## 3 Numerical Investigation  
### General Description  
A five-story shearing building model as shown in Figure 2 is employed in this study for the investigation of the effectiveness of the proposed iterative identification procedure. The structural parameters are set as \(m_{i}\) = 500 kg and \(k_{i}\) = 6.0\(\times\)10\({}^{5}\) N/m (\(i\) = 1, 2,..., 5). The Rayleigh damping coefficients are set as \(\alpha\) = 0.6 and \(\beta\) = 8.0 \(\times\) 10\({}^{-5}\) providing approximately 3% damping ratio for the first mode of vibration. To simulate the structural damage, the stiffness of the first floor is assumed to be 80% of its designed value in the healthy state (i.e., \(k_{1}\) = 4.8 \(\times\) 10\({}^{5}\) N/m). In this numerical example, the building structure is assumed to be subject to El-Centro earthquake with a peak ground acceleration (PGA) of 0.34 g. The corresponding structural responses are calculated  
Figure 1: Flowchart of the proposed iterative identification procedure. Figure 2: Five-story building model in the numerical example. and damping coefficient \(\alpha\). The maximum identification errors for the stiffness in the case of 5% noise and 10% noise are 0.10% and 0.41%, respectively. The errors for identified damping coefficients are relatively large, but even with 10% noise the maximum error is only 4.33%. The comparison of the effect of noise intensity on the iteration process is given in Figure 3. Only the identified stiffness of the 5 th floor (\(k_{5}\)) is plotted in Figure 3 as an example. Similar results can be found for the remaining parameters. It can be observed from Figure 3 that the larger the noise intensity involved, the more iterations are required. For the cases of noise-free, 5% noise, and 10% noise, the number of iterations is 147, 172, and 248, respectively. To further show the convergence process during the iteration, the identified stiffness in the case of 10% noise is given in Figure 4 as an example. Obviously, although the iteration is conducted for 248 times due to the relatively stringent tolerance defined before (i.e., \(\varepsilon=1\%\)), the identified stiffness parameters are stably converged to actual ones, the number of iterations being approximately 50. On the basis of the identified stiffness and the element-wise coefficient \(q_{i}\) in matrix \(\mathbf{Q}\), the damping coefficient \(\beta\) can be calculated according to Equation (16) as 7.556 \(\times\) 10\({}^{-5}\), 6.737 \(\times\) 10\({}^{-5}\), and 4.103 \(\times\) 10\({}^{-5}\) for the cases of noise-free, 5% noise, and 10% noise, respectively. Since the value of \(\beta\) is extremely small and difficult to assess, these identified results for \(\beta\), to some extent, are acceptable. Since only the standard LSE technique and the input modification process are involved, the computational time for the proposed approach is very short. In this example, only 10 seconds are required for the identification. Besides the identification of structural parameters, the unknown ground motion can be estimated by the proposed approach as well. Similarly, taking the case of 10% noise as an example, the comparison of the identified ground acceleration with the actual one is given in Figure 5a. The dashed line represents the identified ground acceleration, whereas the solid line is its actual value. For clarity of comparison, the time segment from 4 s to 6 s is plotted in Figure 5b. Clearly, the proposed approach can identify the ground acceleration very well, even when the signals are contaminated by 10% noise. Based on these identified parameters and ground motion, the structural responses can also be calculated using the equation of motion. Figure 6 gives the comparison of the actual structural responses with the estimated ones. Due to the limitation of the length of this paper, only the responses on the top floor are shown as examples. It is obvious that the estimated responses are in good agreement with the actual ones.