# Analysis of the Vibration Characteristics of Ballastless Track on Bridges Using an Energy Method
## 4 Conclusions  
In this paper we presented an energy method based on power flow theory to investigate the vibration characteristics of the ballastless track on bridges. Firstly, the methodology of the numerical model of the vehicle-track-bridge coupled system, power flow theory, and the corresponding evaluation methods were presented. Then, the vibration characteristics under different running speeds and various stiffness levels of the fasteners and slab mat layer were investigated. The major conclusions of this work are as below:  
1. The vibration energy declined obviously from top to bottom of the track-bridge system at the frequencies investigated. This is because the energy gradually attenuates from top to bottom in the transfer process. Moreover, the attenuation effects are mainly the result of the elasticity and damping effects of the fasteners and the slab mat layer.
2. With increasing slab mat layer stiffness, the vibration energy of the rail decreased slightly; on the contrary, that of the slab track and the bridge increased obviously. That is to say, the vibration  
Figure 16: The PFTRs from (**a**) the rail to the slab track and (**b**) the slab track to the bridge.  
Figure 17: The AELVs of the track-bridge system.  
energy of the rail is less sensitive than that of the slab tack and the bridge to changes in the slab mat layer stiffness. Therefore, the stiffness of the slab mat layer employed ought to be a suitable value; a larger one would cause more environmental vibration, and a smaller one would cause rail defects, even polygonal wheels.
3. With increasing fastener stiffness, the vibration energy of the entire track-bridge system increased, as the track-bridge system stores more vibration energy under such conditions. It can be inferred that variation in the fasteners' stiffness plays an important part in the vibration energy of the track-bridge system. To sum up, the stiffness of the fasteners employed ought to be a suitable value as well; larger stiffness would cause an increase in the ambient vibration, while smaller stiffness would affect the running safety of the vehicle.
4. With increasing running speed, the vibration energy of the track-bridge system increased obviously. That is to say, the running speed is a sensitive index for the vibration energy of the track-bridge system. Therefore, it can be concluded that the running speed plays an important part in the vibration energy of the track-bridge system as well.  
The results reveal that reasonable stiffness levels of the fasteners and the slab mat layer are 40 to 60 kN/mm and 40 to 60 MPa/m, respectively, under the conditions investigated in this work. Further, the findings in this work provide a novel way to investigate the vibration characteristics of track-bridge systems in terms of energy, and the developed model can be adopted to study the vibration characteristics of the ballastless track on bridges of HSRs once the corresponding parameters have been determined.  
Conceptualization, [PERSON] and L.G.; methodology, [PERSON] and L.G.; validation, [PERSON] and L.G.; formal analysis, [PERSON] and L.G.; investigation, [PERSON] and L.G.; data curation, [PERSON] and L.G.; writing--original draft preparation, [PERSON] and L.G.; writing--review and editing, [PERSON] and L.G.; visualization, [PERSON] and L.G.; funding acquisition, L.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China under Grant 51827813, and by the Key Program of National Natural Science Foundation of China under Grant U1734206.  
The authors express their gratitude to the anonymous reviewers and [PERSON] for their valuable advice.  
The authors declare no conflict of interest.