# Dynamic Impact Factor and Resonance Analysis of Curved Intercity Railway Viaduct
## 4 Numerical Analysis
### Time-History Response under Resonance  
The vertical displacement response of a bridge spanning 30 m at different resonant speeds of 246 km/h (theoretical value 262 km/h), 131 km/h, and 87 km/h were calculated. Likewise, the radial displacement responses of the bridge when the resonant speeds were 680 km/h, 340 km/h, and 227 km/h, respectively, were also determined. The results are shown in Figure 8. It can be seen that in the scenarios of the vehicle crossing and leaving the bridge, the amplitudes of vertical displacements corresponding to the speed of 246 km/h are both greater than those corresponding to the speed of 262 km/h. For the vertical and radial displacements of the bridge, the dynamic response trend of Figure 8a,b is relatively regular at the first resonant speed, with six large peaks. In addition, there are large vibration displacements after the vehicle leaves the bridge.  
Figure 8: Time-history response of the bridge spanning 30 m under resonant velocity: (**a**) 1 st vertical resonance; (**b**) 1 st radial resonance; (**c**) 2 nd vertical resonance; (**d**) 2 nd radial resonance; (**e**) 3 rd vertical resonance; (**f**) 3 rd radial resonance.  
It is also observed from Figure 8c-f that there is no apparent relation for the high-order speed dynamic response trend of vertical displacement and radial displacement. For instance, when the speed was 131 km/h and 340 km/h, there were ten prominent peaks in the vertical displacement and radial displacement. This is because there were five carriages of vehicle, including ten steering frames. At this speed, the action of the two wheelsets of the same bogie equals an independent load. However, when the resonance speed is 87.3 km/h and 227 km/h, there are no less prominent large peaks in vertical and radial displacements.  
The results of the vertical displacement response of the bridge spanning 25 m at the resonant speeds of 377 km/h, 189 km/h, and 126 km/h are shown in Figure 9. When the vehicle passes through the bridge at the first-order vertical resonant speed of 377 km/h (Figure 9a), the bridge experiences a large vertical displacement, even after the vehicle leaves the bridge. There are five prominent large peaks vertically down the bridge, and the values of the peaks increase gradually with the vehicle passing. If the vehicle is running through the curved bridge at the speed of 189 km/h (Figure 9b), the bridge displacement dynamic response has five pronounced peaks vertically downward, where each \"peak range\" contains two sub-peaks and one sub-trough. This is because there are five relatively closer bogie combinations in the six cars. When the vehicle passes the curved bridge at the speed of 126 km/h, the vertical displacement trend of the bridge is close to the trend as was seen for 189 km/h speed (Figure 9c). However, the upper and lower vibration range between the wave crest and trough is wider. When the vehicle crosses the bridge at 163 km/h, the dynamic response of radial displacement presents five significant transverse displacement peaks, while the whole process is rough. Once the vehicle leaves the bridge, its free vibration amplitude is small.  
Figure 9: Time-history response of bridge spanning 25 m under resonant velocity: **(a)** 1 st vertical resonance of 1 st mode; **(b)** 2 nd vertical resonance of 1 st mode; **(c)** 3 rd vertical resonance of 1 st mode; **(d)** Radial resonance.