## 2 Theoretical Analysis Model and Calculation Method
### Method for Calculating Vibration Power Flow  
Since this paper mainly studies the vibration energy transfer of the vibration-damping track on the bridge line, the frequency domain average power is selected to evaluate its transmission characteristics. The average power calculation formula is as follows. \[P(w)=\frac{1}{2}\text{Re}(FV^{*}) \tag{1}\]  
where \(P(w)\) is the power flow, and its unit is N\(\cdot\)m/s, that is, \(W\). The complex values of \(F\) and \(V\) represent the velocity in the frequency domain. Re represents the real part of a complex number. The superscript * represents conjugation. The velocity and force of the track, integral ballast bed, and box girder bridge joints obtained from the vehicle-track-bridge coupling dynamics model are used as input conditions for power flow analysis. However, in the dynamic analysis modeling presented in this paper, beam elements and plate shell elements are used for the track and box girder bridge, respectively. This approach makes it impossible to directly extract the nodal forces \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Track Component** & **Cell Type** & \begin{tabular}{c} **Correlation** \\ **Parameter** \\ \end{tabular} & **Numerical Value** \\ \hline \multirow{4}{*}{Track} & \multirow{4}{*}{Beam188} & Cross-sectional & \multirow{4}{*}{2.1 \(\times\) 10\({}^{11}\)} \\  & & \begin{tabular}{c} \(\text{inertia}/(\text{m}^{4})\) \\ Elastic modulus/(Pa) \\ Poisson’s ratio \\ \end{tabular} & 3.215 \(\times\) 10\({}^{5}\) \\ \hline \multirow{4}{*}{Fastener} & \multirow{4}{*}{Combin14} & Linear & \multirow{4}{*}{60.64} \\  & & density/(kg\(\cdot\)m\({}^{-1}\)) & 4 \(\times\) 10\({}^{7}\) \\ \cline{1-1} \cline{3-4}  & & \
\begin{tabular}{c} \(\text{Damp}/(\text{N}\cdot\text{s}\cdot\text{m}^{-1})\) \\ Elastic modulus/GPa \\ \end{tabular} & 2.26 \(\times\) 10\({}^{4}\) \\ \hline \multirow{2}{*}{Integrated ballast bed} & \multirow{2}{*}{Solid45} &
\begin{tabular}{c} \(\text{Density}/(\text{kg}\cdot\text{m}^{-3})\) \\ Elastic modulus/GPa \\ \end{tabular} & 2251 \\ \hline \multirow{2}{*}{Bridge} & \multirow{2}{*}{Shell63} & Elastic modulus/Pa & 3 \(\times\) 10\({}^{10}\) \\ \cline{1-1} \cline{3-4}  & & Poisson’s ratio & 0.25 \\ \cline{1-1} \cline{3-4}  & & Density/(kg\(\cdot\)m\({}^{-3}\)) & 2500 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Track structural parameters and bridge structural parameters. Figure 2: Track-integral ballast bed–bridge finite-element model. of each node. Therefore, the internal stress of the node is extracted from the element in which the node is situated, rather than directly from the node force. In the power flow analysis, if the entire span of the overhead line is used for solving, a significant number of nodes are involved in the calculation process, leading to a notable reduction in solving efficiency. Given that this paper primarily analyzes the power flow transmission characteristics of elevated lines, a finite-element model of 1/6 of the lines can be selected for analysis. In other words, the 5 m track and bridge structure can be calculated to effectively improve the calculation efficiency while ensuring the accuracy of the calculation structure. The total power flow can be expressed as follows. \[P_{w}(k)=\frac{1}{2}\sum_{i=1}^{n}\mathrm{Re}\left[F_{i}(k)V_{k}^{i}(k)\right] \tag{2}\]  
where \(i\) is the node number of the desired power flow. n is the total number of nodes. \(k\) is the calculation frequency point, the unit is Hz. The power flow characteristics of the track, integral ballast bed, and box girder bridge can be obtained individually. The power flow characteristics of the entire system can then be determined by summing the characteristics of each subsystem. Figure 3: Numerical simulation utilizing the sixth-level spectrum of American railroads: (**a**) Random vertical irregularity of the left rail; (**b**) Random vertical irregularity of the right rail; (**c**) Random lateral irregularity of the left rail; (**d**) Random lateral irregularity of the right rail.