# Analysis of Train Car-Body Comfort Zonal Distribution by Random Vibration Method
## 4 Case Study
### Analysis of Influence of the Train Speed on the Zonal Distribution Characteristics of Train Comfort  
A vehicle element was simulated using the PEM to run a train with a track irregularity of Grade-2 at speeds of 150 km/h, 250 km/h, and 350 km/h to compare the influence of the different train speeds on the zonal distribution of the Car-Body Sperling index of the car body, as shown in Figure 8. In Figure 8, it can be seen that:  
1. The symmetrical characteristic of the zonal distribution of the car-body Sperling index of the car body was identical to that shown in Section 4.4 for both the horizontal and vertical components;
2. As the train accelerated, the values of the car-body Sperling index for the entire network increased significantly for both the horizontal and vertical components;
3. In general, the train comfort deteriorated when the train traveled too fast. This was because the amplitude of the vibration velocity and the acceleration of the track irregularity contained the linear and quadratic terms of the train speed V, respectively. When the train accelerated, the amplitudes increased rapidly, which increased the input excitation and led to a significant increase in the vibration response of the car body. Therefore, appropriate control of a train's speed can help to improve passenger comfort. Analysis of Influence of the Car-Body Mass on the Zonal Distribution Characteristics of Train Comfort  
In rigid dynamics, the moment of inertia of a car-body center is equivalently linear to the car-body mass. Considering the that center of mass is overlapped by the car-body center, the mass matrix of the vehicle element in Equation (4) can be respectively redefined as Equation (40) where \(\alpha\) is the mass factor:  
\[M_{\text{ve}}=\text{diag}(\alpha m_{c},\alpha m_{c},\alpha I_{\text{sc}},\alpha I _{\text{yc}},\alpha I_{\text{zc}},m_{\text{b}},m_{\text{b}},I_{\text{sb}},I_{ \text{yb}},I_{\text{zb}},m_{\text{b}},m_{\text{b}},I_{\text{yb}},I_{\text{zb}}) \tag{40}\]  
A vehicle element was simulated using the PEM to run on a track with a Grade-6 track irregularity for 5 s at a speed of 250 km/h to compare the influence of a full load (Full rated, \(\alpha=1.2\)) and an empty load (Unladen, \(\alpha=1\)) on the zonal distribution of the car body's Sperling index, as shown in Figure 9. In Figure 9, it can be seen that:  
Figure 8: Zonal distribution of car-body Sperling index influenced by the train speed according to PEM: (**a**) horizontal Sperling index; (**b**) vertical Sperling index. 1. The symmetrical characteristics of the zonal distribution of the car body's Sperling index for the horizontal and vertical components were identical to those shown in Section 4.4;
2. With the additional mass of the car body, the values of the Sperling index for the entire mesh decreased significantly for both the horizontal and vertical components;
3. In general, the comfort of the train deteriorated with the loss in the car-body mass. Therefore, it is important to reasonably distribute the number of passengers during transfer and optimize the original mass design of the car body. Analysis of Influence of the Damage of Secondary Suspension System on the Zonal Distribution Characteristics of Train Comfort  
The comfort of a train is highly dependent on the performance of the secondary suspension system. Reducing the stiffness and increasing the damping can help to simulate the damage of the secondary suspension system. In this way, one of the vertical springs and dampers in the secondary suspension system connecting the car body to the front bogie was multiplied by the damage factors \(d_{\text{fk}}\) and \(d_{\text{fc}}\), respectively. Therefore, the corresponding stiffness submatrix of the vehicle element including Equations (3) and (5)-(7) was updated in Equations (41)-(44). \[\mathbf{K}_{\text{ve}}=