## 3 Numerical Simulations and Experimental Verification  
### Six DOF Chain Model of a Cantilever Beam  
To illustrate and confirm the validity of the proposed method in this paper, we first consider a numerical example of a chain model with six degrees of freedom (6-DOF) to simulate a cantilever beam, as shown schematically in Figure 1. The masses \(m_{1}\), \(m_{2}\), \(m_{3}\), \(m_{4}\), \(m_{5}\), and \(m_{6}\) for the 6-DOF chain model are equal to 2, 2, 2, 2, 3, and 4 kg. The stiffnesses \(k_{1}\), \(k_{2}\), \(k_{3}\), \(k_{4}\), \(k_{5}\), and \(k_{6}\) are equal to 1, 1, 1, 1, 2, and 3 N/m, respectively. The damping matrix \([C]\) of the system is in the form of \([C]=0.1[M]+0.001[K]\frac{\mathrm{N}}{\mathrm{m}}\), which indicates that this structure contains proportional damping because of damping matrix \([C]\), expressed as the linear combination of the mass matrix \([M]\) and stiffness matrix \([K]\). The excitation force is simulated as stationary white noise, which is approximately generated as a zero-mean band-pass noise [21], whose frequency range is from 0 to 50 Hz, and the standard deviation, i.e., power spectrum density, is 0.04 N\({}^{2}\cdot\)s/rad. The sampling interval is chosen as \(\Delta t=0.01\) s, and the sampling period, as shown in Figure 2, is \(T=N_{t}\cdot\Delta t=1310.72\) s, where \(N_{t}\) was chosen as \(2^{17}\). The cut-off frequency, \(\omega_{c}\) is 314.15 rad/s, and the resolution in frequency domain \(\Delta\omega\) is 4.79 \(\times\) 10\({}^{-3}\) rad/s. The sampling interval is chosen as 0.01 s, and the sampling period is 150 s. Through [PERSON]'s method, the displacement responses of the system are obtained and then employed for modal estimation through the modified SSI method. By introducing the procedure of solving the system matrix in SSI-COV, in conjunction with SSI-DATA, we can increase the computational efficiency without using the original large-dimension data matrix, through the singular value decomposition of the improved projection matrix. To make the number of modes evaluated through the SSI algorithm equal to or greater than the number of modes to be identified, the dimensions of the Hankel matrix must be not less than the system order to be identified (\(li\geq 2n\)). Through the channel expansion technique, we set the number of expansion channels to 20. In practical engineering analysis, a continuum structure has an infinite number of degrees of freedom and modes; thus, the dimensions of the Hankel matrix depend on the number of modes to be identified. The projection matrix, \([\Omega]\) is obtained by Hankel matrix calculation. The SVD analysis of the projection matrix \([\Omega]\) can then be performed, and the number of singular values is employed to determine the order of system to be identified. In Figures 3 and 4, the distribution of the singular values of \([\Omega][\Omega]^{T}\) shows a relatively more obvious drop than those of \([\Omega]\) around the singular value number 12 and can be further employed to estimate the system order and the number of modes to be identified. In addition, the SVD analysis of \([\Omega][\Omega]^{T}\) can be employed to reduce a greater number of calculations, rather than performing the SVD of \([\Omega]\) or system matrix \([A]\). A comparison of the computational efficiency of SVD between \([\Omega]\) and \([\Omega][\Omega]^{T}\) for different sampling  
Figure 1: Schematic plot of a 6-DOF chain model of a cantilever beam. Figure 2: A sample function of stationary white noise in the time domain. points is shown in Table 1, in which we clearly see the computation time of SVD of \([\Omega]\) is much longer than that of \([\Omega][\Omega]^{T}\), especially for \([\Omega]\) with high dimensions. In addition, through the stabilization diagram of different polynomial orders constructed by identified natural frequency, corresponding to eigenvalues from the frequency response function matrix with different polynomial orders, estimation results can then be sorted as either structural or fictitious modes.