# Vibration Response Analysis of Overhead System Regarding Train-Track-Bridge Dynamic Interaction
## 3 Coupling Vibration Analysis of Train-Track-Overhead System
### Influence of Track Irregularity on the Dynamic Response of Train Track Overhead System  
Orbital irregularities are variations in orbital geometry, space position, and size from their typical states. Direction, gauge, height, and level irregularity are the four fundamental types of track irregularity that are typically discussed [20; 21; 22]. Short wave irregularity, medium wave irregularity, and long wave irregularity are the three levels of irregularity that can be classified based on wavelength [23; 24; 25]. The North American Railway Association (AAR) obtained the power spectral density curve of track irregularity based on a significant amount of observed data, and fitted it into an even function indicated by roughness constant and cutoff frequency. The track level is separated into six levels, with the first level being the worst and the sixth level being the best. The wavelength range is 1.524-304.8 m.  
(1) Uneven height:  
\[S_{v}(\Omega)=\frac{kA_{v}\Omega_{c}^{2}}{\Omega^{2}(\Omega^{2}+\Omega_{c}^{2 })} \tag{8}\]  
(2) Uneven direction  
\[S_{a}(\Omega)=\frac{kA_{a}\Omega_{c}^{2}}{\Omega^{2}(\Omega^{2}+\Omega_{c}^{2 })} \tag{9}\]  
(3) Uneven level and gauge: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Classification** & **Index** & **Passenger Train** & **Freight Train** \\ \hline \multirow{3}{*}{Security Index} & \multicolumn{2}{c}{deraliment coefficient} & 0.8 & 0.8 \\ \cline{2-4}  & Wheel load reduction rate & 0.9 & 0.9 \\ \cline{2-4}  & \multicolumn{2}{c}{Axle lateral force (kN)} & 48 & 84 \\ \hline \multirow{3}{*}{Riding index} & Body acceleration (g) & transverse & 0.10 & 0.5 \\ \cline{2-4}  & vertical & 0.13 & 0.7 \\ \cline{2-4}  & \multirow{2}{*}{Sperling index} & transverse & 3.0 & 4.25 \\ \cline{2-4}  & vertical & 3.0 & 4.25 \\ \hline \multirow{3}{*}{Bridge dynamic performance} & Lateral amplitude (mm) & & 2.00 \\ \cline{2-4}  & Vertical amplitude (mm) & & 10.00 \\ \cline{1-1} \cline{2-4}  & Lateral acceleration (g) & & 0.20 \\ \cline{1-1} \cline{2-4}  & vertical acceleration (g) & & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Standard limits for dynamic performance evaluation of train-bridge systems. \[S_{c}(\Omega)=S_{g}(\Omega)=\frac{4 kA_{\varphi}\Omega_{c}^{2}}{(\Omega^{2}+\Omega_{c }^{2})(\Omega^{2}+\Omega_{s}^{2})} \tag{10}\]  
where: \(S(\Omega)\) is the power spectral density (\(W/\mathrm{Hz}\)); \(\Omega\) is the spatial circular frequency of the track smoothness (rad/m); \(A_{\varphi}\), \(A_{a}\) is the roughness constant (cm\({}^{2}\cdot\)rad/m); \(\Omega_{c}\), \(\Omega_{s}\) is the cutoff frequency (rad/m); \(k\) is the safety factor, and \(A\) is the safety factor, which can range from 0.25 to 1.0 as needed, but is usually 0.25. Table 6 contains the parameter values. For simulation purposes, the American orbital irregularity spectrum was used [26; 27]. The track smoothness wavelength is set at 1-80 m, and the number of harmonics is set at 3000, taking into account the influence of medium and long waves and ignoring the influence of short waves. The train load takes some time to reach steady-state balance. In the model calculation, the first distance of the overhead system was set at 20 m. Simulation analysis was performed on the passenger train for the I and II overhand systems, and six different levels of irregularity for the train's 45 km/h design speed were chosen. Since the dynamic response therein is fairly obvious, this section concentrates on the dynamic response in the middle of the span. Tables 7 and 8 list the maximum value for each response, and a \"*\" is displayed for any data that do not meet the specifications.