# Effect of the Anti-Yaw Damper on Carbody Vertical Vibration and Ride Comfort of Railway Vehicle
## 3 Equations of Motion
 \tag{12}\]  
If the relations (4)-(6) are replaced in the motion equations of the carbody, they become:  
\[ \begin{array}{l}m_{c}\ddot{z}_{c}+2c_{zc}[2\dot{z}_{c}+2\lambda_{1}\dot{T}_{ c}-(\dot{z}_{10}-\dot{z}_{b2})]+2k_{zc}[2z_{c}+2\lambda_{1}T_{c}-(z_{b1}+z_{b2})]+ \\ +k[[4z_{c}+2(\lambda_{2}+\lambda_{3})]\sin\alpha+2h(\lambda_{4}-\lambda_{5})T_{ c}\cos\alpha-(\Delta l_{11}+\Delta l_{12}+\Delta l_{21}+\Delta l_{22})]\sin\alpha=0 \end{array}  \tag{13}\]  
\[ \begin{array}{l}J_{c}\ddot{\phi}_{c}+2c_{zc}a_{c}[2a_{c}\dot{\phi}_{c}-(\dot {z}_{b1}-\dot{z}_{b2})]+2k_{zc}a_{c}[2a_{c}\dot{\phi}_{c}-(z_{b1}-z_{b2})]+\\ +k[(a_{c}+b)\sin\alpha+h\cos\alpha][2(a_{c}+b)\phi_{c}\sin\alpha+2h\theta_{c} \cos\alpha-(\Delta l_{11}-\Delta l_{22})]+\\ +k[(a_{c}-b)\sin\alpha-h\cos\alpha][2(a_{c}-b)\phi_{c}\sin\alpha-2h\phi_{c} \cos\alpha-(\Delta l_{12}-\Delta l_{21})]=0\end{array}  \tag{14}\]  
\[ \begin{array}{l}m_{mc}\ddot{T}_{c}+c_{mc}\dot{T}_{c}+k_{mc}T_{c}+2c_{zc}\lambda _{1}[2\dot{z}_{c}+2\lambda_{1}\dot{T}_{c}-(\dot{z}_{b1}-\dot{z}_{b2})]+2k_{zc} \lambda_{1}[2z_{c}+2\lambda_{1}T_{c}-(z_{b1}+z_{b2})]+\\ +k(\lambda_{2}\sin\alpha+h\lambda_{4}\cos\alpha)[2(z_{c}+\lambda_{2}T_{c})\sin \alpha+2h\lambda_{4}T_{c}\cos\alpha-(\Delta l_{11}+\Delta l_{22})]+\\ +k(\lambda_{3}\sin\alpha-h\lambda_{5}\cos\alpha)[2(z_{c}+\lambda_{3}T_{c})\sin \alpha-2h\lambda_{5}T_{c}\cos\alpha-(\Delta l_{12}+\Delta l_{21})]=0.\end{array} \] (The motion equations of the bounce and pitch of the bogies are  
\[m_{b}\ddot{z}_{bi}=\sum_{j=2i-1}^{2i}F_{zbj}-2F_{zci}-\sum_{i=1}^{2}\sum_{k=1}^{2 }F_{zik},\text{ for }i=1,\;2, \tag{16}\]  
\[J_{b}\ddot{\theta}_{bi}=a_{b}\sum_{j=2i-1}^{2i}(-1)^{j+1}F_{zbj},\text{ for i}=1,\;2, \tag{17}\]  
where \(m_{b}\) represents the bogie suspended mass, \(J_{b}\) is the inertia moment of the bogie, and \(2a_{b}\) is the bogie wheelbase; \(F_{zbj}\) represent the forces due to the primary suspension corresponding to the wheelset \(j\), namely (\(j+1\)). After processing, the Equations (16) and (17) are as such,  
\[ \begin{array}{l}m_{b}\ddot{z}_{b1}+2c_{zb}[2\dot{z}_{b1}-(\dot{z}_{w1}+\dot{ z}_{w2})]+2k_{zb}[2z_{b1}-(z_{w1}+z_{w2})]+\\ +2c_{zc}(\dot{z}_{b1}-\dot{z}_{c}-a_{c}\ddot{\theta}_{c}-\lambda_{1}\dot{T}_{c })+2k_{zc}(z_{b1}-z_{c}-a_{c}\ddot{\theta}_{c}-\lambda_{1}T_{c})-\\ -k[2z_{c}+2a_{c}\ddot{\theta}_{c}+(\lambda_{2}+\lambda_{3})T_{c}]\sin\alpha+h( \lambda_{4}-\lambda_{5})T_{c}\cos\alpha-(\Delta l_{11}-\Delta l_{12}))\sin \alpha=0\\ \\ m_{b}\ddot{z}_{b2}+2c_{zb}[2\dot{z}_{b2}-(\dot{z}_{w3}+\dot{z}_{w4})]+2k_{zb}[ 2z_{b2}-(z_{w3}+z_{w4})]+\\ +2c_{zc}(\dot{z}_{b2}-\dot{z}_{c}+a_{c}\ddot{\theta}_{c}-\lambda_{1}\dot{T}_{c })+2k_{zc}(z_{b2}-z_{c}+a_{c}\ddot{\theta}_{c}-\lambda_{1}T_{c})-\\ -k[[2z_{c}-2a_{c}\ddot{\theta}_{c}+(\lambda_{2}+\lambda_{3})T_{c}]\sin\alpha+h( \lambda_{4}-\lambda_{5})T_{c}\cos\alpha-(\Delta l_{21}-\Delta l_{22})]\sin \alpha=0\\ \\ J_{b}\ddot{\theta}_{b1}+2a_{b}c_{zb}[2a_{b}\dot{\theta}_{b1}-(\dot{z}_{w1}- \dot{z}_{w2})]+2a_{b}k_{zb}[2a_{b}\dot{\theta}_{b1}-(z_{w1}-z_{w2})]=0\\ \\ J_{b}\ddot{\theta}_{b2}+2a_{b}c_{zb}[2a_{b}\dot{\theta}_{b2}-(\dot{z}_{w3}- \dot{z}_{w4})]+2a_{b}k_{zb}[2a_{b}\dot{\theta}_{b2}-(z_{w3}-z_{w4})]=0.\\ \end{array}  \tag{19}\]  
The equations of the Maxwell systems are as below,  
\[c\dot{\Delta l}_{11,22}+k\Delta l_{11,22}-c\dot{z}_{b1,2}\sin\alpha-k[[z_{c} \pm(a_{c}+b)\theta_{c}+\lambda_{2}T_{c}]\sin\alpha\pm h(\theta_{c}\pm\lambda_ {4}T_{c})\cos\alpha]=0, \tag{22}\]  
\[c\dot{\Delta l}_{12,21}+k\Delta l_{12,21}-c\dot{z}_{b1,2}\sin\alpha-k[[z_{c} \pm(a_{c}-b)\theta_{c}+\lambda_{3}T_{c}]\sin\alpha\mp h(\theta_{c}\pm\lambda_ {5}T_{c})\cos\alpha]=0. \tag{23}\]  
**4. The Vehicle Dynamic Response to the Track Vertical Irregularities**  
The track vertical irregularities are considered to be represented as a stationary stochastic process, which can be described via the power spectral density.