## 4 Numerical Analysis  
### Numerical Analysis Summary  
#### 4.1.1 Vehicle Model Including Suspension System  
A model for analyses of a vehicle load including a suspension system is as shown in Figure 12. The railroad vehicle consists of a car body, bogies, and axles, which are connected by a bolster spring and the suspension system on the axle and the axle journal. A sprung mass and an unsprung mass generate translational vibrations and rotational motions while the train runs. Of the six vibrations, three vibrations are related to the unsprung mass and the bogie and the other three vibrations are related to the motion of the sprung and unsprung mass [2]. Thus, the rigid motion of the vehicle running at a constant speed consists of six degrees of freedom, which include the translational motions in the \(x\), \(y\), \(z\) directions and the rotational motions in the \(\phi\) (Rolling), \(\theta\) (Yawing), and \(\psi\) (Pitching) directions, as shown in Figure 13.  
Figure 11: Results of measurement. Figure 12: Moving load of the vehicle including suspension system. The motion generated by \(\phi\) is considered as being the vertical direction only, but it is in fact influenced transversely as well. Thus, vertical and transverse springs and dampers are compressed or tensioned by rolling. The motion generated by \(\theta\) is also considered the transverse and longitudinal direction according to same principle. The motion generated by \(\psi\) is considered a vertical and longitudinal displacement. A vehicle-track interaction system to the vertical motion can be modeled by the semi-mass of the car body, the bogie, and the unsprung mass [18]. In such a case, an equation of the vertical motion of the car body is as in Equation (6):  
\[m_{c}\ddot{y_{c}}+c_{s}\Big{(}\dot{y_{c}}-\dot{y_{b}}\Big{)}+k_{s}(y_{c}-y_{b}) =P_{c} \tag{6}\]  
where \(m_{c}\): 1/2 of car body mass; \(c_{s}\): Damping coefficient of secondary suspension system; \(k_{s}\): Spring coefficient of secondary suspension system; \(P_{c}\): 1/2 of car body weight. Equations of the vertical and rotational motions of the bogie can be represented by Equations (7) and (8):  
\[m_{b}\ddot{y_{b}}+c_{s}\Big{(}\dot{y_{b}}-\dot{y_{c}}\Big{)}+c_{p}\Big{(}2\dot {y_{b}}-\dot{y_{w1}}-\dot{y_{w2}}\Big{)}+k_{s}(y_{b}-y_{c})+k_{s}\Big{(}2\dot {y_{b}}-\dot{y_{w1}}-\dot{y_{w2}}\Big{)}=P_{b}, \tag{7}\]  
\[I_{b}\ddot{O_{b}}+c_{p}d\Big{(}2d\dot{\theta_{b}}+\dot{y_{w1}}-\dot{y_{w2}} \Big{)}+k_{p}d(2d\theta_{b}+y_{w1}-y_{w2})=0, \tag{8}\]  
where \(y_{b}\): Vertical displacement of bogie; \(\theta_{b}\): Rotational displacement of bogie; \(m_{b}\): Mass of bogie; \(I_{b}\): Rotational inertia moment of bogie; \(c_{p}\): Damping coefficient of primary suspension system; \(k_{p}\): Spring coefficient of primary suspension system; \(c_{s}\): Damping coefficient of secondary suspension system; \(k_{s}\): Spring coefficient of secondary suspension system; \(P_{b}\): Mass of bogie; \(d\): 1/2 of wheelbase. Equations of motion by two unsprung masses of the wheel and axle can be represented as follows:  
\[ \begin{split}& m_{w}y_{w1}^{-}+c_{p}\Big{(}y_{w1}-\Big{(}\dot{y_{b}}-d \dot{\theta}\Big{)}\Big{)}+k_{p}\big{[}y_{w1}-(y_{b}-d\theta)\big{]}\\ &\qquad+k_{c}\big{[}y_{w1}-y(x_{w1})\big{]}=P_{w1}+k_{c}y_{w/r}( x_{w1})\end{split}  \tag{9}\]  
\[ \begin{split}& m_{w}y_{w2}^{-}+c_{p}\Big{(}y_{w2}-\Big{(}\dot{y_{b }}-d\dot{\theta}\Big{)}\Big{)}+k_{p}\big{[}y_{w2}-(y_{b}-d\theta)\big{]}\\ &\qquad+k_{c}\big{[}y_{w2}-y(x_{w2})\big{]}=P_{w2}+k_{c}y_{w/r}( x_{w2})\end{split}  \tag{10}\]  
where \(P_{w1}\): Weight of first unsprung mass; \(P_{w2}\): Weight of second unsprung mass; \(y_{w/r}(x_{w1})\): Roughness of the bottom of first unsprung mass; \(y_{w/r}(x_{w2})\): Roughness of the bottom of second unsprung mass. Figure 13: Rigid motion of vehicle.