## Appendix A  
The forces and moments, respectively, due to the secondary suspension of bogie \(i\) (with \(i\) = 1, 2) are as below  
\[F_{zci}=-2\varepsilon_{zc}\bigg{(}\frac{\partial w_{c}(l_{i},t)}{\partial t}- \dot{z}_{bij}\bigg{)}-2k_{zc}[w_{c}(l_{i},t)-z_{bi}] \tag{1}\]  
\[F_{xci}=2\varepsilon_{xc}\bigg{(}h_{c}\frac{\partial^{2}w_{c}(l_{i},t)}{ \partial x\partial t}+h_{b}\dot{\theta}_{bi}\bigg{)}+2k_{xc}\bigg{(}h_{c} \frac{\partial w_{c}(l_{i},t)}{\partial x}+h_{b}\theta_{bi}\bigg{)} \tag{2}\]  
\[M_{cl}=-2\varepsilon_{bc}\bigg{(}\frac{\partial^{2}w_{c}(l_{i},t)}{\partial x \partial t}-\dot{\theta}_{bi}\bigg{)}-2k_{bc}\bigg{[}\frac{\partial w_{c}(l_{ i},t)}{\partial x}-\theta_{bi}\bigg{]} \tag{3}\]The carbody motion equations are in the form of  
\[m_{c}\ddot{z}_{c}+2c_{zc}[2\dot{z}_{c}+2\epsilon\dot{T}_{c}-(\dot{z}_{b1}+\dot{z} _{b2})]+2k_{zc}[2z_{c}+2\epsilon T_{c}-(z_{b1}+z_{b2})]=0 \tag{100}\]  
\[\begin{array}{l}J_{c}\ddot{\Theta}_{c}+2c_{zc}a_{c}[2a_{c}\dot{\Theta}_{c}-( \dot{z}_{b1}-\dot{z}_{b2})]+2k_{zc}a_{c}[2a_{c}\dot{\Theta}_{c}-(z_{b1}-z_{b2}) ]+\\ +2c_{zc}h_{c}[2h_{c}\dot{\Theta}_{c}+h_{b}(\dot{\Theta}_{b1}+\dot{\Theta}_{b2}) ]+2k_{xc}h_{c}[2h_{c}\dot{\Theta}_{c}+h_{b}(\Theta_{b1}+\dot{\Theta}_{b2})]+\\ +2c_{bc}[2\dot{\Theta}_{c}-(\dot{\Theta}_{b1}+\dot{\Theta}_{b2})]+2k_{\Theta c} [2\theta_{c}-(\theta_{b1}+\theta_{b2})]=0\end{array} \tag{101}\]  
\[\begin{array}{l}m_{mc}\ddot{T}_{c}+c_{mc}\dot{T}_{c}+k_{mc}T_{c}+\\ +2c_{zc}\epsilon[2\dot{z}_{c}+2\epsilon\dot{T}_{c}-(\dot{z}_{b1}+\dot{z}_{b2}) ]+2k_{zc}\epsilon[2z_{c}+2\epsilon T_{c}-(z_{b1}+z_{b2})]+\\ +2c_{xc}h_{c}\lambda[2h_{c}\dot{\lambda}T_{c}+h_{b}(\dot{\Theta}_{b1}-\dot{ \Theta}_{b2})]+2k_{xc}h_{c}\lambda[2h_{c}\lambda T_{c}+h_{b2}(\theta_{b1}- \theta_{b2})]+\\ +2c_{\Theta c}\lambda[2\lambda\dot{T}_{c}-(\dot{\Theta}_{b1}-\dot{\Theta}_{b2} )]+2k_{\Theta c}\lambda[2\lambda T_{c}-(\dot{\Theta}_{b1}-\theta_{b2})]=0\end{array} \tag{102}\]  
where the following notations were introduced, based on the symmetry properties of the eigenfunction \(X_{c}(x)\).  
\[\epsilon=X_{c}(l_{1})=X_{c}(l_{2}) \tag{103}\]  
\[\lambda=\frac{\mathrm{d}X_{c}(l_{1})}{\mathrm{d}x}=-\frac{\mathrm{d}X_{c}(l_ {2})}{\mathrm{d}x} \tag{104}\]  
The forces due to the primary suspension are  
\[F_{zb1,2}=-2c_{zb}(\dot{z}_{b1}+a_{b}\dot{\Theta}_{b1}-\dot{z}_{w1,2})-2k_{ zb}(z_{b1}+a_{b}\theta_{b1}-z_{w1,2}) \tag{105}\]  
\[F_{zb3,4}=-2c_{zb}(\dot{z}_{b2}-a_{b}\dot{\theta}_{b2}-\dot{z}_{w3,4})-2k_{zb} (z_{b2}-a_{b}\theta_{b2}-z_{w3,4}) \tag{106}\]  
The final form of the bogie motion equations is  
\[\begin{array}{l}m_{b}\ddot{z}_{b1}+4c_{zb}\dot{z}_{b1}+4k_{zb}z_{b1}+2c_{zc }(\dot{z}_{b1}-\dot{z}_{c}-a_{c}\dot{\Theta}_{c}-\epsilon\dot{T}_{c})+\\ +2k_{zc}(z_{b1}-z_{c}-a_{c}\dot{\Theta}_{c}-\epsilon T_{c})=2c_{zb}(\dot{z}_{ w1}+\dot{z}_{w12})+2k_{zb}(z_{w1}+z_{w2})\end{array} \tag{107}\]  
\[\begin{array}{l}J_{b}\ddot{\Theta}_{b1}+4c_{zb}a_{b}^{2}\dot{\Theta}_{b1}+4 k_{zb}a_{b}^{2}\theta_{b1}+2c_{xc}h_{b}[h_{b}\dot{\Theta}_{b1}+h_{c}(\dot{ \Theta}_{c}+\epsilon\dot{T}_{c})]+\\ +2k_{xc}h_{b}[h_{b}\theta_{b1}+h_{c}(\dot{\Theta}_{c}+\epsilon\dot{T}_{c})]=2 c_{zb}a_{b}(\dot{z}_{w1}-\dot{z}_{w2})+2k_{zb}a_{b}(z_{w1}-z_{w2})\end{array} \tag{108}\]  
\[\begin{array}{l}m_{b}\ddot{z}_{b2}+4c_{zb}\dot{z}_{b2}+4k_{zb}z_{b2}+2c_{zc} (\dot{z}_{b2}-\dot{z}_{c}+a_{c}\dot{\Theta}_{c}-\epsilon\dot{T}_{c})+\\ +2k_{zc}(z_{b2}-z_{c}+a_{c}\dot{\Theta}_{c}-\epsilon T_{c})=2c_{zb}(\dot{z}_{ w3}+\dot{z}_{w14})+2k_{zb}(z_{w3}+z_{w4})\end{array} \tag{109}\]  
\[\begin{array}{l}J_{b}\ddot{\Theta}_{b2}+4c_{zb}a_{b}^{2}\dot{\Theta}_{b2}+4 k_{zb}a_{b}^{2}\theta_{b2}+2c_{xc}h_{b}[h_{b}\dot{\Theta}_{b2}+h_{c}(\dot{ \Theta}_{c}-\epsilon\dot{T}_{c})]+\\ +2k_{xc}h_{b}[h_{b}\dot{\Theta}_{b2}+h_{c}(\theta_{c}-\epsilon T_{c})]=2c_{zb }a_{b}(\dot{z}_{w3}-\dot{z}_{w4})+2k_{zb}a_{b}(z_{w3}-z_{w4})\end{array} \tag{110}\]