# Experimental and Mathematical Study of Flexible-Rigid Rail Vehicle Riding Comfort and Safety
## 2 Rail Vehicle Mathematical Modelling
### Equation of Motion for Combined Rigid and Flexural Motion  
The equations for the combined flexural-rigid motion of a vehicle body are formulated using three degrees of freedom (DOF). A schematic representation of the vehicle is shown in Figure 1. Presumably, the vehicle body and each bogie frame have three DOF of rigid body motion. These DOF are called pitch, vertical, and roll motions, respectively. In addition, flexible body motion is considered for the vehicle's body, as described in the preceding section. The wheelsets are considered rigid bodies, since they only have two DOF: the ability to move in a vertical and roll motion. The normal contact between the wheels and the rail is represented in this model as a linear spring. Consider a railway vehicle consisting of a car body with mass \(m_{c}\) and moment of inertia \(l_{cx}\) and \(l_{cy}\) along the \(x\) and \(y\) axes, respectively. The car body is supported by two bogies, each with a frame of mass \(m_{b}\), moment of inertia \(l_{bx}\) and \(l_{yy}\) along the \(x\) and \(y\) axes, respectively, and a wheelset with mass \(m_{w}\) and moment of inertia \(l_{wx}\) along the x axis. \(L_{b}\) is the half distance between two bogies, \(L_{w}\) is the half distance among two wheelsets of the bogie, the car body has a length \(L\), and \(L_{c}\) is centered at a longitudinal position. The front and rear secondary suspensions are located at longitudinal positions \(L_{f}\) and \(L_{r}\), respectively. The lateral displacement of the secondary suspension is \(b\), and \(b_{w}\) is the track gauge (half). The primary suspension axle has a vertical stiffness of \(k_{p}\) and a vertical damping of \(c_{p}\). The secondary suspension for each side of the bogie has a vertical stiffness of \(k_{s}\) and a vertical damping of \(c_{s}\). The equivalent linear spring of the Hertzian contact is \(k_{hz}\). The railway vehicle has two flexible bending modes with damping constants \(\zeta_{1z}\) and \(\zeta_{2z}\) for the first and second modes, respectively, and a torsional mode with damping constant \(\zeta_{1\phi}\). Taking into account the model presented in Figure 1, the EOM that describes the behaviors of the vehicle body may be stated as follows:  
\[m_{c}\ddot{z}_{c}=-f_{sfr}-f_{sfl}-f_{srr}-f_{srl} \tag{6}\]  
\[I_{cy}\ddot{\theta}_{c}=L_{b}f_{sfr}+L_{b}f_{sfl}-L_{b}f_{srr}-L_{b}f_{srl} \tag{7}\]  
\[I_{cx}\dot{\theta}_{c}=-bf_{sfr}+bf_{sfl}-bf_{srr}+bf_{srl} \tag{8}\]  
where \(f_{sij}\) is secondary suspension forces on the bogie frames and the vehicle body, respectively [18]. These forces may be defined using Equations (9)-(17). \[f_{sfr}=k_{s}\Big{[}\Big{(}z_{c}+\sum\ X_{nz}\Big{(}L_{f}\Big{)}q_{nz}-L_{b} \theta_{c}+b\Big{(}\phi_{c}+\sum\ X_{n\phi}\Big{(}L_{f}\Big{)}q_{n\phi}\Big{)} \Big{)}-\Big{(}z_{bf}+b\phi_{bf}\Big{)}\Big{]} \tag{9}\] \[f_{sfr}=k_{s}\Big{[}\Big{(}z_{c}+\sum\ X_{nz}\Big{(}L_{f}\Big{)}q_ {nz}-L_{b}\theta_{c}+b\Big{(}\phi_{c}+\sum\ X_{n\phi}\Big{(}L_{f}\Big{)}q_{n \phi}\Big{)}\Big{)}-\Big{(}z_{bf}+b\phi_{bf}\Big{)}\Big{]}\] (10) \[f_{sfr}=k_{s}\Big{[}\Big{(}z_{c}+\sum\ X_{nz}\Big{(}L_{f}\Big{)}q_ {nz}-L_{b}\theta_{c}+b\Big{(}\phi_{c}+\sum\ X_{n\phi}\Big{(}L_{f}\Big{)}q_{n \phi}\Big{)}\Big{)}-\Big{(}z_{bf}+b\phi_{bf}\Big{)}\Big{]}\] (11) \[f_{sfr}=k_{s}\Big{[}\Big{(}z_{c}+\sum\ X_{nz}\Big{(}L_{f}\Big{)}q_ {nz}-L_{b}\theta_{c}+b\Big{(}\phi_{c}+\sum\ X_{n\phi}\Big{(}L_{f}\Big{)}q_{n \phi}\Big{)}\Big{)}-\Big{(}z_{bf}+b\phi_{bf}\Big{)}\Big{]}\] (12) \[\qquad\qquad\qquad+\varepsilon_{s}\Big{[}\Big{(}z_{c}+\sum\ X_{nz }\Big{(}L_{f}\Big{)}q_{nz}-L_{b}\theta_{c}+b\Big{(}\phi_{c}+\sum\ X_{n\phi} \Big{(}L_{f}\Big{)}q_{n\phi}\Big{)}\Big{)}-\Big{(}z_{bf}+b\phi_{bf}\Big{)} \Big{]}\]  
where  
\[\ddot{q}_{nz}+2\zeta_{nz}\omega_{nz}\dot{q}_{nz}+\omega_{nz}^{2}q_{nz}=\frac{ 2}{\rho AL}\Big{[}-f_{sfr}X_{nz}\Big{(}L_{f}\Big{)}-f_{sfl}X_{nz}\Big{(}L_{f} \Big{)}-f_{srr}X_{nz}(L_{r})-f_{srt}X_{nz}(L_{r})\Big{]} \tag{13}\]  
\[m_{c}\ddot{z}_{c}=-f_{sfr}-f_{sfl}-f_{srr}-f_{srl} \tag{6}\]0  
The bogie frame EOM is given in Equations (15)-(17).