# Time Domain Damage Identification Approach of the Newly Hollow Floor Structure with TRMD
## 3 Numerical Example  
A six-story frame structure with a lumped mass model subject to earthquake excitation is taken as an example to verify the accuracy of the proposed identification method. In this numerical example, each lumped mass is selected to be the same at 16.32 \(\times\) 10\({}^{3}\) kg, the stiffness coefficients of each story are assumed to be 13.51, 12.86, 11.58, 9.65, 7.07, and 3.86 \(\times\) 10\({}^{3}\) kN/m, respectively, and the damping coefficients of each story are 27.9, 23.76, 21.39, 17.87, 13.07 and 10.04 kN-s/m. The first six natural frequencies of the controlled structure are 6.28, 15.38, 24.32, 33.23, 42.13, and 51.02 rad/s. In order to achieve the optimum vibration control effect, the natural frequency of the damper needs to be resonant with the natural frequencies of the structure. Therefore, the radius difference between the arc track and oscillator \(\rho\) is calculated as 0.178 m.  
It is assumed that the oscillator located in the top story of the structure is made of steel with a mass density of 7850 kg/m\({}^{3}\). The mass ratio of the tuned rolling mass damper system mass to the total mass of the main structure is selected as 5%, so the oscillator has a physical mass of 395.3 kg and a radius of 0.230 m. The external excitation, in this case, can be written as \(F_{i}=-(M_{ii}+\sum_{j=1}^{n_{i}}m_{ii})\bar{x}_{g}(t)\), where \(\bar{x}_{g}(t)\) is the El-Centro earthquake acceleration, with a peak value of 1.37 m/s\({}^{2}\). Figure 5 shows a comparison of story displacement with and without the TRMD of structures subject to El-Centro excitation. As can be observed, although the effectiveness of the TRMD at controlling the vibration for the peak responses was not significant, the responses after the peak had been reached were reduced to a large degree because of the energy dissipation mechanism of the TRMD. In order to study the factors affecting the accuracy of the time domain damage identification approach of the controlled structure with TRMD, Table 1 shows four different cases of damage identification considering damage location, damage severity, and noise severity. It is shown in Table 1 that the stiffness of the 3 rd floor is reduced by 15% in Case 1 and Case 2, and the stiffness of the 3 rd floor and 5 th floor are respectively reduced by 15% and 10% in Case 3 and Case 4. In Case 1 and Case 3, the measurement responses with no noise are considered, and 5% measurement noise is considered in Case 2 and Case 4. Considering the acceleration responses of the controlled structure with the TRMD directly as the damage  
Figure 4: The damage identification flow chart of the controlled structure with TRMD. identification index, the sensitivity of the acceleration response with respect to elemental stiffness factors is employed to identify local damage by minimizing the difference between the intact state response and the damage state response. In this numerical example, the dynamic acceleration response difference at the 3 rd floor is selected as the objective function for damage identification. Figure 6 shows the dynamic acceleration response at the 3 rd floor for the intact state and the damage state of Case 1. Local damages in the six-story frame structure are identified by the iterative model updating technique based on the dynamic response sensitivity illustrated in the previous section. Figure 7 shows the identified damage results in the four different cases. It is shown that the identified damage location and extent are consistent with real damages in four cases from Figure 7a-d. The identified stiffness and stiffness variation ratio of each story in the four damage cases are listed in Table 2. It is shown in Table 2 that the measurement responses without noise give better accuracy damage identification results with a relative difference of 0.23% in Case 1. In Case 2, the location and extent of the damage can also be identified with a relative difference of 0.50% under the measurement responses with 5% noise.