## 4 Dynamic Response of Unsupported Track
### Unsupported Sleeper on One Side  
This section analyses the dynamic response of the unsupported sleeper on one side. The front view of the unsupported sleepers on one side is the same as Figure 5a, and the lateral view of the unsupported left rail is shown in Figure 11.  
Figure 10: Effect of different speeds on dynamic response of point A under bilateral unsupported sleeper: (**a**) maximum displacement of the track and (**b**) maximum acceleration of the track. As can be seen from Figure 12, the effect of one unilateral unsupported sleeper can be ignored in the displacement of the bridge, which is the same as that of the bilateral unsupported sleeper. However, the maximum acceleration of one unsupported sleeper on one side is less than that on both sides. The maximum acceleration of the left rail with one unilateral unsupported sleeper is 0.3951 m/s\({}^{2}\), while that with one bilateral unsupported sleeper is 0.4928 m/s\({}^{2}\). Figure 13 shows that the maximum wheel-rail interaction force on the left and right sides are 172.3 kN and 142.4 kN respectively, which are about 1.25 and 1.04 times higher than that of normal support. It indicates that the rolling force of the vehicle leads to the change of the wheel-rail force on the intact side due to the inconsistency of the supporting stiffness of the left-side rail and the right-side rail. Figure 11: The lateral view of defect location on the left-side rail. Figure 12: Dynamic responses under the unilateral unsupported sleepers at point B: (**a**) midspan displacement of the bridge and (**b**) midspan acceleration of the bridge. Figure 14 shows the bridge acceleration and wheel-rail vertical force of the left-side and right-side rail under the different numbers of unsupported sleepers at point B. When there are two unilateral unsupported sleepers, the maximum acceleration is 0.4205 m/s\({}^{2}\), which is about 1.35 times higher than that of normal support. The wheel-rail interaction force of the left-side and right-side rail is 182 kN and 143.7 kN, respectively, which are 1.32 and 1.05 times higher than those of normal support. It indicates that the acceleration of the bridge under the unilateral unsupported sleepers is smaller than the acceleration of the bilateral unsupported sleepers. The wheel-rail contact force on the unsupported side of the unilateral unsupported sleeper is slightly smaller than that of the bilateral unsupported sleeper, while the wheel-rail contact force on the supporting side of the unilateral unsupported sleeper is slightly larger than that of the normal supporting sleeper. As can be seen from Figure 15, the maximum displacement of the left-side and right-side rails under one unilateral unsupported sleeper are 0.5355 mm and 0.3729 mm respectively, which are about 1.47 times and 1.02 times higher than those under normal support. The maximum acceleration of the left-side and right-side rail under one unilateral unsupported sleeper are 31.92 m/s\({}^{2}\) and 24.2 m/s\({}^{2}\) respectively, which are about 1.36 times and 1.03 times higher than those under normal support. It indicates that for the unilateral supported sleeper, the dynamic response of the track has a significant effect on the unsupported side, but the effect on the other supporting side can be ignored. Figure 13: Wheel–rail vertical contact forces under the unilateral unsupported sleepers at point B: (**a**) left side and (**b**) right side. Figure 14: The effect of unsupported track at point B.  
Figure 15: _Cont._  
Figure 16 shows the dynamic response of the left-side and right-side rail at point A. When there are two unilateral unsupported sleepers and the driving speed is 100 m/s, the maximum displacement and acceleration of the left-side rail are respectively 0.6014 mm and 38.74 m/s\({}^{2}\), which are about 1.65 and 1.65 times higher than those of normal support. The maximum displacement and acceleration of the right-side rail are, respectively, 0.3767 mm and 24.74 m/s\({}^{2}\), which are about 1.04 and 1.06 times higher than those of normal support.