]  
When the vehicle runs on roads with different grades, the input of the road surface spectrum can be regarded as white noise [34], namely:  
\[G_{\dot{q}}(f)=4\pi^{2}G_{q}(n_{0})n_{0}^{2}v \tag{5}\]  
where \(n_{0}\) is the reference space frequency, \(n_{0}=0.1\) m\({}^{-1}\), and \(f=\omega/2\pi\).  
According to the vehicle random vibration theory, the mean square value of the car body vertical acceleration can be expressed as  
\[\sigma_{\tilde{z}_{2}}^{2}=\int_{0}^{+\infty}\left|H(\mathrm{j}\omega)_{\tilde{z }_{2}\sim\tilde{q}}\right|^{2}G_{\tilde{q}}(f)\mathrm{d}f \tag{6}\]  
Substituting Equations (3)-(5) into Equation (6), and using the Residue theorem, Equation (6) can be further expressed as  
\[\sigma_{\tilde{z}_{2}}^{2}=\pi^{2}G_{q}(n_{0})n_{0}^{2}v\frac{\omega_{0}^{3}(1+ r_{\mathrm{m}}+4r_{\mathrm{m}}r_{\mathrm{k}}\tilde{v}^{2})}{2\tilde{z}r_{ \mathrm{m}}} \tag{7}\]  
Equation (7) indicates that the RMS (root mean square) value of the car body vertical acceleration \(\sigma_{\tilde{z}_{2}}\) is related to \(v\), \(G_{q}(n_{0})\), \(\omega_{0}\), \(r_{\mathrm{k}}\), \(r_{\mathrm{m}}\), and \(\tilde{z}\). In other words, the functional relationship among the current road condition, the vehicle response, and the vehicle parameters can be established by Equation (7). Thus, Equation (7) is the basis for establishing the road identification method.