## 3 Numerical Simulations and Experimental Verification
### Six DOF Railway Vehicle Model with Modal Interference  
In the previous examples, only a proportionally damped structure is considered. Actually, the hypothesis of proportional damping, although frequently considered in the literature, is difficult to be found in practice in real mechanical systems [23,24]. To study the feasibility and validity of the proposed method for relatively complex structural systems, we consider a linear 6-DOF railway vehicle model with viscous damping [25], as shown schematically in Figure 8. The mass matrix \(M\), stiffness matrix \(K\), and the damping matrix \(C\) of the system are given as follows [26]:  
\[M=\left[ \begin{array}{cccccc}1200&0&0&0&0\\ 0&850&0&0&0&0\\ 0&0&4125&0&0&0\\ 0&0&0&125000&0&0\\ 0&0&0&0&850&0\\ 0&0&0&0&0&1220\\ \end{array} \right]\mathrm{kg}\]  
Figure 6: Comparison between the identified mode shapes and the exact mode shapes of a 6-DOF chain model of a cantilever beam subjected to stationary white noise: (**a**) 1 st bending mode shape; (**b**) 2 nd bending mode shape; (**c**) 3 rd bending mode shape; (**d**) 4 th bending mode shape; (**e**) 5 th bending mode shape; (**f**) 6 th bending mode shape. Figure 7: Typical plot of the Modal Assurance Criterion (MAC) for the identified mode shapes and exact mode shapes of a 6-DOF chain model of a cantilever beam. \[K=\left[ \begin{array}{cccccc}k_{1}+k_{2}&-k_{2}&0&0&0&0\\ -k_{2}&k_{2}+k_{3}&-k_{3}&-k_{3}L&0&0\\ 0&-k_{3}&k_{3}+k_{4}&k_{3}L-k_{4}L&-k_{4}&0\\ 0&-k_{3}L&k_{3}L-k_{4}L&k_{3}L^{2}+k_{4}L^{2}&k_{4}L&0\\ 0&0&-k_{4}&k_{4}L&k_{4}+k_{5}&-k_{5}\\ 0&0&0&0&-k_{5}&k_{5}+k_{6}\\ \end{array} \right]\text{N/m}\]  
\[C=\left[ \begin{array}{cccc}c_{1}+c_{2}&-c_{2}&0&0&0&0\\ -c_{2}&c_{2}+c_{3}&-c_{3}&-c_{3}L&0&0\\ 0&-c_{3}&c_{3}+c_{4}&c_{3}L-c_{4}L&-c_{4}&0\\ 0&-c_{3}L&c_{3}L-c_{4}L&c_{3}L^{2}+c_{4}L^{2}&c_{4}L&0\\ 0&0&-k_{4}&c_{4}L&c_{4}+c_{5}&-c_{5}\\ 0&0&0&0&-c_{5}&c_{5}+c_{6}\\ \end{array} \right]\]  
in which \(K_{1}=3\times 10^{7}\) N/m, \(K_{2}=10^{6}\) N/m, \(K_{3}=6\times 10^{6}\) N/m, \(K_{4}=6\times 10^{6}\) N/m, \(K_{5}=10^{6}\) N/m, \(K_{6}=3\times 10^{7}\) N/m, \(L=8.53\) m, \(c_{1}=0\), \(c_{2}=6\times 10^{3}\) N-s/m, \(c_{3}=1.8\times 10^{4}\) N-s/m, \(c_{4}=1.8\times 10^{4}\) N-s/m, \(c_{5}=6\times 10^{3}\) N-s/m, and \(c_{6}=0\). This railway structural system has the features of relatively high modal damping levels for third and fourth modes, and a pair of closely spaced fifth and sixth modes. Note that this railway structure has non-proportional damping, because damping matrix, \(C\), cannot be expressed as the linear combination of the mass matrix, \(M\), and stiffness matrix, \(K\). The stationary white noise in the previous numerical example is still used as the excitation force acting the 6 th mass of the railway vehicle system. The sampling interval is set as 0.01 s, and the sampling period is 110 s. Then, using [PERSON]'s method, the displacement responses obtained are employed for modal estimation using the modified SSI method proposed in this paper. It should be noted that the number of modes to be identified serves to determine the dimensions of the Hankel matrix. Using the channel expansion technique, we set the number of expansion channels at 20. The dimensions of the Hankel matrix are not less than the order of the system to be identified, thus satisfying the condition that the number of modes evaluated in the SSI algorithm is not less than the number of modes to be identified. The SVD analysis of the projection matrix, which is obtained from the Hankel matrix calculation, can then be implemented, and the num  
Figure 8: Schematic plot of the railway vehicle model. employed to determine the order of the system to be identified. Around the 12 th singular value, an obvious drop showed in the distribution of singular values associated with the data matrix \([\Omega][\Omega]^{T}\) constructed by projection matrix \([\Omega]\) from stationary response data, as shown in Figure 9. It can be estimated that the order of the system, i.e., the number of modes to be identified, is 12.