## 2 Construction and Verification of MBD Model
### Verification of MBD Model Based on Euler-Lagrange Method
#### 2.2.1 Simplification of Symmetrical Suspension  
According to the study of [PERSON] et al. [16], the symmetrical suspension, shown in Figure 7a, could be simplified as shown in Figure 7b.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Road Classification**} & \multicolumn{3}{c}{**Geometric Mean**} \\ \cline{2-3}  & \multicolumn{1}{c}{_Gd(\(n_{0}\))/m\({}^{3}\)_} & \(\sigma_{q}\)/m_ \\  & **(\(n_{0}\) = 0.1 m\({}^{-1}\))** & **(0.011 m\({}^{-1}\) \(<n\) \(<\) 2.83 m\({}^{-1}\))** \\ \hline A & \(16\times 10^{-6}\) & 3.81 \(\times 10^{-3}\) \\ B & \(64\times 10^{-6}\) & 7.61 \(\times 10^{-3}\) \\ C & \(256\times 10^{-6}\) & 15.23 \(\times 10^{-3}\) \\ D & \(1024\times 10^{-6}\) & 30.45 \(\times 10^{-3}\) \\ E & \(4096\times 10^{-6}\) & 60.90 \(\times 10^{-3}\) \\ F & 16,384 \(\times 10^{-6}\) & 121.80 \(\times 10^{-3}\) \\ G & 65,536 \(\times 10^{-6}\) & 243.61 \(\times 10^{-3}\) \\ H & 262,144 \(\times 10^{-6}\) & 487.22 \(\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Power spectral density of standard road surfaces.  
Figure 6: An MBD model of the tracked vehicle and the road surface.  
The actual stiffness and damping of the shock absorbers are denoted as \(k_{r}\) and \(c_{r}\), respectively, while the corresponding equivalent stiffness and equivalent damping are denoted as \(k_{e}\) and \(c_{e}\), respectively. The distance that the wheel moves along the Y-axis is denoted as \(\Delta y\), and the center of the wheel \(D\) could rotate and rise around the E-axis. The compress distance of the smaller shock absorbers in Figure 7a is \(\Delta l\). According to the principle of equivalence, it can be derived as follows:  
\[k_{e}\cdot\Delta y=k_{r}\cdot\Delta l\cos(\beta+\Delta\beta) \tag{4}\]  
where \(\Delta\beta\) is the variation in the angle \(\beta\). Compared with the \(L_{0}+L_{HD}\), \(h_{0}\) is too small to be ignored. Thus, points F, H, and D can be treated as in one line. Thereby, according to Figure 7a, the values of \(\Delta y\) and \(\Delta L\) can be obtained from the trigonometric functions as follows:  
\[\Delta y=\frac{L_{ED}\cdot\cos\alpha}{\tan\beta}-\frac{L_{ED}\cos(\alpha- \Delta\beta)}{\tan(\beta+\Delta\beta)} \tag{5}\]  
\[\Delta l=\frac{L_{ED}\cdot\cos\alpha}{\sin\beta}-\frac{L_{ED}\cos(\alpha- \Delta\beta)}{\sin(\beta+\Delta\beta)} \tag{6}\]  
When the tracked vehicle moves, compared with \(L_{1}\), \(\Delta l\) is too small to be ignored. According to Equations (4)-(6), the relationship of \(k_{e}\) and \(k_{r}\) could be expressed as Equation (7) as follows:  
\[k_{e}=k_{r}.\frac{\cos\alpha\sin(\beta+\Delta\beta)-\sin\beta\cos(\alpha- \Delta\beta)}{\cos\beta[\cos\alpha\tan(\beta+\Delta\beta)-tan\beta\cos(\beta+ \Delta\beta)]} \tag{7}\]  
Figure 7: Simplification of symmetrical suspension.  
Similarly, the relationship of \(c_{e}\) and \(c_{r}\) could be expressed as Equation (8).  
\[c_{e}\cdot\dot{\Delta y}=c_{r}\dot{\Delta l}\cos(\beta+\Delta\beta) \tag{8}\]  
\[c_{e}=c_{r}\frac{\sin(\Delta\beta-\alpha)\sin(\Delta\beta+\beta)+\cos(\Delta \beta-\alpha)\cos^{2}(\Delta\beta+\beta)}{\sin(\Delta\beta-\alpha)\cos(\Delta \beta+\beta)\sin(\Delta\beta+\beta)+\cos(\Delta\beta-\alpha)} \tag{9}\]