# A New Top-Mounted Shear-Hinge Structure Based on Modal Theory and Rubber-Pad Damping Theory
## 5 Installation Modal Testing
### Modal Test under Installed Conditions  
For this modal test, the Eastern Institute INV-3018 synchronous acquisition card was used due to the finite-element calculation results of the modal frequency being high. To ensure that the modal test did not lose order, we selected two ranges of 50 g and a range of 100 g for the PCB acceleration sensor via external tooling to form three acceleration sensors (the specific parameters are shown in Table 11). We used multi-point excitation and multi-point response (MIMO) with 2 preset response points and 67 excitation points. The shear hinge's natural frequency of concern was concentrated in the middle-to-high range, and a metal hammer head was used to generate pulse excitation. Due to the limitation of the test conditions, the modal test was carried out indoors, and a standard C55 concrete test block and rubber pad were used to simulate the FST to imitate actual floating-slab working conditions. According to [4], in a floating-slab structure that does not consider the subway train load, the FST's natural frequency should comply with the following formula:  
\[f_{0}=\frac{1}{2\pi}\sqrt{\frac{k_{f}}{m_{f}}} \tag{19}\]  
where:  
\(f_{0}\) is the floating slab's natural frequency (Hz);  
\(k_{f}\) is the floating slab's support stiffness per linear meter (N/m);  
\(m_{f}\) is the mass of the floating slab per linear meter (kg). The floating slab is mainly excited by the train load from the vertical direction during operation, and the support stiffness mentioned in the formula is understood as vertical stiffness. The first-order natural frequency of the prefabricated floating slab is about 13 Hz [36]. Considering the space limitation of the test chamber and the size of the shear hinge, the planned size of the cast concrete support was 0.5 m \(\times\) 0.3 m \(\times\) 0.18 m, its density was 2490 kg/m\({}^{3}\), and its calculated mass was 67.23 kg. Substituting these values into Equation (19) gives \(k_{f}\) = 448,548.65 Hz. According to the formula for the rubber pad vertical stiffness (see Section 2.4), considering the stability of the concrete support and to ensure that the natural frequency was close to that of a real floating slab, four circular rubber pads were used for support. The principle was the same as that for four springs in parallel, so each rubber pad's vertical stiffness was required to be 112,137.16 Hz, which was substituted into the specific parameters of Formula (2) (see Table 12). The formula has two unknowns at the same time: the thickness of the rubber \(t_{r}\) and the rubber pad diameter D. These have an obvious positive relationship with the vertical stiffness, but through MATLAB R2022b software analysis, it was found that an increase in both of these factors at the same time will increase \(K_{v}\). In order to obtain the target vertical stiffness and ensure that it is in line with the actual value, we assumed that \(t_{r}\) and D were 30 mm, 40 mm, and 50 mm at vertical stiffness values of 134,120 N/m, 178,820 N/m, and 223,530 N/m, respectively. Obviously, with a thickness of 30 mm, when the vertical stiffness is closer to the target value, the thickness cannot be greater than the diameter, or the support will be unstable. Therefore, the diameter was reduced and calculated using Formulas (2)-(4). Finally, for each rubber pad, we assumed that D = 28 mm and \(t_{r}\) = 30 mm. \begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Sensitivity (mv/g)** & **Frequency Range Hz** \\ \hline PCB 352C33/ICP & 100 & 0.5–10,000 Hz \\ PCB 353B31/ICP & 50 & 1–5000 Hz \\ \hline \end{tabular}
\end{table} 
Table 11: Sensor parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & **Experimental** & \multirow{2}{*}{**Experimental Nephogram**} & **Simulation** & \multirow{2}{*}{**Simulation Nephogram**} & **Error** \\
**Order** & **Mode/Hz** & & & & \\ \hline
1 & 1763.5 & & 1947.4 & & 9.4\% \\
2 & 2022.6 & & 2248.8 & & 10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Calculated parameters of vertical stiffness of rubber pad.