# 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 Shear-Hinge Frequency-Response Function When Metal Padding Is Added  
To further investigate the resonant frequency of the TMSH installation state, frequency-response analysis was performed on the TMSH. According to the peak of the excitation frequency of the floating slab, the sweeping frequency range was determined to be 20-3000 Hz for sinusoidal excitation, and the vibration amplitude was extracted when the steady state was reached. It was found that the modal vibration pattern had a certain symmetry along the longitudinal direction. Considering the damage found during the field research, the frequency-response function curves of a total of four damage-prone areas under vertical and transverse load excitations were extracted from the connection between the shear rod and the sleeve, and the root of the head of the front, middle, and rear three bolts (see figure for definition), respectively. The extraction results are shown in Figure 13.  
Using the resonance peak position of the frequency-response function curve, as shown in Figure 13a, it was found that the shear rod connected with the sleeve was more sensitive to transverse excitation near the second order (2184 Hz), and it was observed that the modal vibration pattern of this order had some influence on the connection between the shear rod and the sleeve. In Figure 13b, it can be seen that under the vertical excitation of the floating slab, the shear rod had a resonance peak in the third order (2265.5 Hz), a resonance peak existed near the seventh order (2710.2 Hz), and the acceleration admittance at 2689.2 Hz was six times higher compared to that at 2253.5 Hz. The third-order vibration pattern mainly manifested as twisting of the front end of the shear hinge and reverse bending of the shear rod, and the seventh-order vibration pattern was characterized by the bending and buckling of both ends of the sleeve, which also caused bending of the shear rod in the opposite direction to the sleeve's bending direction. Based on this, it can be speculated that under long-term cyclic high-frequency excitation, the second order, the third order, and, especially, the seventh order may cause the shear rod to develop cracks and eventually fracture. Figure 12: Comparison of vibration levels under four operating conditions before and after adding rubber pads to the shear hinge. Compared with the shear rod, the effect of external excitation on the bolt was small, but there were multiple resonance peaks. Figure 13a shows that the front-end, middle-end, and rear-end bolts all had resonance peaks at 2260.5 Hz during lateral excitation, i.e., the third order (2265.5 Hz), and this order frequency had a greater effect on the front-end bolt. The rear-end bolt and middle-end bolt had resonance peaks at 2443.5 Hz, i.e., a peak also appeared in the fifth-order mode (2444.5 Hz), and this order mode's rear-end bolt response was greater compared to that of the third-order mode. In the middle-end bolt, despite having a resonance peak in both mode orders, compared to the front-end and rear-end bolts, its peak was not obvious. In the vertical excitation of the front-end bolts in the third order and the seventh order, there were two resonance peaks. The seventh-order peak was higher, the rear-end bolts in the second-, third, and seventh-order modes had weak resonance peaks, and the middle-end bolts were not excited. The front-end bolt was prone to damage in the third-order mode and seventh-order mode, the middle-end bolt was prone to damage in the third-order mode, and the rear-end bolt was prone to damage in the second-order mode and fifth-order mode. In summary, under train-load excitation, the TMSH's natural frequency resonates with the high-excitation frequency of the floating slab, and the resonant frequency intensifies the fatigue striations generated during the actual vibration process. This eventually accelerates the rate of fatigue fracture of the member from high-frequency vibration, which, in turn, leads to broken bolts and shear-rod fracture problems in the TMSH.