## 4 Dynamic Response of Unsupported Track
### Unsupported Sleeper on Both Sides  
The defect of track support belongs to dynamic irregularity. Its effect on the dynamic response of the train-track-bridge interaction system is greater than that of geometric irregularity. Therefore, the dynamic analysis of this paper only considers the effect of dynamic irregularity but does not consider the effect of geometric irregularity. As shown in Figure 5a, continuous unsupported and normally supported sleepers in the bridge and subgrade sections are selected for numerical analysis. The dynamic analysis is carried out at midpoint A of the transition section and point B of the mid-span of the bridge, which are adjacent to the defect area. Figure 5b shows that the supports on the left and right sides of the same sleeper are invalid. Figure 5: The defect location of the unsupported track on both sides: (**a**) front view and (**b**) lateral view. As can be seen from Figure 6, compared with the normal support under the track, the effect of one bilateral unsupported sleeper on the displacement of the bridge can be ignored. However, the effect of one bilateral unsupported sleeper increases the acceleration in the midspan of the bridge, resulting in high-frequency oscillations significantly. The maximum acceleration under one bilateral unsupported sleeper increases from 0.3112 m/s\({}^{2}\) to 0.4928 m/s\({}^{2}\), which is about 1.58 times higher than that of normal support. It suggests that the force at the original coupled location is dispersed to nearby area due to the loss of bridge-rail connection, resulting in a significant increase in the acceleration of the bridge. As can be seen from Figure 7, when both the left and right sides of the same sleeper are not supported, the vertical wheel-rail contact forces on the left and right sides are the same. The vertical wheel-rail contact force exhibits periodic changes when the vehicle runs smoothly. The reason for the irregular change in the wheel-rail interaction force at the beginning is that the vehicle suddenly exerts a load on the track. The maximum wheel-rail interaction force under one bilateral unsupported sleeper increases from 137.4 kN to 173.9 kN, which is 1.27 times higher than that of normal support. When the train passes through the support defect under the track, the uneven cosine settlement causes the sudden change of the relative displacement between the wheel and the track, which leads to the aperiodic change of the wheel-rail contact force. Figure 6: Dynamic responses under the bilateral unsupported sleeper at point B: (**a**) midspan displacement of the bridge and (**b**) midspan acceleration of the bridge. Figure 8 shows the bridge acceleration and wheel-rail vertical force under different numbers of the unsupported sleeper at point B. As the number of unsupported sleepers increases, the effect of defects becomes more and more serious. When two sleepers under the track fail to be supported on both sides, the maximum acceleration and wheel-rail interaction force are respectively 0.5458 m/s\({}^{2}\) and 184.6 kN, which are 1.75 and 1.34 times higher than that of normal support. It indicates that the support defect under the track leads to the loss of the buffering effect in the defect area of the track. The greater the wheel-rail contact force is, the greater the dynamic response in the adjacent defect area is. The track is directly supported by the subgrade in the transition section. Figure 9 shows the displacement and the acceleration at point A, and the driving speed of the train is also 100 m/s. The maximum displacement and the acceleration under one bilateral unsupported sleeper increase from 0.3639 mm to 0.5479 mm and 23.43 m/s\({}^{2}\) to 32.62 m/s\({}^{2}\) respectively, which are about 1.51 times and 1.39 times higher than that of normal support. It indicates that when the track vibration is caused by the high-speed train, the subgrade can significantly reduce the impact on the track.