## 2 Construction and Verification of MBD Model
### Verification of MBD Model Based on Euler-Lagrange Method
#### 2.2.2 Mathematical Kinematic Equations of Symmetrical Suspension Based on Euler-Lagrange Method  
According to the Euler-Lagrange method, the kinematic problems of a motion system can be simplified significantly [17]. Based on the Euler-Lagrange method, \(Z\) was defined as the vertical displacement of the jointed point of the two pendulum arms, while \(\theta_{1}\) and \(\theta_{2}\) were defined as the angular increments of the pendulum arms 1 and pendulum arms 2, respectively. The energy of the symmetrical suspension consists of the kinetic energy \(T\), potential energy \(V\), and dissipated energy \(D\), caused by the dampers. The corresponding energy could be expressed as Equations (10)-(12), respectively, as follows:  
\[T=\frac{1}{2}M\dot{Z}^{2}+\frac{1}{2}J_{1}\dot{\theta_{1}}^{2}+\frac{1}{2}J_{ 2}\;\dot{\theta_{2}}^{2} \tag{10}\]  
\[V=\frac{1}{2}k_{e1}X_{1}^{2}+\frac{1}{2}k_{e2}X_{2}^{2}+\frac{1}{2}k_{3}X_{3} ^{2} \tag{11}\]  
\[D=\frac{1}{2}c_{e1}X_{1}^{2}+\frac{1}{2}c_{e2}\dot{X_{2}}^{2}+\frac{1}{2}c_{3 }\dot{X_{3}}^{2} \tag{12}\]  
where \(M\) is the 1/4 mass of the tracked vehicle (there are four symmetrical suspensions to support the vehicle), \(J_{1}\) and \(J_{2}\) are the rotational inertia of the mentioned two arms, respectively, \(X_{1}\), \(X_{2}\), and \(X_{3}\) are the displacements of the three dampers, respectively, which could be obtained by Equations (13)-(15), respectively. \[X_{1}=Z-\frac{l_{1}}{2}\cos\theta+\frac{l_{1}}{2}\cos(\theta+\theta_{1})-Q_{1 }(t) \tag{13}\]  
\[X_{2}=Z-\frac{l_{2}}{2}\cos\theta+\frac{l_{2}}{2}\cos(\theta+\theta_{2})-Q_{2 }(t) \tag{14}\]  
\[X_{3}=\left[\frac{l_{2}}{2}\sin\theta-\frac{l_{2}}{2}\sin(\theta+\theta_{2}) \right]+\left[\frac{l_{1}}{2}\sin\theta-\frac{l_{1}}{2}\sin(\theta+\theta_{1 })\right] \tag{15}\]  
where \(l_{1}\) and \(l_{2}\) are the lengths of the two swing arms, respectively, \(\theta\) is the initial angle between the two swing arms, \(Q_{1}(t)\) and \(Q_{2}(t)\) are the road excitation of the front and rear wheels of symmetrical suspension, respectively. According to the Euler-Lagrange equation in a non-conservative state, the partial differential of Equations (10)-(12) can be obtained as follows:  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial\dot{Z}}\right) -\frac{\partial T}{\partial Z}+\frac{\partial V}{\partial Z}+\frac{\partial D }{\partial\dot{Z}}=Q_{\mathrm{1i}} \tag{16}\]  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial\dot{\theta_{1}}} \right)-\frac{\partial T}{\partial\theta_{1}}+\frac{\partial V}{\partial \theta_{1}}+\frac{\partial D}{\partial\dot{\theta_{1}}}=Q_{\mathrm{2i}} \tag{17}\]  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial\dot{\theta_{2}} }\right)-\frac{\partial T}{\partial\dot{\theta_{2}}}+\frac{\partial V}{ \partial\theta_{2}}+\frac{\partial D}{\partial\dot{\theta_{2}}}=Q_{\mathrm{3i}} \tag{18}\]  
where \(Q_{\mathrm{1i}}\), \(Q_{\mathrm{2i}}\) and \(Q_{\mathrm{3i}}\) are the generalized forces acting on the components, respectively. Any motion system will be in a relatively stable state finally, which will result in the generalized forces being zero. Thus, the kinematic equations of the symmetrical suspension system would be obtained as described in Equations (19)-(21). \[ \begin{array}{ll}m\ddot{z}+(k_{e1}+k_{e2})z_{1}-\frac{k_{e1}l_{1}+k_{e2}l_{2}}{ 2}cos\theta\\ \quad+\frac{k_{e1}l_{1}}{2}cos(\theta+\theta_{1})-k_{e1}Q_{1}(t)-k_{e2}Q_{2}(t )+\dot{z}(l_{1}+l_{2})\\ \quad-\frac{c_{e2}l_{1}}{2}sin(\theta+\theta_{1})\dot{\theta}_{1}-\frac{c_{e2}l _{2}}{2}sin(\theta+\theta_{2})\dot{\theta}_{2}-c_{e1}\dot{Q_{1}}(t)\\ \quad-c_{e2}Q_{2}(t)=0\\ \end{array}  \tag{19}\]  
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