# Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method
## 4 Case Study
### Demonstration of the Methodological Correctness
\begin{table}
\begin{tabular}{c c} \hline
**Items** & **Values** \\ \hline Half of longitudinal distance of wheel sets \(d_{1}\) & 1.25 m \\ Half of longitudinal distance of bogies \(d_{2}\) & 8.75 m \\ Half of lateral distance of springs in 1 st suspension system \(b_{1}\) & 1.00 m \\ Half of lateral distance of springs in 2 nd suspension system \(b_{2}\) & 1.00 m \\ Vertical distance, car body center to 2 nd suspension system \(h_{1}\) & 0.80 m \\ Vertical distance, 2 nd suspension system to bogie center \(h_{2}\) & 0.20 m \\ Vertical distance, bogie center to 1 st suspension system \(h_{3}\) & 0.10 m \\ Mass of wheel set \(m_{\mathrm{w}}\) & 2000 kg \\ Mass of bogie \(m_{\mathrm{b}}\) & 3000 kg \\ Moment of inertia of bogie in longitudinal direction \(l_{\mathrm{bx}}\) & 3000 kg\(\cdot\)m\({}^{2}\) \\ Moment of inertia of bogie in lateral direction \(l_{\mathrm{by}}\) & 3000 kg\(\cdot\)m\({}^{2}\) \\ Moment of inertia of bogie in vertical direction \(l_{\mathrm{bx}}\) & 3000 kg\(\cdot\)m\({}^{2}\) \\ Mass of car body \(m_{\mathrm{b}}\) & 40 t\(\cdot\)m\({}^{2}\) \\ Moment of inertia of car body in longitudinal direction \(l_{\mathrm{cx}}\) & 100 t\(\cdot\)m\({}^{2}\) \\ Moment of inertia of car body in lateral direction \(l_{\mathrm{cy}}\) & 2000 t\(\cdot\)m\({}^{2}\) \\ Moment of inertia of car body, about vertical direction \(l_{\mathrm{cz}}\) & 200 t\(\cdot\)m\({}^{2}\) \\ Longitudinal damping of 1 st suspension system/bogie side \(c_{\mathrm{x1}}\) & 1 kN\(\cdot\)s/m \\ Lateral damping of 1 st suspension system/bogie side \(c_{\mathrm{y1}}\) & 1 kN\(\cdot\)s/m \\ Vertical damping of 1 st suspension system/bogie side \(c_{\mathrm{z1}}\) & 20 kN\(\cdot\)s/m \\ Longitudinal damping of 2 nd suspension system/car-body side \(c_{\mathrm{x2}}\) & 60 kN\(\cdot\)s/m \\ Lateral damping of 2 nd suspension system/car-body side \(c_{\mathrm{y2}}\) & 60 kN\(\cdot\)s/m \\ Vertical damping of 2 nd suspension system/car-body side \(c_{\mathrm{z2}}\) & 10 kN\(\cdot\)s/m \\ Longitudinal stiffness of 1 st suspension system/bogie side \(k_{\mathrm{x1}}\) & 1000 kN/m \\ Lateral stiffness of 1 st suspension system/bogie side \(k_{\mathrm{y1}}\) & 1000 kN/m \\ Vertical stiffness of 1 st suspension system/bogie side \(k_{\mathrm{z1}}\) & 1000 kN/m \\ Longitudinal stiffness of 2 nd suspension system/car-body side \(k_{\mathrm{x2}}\) & 200 kN/m \\ Lateral stiffness of 2 nd suspension system/car-body side \(k_{\mathrm{y2}}\) & 200 kN/m \\ Vertical stiffness of 2 nd suspension system/car-body side \(k_{\mathrm{z2}}\) & 200 kN/m \\ Lower limit of angular wave number \(n_{\mathrm{min}}\) & \(\tau\)/\(V\) rad/m \\ Upper limit of angular wave number \(n_{\mathrm{max}}\) & 80\(\tau\)/\(V\) rad/m \\ Sampling rate of angular wave number \(d_{\mathrm{tr}}\) & 2\(\tau\)/\(5V\) rad/m \\ Lower limit of time \(t_{\mathrm{min}}\) & 0 s \\ Upper limit of time \(t_{\mathrm{max}}\) & 10 s \\ Time sampling rate \(d_{\mathrm{tr}}\) & 1/80 s \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters of the train operation. variables corresponds to the time average of a single random time sample, as shown in Equation (37), where \(v_{j}\) represents a particular random time sample of the family of random variables \(V\). For the stationary Gaussian random process with a zero mean, the variance can be expressed as Equation (38). By combining Equation (25) with the time lag \(\tau=0\), the autocorrelation of the signal is equal to its variance. \[\mu=\mathrm{E}[\mathbf{V}(t)]=\mathrm{E}[v_{j}(t)]=\sum_{i=1}^{n}\frac{v_{j}(t)}{n }=\lim_{T\to\infty}\frac{1}{T}\int_{-\frac{\tau}{2}}^{\frac{T}{2}}v_{j}(t)dt \tag{37}\]  
\[\sigma^{2}=\mathrm{E}[\mathbf{V}(t)-\mu]^{2}=\mathrm{E}[v_{j}^{2}(t)]=\lim_{T\to \infty}\frac{1}{T}\int_{-\frac{\tau}{2}}^{\frac{T}{2}}v_{j}^{2}(t)dt=R_{vv}(0) \tag{38}\]  
The track irregularities as input of the linear train system are a stationary Gaussian random process with a zero mean, which theoretically has the same properties of probability distribution as the train oscillation response.