## 3 Time and Frequency Domain Analysis of Track Dynamic Response
### Track Dynamics in Time Domain  
Figures 8-11 show the wheel displacement and rail, sleeper and ballast acceleration in the time domain for the ballasted track with and without the ballast mat. In Figure 8, it is observed that when no ballast mat is considered, the wheel movement comprises four major jumping and bouncing processes on the ballasted track. When the ballast mat is installed, the wheel exhibits only two jumping and bouncing phenomena, indicating that the ballast mat greatly improves the track elasticity characteristics. Moreover, the height of the first bouncing of the wheel for the track without the ballast mat is 9.33 mm, which decreases by nearly 59.1% to 3.82 mm when the ballast mat is installed.  
Figure 8: Wheel displacement for tracks with and without ballast mat.  
Figure 7: Acceptance and phase angle above sleopers.  
Figure 9 compares the rail acceleration for tracks with and without the ballast mat. In the first impact process, the first wheel impact causes first rail acceleration peaks of 2960.32 and 2919.02 m/s\({}^{2}\) for the tracks with and without the ballast mat, respectively, with both the peaks occurring at almost the same time. In the second wheel impact, the first rail acceleration peak of the track without the mat is 1644.15 m/s\({}^{2}\), which is considerably greater than that of the track with the ballast mat (1281.84 m/s\({}^{2}\)). The secondary peaks of the second wheel impact show a similar trend. It can also be seen that the second impact for the tracks with the ballast mat occurs approximately 0.03 s earlier than that for the tracks without the ballast mat. Furthermore, it takes only two impact processes for the wheel to land on the tracks with the ballast mat, whereas four impact processes occur before the wheel can land on the tracks without the ballast mat.  
Figure 11: Ballast acceleration for tracks with and without ballast mat.  
Figure 10: Sleeper acceleration for tracks with and without ballast mat.  
Figure 9: Rail acceleration for tracks with and without ballast mat.  
Figures 10 and 11 compare the sleeper acceleration and ballast acceleration for tracks with and without the ballast mat. The first acceleration peak is a direct indicator of the impact force. The first acceleration peaks caused by the wheel impact are nearly the same for both cases. Nevertheless, the second acceleration peak for the case with the ballast mat appears smaller than that without the mat.  
The ratios between the first acceleration peak on the first and second wheel impacts are shown in Figure 11 and presented in Table 2, where a larger ratio indicates a better vibration reduction. The results from Table 2 indicate that the ratios for the case with ballast mat are all higher than those for the case without mat, which demonstrates that the installation of the ballast mat can provide higher resilience and elasticity for the ballasted track. Figure 12 also shows that the ratio increases less for the sleeper, indicating that the ballast mat is better to reduce vibration on the rail and ballast than on the sleeper.  
Figure 14: Sleeper acceleration power spectra for tracks with and without ballast mat.  
Figure 15: Ballast acceleration power spectra for tracks with and without ballast mat.  
Figure 13: Rail acceleration power spectra for tracks with and without ballast mat.  
the numerical analysis shows the best vibration reduction in the frequency range of 40 to 125 Hz. The vibration level is reduced by 15, 17 and nearly 13 dB for the rail, sleeper, and ballast, respectively. For the high-frequency range, i.e., above 125 Hz, the ballast matches reduces the vibration on average by 3 to 4 dB.  
Figure 16: One–third octave analysis of rail vibration level.  
Figure 17: One–third octave analysis of sleeper vibration level.  
Figure 18: One–third octave analysis of ballast vibration level.