# Effect of the Anti-Yaw Damper on Carbody Vertical Vibration and Ride Comfort of Railway Vehicle
## 3 Equations of Motion  
Figure 2 features the forces due to the secondary suspension and the anti-yaw damper acting upon the vehicle carbody. In the carbody supporting points on the secondary suspension, \(A_{i}\) and \(A_{i}^{\prime}\), operate the forces \(F_{zci}\) (for \(i=1,2\)),  
\[F_{zci}=-c_{zc}\left(\frac{\partial w_{c}(a_{i},t)}{\partial t}-z_{bi}\right) -k_{zc}[w_{c}(a_{i},t)-z_{bi}], \tag{4}\]  
where distances \(a_{i}=L_{c}/2\pm a_{c}\) fix the position of the supporting point of the carbody on the secondary suspension, with \(2a_{c}\)--carbody wheelbase. The anti-yaw dampers are fixed to the vehicle carbody in the points \(A_{ik}\), where \(k=1,2\). In these points, the forces \(F_{ik}\) are acting, with the following components on vertical and longitudinal directions:  
\[F_{zik}=-k\left[w(a_{ik},t)\sin\alpha+(-1)^{k-1}h\frac{\partial w(a_{ik},t)}{ \partial x}\cos\alpha-\Delta l_{ik}\right]\sin\alpha, \tag{5}\]  
\[F_{xik}=-k\left[w(a_{ik},t)\sin\alpha+(-1)^{k-1}h\frac{\partial w(a_{ik},t)}{ \partial x}\cos\alpha-\Delta l_{ik}\right]\cos\alpha, \tag{6}\]  
Figure 2: Forces acting upon the carbody. where \(\Delta l_{ik}\) are the internal displacements of the elements in the Maxwell systems; distances \(a_{ik}=L_{c}/2+(-1)^{k-1}a_{c}+(-1)^{k-1}b\) fix the position of the clamping points to the carbody for the anti-yaw dampers; \(b\) represents the distance between the clamping point to the carbody of the anti-yaw damper and the carbody supporting point onto the secondary suspension, while \(h\) is the distance between the carbody medium fiber and the clamping plan of the anti-yaw dampers onto the carbody. The motion equation of the carbody vehicle has the general form,  
\[ \begin{array}{ll}EI\frac{\partial^{2}w_{c}(x,t)}{\partial x^{4}}+\mu I\frac{ \partial^{5}w_{c}(x,t)}{\partial x^{4}\partial t}+\rho_{c}\frac{\partial^{2}w_ {c}(x,t)}{\partial t^{2}}=&\sum\limits_{i=1}^{2}\left(2F_{xi}\delta(x-a_{i}) +\sum\limits_{k=1}^{2}F_{zik}\delta(x-a_{ik})\right)-\\ &-\sum\limits_{i=1}^{2}\sum\limits_{k=1}^{2}(-1)^{-1}F_{xik}\delta^{\prime}( x-a_{ik})\end{array}  \tag{7}\]  
where \(EI\) is bending modulus (\(E\)--longitudinal modulus of elasticity, and \(I\)--inertia moment of the beam's transversal section), \(\mu\)--structural damping coefficient, and \(\delta(. )\) is Dirac's delta function. By applying the modal analysis method and considering the orthogonality property of the eigenfunction of the carbody vertical bending, Equation (7) turns into three two-order differential equations with ordinary derivatives, to describe the motions of bounce, pitch, and bending of the carbody,  
\[m_{c}\ddot{z}_{c}=\sum\limits_{i=1}^{2}\left(2F_{zci}+\sum\limits_{k=1}^{2}F_{ zik}\right), \tag{8}\]  
\[J_{c}\ddot{\phi}_{c}=\sum\limits_{i=1}^{2}\left[2F_{zci}\left(a_{i}-\frac{L_{c }}{2}\right)+\sum\limits_{k=1}^{2}F_{zik}\left(a_{ik}-\frac{L_{c}}{2}\right) \right]+h\sum\limits_{i=1}^{2}\sum\limits_{k=1}^{2}(-1)^{k}F_{xik}, \tag{9}\]  
\[m_{mc}\ddot{T}_{c}+c_{mc}\ddot{T}_{c}+k_{mc}T_{c}=\sum\limits_{i=1}^{2}\left(2 F_{zci}X_{c}(a_{i})+\sum\limits_{k=1}^{2}F_{zik}X_{c}(a_{ik})\right)+h\sum \limits_{i=1}^{2}\sum\limits_{k=1}^{2}(-1)^{k}F_{xik}\frac{\mathrm{d}X_{c}(a_ {ik})}{\mathrm{d}x}, \tag{10}\]  
where \(J_{c}\) is the carbody inertia moment, and \(k_{mc}\), \(c_{mc}\) and \(m_{mc}\), are the stiffness, damping and the carbody modal mass,  
\[k_{mc}=EI\int\limits_{0}^{L}\left(\frac{\mathrm{d}^{2}X_{c}}{\mathrm{d}x^{2}} \right)^{2}\mathrm{d}x,\ c_{mc}=\mu I\int\limits_{0}^{L}\left(\frac{\mathrm{d }^{2}X_{c}}{\mathrm{d}x^{2}}\right)^{2}\mathrm{d}x,\ m_{mc}=\rho_{c}\int \limits_{0}^{L}X_{c}^{2}\mathrm{d}x. \tag{11}\]  
Based on the property of symmetry of the eigenfunction \(X_{c}(x)\), the following notations will be used:  
\[ \begin{array}{l}X_{c}(a_{1})=X_{c}(a_{2})=\lambda_{1},\\ X_{c}(a_{11})=X_{c}(a_{22})=\lambda_{2},\ X_{c}(a_{12})=X_{c}(a_{21})=\lambda _{3},\\ X_{c}^{\prime}(a_{11})=-X_{c}^{\prime}(a_{22})=\lambda_{4},\ X_{c}^{\prime}(a_ {12})=-X_{c}^{\prime}(a_{21})=\lambda_{5}.\end{array}