## 3 Results and Discussion
### Six-DOF Chain Model of a Cantilever Beam  
A six-DOF linear chain model system of a cantilever beam is shown in Figure 1. The mass matrix \(M\), stiffness matrix \(K\), and damping matrix \(C\) of the system are expressed as follows:  
\[M=\left[\begin{array}{cccccc}2&0&0&0&0&0\\ 0&2&0&0&0&0\\ 0&0&2&0&0&0\\ 0&0&0&2&0&0\\ 0&0&0&0&3&0\\ 0&0&0&0&0&4\end{array}\right]kg,\ K=\left[\begin{array}{cccccc}1&-1&0&0&0&0 \\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&0\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&3&-2\\ 0&0&0&0&-2&5\end{array}\right]N/m,\]  
The damping matrix \(C\) of the system is proportional damping (i.e., the linear combination of mass matrix \(M\) and stiffness matrix \(K\)). It should be mentioned that the proportional damping matrices are suitable for modeling the behavior of most structural systems, in which the damping mechanism is distributed rather uniformly throughout the structures. The six DOF are excited by nonstationary white noise using the Newmark method and the initial state is stationary. To obtain the displacement response of the time domain, the time-domain sampling interval is 0.01 s, the number of sampling points is 131,072.  
In the course of system identification and to avoid omission of modal identification, a higher order is used for identification. There will be a fictitious mode in the process. Therefore, we can employ the values of modal assurance criterion (MAC) to check the agreement between the identified and the theoretical mode shapes, and use the stabilization diagram to filter the fictitious mode and to decide the order for calculation.  
Figure 5 is the stabilization diagram of the six-DOF chain model system. It is roughly observed that the frequencies 5, 13, 20, 27, 31, and 33 rad/s can be identified steadily when the order number is 6. The six modes of corresponding frequencies are identified as system modes, and the identification order number \(\alpha\) is 6.  
The displacement response of each DOF is shown in Figure 6. Table 1 shows the result of modal parameter identification using ERA/DC. As seen, the frequency errors are less than 1%, and the maximum error of damping ratio is 5.92%. Figure 7 shows the identified mode shapes, which are approximately coincident with theoretical mode shapes, and the MAC values are larger than 0.99, meaning the method is effective on identification.  
Figure 4: Sample function of nonstationary excitation.  
Figure 5: Stabilization diagram of six-DOF chain model of a cantilever beam.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{3}{c}{**Natural Frequency (rad/s)**} & \multicolumn{3}{c}{**Damping Ratio (\%)**} & \multirow{2}{*}{**MAC**} \\ \cline{2-2} \cline{4-7}  & **Exact** & **ERA/DC** & **Error (\%)** & & **Exact** & **ERA/DC** & **Error (\%)** \\ \hline
1 & 5.03 & 5.04 & 0.04 & 1.25 & 1.19 & 4.66 & 1.00 \\
2 & 13.45 & 13.43 & 0.14 & 1.04 & 1.08 & 3.63 & 1.00 \\
3 & 19.80 & 19.73 & 0.32 & 1.24 & 1.21 & 2.95 & 1.00 \\
4 & 26.69 & 26.53 & 0.58 & 1.52 & 1.50 & 1.44 & 1.00 \\
5 & 31.66 & 31.47 & 0.60 & 1.74 & 1.84 & 5.92 & 0.99 \\
6 & 33.73 & 33.44 & 0.86 & 1.83 & 1.88 & 2.45 & 0.99 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Modal parameter identification result of six-DOF chain model of a cantilever beam.  
Figure 6: Nonstationary response signal of six-DOF chain model of a cantilever beam.