# A Multibody Model Calibration Method for Vehicle Misuse Testing
## 4 Nonlinear Parameter Calibration Using the Modified Quarter-Vehicle Model Analysis
### Equivalent Stiffness Calibration by Tire Detaching Stage
\tag{16}\]  
The corresponding eigen circular frequencies are  
\[\omega_{\pm}^{2}=\frac{\left(c_{\rm u}+c_{\rm e}\right)\mp\sqrt{\left(c_{\rm u }-c_{\rm e}\right)^{2}+4c_{\rm u}}}{2c_{\rm u}}\omega_{0}^{2}, \tag{17}\]  
wherein \(\omega_{+}^{2}=0\) only if \(c_{\rm e}=1\). For the traditional quarter-vehicle model without, i.e., the synclastic eigenvector and frequency are, respectively,  
\[z_{+,f}= \begin{bmatrix}1\\ 1\end{bmatrix} \text{with}\;\omega_{+,f}^{2}=0, \tag{18}\]  
and in the opposite direction, they are, respectively,  
\[z_{-,f}= \begin{bmatrix}-\frac{1}{c_{\rm u}}\\ 1\end{bmatrix} \text{with}\;\omega_{-,f}^{2}=\left(1+\frac{1}{c_{\rm u}}\right) \frac{k_{\rm s}}{m_{\rm w}}. \tag{19}\]  
Mode results (18) and (19) show that in the traditional quarter-vehicle model, synclastic vibration of \(z_{\rm u}\) and \(z_{\rm w}\) only happens in the rigid-body movement (18). Therefore, the traditional quarter-vehicle model is not applicable to the misuse tests. However, with \(c_{\rm e}>1\) in the modified quarter-vehicle model, Formula (16) gives the synclastic eigenmode with \(z_{\rm u}\) and \(z_{\rm w}\) as  
\[z_{+,\textbackslash{\textbackslash{\textbackslash{\textbackslash{\textbackslash{ \textbackslash{\textbackslash{\textbackslash{\textbackslash{\textbackslash{\textbackslash \textbackslash{\textbackslash{\textbackslash{\textbackslash{\textwith the corresponding non-zero circular frequency  
\[\omega_{+,\textc2}^{2}=\left(\frac{\left(c_{\text{u}}+c_{\text{e}}\right)-\sqrt{ \left(c_{\text{u}}-c_{\text{e}}\right)^{2}+4c_{\text{u}}}}{2c_{\text{u}}} \right)\frac{k_{\text{s}}}{m_{\text{w}}}. \tag{21}\]  
Let \(c_{\textc2}^{\textc2}\triangleq z_{\text{u}}/z_{\text{w}}\) be the ratio of the synclastic displacements of the sprung mass and the wheel center in stage 2; Equation (20) suggests  
\[\frac{\left(c_{\text{u}}-c_{\text{e}}\right)+\sqrt{\left(c_{\text{u}}-c_{\text {e}}\right)^{2}+4c_{\text{u}}}}{2c_{\text{u}}}=c_{\textc2}, \tag{22}\]  
so solving this and abandoning the unreasonable solution (\(z_{\text{u}}=0\)), the ratio \(c_{\text{e}}\) is expressed by the ratio \(c_{\text{u}}\) as  
\[c_{\text{e}}=\left(1-c_{\textc2}\right)c_{\text{u}}+\frac{1}{c_{\textc2}}, \tag{23}\]  
and substituting Equation (22) into Equation (21), the ratio between the suspension spring's stiffness and the wheel's center mass is written as  
\[\frac{k_{\text{s}}}{m_{\text{w}}}=\frac{\omega_{+,\textc2}^{2}}{1-c_{\textc2 }}. \tag{24}\]  
In the present vehicle, \(c_{\textc2}^{\textc2}=0.27\) is read from the result (12), and it produces \(c_{\text{e}}=0.73c_{\text{u}}+3.70\). It is important to note that only the synclastic mode is activated during stage 2. Given that the obstacle impact force originates from one side of the mass-spring system, specifically from the tire side, the opposing mode remains inactive. This is evidenced by the alignment of the displacement curve during the half-period of 0.0675 s in stage 2, as illustrated in Figure 7. One might also wonder whether the rapid vibration component of the acceleration response indicates the opposing eigenmode with a higher frequency. However, the answer is no, and the detailed proof is provided in Appendix A.