# Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method
## 2 Establishment and Solution of the Motion Equation  
### Motion Equation of Train System Based on MCM  
In the analysis of the train vibration, it was helpful to generate different series of random track irregularity excitations repeatedly. Later, each of them was solved using the Newmark-\(\beta\) method (N\(\beta\)M) to obtain the corresponding series of the random vibratory response of the train. When the solving process for one of series of random excitation is terminated, it is called once-sampling of MCM. The entire programming procedure was realized in MATLAB software (version R2022a). Before that, the relevant motion equation of train system needed to be established in accordance with the following assumptions [19]:  
1. The interaction is nonexistent among vehicle elements. 2. Each vehicle element is composed of a rigid car-body, bogies, and wheel sets. 3. The springs of a train are linearly elastic, while the dampers are linearly viscous. 4. Each vehicle element is a linear stationary system; namely, the mass, damping, and stiffness matrices of train are constant. 5. The train moves at a constant speed. 6. The wheels and rails fit snugly. The relative displacement between the track and the bridge deck, as well as the elastic effect of the track-bridge system, are neglected. 7. The track irregularity is a zero-mean stationary Gaussian random process. Based on the assumptions above, a train subsystem is made up of a certain number of mutually independent 3D vehicle elements. Each of them consists of a car-body, two bogies, four wheel sets, and a two-layer spring-damper suspension system. Each vehicle element contains 15 independent DOFs for the car-body and bogies, and three dependent DOFs for each wheel set coupled respectively with the vertical, horizontal, and torsional motion states of 4 contact points. The DOFs of each vehicle element are listed in Table 1 and correspond to those in Figure 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Vehicle Element \(i\)** & **DOF** \\ \hline Car body & 5 DOFs: \(y_{ci}\), \(z_{ci}\), \(\theta_{ci}\), \(\theta_{ci}\), \(\psi_{ci}\), \(\psi_{ci}\)1 \\ Bogies & 10 DOFs: \(y_{bi1}\), \(z_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \(\theta_{bi1}\), \\ Wheel sets & 12 DOFs: \(y_{bi1}\), \(z_{bi1}\), \(\theta_{bi1}\), \(y_{bi2}\), \(z_{bi2}\), \(\theta_{bi2}\), \(y_{bi3}\), \(z_{bi3}\), \(\theta_{bi3}\), \(y_{bi3}\), \(y_{bi3}\), \(y_{bi3}\), \(y_  
According to the D'Alembert principle, the motion equation of a train can be expressed as:  
\[\mathbf{M_{\mathrm{t}}}\ddot{\mathbf{u}}_{\mathrm{t}}+\mathbf{C_{\mathrm{t}}}\dot{\mathbf{u}}_{ \mathrm{t}}+\mathbf{K_{\mathrm{t}}}\mathbf{u}_{\mathrm{t}}=\mathbf{T_{\mathrm{s}}}\mathbf{u}_{ \mathrm{w}}+\mathbf{T_{\mathrm{d}}}\dot{\mathbf{u}}_{\mathrm{w}} \tag{1}\]  
where \(\mathbf{M_{\mathrm{t}}}\), \(\mathbf{C_{\mathrm{t}}}\), and \(\mathbf{K_{\mathrm{t}}}\) are the mass, the damping, and the stiffness matrix, respectively. \(\mathbf{T_{\mathrm{s}}}\) and \(\mathbf{T_{\mathrm{d}}}\) are the projector matrices for the spring and the damper, respectively.