## 3 Numerical Simulations and Experimental Verification
### Six DOF Chain Model of a Cantilever Beam
In addition, the number of structural modes can clearly be seen to be six, as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Sampling Points**} & \multicolumn{2}{c}{**Computation Time (s) of SVD**} & \multicolumn{2}{c}{**Matrix Multiplication**} \\ \cline{2-4}  & \([\Omega]\) & \([\Omega][\Omega]^{T}\) & **Processing Time for \([\Omega][\Omega]^{T}\)** \\ \hline \(2^{12}\) & 0.570 & 0.000574 & 0.0003616 \\ \hline \(2^{13}\) & 2.217 & 0.000760 & 0.0005376 \\ \hline \(2^{14}\) & 9.074 & 0.001219 & 0.0009967 \\ \hline \(2^{15}\) & 37.197 & 0.003244 & 0.0029074 \\ \hline \(2^{16}\) & Out of memory & 0.003264 & 0.0030135 \\ \hline \(2^{17}\) & Out of memory & 0.006189 & 0.0059404 \\ \hline \end{tabular} Central processing unit (CPU) is Intel® CoreTM 17-9700K CPU @ 3.60 GHz. Random-access memory (RAM) is 64.0 GB.  
\end{table} 
Table 1: Comparison of computation time of SVD of \([\Omega]\) and \([\Omega][\Omega]^{T}\) for different sampling points as well as matrix multiplication processing time for \([\Omega][\Omega]^{T}\). Figure 3: Distribution of the singular values of projection matrix \([\Omega]\) from the stationary responses of a 6-DOF chain model of a cantilever beam. In the stabilization diagram, a clear \"location\" of modal frequency obviously exists, even though no obvious peaks appear among close modes in the frequency response function because of modal interference. The system matrix, \([A]\), can be obtained from Equations (9)-(11), and the modal parameters can then be estimated through the eigenvalue analysis of system matrix, \([A]\), as summarized in Tables 2 and 3, where we clearly see the well-estimated structural modal parameters through the Modified Stochastic Subspace Identification (MSSI) method of both \([\Omega]\) and \([\Omega][\Omega]^{T}\). Note that the MSSI method is a combination of the conventional SSI method and SVD analysis. A comparison between the exact and identified mode shapes is shown in Figure 6, and the corresponding Modal Assurance Criterion (MAC) [22] values evaluated are shown in Figure 7, where good agreement is observed. Figure 4: Distribution of the singular values of the data matrix, \([\Omega][\Omega]^{T}\), constructed by projection matrix, \([\Omega]\), from the stationary responses of a 6-DOF chain model of a cantilever beam. Figure 5: Typical plot of the stabilization diagram of stationary response of first DOF of a 6-DOF chain model of a cantilever beam. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{3}{c}{**Natural Frequency (rad/s)**} & \multicolumn{3}{c}{**Damping Ratio (\%)**} \\ \cline{2-7}  & **Exact** & **MSSI** & **Error (\%)** & **Exact** & **MSSI** & **Error (\%)** \\ \hline
1 & 5.03 & 5.04 & 0.09 & 1.25 & 1.38 & 10.39 \\ \hline
2 & 13.45 & 13.39 & 0.41 & 1.04 & 1.17 & 11.73 \\ \hline
3 & 19.80 & 19.71 & 0.41 & 1.24 & 1.23 & 0.92 \\ \hline
4 & 26.68 & 26.56 & 0.44 & 1.52 & 1.47 & 3.59 \\ \hline
5 & 31.65 & 31.18 & 1.48 & 1.74 & 1.79 & 2.78 \\ \hline
6 & 33.72 & 33.46 & 0.78 & 1.84 & 1.89 & 3.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Identification results of 6-DOF chain model through Modified Stochastic Subspace Identification (MSSI) method from projection matrix \([\Omega]\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{3}{c}{**Natural Frequency (rad/s)**} & \multicolumn{3}{c}{**Damping Ratio (\%)**} \\ \cline{2-7}  & **Exact** & **MSSI** & **Error (\%)** & **Exact** & **MSSI** & **Error (\%)** \\ \hline
1 & 5.03 & 5.04 & 0.08 & 1.25 & 1.38 & 8.97 \\ \hline
2 & 13.45 & 13.39 & 0.42 & 1.04 & 1.17 & 10.11 \\ \hline
3 & 19.80 & 19.71 & 0.42 & 1.24 & 1.23 & 2.47 \\ \hline
4 & 26.68 & 26.56 & 0.47 & 1.52 & 1.47 & 4.88 \\ \hline
5 & 31.65 & 31.18 & 1.50 & 1.74 & 1.79 & 1.15 \\ \hline
6 & 33.72 & 33.46 & 0.78 & 1.84 & 1.89 & 1.19 \\ \hline \hline \end{tabular}
\end{table}