# Research on the Influence of Loading Frequency on the Dynamic Response of Concrete Sleopers
## 3 Dynamic Responses of Sleepers at Different Loading Frequencies
### Influence of Loading Frequency on Rail Dynamic Response  
The trainload is transmitted to the sleeper through the rail, so the displacement, velocity, and acceleration of the rail under the action of the cyclic load can reflect the dynamic response of the sleeper to a certain extent. Under the four loading frequencies, the recorded rail displacement, velocity, and acceleration changes with the periodic load are all shown in Figures 12-14.  
The contact between the rail and the sleeper in the initial state will affect the results of subsequent loading. Therefore, in the discrete element model of the rail-sleeper, the rail and sleeper in the initial state were not in direct contact, and there was a small distance between them to ensure that the contact force of the rail-sleeper in the initial stage was zero. Figure 12 shows that when the rail moved down by approximately 2 mm, the rail and the sleeper were in contact, and the load was transmitted between the two, which led to a large dynamic response due to the sudden contact between the rail and the sleeper in the initial stage. Subsequently, as the cyclic loading progressed, the system gradually reached a steady state. After the interaction between the rail and the sleeper, the vertical displacements of the rail under different working conditions were all within 0.25 mm.  
Comparing the vertical displacement curves of the rails under different loading frequencies in Figure 12, it can be seen that the change trends of the left and right rails were the same, the change curves were nearly overlapped, and the vertical displacement change  
Figure 12: The vertical displacement curve of the rail. (**a**) 5 Hz. (**b**) 10 Hz. (**c**) 20 Hz. (**d**) 50 Hz.  
curves of the left and right rails changed periodically with the cyclic load. The vertical displacement of the rail corresponding to 5 Hz was the smallest, approximately 2 mm. The vertical displacement value of the left and right strand rails corresponding to 50 Hz was the largest, close to 2.25 mm. The vertical displacement of the rail increased with the increase in the loading frequency, and the vertical displacement of the rail increased slowly with the increase in the number of cycles until it tended to a stable value.  
As shown in Figure 13, it can be seen that when the loading frequency was 5 Hz, the maximum speed of the left and right rails was approximately 0.01 m/s; the maximum rail speed corresponding to a 10 Hz loading frequency was approximately 0.015 m/s; the maximum speed of the left and right rails corresponding to a 20 Hz loading frequency was approximately 0.03 m/s; and the maximum rail speed was 0.04 m/s at a loading frequency of 50 Hz. The velocity changes of the left and right strand rails were similar, with periodic variation with the cyclic load, and the velocity amplitude increased with increasing loading frequency.  
Figure 13: The vertical speed change curve of the rail. (**a**) 5 Hz. (**b**) 10 Hz. (**c**) 20 Hz. (**d**) 50 Hz.  
Observing the variation of the rail acceleration, it can be seen that when the loading frequencies were 5 Hz, 10 Hz, 20 Hz, and 50 Hz, the corresponding maximum rail accelerations were 20 m/s\({}^{2}\), 30 m/s\({}^{2}\), 60 m/s\({}^{2}\), and 150 m/s\({}^{2}\), respectively. The vertical acceleration of the rail changed periodically with cyclic loading.  
In summary, under loading frequencies of 5 Hz to 50 Hz, the vertical displacement, maximum velocity, and acceleration amplitude of the rail all increased with increasing loading frequency. This shows that the greater the loading frequency, the greater the increase in the dynamic response of the rail, thereby increasing the interaction force between the rail and the sleeper, which indirectly causes the dynamic response of the sleeper to increase, which in turn increases the possibility of sleeper damage and reduces the service life of the sleeper.