# Development of a New Vertical Dynamic Model of a Rail Vehicle for the Analysis of Ride Comfort
## 2 Modeling of Rail Vehicles
### Assembling of Rigid Bodies  
In this study, two rigid bodies were connected by torsional spring to simulate the vertical bending motion of the car body (see Figure 6). The flexible model of the car body had two rigid masses (long and slender beam) with their moments of inertia. Also, one generalized coordinate, \(\theta\), was assumed because of the symmetry in the longitudinal direction. The equation of motion was derived using Lagrange's equation [23]. The position vectors of the front and rear car body halves were calculated as follows:  
\[\begin{array}{l}\overset{\rightarrow}{r}_{F}=\frac{L_{h}}{2}\cos\theta \mathbf{i}+\frac{L_{h}}{2}\sin\theta\mathbf{j}\\ \overset{\rightarrow}{r}_{R}=\frac{3L_{h}}{2}\cos\theta\mathbf{i}+\frac{L_{h}} {2}\sin\theta\mathbf{j}\end{array} \tag{1}\]  
where \(\overset{\rightarrow}{r}_{i}\) is the position vector for the center of the front and rear half of car body, the subindex \(h\) is used for the word \"half\" and \(L\) is the length of the car body. The kinetic energy and potential energy of the whole car body, using the position vectors (1) and the definition of the moment of inertia, is calculated as follows:  
\[\begin{array}{l}T=\frac{1}{3}m_{c,h}\;L_{h}^{2}\left(\dot{\theta}\right)^{2} +\;\;m_{c,h}\;L_{h}^{2}\left(\dot{\theta}\right)^{2}\sin^{2}\theta,\;\;I_{h}= \frac{m_{c,h}\;L_{h}^{2}}{12}\end{array}\]  
\[\begin{array}{l}V=\frac{1}{2}k_{\theta}(2\theta)^{2}=2k_{\theta}\;\theta^{2} \end{array} \tag{2}\]  
where \(T\) is the kinetic energy, \(V\) is the potential energy of the system, \(m_{c}\) is the mass of car body and \(I\) is the moment of inertia. Due to the symmetry, a rotation of \(\theta\) caused a \(2\theta\) change in potential energy in total. This assumption was used in the potential energy equation.  
Substituting the energy terms into the Lagrangian formulation yields the following equation of motion:  
\[\begin{array}{l}\left(\frac{2m_{c,h}\;L_{h}^{2}}{3}+\;\;2m_{c,h}\;L_{h}^{2} \;\;\sin^{2}\theta\right)\ddot{\theta}+\;2m_{c,h}\;L_{h}^{2}\left(\dot{\theta} \right)^{2}\sin\theta\;\;\cos\theta+4k_{\theta}\;\theta=0\end{array} \tag{3}\]  
Figure 6: Schematic representation of the assembling of rigid bodies.  
where \(k_{\theta}\) is the stiffness of the torsional spring connecting the car body halves. After linearization, the nonlinear terms, with the assumption of small displacement, were arranged in the equation of motion (3); the linearized equation of motion became, in implicit form, as follows:  
\[\ddot{\theta}+\ \frac{6k_{\theta}}{m_{c,h}\ L_{h}^{2}}\theta=0 \tag{4}\]  
The first bending frequency of the car body was used to determine the stiffness of the torsion spring in the flexible rail vehicle model in Dymola. Therefore, the influence of flexible body on vertical ride comfort was included in the simulation.  
The damping coefficient was quite complex to determine, and, for rail vehicle car bodies, whose natural frequency is between 8-12 Hz, the structural damping is negligible, according to a previous study [18]. Therefore, it was neglected in this study.