## 5 Analysis of Dynamic Effects of Train-Bridge System
### Effect of Traveling Speed  
According to the CRH2 model established earlier, a total of seven kinds of speed conditions of 150, 180, 200, 230, 250, 280, and 300 km/h were selected to calculate the dynamic response of the vehicle and bridge with the integral step of 0.0001 s when the train runs upstream. Before the calculation, we set the train running smoothly and then drove the bridge after a certain distance; after the train was completely out of the bridge, the calculation was finished.  
The calculated dynamic responses included the index for safety of the train operation (derailment coefficient, wheel unloading rate, wheel rail lateral force) and index for stability of the train operation (lateral acceleration of the train, vertical acceleration of the train, lateral Sperling index, vertical Sperling index). The calculated bridge dynamic responses included lateral displacement (pier top, midspan), vertical displacement (midspan), lateral acceleration (pier top, midspan), and vertical acceleration (midspan).  
In the speed range of 150-300 km/h, the maximum dynamic response indices of the vehicle vary with train speed, as shown in Figure 16.  
As illustrated in Figure 16, the dynamic responses of the train increase with the train speed. The derailment coefficient, wheel unloading rate, and wheel-rail lateral force all rise as the train speed increases, but the maximum is within the limit of the standard requirement. With the increase in the train speed, the stability indexes, such as lateral and vertical acceleration and lateral and vertical Sperling index, gradually increase, indicating that the ride comfort is gradually decreasing, but the indexes do not exceed the established standard requirements. In addition, lateral and vertical comfort indexes present excellent grades.  
Similarly, in the speed range of 150-300 km/h, the maximum dynamic response of the bridge varies with the train speed, as shown in Figures 17 and 18.  
Figure 16: Relationship between stability index and speed of train operation: (**a**) derailment coefficient; (**b**) wheel unloading rate; (**c**) train body acceleration; (**d**) Sperling index.  
Comparing and analyzing the results of the dynamic response of the train at each pier height, the following can be seen:  
1. Within the range of pier heights studied, the impact of pier height on train operating safety is minimal. The train safety indicators show almost no change with increasing pier heights.
2. Within the range of pier heights studied, both vertical and lateral indicators increase with the increase in pier height, which shows that the smoothness of the train operation on high-pier bridges is poorer than that on low-pier bridges. Comparing and analyzing the results of the dynamic response of the bridge at each pier height, the following can be seen: 1. Over the range of pier heights studied, the lateral displacement increases as the height of the pier increases, both at the midspan of the main span (Section C) and the pier's top (Section B). Especially for Case 1, the lateral displacement of the bridge structure increases significantly
2. Vertical and lateral accelerations at the midspan of the main span (Section C) and midspan of the side span (Section A) up to 0.731 m/s\({}^{2}\), 0.445 m/s\({}^{2}\), 0.634 m/s\({}^{2}\), and 0.365 m/s\({}^{2}\), respectively, which are less than the specified limits (0.35 g and 0.14 g).