# A New Top-Mounted Shear-Hinge Structure Based on Modal Theory and Rubber-Pad Damping Theory
## 4 Modal Analysis and Frequency-Response Analysis under Assembly Conditions
### Analysis of TMSH Modalities When Using a Rubber Pad  
According to the new structure proposed in Section 2.4, the rubber gasket is placed between the TMSH and the metal pad, and the modal calculation results are shown in Figure 11, and the vibration patterns are described in Table 6.  
Only the third-order-mode TMSH was found to occur in vibration mode. According to the theoretical calculation, the natural frequency of the new structure should be about 100 Hz, but the first few orders of vibration-mode panning are mostly considered to be false, so they were not considered. A mode of more than 313 Hz on the TMSH is considered weak, so the third-order mode was extracted. It can be seen in Table 7 that the TMSH's natural frequency decreased significantly after adding the rubber pad, and the average modal improvement rate was 86-87%, which proves that the rubber pad can effectively reduce the TMSH's natural frequency. According to the results, the TMSH's natural frequency after the installation of the rubber pad was in the operating low-frequency region of 250-315 Hz for the floating slab. According to the currently available data [29], four different working conditions were established, as shown in Table 8, which lists the differences in vibration levels when the TMSH was placed on the rubber pad and the metal pad, respectively. Moreover, the magnitudes of the vibration levels were compared when the TMSH's natural frequency coincided with the excitation frequency of the SSFST before and after the addition of the rubber pad, and the results are shown in Figure 12.  
In Figure 12 and Table 7, it can be seen that the vibration levels increased considerably from Case 2 to Case 3, which is due to the fact that Case 1 and Case 2 represent the results of the hammer test conducted indoors, whereas Case 3 and Case 4 represent the field measurements of the running line, and the field excitation was larger compared to the hammer excitation indoors. Meanwhile, it can be seen that the vibration level of the TMSH after adding the rubber pad was lower compared to that of the structure without the TMSH, except in the vibration test of the linear section. Moreover, the vibration levels at the location of the natural frequency under the other three working conditions were reduced, with the vibration levels of working condition 2 and working condition 4 reduced by 9 dB, and the vibration level of working condition 1 reduced by 3 dB. This proves that the installation of the rubber pad can effectively reduce the vibration level of the TMSH in the curved sections of the 4.8 m floating slab and 3.6 m floating slab. It is proven that the installation of rubber pads can effectively reduce the vibration levels of the 4.8 m floating slab and the 3.6 m floating slab when the TMSH's natural frequency and the excitation frequency of the floating slabs coincide so that the amplitude under the cyclic load is reduced and the fatigue-damage time is slowed down. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **Frequency/Hz** & **Description of the Vibration Mode** \\ \hline First & 261.83 & The TMSH is rotated along the \(x\)-axis and slightly bent along the \(z\)-axis. \\ Second & 293.22 & The TMSH is bent along the \(z\)-axis \\ Third & 313.73 & The TMSH is bent along the \(y\)-axis \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Description of the vibration pattern when the TMSH is placed on the rubber pad. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Metal Pad** & **Rubber Pad** & **Modal Improvement Rate** \\ \hline First & 1947.4 & 261.8 & 86.6\% \\ Second & 2248.8 & 293.2 & 87\% \\ Third & 2274.1 & 313.7 & 86.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of the shear force before and after the addition of rubber pads in relation to the natural frequency of the structure.