# Analysis of the Vibration Characteristics of Ballastless Track on Bridges Using an Energy Method
## 2 Methods
### Description of Power Flow Theory  
The average power flow method in the frequency domain was adopted in this paper to conduct analysis of the vibration energy. The mathematical expression [19] of the average power flow method is shown in Equation (11):  
\[P(\omega)=\frac{1}{2}\mathrm{Re}(F\cdot v^{*}), \tag{11}\]  
where \(F\) is the complex number value of forces in the frequency domain, that is, \(\mathrm{F}=F_{0}\cdot e^{\mathrm{i}\omega}\); \(v\) denotes the complex number value of velocities in the frequency domain, that is, \(\mathrm{v}=v_{0}\cdot e^{\mathrm{i}\omega}\); \(\mathrm{Re}(.)\) represents the real part of a complex number; and superscript \({}^{*}\) refers to the complex conjugate.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Dynamic Response**} & **Rail vertical** & **Track Slab Vertical** & **Bridge Vertical** \\  & **Acceleration (g)** & **Acceleration (g)** & **Acceleration (g)** \\ \hline [PERSON] et al. [18] & 139.2 & 3.6 & 0.24 \\ This paper & 136.8 & 3.5 & 0.21 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparisons of the maximum dynamic responses.  
To describe the influence of the impact effect on the vibration energy of the track-bridge system, the node power flows of the research area need to be summed up. The mathematical expression of the total power flow is presented in Equation (12):  
\[P(f)=\frac{1}{2}{\sum\limits_{t=1}^{n}{\rm{Re}}(F_{i}(f)\cdot v_{i}^{*}(f))}, \tag{12}\]  
where \(f\) denotes the calculated frequency and \(n\) is the number of nodes.  
The power flow of the track-bridge system reflects the transfer of vibration energy in the components of the track-bridge system, and the vertical transfer of vibration energy is illustrated in Figure 4. In order to balance the accuracy and calculation cost and eliminate the influence of the boundary effect, a proper length of research area needs to be chosen. In this work, a length of track slab in the middle of the vehicle-track-bridge system was selected. Furthermore, the power flows of the rail and connecting elastic elements of the fasteners were regarded as the input vibration energy of the track-bridge system with the symbol \(P_{in}\). The power flows of the slab track and connecting elastic elements of the slab mat layer were regarded as the input vibration energy of the slab track with the symbol \(P_{1}\). The power flows of the bridge body and the connecting elastic elements of the slab mat layer were then summed up as the input vibration energy of the bridge with the symbol \(P_{2}\). Finally, the power flow transfer from the bridge to the ground was denoted \(P_{out}\).  
After the total power flow of the track-bridge system is obtained, the relative power flow needs to be calculated for obvious comparisons. The mathematical expression is depicted in Equation (13):  
\[P_{\rm{re}}(f)=20\cdot\log_{10}(\frac{P(f)}{P_{0}}), \tag{13}\]  
where \(P(f)\) is the total power flow, which is determined by Equation (12); \(P_{0}\) denotes the reference power flow with a value of \(1\times 10^{-8}\) N\(\cdot\)m/s.