# A Multibody Model Calibration Method for Vehicle Misuse Testing
## 4 Nonlinear Parameter Calibration Using the Modified Quarter-Vehicle Model Analysis
### Method Validation via Numerical Simulation  
The complete progression of the present misuse scenario has been divided into three stages according to different tire-ground contact conditions, where stages 1 through 3 are described by ODEs (32), (13), and (25), respectively. The systematic ODEs are summarized here in a combined formulation as follows:  
\[ \begin{bmatrix}m_{\text{u}}&0\\ 0&m_{\text{w}}\end{bmatrix}\begin{bmatrix}\tilde{z}_{\text{u}}(t)\\ \tilde{z}_{\text{w}}(t)\end{bmatrix} + \begin{bmatrix}k_{\text{e}}+k_{\text{s}}& -k_{\text{s}}\\ -k_{\text{s}}&k_{\text{s}}\end{bmatrix}\begin{bmatrix}z_{\text{u}}(t)\\ z_{\text{w}}(t)\end{bmatrix} =\left\{ \begin{array}{ll}\left[\begin{array}{ll} 0\\ k_{\text{t}}v_{\text{e}}t-k_{\text{t}}z_{\text{w}}(t)\end{array}\right],& \text{for $t_{1}\leq t<t_{2}$,}\\ \left[\begin{array}{ll}0\\ 0\\ 0\end{array}\right],&\text{for $t_{2}\leq t<t_{3}$,}\\ \left[\begin{array}{ll}0\\ 0\\ -k_{\text{t}}z_{\text{w}}(t)\end{array}\right],&\text{for $t_{3}\leq t$.}\end{array} \right. \tag{40}\]  
wherein \(t\) represents the time variable; \(z_{\text{u}}(t)\) and \(z_{\text{w}}(t)\) represent the vertical displacement of the sprung mass and the wheel center, respectively; \(m_{\text{u}}\) represents the sprung mass; \(m_{\text{w}}\) represents the wheel's center mass; \(k_{\text{s}}\) represents the suspension spring's stiffness; \(k_{\text{t}}\) represents the tire's stiffness; \(k_{\text{e}}\) represents the equivalent stiffness between the sprung mass and the imagined ceiling; and \(v_{\text{e}}\) represents the equivalent vertical velocity of the ground during tire-obstacle contact. Divide the two sides by \(m_{\text{w}}\) and take \(c_{\text{u}}=m_{\text{u}}/m_{\text{w}}\geq 0\), \(c_{\text{e}}=(k_{\text{e}}+k_{\text{s}})/k_{\text{s}}\geq 1\), and \(c_{\text{t}}=(k_{\text{t}}+k_{\text{s}})/k_{\text{s}}\geq 1\), as in Sections 4.2-4.4; i.e., with the ODEs (33), (14), and (26), the systematic ODEs to be solved are combined as follows:  
\[ \begin{bmatrix}c_{\text{u}}&0\\ 0&1\end{bmatrix}\begin{bmatrix}\tilde{z}_{\text{u}}(t)\\ \tilde{z}_{\text{w}}(t)\end{bmatrix} +\frac{k_{\text{s}}}{m_{\text{w}}} \begin{bmatrix} c_{\text{e}}&-1\\ -1&1\end{bmatrix}\begin{bmatrix}z_{\text{u}}(t)\\ z_{\text{w}}(t)\end{bmatrix} =\left\{ \begin{array}{ll}\frac{k_{\text{t}}}{m_ {\text{w}}}\begin{bmatrix}0\\ v_{\text{e}}t-z_{\text{w}}(t)\end{bmatrix},&\text{for $t_{1}\leq t<t_{2}$,}\\ \left[\begin{array}{ll}0\\ 0\end{array}\right],&\text{for $t_{2}\leq t<t_{3}$,}\\ -\frac{k_{\text{t}}}{m_{\text{w}}}\begin{bmatrix}0\\ z_{\text{w}}(t)\end{bmatrix},&\text{for $t_{3}\leq t$.}\end{array} \right.\end{bmatrix} \tag{41}\]  
In Equations (40) and (41), the nonlinearity resulting from varying tire-ground contact conditions is clearly evident and ultimately leads to nonlinearity in the vehicle's response. The key parameters of the modified quarter-vehicle model, namely \(c_{\text{e}}\), \(c_{\text{t}}\), and \(v_{\text{e}}\), have been provided by Expressions (23), (31), and (38), respectively. Then, the systematic responses of the quarter-vehicle model can be computed using a time integration function in software, such as MATLAB(2022)'s ode15s, a numerical integration solver for stiff problems with 1-5-order polynomials, or MATLAB(2022)'s ode45, a widely used efficient solver for non-stiff problems using the Runge-Kutta method. The systematic parameters for numerical simulation are listed in Table 4, where a Rayleigh damping of 0.002, proportional to the stiffness, is added to the simulation model to obtain the damped results. The numerical results were computed using ode15s in MATLAB and subsequently compared with the actual vehicle experiment data (test data), as shown in Figure 9. For the shock absorber's compression \(d(t)\), the results are well coincident in stage 2 and the initial part of stage 3.This is ensured by the calibration processes of the parameters \(c_{\mathrm{e}}c_{\mathrm{t}}\), respectively.