# 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
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{8}{c}{**Line Grade**} \\ \cline{2-9}  & **Class One** & **Class Two** & **Class Three** & **Class Four** & **Class Five** & **Class Six** \\ \hline \(A_{\varphi}\) (cm\({}^{2}\cdot\)rad/m) & 1.2107 & 1.0181 & 0.6816 & 0.5376 & 0.2095 & 0.0339 \\ \hline \(A_{a}\) (cm\({}^{2}\cdot\)rad/m) & 3.3634 & 1.2107 & 0.4128 & 0.3027 & 0.0762 & 0.0339 \\ \(\Omega_{s}\) (rad/m) & 0.6046 & 0.9308 & 0.8520 & 1.1312 & 0.8209 & 0.4380 \\ \(\Omega_{c}\) (rad/m) & 0.8245 & 0.8245 & 0.8245 & 0.8245 & 0.8245 & 0.8245 \\ \hline Maximum allowable speed (km/h) & Truck & 16 & 40 & 64 & 96 & 128 & 176 \\ \hline \end{tabular}
\end{table} 
Table 6: US AAR orbital spectral parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Overhead**} & **Grade of** & **Deraliment** & **Horizontal** & **Acceleration** & **Lateral** & **Wheel Weight** & **Sperling Index** \\ \cline{3-8}  & **Irregularity** & **Coefficient** & **Force (kN)** & **(m/s2)** & **(m/s2)** & **Rate** & **Vertical** & **Lateral** \\ \hline \multirow{8}{*}{I} & Level 1 & *1.98* & *154.91* & 0.87 & *2.41* & *2.48* & 2.38 & 2.69 \\  & Level 2 & *1.37* & *126.78* & 0.22 & *2.53* & *1.92* & 1.56 & 2.54 \\  & Level 3 & *1.63* & *71.93* & 0.30 & *1.44* & *1.27* & 1.34  
Tables 7 and 8 show that the dynamic performance of the two types of overhead systems gradually improves as the track irregularity level decreases. The two overhead systems' performance improvements are minimal but fall within the normative limit listed in Table 5, demonstrating that the overhead systems' sensitivity to the level of track irregularity is low. Under the same irregularity level, the dynamic deflection of the I overhead system is less than the II aerial system response. The dynamic response of trains is significantly influenced by the unevenness level, and the dynamic response indices gradually rise as the unevenness level decreases. Table 7 demonstrates that the risk of a derailment is decreased when the level of track irregularities is controlled above AAR5. The safety indexes of train operation are primarily impacted by track irregularity, and the Sperling indexes of train bodies satisfy all levels of the requirements. In order for the smoothness of the track to meet the requirements of the AAR5 standard, the change in track line must be strictly monitored during the construction of this road bridge. To assess the system's safety, the dynamic response of the overhead system under a load of freight trains was computed. The target speed was set at 45 km/h, and AAR5 was used to adjust for track unevenness. Figure 14: I Overhead system dynamic response: (**a**) Mid\(-\)span vertical deflection (unit: mm). (**b**) Mid\(-\)span vertical acceleration (unit: m/s\({}^{2}\)). (**c**) Mid\(-\)span lateral displacement (unit: mm). (**d**) Mid\(-\)span lateral acceleration (unit: m/s\({}^{2}\)). Figure 15: II Overhead system dynamic response: (**a**) Mid\(-\)span vertical deflection (unit: mm). (**b**) Mid\(-\)span vertical acceleration (unit: m/s\({}^{2}\)). (**c**) Mid\(-\)span lateral displacement (unit: mm). (**d**) Mid\(-\)span lateral acceleration (unit: m/s\({}^{2}\)). As seen in Table 9, when there is a significant wheel/rail lateral force that is close to the specification limit, freight trains can cross the I aerial system. In general, both overhead systems can carry the necessary loads at their intended speeds.