## 3 Numerical Simulations and Experimental Verification
### Six DOF Railway Vehicle Model with Modal Interference
Compared with the efficiency when performing the SVD of the system matrix \([A]\), computation can be reduced by implementing SVD analysis of data matrix \([\Omega][\Omega]^{T}\) constructed from the projection matrix \([\Omega]\). As shown in Figure 10, from the stabilization diagram corresponding to different modal orders, it can be observed that the number of structural modes to be identified is six, and serious modal interference can be observed between the last two close modes. The system matrix, \([A]\), can be found using Equations (9)-(11), and the modal parameters can then be determined through Equations (12)-(15). Table 4 presents the well-implemented modal estimation through eigenvalue analysis of the system matrix, \([A]\). A comparison between the exact and identified mode shapes is shown in Figure 11, and the corresponding MAC values evaluated are shown in Figure 12, where good agreement is observed. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{3}{c}{**Natural Frequency (Hz)**} & \multicolumn{3}{c}{**Damping Ratio (\%)**} \\ \cline{2-7}  & **Exact** & **MSSI** & **Error (\%)** & **Exact** & **MSSI** & **Error (\%)** \\ \hline
1 & 2.79 & 2.79 & 0.00 & 4.89 & 4.89 & 0.01 \\ \hline
2 & 3.71 & 3.71 & 0.00 & 6.62 & 6.62 & 0.01 \\ \hline
3 & 16.55 & 16.53 & 0.09 & 16.65 & 16.62 & 0.17 \\ \hline
4 & 19.27 & 19.25 & 0.11 & 18.78 & 18.74 & 0.24 \\ \hline
5 & 25.36 & 25.30 & 0.21 & 1.74 & 1.74 & 0.42 \\ \hline
6 & 25.57 & 25.51 & 0.21 & 1.75 & 1.75 & 0.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Identification results of railway vehicle through Modified Stochastic Subspace Identification (MSSI) method. Figure 9: Distribution of the singular values associated with a data matrix, \([\Omega][\Omega]^{T}\), constructed by projection matrix, \([\Omega]\), from stationary responses of a railway vehicle model..  
Figure 11: Comparison between the identified mode shapes and the exact mode shapes of 6-DOF chain model of a railway vehicle subjected to stationary white noise: (**a**) 1 st mode shape; (**b**) 2 nd mode shape; (**c**) 3 rd mode shape; (**d**) 4 th mode shape; (**e**) 5 th mode shape; (**f**) 6 th mode shape. Figure 10: Typical plot of the stabilization diagram of stationary responses of 1 st DOF of a 6-DOF chain model of a railway vehicle. Due to the experimental restrictions of economic cost and structural geometry in the practical measurement of structural response, the number of sensors sited on the structures to record response data is not usually sufficient for the overall degrees of freedom of a structure to be identified. This may cause problems of incomplete measurement of the degrees of freedom of the identified mode-shape vector. To address this issue, we also implement modal estimation from the incomplete modal-information data obtained due to insufficient measurement channels; thus, only response data of the first, third, and fifth DOF of the railway system subjected to ambient excitation in the previous numerical example are employed to implement modal estimation. The modal estimation results are shown in Table 5. The modal parameters estimated by the proposed method and the exact results are in good agreement, because the errors in both natural frequencies and damping ratios are less than 1%. Thus, we confirm the effectiveness of the proposed method under the likely practical conditions of insufficient measurement information.