# Analysis of the Vibration Characteristics of Ballastless Track on Bridges Using an Energy Method
## 3 Results and Discussion
### Influence of the Slab Mat Layer's Stiffness  
With the running speed of the vehicle set to the typical value of 300 km/h, conditions of the stiffness of the slab mat layer of 20 MPa/m, 40 MPa/m, 60 MPa/m, and 80 MPa/m were included. The influence of the slab mat layer's stiffness on the vibration energy was thus investigated.  
Figure 9 presents the influence of different stiffness levels of the slab mat layer on the power flows. It can be observed from Figure 9a that the changes in the slab mat layer's stiffness mainly influenced the power flows of the rail below 80 Hz and scarcely affected the power flows above 80 Hz. In addition, the rail's power flows decreased with increasing slab mat layer stiffness, except for at certain frequencies. Further, the changes from 20 MPa/m to 40 MPa/m are more obvious than those from 60 MPa/m to 80 MPa/m among the frequency ranges of 6 Hz to 28 Hz and 44 Hz to 66 Hz.  
From Figure 9b, it can be observed that the variations in the slab track's power flows were complicated with increasing slab mat layer stiffness. However, the slab track's power flows for 20 MPa/m reached minima at frequencies of 60 Hz to 82 Hz and 161 Hz to 211 Hz, whereas they reached maxima at frequencies of 4 Hz to 9 Hz, 12 Hz to 28 Hz, and 37 Hz to 55 Hz. There were small differences in the power flows at other frequencies. From the above analysis, the influence of the stiffness of the slab mat layer on the power flows of the slab track mainly occurs below 80 Hz, that is, the low frequency range.  
Figure 9c demonstrates that the bridge's power flows for 80 MPa/m attained maxima at frequencies of 5 Hz to 12 Hz, 28 Hz to 48 Hz, 57 Hz to 150 Hz, and 217 Hz to 267 Hz. Further, the bridge's power flows rose with increasing stiffness of the slab mat layer within the frequency ranges of 5 Hz to 12 Hz and 87 Hz to 139 Hz, and in those frequency ranges, greater stiffness of the slab mat layer was not in  
Figure 9: The power flows of (**a**) the rail, (**b**) the slab track, and (**c**) the bridge.  
favor of reducing vibration. Besides this, differences in the stiffness of the slab mat layer had little effect on the power flows of the bridge above 270 Hz.  
Figure 10 demonstrates the PFTs of the track-bridge system with different stiffness levels of the slab mat layer. It can be observed from Figure 10a that the PFTs were all below 1.0 when the vibration energy transferred from the rail to the slab track. In addition, the PFTs for 20 MPa/m reached maxima at frequencies of 12 Hz to 30 Hz and 37 Hz to 46 Hz, whereas they reached minima at frequencies of 58 Hz to 92 Hz and 160 Hz to 212 Hz. There were few differences in the remaining frequency ranges.  
Figure 10b shows that the PFTs from the slab track to the bridge were all below 1.0, except for at some frequencies. Moreover, the PFTs basically rose with increasing slab mat layer stiffness below 158 Hz; that is, the stiffness mainly plays a role in the PFTs at a low frequency range. Therefore, if we employ greater stiffness of the slab mat layer in the low frequency range, more vibration energy will transfer to the bridge, which will go against the reduction of ambient vibration.  
Figure 11 indicates the AELVs of the track-bridge system. It can be observed that although the variations of the AELVs of the rail were slight, the AELVs of the rail increased when the slab mat layer stiffness decreased. On the contrary, the changes in the AELVs of the slab track and the bridge are obvious, and the AELVs of the slab track and the bridge increased when the slab mat layer stiffness increased. Therefore, the AELVs of the slab tack and the bridge are more sensitive to changes in the slab mat layer's stiffness than are those of the rail.  
Figure 11: The AELVs of the track–bridge system.  
Figure 10: The PFTs from (**a**) the rail to the slab track and (**b**) the slab track to the bridge.