# Influence of Fastener Failure on Dynamic Performance of Subway Vehicle
## 2 Model of Vehicle-Structure System
### System Model
#### 2.1.2 Model of Vehicle  
The vehicle model is simulated using a multi-rigid, body-spring-damping system, with each vehicle consisting of one vehicle car body, two bogies, and four wheel sets; all of which are rigid bodies. Spring stiffness viscous damping is used to simulate the suspension systems between the vehicle car body and the bogie (secondary suspension), and the suspension system between the bogie and the wheel set (primary suspension), as shown in Figure 2. The dynamic equation of the vehicle can be written as the following equation using the principle of elastic potential energy invariance [25]:  
\[\mathbf{M}_{v}\ddot{\mathbf{X}}_{v}+\mathbf{C}_{v}\dot{\mathbf{X}}_{v}+ \mathbf{K}_{v}\mathbf{X}_{v}=\mathbf{F}_{v} \tag{2}\]  
where \(\mathbf{M}_{v}\), \(\mathbf{C}_{v}\), and \(\mathbf{K}_{v}\) denote the mass matrix, damping matrix and stiffness matrix of vehicle, respectively. The mass matrix of vehicle can be written as:  
\[\mathbf{M}_{v}=\text{diag}[\mathbf{M}_{v1}\ \mathbf{M}_{v2}\dots\mathbf{M}_{vn}] \tag{3}\]  
where \(n\) denotes the number of vehicle carriage; \(\mathbf{M}_{vi}\) denotes \(i\)th mass matrix of vehicle, and it can be represented as:  
\[\mathbf{M}_{vi}=\text{diag}[\mathbf{M}_{c}\ \mathbf{M}_{b1}\ \mathbf{M}_{b2}\ \mathbf{M}_{w1}\ \mathbf{M}_{w2}\ \mathbf{M}_{w3}\ \mathbf{M}_{w4}] \tag{4}\]  
Figure 1: Rail–slab system considering fastener failure. With  
\[\mathbf{M}_{c} =\text{diag}\big{[}m_{c}\ m_{c}\ J_{cx}\ J_{cy}\ J_{cz}\big{]} \tag{5}\] \[\mathbf{M}_{bi} =\text{diag}\big{[}m_{b}\ m_{b}\ J_{bx}\ J_{by}\ J_{bz}\big{]}\] (6) \[\mathbf{M}_{wi} =\text{diag}\big{[}m_{w}\ m_{w}\ J_{wx}\ J_{uw}\ J_{wz}\big{]} \tag{7}\]  
where \(m_{c}\), \(m_{t}\) and \(m_{w}\) denote mass of car body, bogie, and wheel set, respectively; \(J\) denotes moment of inertia, and the first subscript \(c\), \(b\) and \(w\) denote car body, bogie, and wheel set, respectively; the second subscript \(x\), \(y\), and \(z\) denotes \(X\)-axis, \(Y\)-axis, and \(Z\)-axis, respectively. The stiffness matrix of vehicle can be written as:  
\[\mathbf{K}_{v}=\text{diag}[\mathbf{K}_{v1}\ \mathbf{K}_{v2}\dots\mathbf{K}_{vn}] \tag{8}\]  
where \(\mathbf{K}_{vi}\) denotes stiffness matrix of \(i\)th carriage, and it can be represented as:  
\[\mathbf{K}_{vi}= \begin{bmatrix}\mathbf{K}_{c}&&&&\text{symm.}\\ \mathbf{K}_{cb1}&\mathbf{K}_{b1t1}&&&&\\ \mathbf{K}_{cb2}&0&\mathbf{K}_{b2t2}&&&&\\ 0&\mathbf{K}_{b1t1}&0&\mathbf{K}_{w1t1}&&\\ 0&\mathbf{K}_{b1t2}&0&0&\mathbf{K}_{w2t2}&&\\ 0&0&\mathbf{K}_{b2w3}&0&0&\mathbf{K}_{w3t3}&&\\ 0&0&\mathbf{K}_{b2w4}&0&0&0&\mathbf{K}_{w4t2w4}\end{bmatrix}  \tag{9}\]  
With  
\[\mathbf{K}_{c}= \begin{bmatrix}4k_{y2}&&&&\text{symm.}\\ 0&4k_{z2}&&&&&\\ 4h_{1}k_{y2}&0&4h_{1}^{2}k_{y2}+4b_{2}^{2}k_{z2}&&&&&\\ 0&0&0&4d_{2}^{2}k_{z2}+4h_{1}^{2}k_{x2}&&&&&\\ 0&0&0&0&4d_{2}^{2}k_{y2}+4b_{2}^{2}k_{x2}&&\\ \end{bmatrix}  \tag{10}\]  
\[\mathbf{K}_{cb1}= \begin{bmatrix}-2k_{y2}&0&-2h_{1}k_{y2}&0&-2d_{2}k_{y2}\\ 0&-2k_{z2}&0&2d_{2}k_{z2}&0\\ 2h_{2}k_{y2}&0&d&0&2h_{2}d_{2}k_{y2}\\ 0&0&0&2h_{1}h_{2}k_{x2}&0\\ 0&0&0&0&-2b_{2}^{2}k_{x2}\end{bmatrix}  \tag{11}\]  
\[\mathbf{K}_{cb2}= \begin{bmatrix}-2k_{y2}&0&-2h_{1}k_{y2}&0&2d_{2}k_{y2}\\ 0&-2k_{z2}&0&-2d_{2}k_{z2}&0\\ 2h_{2}k_{y2}&0&d&0&-2h_{2}d_{2}k_{y2}\\ 0&0&0&2h_{1}h_{2}k_{x2}&0\\ 0&0&0&0&-2b_{2}^{2}k_{x2}\end{bmatrix}  \tag{12}\]  
\[\mathbf{K}_{b1b1}=\mathbf{K}_{b2b2}= \begin{bmatrix}2k_{y2}+4k_{y1}&&\text{ symm.}\\ 0&2k_{z2}+4k_{z1}&&\\ 4h_{3}k_{y1}-4k_{y1}&0&a&\\ 0&0&0&b&\\ 0&0&0&0&c\end{bmatrix}  \tag{13}\]  
\[\mathbf{a}=2b_{2}^{2}k_{z2}+2h_{2}^{2}k_{y2}+4b_{1}^{2}k_{z1}+4h_{3}^{2}k_{y1} \tag{14}\]  
\[\mathbf{b}=2h_{2}^{2}k_{x2}+4d_{1}^{2}k_{z1}+4h_{3}^{2}k_{x1} \tag{15}\]  
\[\mathbf{c}=4b_{1}^{2}k_{x1}+4d_{1}^{2}k_{y1}+2b_{2}^{2}k_{x2} \tag{16}\]\[\mathbf{K}_{b1w1}=\mathbf{K}_{b2w3}= \begin{bmatrix}-2k_{y1}&0&-2h_{3}k_{y1}&0&2d_{1 }k_{y1}\\ 0&-2k_{z1}&0&-2d_{1}k_{z1}&0\\ 0&0&-2b_{1}^{2}k_{z1}&0&0\\ 0&0&0&0&-2b_{1}^{2}k_{x1}\end{bmatrix}  \tag{17}\]  
\[\mathbf{K}_{w1w1}=\mathbf{K}_{w2w2}=\mathbf{K}_{w3w3}=\mathbf{K}_{w4w4}=