## 2 Dynamic Model of DCT Vehicle with Mount
### Vehicle Dynamic Model with Mount  
#### 2.5.1 Powertrain Mounting System Model  
In this paper, the powertrain mounting system is a three-point mount, and the engine is placed longitudinally. A coupling dynamic model of vehicle considering mount was established [27; 28; 29]. Considering the translational and torsional motion of the powertrain in the three axis directions, the powertrain mounting system is simplified into a 6-DOF dynamic model, as shown in Figure 3.  
In Figure 3, \(O\)-\(XYZ\) is the moving coordinate system; \(O\) is the center of mass of the powertrain; \(X\) is the forward direction of the vehicle; \(Y\) is determined by the right-hand rule and \(Z\) is the vertical direction of the vehicle. The coordinate system \(o\)-\(xyz\) is a static coordinate system and the origin coincides with the center of mass of the powertrain. 1, 2 and 3 are the three mounting points on the powertrain. q (\(X\)\(Y\)\(Z\)\(\theta_{X}\)\(\theta_{Y}\)\(\theta_{Z}\))\({}^{\rm T}\) represents the generalized coordinates of powertrain mounting system. Using the Lagrange method, the motion equation of the powertrain mounting system is established as follows:  
\[\frac{d}{dt}\left(\frac{\partial E_{T}}{\partial\dot{q}}\right)-\frac{\partial E _{T}}{\partial\dot{q}}+\frac{\partial E_{V}}{\partial q}+\frac{\partial E_{D }}{\partial\dot{q}}=F, \tag{10}\]  
where \(E_{T}\) is the kinetic energy of the system; \(E_{D}\) is the dissipated energy of the system; \(E_{V}\) is the potential energy of the system and \(F\) is the generalized force of the system. The kinetic energy of the powertrain includes translational kinetic energy and rotational kinetic energy as follows:  
\[E_{T}=\frac{1}{2}\left[\dot{X}\dot{Y}\dot{Z}\dot{\theta}_{X}\)\(\dot{\theta}_{Y}\)\(\dot{\theta}_{Z}\right] \begin{bmatrix}m_{p}&0&0&0&0&0\\ 0&m_{p}&0&0&0&0\\ 0&0&m_{p}&0&0&0\\ 0&0&0&I_{m,xx}&-I_{m,xy}&-I_{m,xz}\\ 0&0&0&-I_{m,yx}&I_{m,yy}&-I_{m,yz}\\ 0&0&0&-I_{m,zx}&-I_{m,zy}&I_{m,zz}\end{bmatrix}\begin{bmatrix}\dot{X}\\ \dot{Y}\\ \dot{Z}\\ \dot{\theta}_{X}\\ \dot{\theta}_{Y}\\ \dot{\theta}_{Z}\end{bmatrix} =\frac{1}{2}\dot{q}^{\top}M\dot{q}, \tag{11}\]  
and the mass matrix M is:  
\[M= \begin{bmatrix}m_{p}&0&0&0&0&0\\ 0&m_{p}&0&0&0&0\\ 0&0&m_{p}&0&0&0\\ 0&0&0&I_{m,xx}&-I_{m,xy}&-I_{m,xz}\\ 0&0&0&-I_{m,yx}&I_{m,yy}&-I_{m,yz}\\ 0&0&0&-I_{m,zx}&-I_{m,zy}&I_{m,zz}\end{bmatrix} , \tag{12}\]  
Figure 3: Dynamic model of a powertrain mount. The system potential energy of the powertrain is:  
\[E_{V}=\frac{1}{2}\sum_{i=1}^{n}\left[\Delta u_{i}\,\Delta v_{i}\,\Delta w_{i} \right] \begin{bmatrix}k_{ui}&0&0\\ 0&k_{vi}&0\\ 0&0&k_{wi}\end{bmatrix} \left[ \begin{array}{c}\Delta u_{i}\\ \Delta v_{i}\\ \Delta w_{i}\end{array} \right]=\frac{1}{2}q^{\mathrm{T}}\mathrm{Kq},K_{i}= \begin{bmatrix}k_{ui}&0&0\\ 0&k_{vi}&0\\ 0&0&k_{wi}\end{bmatrix} . \tag{13}\]  
The system dissipated energy of the powertrain is:  
\[E_{D}=\frac{1}{2}\sum_{i=1}^{n}\left[\Delta\dot{u}_{i}\,\Delta\dot{v}_{i}\, \Delta\dot{w}_{i}\right] \begin{bmatrix}c_{ui}&0&0\\ 0&c_{vi}&0\\ 0&0&c_{wi}\end{bmatrix}\begin{bmatrix}\Delta\dot{u}_{i}\\ \Delta\dot{v}_{i}\\ \Delta\dot{w}_{i}\end{bmatrix} =\frac{1}{2}\tilde{q}^{\mathrm{T}}C\dot{q},C_{i }= \begin{bmatrix}c_{ui}&0&0\\ 0&c_{vi}&0\\ 0&0&c_{wi}\end{bmatrix} . \tag{14}\]  
The excitation force received by the powertrain mounting system is:  
\[\left[F\right]=\left[F_{x}\,F_{y}\,F_{z}\,M_{x}\,M_{y}\,M_{z}\right], \tag{15}\]  
where  
\[ \begin{array}{l}F_{x}=F_{\mathrm{mou}},\\ F_{y}=4m_{2}r\lambda{w_{e}}^{2}\sin\theta\cos 2w_{e}t,\\ F_{z}=4m_{2}r\lambda{w_{e}}^{2}\cos\theta\cos 2w_{e}t,\\ M_{x}=T_{e}(1+1.3\sin 2w_{e}t),\\ M_{y}=F_{z}A-T_{e},\\ M_{z}=F_{y}A,\end{array}