## 2 Dynamic Modeling of a Railway Vehicle
### Vehicle Model
\[F_{zab,j}=K_{b}(z_{a,j}-z_{w,j})+C_{b}\big{(}\dot{z}_{a,i}-\dot{z}_{w,i}\big{)}, \tag{4b}\]  
The support force of the pinion side bearing can be written by  
\[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]0 \[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]1  
The support force of the gear side bearing can be written by  
\[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]2 \[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]3  
The support force of the motor bearing can be written by  
\[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]4 \[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]5  
The coupling force between the motor and the pinion can be given by  
\[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]6 \[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]7 (8b) \[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]8  
The suspender force of the gearbox housing can be written by  
\[F_{xcp,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{epkb}\beta_{b,j}-H_{epka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{epkb}\dot{\beta}_{b,j}-H_{ epka}\dot{\beta}_{a,i}\Big{)}, \tag{1b}\]9 \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]0  
The suspender force of the motor can be given by  
\[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]1 (10a) \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]2 (10b) \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]3 (10c) \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]4 (10d) where \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]5 is the longitudinal distance from the external primary suspension to the center of the bogie frame; \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]6 is the longitudinal distance from the external primary suspension to the center of the axlebox; \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]7 is the longitudinal distance from the inner primary suspension to the center of the bogie frame; \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]8 is the longitudinal distance from the inner primary suspension to the center of the axlebox; \[F_{zip,i}=K_{zp}\Big{[}z_{b,j}-z_{a,i}+(-1)^{i+1}l_{ipkb}\beta_{b,j}+(-1)^{i+1} l_{ipka}\beta_{a,i}\Big{]}+C_{zp}\Big{[}\dot{z}_{b,j}-\dot{z}_{a,i}+(-1)^{i+1}l_{ ipkb}\dot{\beta}_{b,j}+(-1)^{i+1}l_{ipka}\dot{\beta}_{a,i}\Big{]}, \tag{1c}\]9 is the longitudinal distance from the center of the axlebox; \[F_{zip,i}=K_{xp}\Big{(}x_{b,j}-x_{a,i}-H_{ipkb}\beta_{b,j}-H_{ipka}\beta_{a,i} \Big{)}+C_{xp}\Big{(}\dot{x}_{b,j}-\dot{x}_{a,i}-H_{ipkb}\dot{\beta}_{b,j}-H_{ ipka}\dot{\beta}_{a,i}\Big{)}, \tag{1d}\]0 is the longitudinal distance between the secondary suspension and the center of the carbody; \(l_{bgb\beta}\) is the longitudinal distance between the pinion side bearing and the center of the gearbox housing; \(l_{bgb\beta}\) is the longitudinal distance between the gear side bearing and the center of the gearbox housing; \(l_{bgb\beta}\) is the longitudinal distance between the gearbox suspender and the center of the gearbox housing; \(l_{gbb}\) is the longitudinal distance between the gearbox suspender and the center of the bogie frame; \(l_{gbfb}\) is the longitudinal distance between the motor suspender and the center of the motor housing; \(l_{mbb}\) is the longitudinal distance between the motor housing and the center of the bogie frame; \(H_{epb\beta}\) is the vertical distance from the external primary suspension to the center of the bogie frame; \(H_{epb\beta}\) is the vertical distance from the external primary suspension to the center of the axlebox; \(H_{plb\beta}\) is the vertical distance from the inner primary suspension to the center of the bogie frame; \(H_{ipb\beta}\) is the vertical distance from the inner primary suspension to the center of the axlebox; \(H_{skc}\) is the vertical distance between the secondary suspension and the center of the carbody; \(H_{skb}\) is the vertical distance from the secondary suspension to the center of the bogie frame; \(H_{trc}\) is the vertical distance between the tractive rod and the center of the carbody; \(H_{trb}\) is the vertical distance between the tractive rod and the center of the bogie frame; \(H_{gbfb\beta}\) is the vertical distance between the gearbox suspender and the center of the gearbox housing; \(H_{gbfb}\) is the vertical distance between the gearbox suspender and the center of the bogie frame; \(H_{mhmh}\) is the vertical distance between the motor suspender and the center of the motor housing; \(H_{mh\beta}\) is the vertical distance from the motor suspender to the center of the bogie frame; \(K_{xp}\) and \(K_{zp}\) are the longitudinal and vertical stiffness of the primary suspension; \(C_{xp}\) and \(C_{zp}\) are the longitudinal and vertical damping of the primary suspension; \(K_{xs}\) and \(K_{zs}\) are the longitudinal and vertical stiffness of the secondary suspension; \(C_{xs}\) and \(C_{zs}\) are the longitudinal and vertical damping of the secondary suspension; \(K_{tr}\) is the stiffness of the tractive rod; and \(K_{ab}\) and \(C_{ab}\) are the support stiffness and damping of the bearing.