# Research on the Influence of Loading Frequency on the Dynamic Response of Concrete Sleopers
## 2 Selection of Model Parameters and Establishment of the Model
### The Initial Model of the Sleeper  
A two-dimensional model diagram was drawn according to the geometric size of the concrete sleeper and imported into the software of discrete element. Circular particles with a particle size of 5 mm were selected to fill the geometric diagram of the sleeper, and mechanical properties were applied to the sleeper according to the mechanical parameters determined after verification. The sleeper was composed of ball elements and combined according to certain arrangement rules. At the same time, to increase the calculation rate of the program, the rail was simulated by the clump element. The generated 2D rail-sleeper initial model is shown in Figure 6. In this paper, the sleeper was the main research object, and the discrete element method was used to simulate and analyze it. In order to make the research object more intuitive, the independent sleeper was directly simulated without considering the interaction between the sleeper and the track bed. Therefore, the main function of the wall at the bottom of the sleeper in the model was to limit the displacement of the sleeper particles and prevent the sleeper from passing through the wall.  
Figure 5: Displacement cloud diagram of concrete specimen after compression. (**a**) 1 mm particle size. (**a1**) Full displacement. (**a2**) X direction—lateral displacement. (**a3**) Y direction—vertical displacement. (**b**) 5 mm particle size. (**b1**) Full displacement. (**b2**) X direction—lateral displacement. (**b3**) Y direction—vertical displacement.  
To analyze the response of sleepers under dynamic loading, cyclic sinusoidal loads of equal magnitude were simultaneously applied to the left and right rails. The calculation formula [21] of the load is shown in Formula (1).  
\[F_{(t)}=-F(1-cos(2\pi ft)) \tag{1}\]  
where \(F\) represents the half-peak value of the load applied to the left and right rails, which is 63 kN [22]; \(t\) represents the loading time, and the total loading time applied to the rail is 1 s; and \(f\) is the frequency of the applied load, with 5 Hz, 10 Hz, 20 Hz, and 50 Hz selected in this study. The cycle loading times corresponding to different loading frequencies in the total calculation time were 5 times, 10 times, 20 times, and 50 times. The loading curves corresponding to different loading frequencies are shown in Figure 7.  
Different loading frequencies correspond to different train speeds. The relationship between train speed and action frequency is shown in Formula (2) [23; 24].  
\[f=V/L \tag{2}\]  
Figure 6: Steel-sleeper two-dimensional discrete element initial model diagram.  
Figure 7: Loading curve of rail-sleeper discrete element model. (**a**) 5 Hz. (**b**) 10 Hz. (**c**) 20 Hz. (**d**) 50 Hz.  
where \(f\) is the action frequency applied to the rail; \(V\) corresponds to the running speed of the train, in m/s; and \(L\) represents the disturbance wavelength, i.e., the wheelbase of the train, in m. In this paper, the C80 freight train is taken as an example, \(L\) is the wheelbase of the C80 freight train (1830 mm) [25], and the four loading frequencies of 5 Hz, 10 Hz, 20 Hz, and 50 Hz correspond to train running speeds of 32.94 km/h, 65.88 km/h, 131.76 km/h, and 329.4 km/h, respectively.  
By studying the influence of different loading frequencies on the mechanical properties of sleepers, the influence of train running speed on the service performance of sleepers can be reflected.