## 3 Numerical Simulation
### User-Defined Parameter  
Before applying SI or RIS, the first and inevitable question is the size of Hankel matrices, such as \(\mathbf{U}_{f}\), \(\mathbf{\Xi}_{p}\), and \(\mathbf{Y}_{f}\). The only user-defined parameter is \(i\) because the left-hand side matrix in Equation (9) is a square matrix. Once \(i\) is assigned, \(j\) can be determined using Equation (11), and the total data size within the current window, \(l_{w}\), called window length, is resolved as  
\[l_{w}=2i\left(m+l+1\right)-1 \tag{28}\]  
In the field application, the window length is very important since the product of the window length and the sampling interval is the time of providing the first identification \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Mode Number** & **1 st** & **2 nd** & **3 rd** & **4 th** \\ \hline Frequency (Hz) & 1.0174 & 3.0176 & 4.9150 & 6.6450 \\ Mode Number & 5 th & 6 th & 7 th & 8 th \\ Frequency (Hz) & 8.1487 & 9.3750 & 10.2820 & 10.8388 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The 8 modal frequencies of numerically simulated structure. Figure 2: The flowchart for RSI implementation using bona fide recursive algorithm. result after an earthquake starts. If the window length is longer than the duration of the mainshock, RSI can only identify the modal parameters belonging to the damaged structure. The results may be useless without comparing with the original state. By setting \(i\) as 10, the modal parameters identified by RSI are shown in Figure 3.  
Obviously, RSI implementation using the bona fide recursive algorithm gives an excellent estimation of the modal parameters. The grids in Figure 3b are drawn according to the 8 modal frequencies listed in Table 1 and, clearly, the different modes can be successfully identified at each step. Note that the identified damping ratios, \(\zeta\), in Figure 3c are multiplied by the number of modes to avoid overlap surrounding 0.02 (2%). Therefore, the identified results evenly distributed from 0.02 to 0.16 illustrated that the damping ratios over 8 modes are all 2%. The identified mode shapes are also correct but not shown here due to limited space. Overall, RSI is capable to provide correct modal parameters during the seismic excitation. As a key parameter, \(i\) is crucial to the performance of SI and RSI and some literature gives suggestions about how to decide it [28; 29; 30]. The concept is that \(i\) shall be larger than the half-cycle of the fundamental period and the fundamental period is usually the first modal period of the target structure. Hence, it can be computed by multiplying the Nyquist frequency, \(N_{f}\) (in Hz), and the fundamental period, \(T_{f}\) (in seconds). However, in the above example, \(i\) is 25 if the measurement is taken via 50 Hz and the window length is 499, which  
Figure 3: The identified modal parameters under the seismic excitation. (**a**) Ground excitation and structural responses; (**b**) identified frequency over time; (**c**) identified damping ratio over time. indicates almost 10 s, for the first identification result. Apparently, the suggested criterion is too conservative and inadequate for the practical implementation of building structures or infrastructures under seismic excitations. Considering the projection is done by the whole part of the matrix, the criterion shall be taken on the data number rather than the row number of the Hankel matrices. To be specific, the proposed criterion is  
\[i\geq C_{E}\frac{2N_{f}T_{f}+1}{2\left(m+l+1\right)} \tag{29}\]  
where \(C_{E}\) is the extension coefficient. \(C_{E}\) shall be at least larger than 1, it can be more than 3 or 4 for stable identified results, and it needs to be further increased while dealing with a noisy measurement. With a 50 Hz measurement, Figure 4 exhibits the identified modal frequencies across various \(C_{E}\). It can be observed that the identification fails when \(i\) is equal to 2 (\(C_{E}=0.78\)) and provide stable results when \(i\) is around 9 or 10 (\(C_{E}=3.53\) or 3.92).