# Analysis of the Vibration Characteristics of Ballastless Track on Bridges Using an Energy Method
## 3 Results and Discussion
### Influence of the Fasteners' Stiffness  
In this subsection, the power flows and corresponding indices are calculated for the running speed of 300 km/h when the fasteners' stiffness is set to 20 kN/mm, 40 kN/mm, 60 kN/mm, and 80 kN/mm.  
The power flows of the track-bridge system under different stiffness conditions are depicted in Figure 12. It can be observed from Figure 12a that the changes in the fasteners' stiffness mainly had an effect on the power flows of the rail below 200 Hz, and there were little differences in the rail's power flows above 200 Hz. In addition, the rail's power flows decreased at frequencies below 200 Hz when the fasteners' stiffness increased. Therefore, it can be said that variation in the fasteners' stiffness mainly influences the power flows of the rail in a low frequency range. For the purpose of reducing rail vibration at the low frequency range, greater stiffness of the fasteners is recommended.  
Figure 12b indicates that the slab track's power flows increased when the fasteners' stiffness rose at frequencies of 14 Hz to 67 Hz. The power flows of the slab track when applying fasteners of 20 kN/mm reached minima at frequencies of 1 Hz to 8 Hz as well. Although the power flows of the slab track showed little differences above 80 Hz, it can be inferred that the power flows with fastener stiffness of 80 kN/mm were basically at maximum.  
From Figure 12c, it can be observed that the bridge's power flows did not simply vary when the fastener stiffness increased. The bridge's power flows for 20 kN/mm reached minima at frequencies of 1 Hz to 7 Hz, 17 Hz to 30 Hz, and 63 Hz to 75 Hz. The bridge's power flows for 40 kN/mm also reached minima at frequencies of 13 Hz to 17 Hz and 50 Hz to 62 Hz. Nevertheless, the bridge's power flows when adopting the 80 kN/mm fasteners were mainly at maximum at the frequencies investigated.  
Figure 12: The power flows of (**a**) the rail, (**b**) the slab track, and (**c**) the bridge.  
The PFTRs of the track-bridge system with different fastener stiffness levels are depicted in Figure 13. It can be seen from Figure 13a that the PFTs were all below 1.0 when the vibration energy transferred from the rail to the slab track. Besides this, the PFTs increased as the fasteners' stiffness rose at frequencies of 9 Hz to 139 Hz, except for some frequencies. In addition, the PFTs for 40 kN/mm reached minima for frequencies below 9 Hz. The PFTs for 80 kN/mm were basically at maximum in the frequency range investigated.  
Figure 13b indicates that the PFTs from the slab track to the bridge were around 1.0 when the frequencies were below 10 Hz and the PFTs were smaller than 1.0, expect for some frequencies. The PFTs of the bridge were maximum at the frequency of 19 Hz, reaching 1.015 with the stiffness of 80 kN/mm. The PFTs under all four stiffness conditions presented little variation when the frequencies were larger than 190 Hz.  
Figure 14 shows the AELVs of the track-bridge system. It can be clearly observed that the AELVs of the track-bridge system all increased as the fasteners' stiffness rose. In other words, the track-bridge system stores more vibration energy when the fasteners' stiffness increases. It can be concluded that the AELVs of the track-bridge system are sensitive to changes in the stiffness of the fasteners.  
Figure 14: The AELVs of the track–bridge system.  
Figure 13: The PFTRs from (**a**) the rail to the slab track and (**b**) the slab track to the bridge.