# Vibration Response Analysis of Overhead System Regarding Train-Track-Bridge Dynamic Interaction
### Overhead System Model  
#### 1.1.1 Overhead System Parameters  
The Jingjiu-Longhai railway overpass is built in the form of a crossing ground bridge. During construction, horizontal and vertical beams are used to reinforce the existing track. Meanwhile, the train passing through this section is limited to a maximum speed of 45 km/h. The overhead railway line is 80 m long, with 10 \(\times\) 8 m continuous steel structural beams used for reinforcement. The longitudinal beam is made of I100 I-beam, and the transverse girder is made of I56b I-beam, with four transverse spans (12 m and 9 m) and four ordinary beret plates forming a group. The transverse spans (12 m and 9 m) are referred to as \"I aerial system\" and \"II aerial system\", respectively. Figures 1 and 2 show cross-sections and elevations.  
Figure 1: Cross-section of an overhead system (unit: mm): (a) Cross-section of type I overhead system. (b) Cross-section of type II overhead system.  
This study uses the T60 rail type, which has the equivalent rectangular section, equivalent height \(h\), and equivalent width \(b\). The T60 rail has an elastic modulus (E) of \(2.059\times 10^{11}\) MPa. Then:  
\[b\;h=A \tag{1}\]  
\[bh^{3}/12=J_{y} \tag{2}\]  
where: \(A\)-rail section area, value \(A=77.45\) cm\({}^{2}\).  
\(J_{y}\)-second moment of area, value \(J_{y}=3217\) cm\({}^{4}\).  
Simultaneous Equations (1) and (2) are obtained. \(b=0.0347\) m, \(h=0.223\) m. The rail equivalent is shown in Figure 3.  
The sleeper type is the speed-up line sleeper (type II), and the basic parameters are shown in Table 1.  
Figure 3: Model of equivalent rectangular rail beam (unit: m).  
Figure 2: Elevation of the overhead system (unit: mm): (**a**) Elevation of type I overhand system. (**b**) Elevation of type II overhand system.  
Table 2 and 3 display the precise dimensional parameters of I100 I-beams and I56 I-beams. Material 16 Mn; chord 2I10a; channel steel (C100 48 5.3/8.5, spacing 8 cm); web 18 (H = 80 mm, B = 50 mm, Tf = 4.5 mm, Tw = 6.5 mm) are the specifications for the Bailey plate transverse beam. The Bailey plate's connection is made with a pin. The \"V\" arrangement of T-shaped steel is used to reinforce the reinforcement system, and the type of T-shaped steel used is T75-150 \(\times\) 7 \(\times\) 10 (\(h\) = 75 mm, \(b\) = 150 mm, Tf = 10 mm, Tw = 7 mm). This ensures the transverse stability of the longitudinal and transverse beams. Figure 4 illustrates specific material dimension information.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Main Type** & **Sleeper Quality** & **Sleeper Length** & **The Average** & **Effective** \\  & \(M_{s}\) (kg) & \(L_{s}\) (m) & **Bottom Surface** & **Support Length** \\  & & & & \(l_{b}\) (m) & \(l_{e}\) (m) \\ \hline Speed-up line & & & & \\ sleeve (type II) & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Basic parameters of sleeper.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Name** & **Type** & \begin{tabular}{c} **Elasticity** \\ **Modulus** \\ **(Pa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Moment of** \\ **Inertia** \\ **(m\({}^{4}\))** \\ \end{tabular} & \begin{tabular}{c} **Acreage** \\ **(m\({}^{2}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Altitude** \\ **(m)** \\ \end{tabular} &
\begin{tabular}{c} **Density** \\ **(kg/m\({}^{3}\))** \\ \end{tabular} \\ \hline I-beam & I100 & \(2.1\times 10^{11}\) & 0.3 & \(9.23\times 10^{-3}\) & \(6.56\times 10^{-2}\) & \(1\) & 7800 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation parameters of the strengthened longitudinal beam.  
Figure 4: _Cont._