## 4 Numerical Simulation
### 3-DOF Mass-Damping-Spring System  
In order to verify the accuracy of the improved SSI method, a spring oscillator system with three-degrees-of-freedom (DOF) was considered [30] (Figure 6). The parameters of the 3-DOF system were \(m_{1}\) = 7 kg, \(m_{2}\) = 10 kg, \(m_{3}\) = 10 kg, \(k_{1}\) = 1000 N/m, \(k_{2}\) = 2000 N/m, and \(k_{3}\) = 100 N/m. The mass matrix, \(M\), and stiffness matrix, \(K\), of the system were calculated, and the damping matrix was determined based on the Rayleigh damping assumption of \(C=aM+bK\). The difference between the conventional SSI method and the improved SSI method with reconstructed displacements was numerically simulated by the 3-DOF system. The coefficients of the damping ratio were \(a\) = 0.005 and \(b\) = 0.002. The natural frequencies of the first three orders of the system were obtained as \(\omega_{1}\) = 0.46 Hz, \(\omega_{2}\) = 1.21 Hz, and \(\omega_{3}\) = 3.84 Hz, with the corresponding damping ratios of \(\xi_{1}\) = 0.59%, \(\xi_{2}\) = 0.33%, and \(\xi_{3}\) = 0.45%, respectively. Gaussian white noises with a power of 60 dBW were then applied as the excitation, which was generated by the WGN function in Matlab. Figure 7 presents  
Figure 5: (a) Comparison between normalized singular values obtained based on acceleration and displacement data. (b) Locally amplified normalized singular values in the order of 0–10. Figure 6: 3-DOF mass-damping-spring system. The modal parameters of the system were identified by calculated accelerations and reconstructed displacements. The frequencies, modals, and damping ratios of the system were identified by its stability diagrams. The \(x\)- and \(y\)-axes of the stability diagrams shown in Figure 10 represent the frequencies and mode orders of the system, respectively. When the difference between the characteristics of high and low orders was within a certain tolerance range, a stable mode appeared. The tolerances for the frequency, the damping ratio, and the modal shape were set to 1%, 5%, and 5% respectively, and the number of Hankel matrices was \(c\times(n-c+1)\), where \(c=500\) and \(n\) is the data length, and the order of the state space model ranged from 2 to 50. Moreover, the appropriate window size was 10 s and the corresponding sliding step was 5 s to satisfy 50% duplication of time-domain signals between adjacent windows. Figure 10 displays the stability diagrams of modal frequencies based on calculated accelerations and reconstructed displacements, where the blue '\(\Circuit\)' marks represent the poles of frequency stability, the red '\(\triangle\)' marks represent the poles of frequency stability and damping ratio stability, and the black '+' marks represent the poles of frequency stability, damping ratio stability, and vibration mode stability. It is noticeable that the first three modes were accurately identified by reconstructed displacements when the modal order was no more than five. On the contrary, the first-order model was identified by calculated accelerations when the modal order was close to 40, causing a loss of modes. Therefore, displacement-based modal parameter identification had better stability for low-frequency modals. Figure 11 presents the variations of the frequency and damping ratio of the system identified by displacement signals in different periods (the first and second orders corresponded to 0-10 and 5-15 s, respectively), where the dashed lines represent the real frequency and damping ratio, and the circles, the boxes, and the triangles represent the frequency and damping of modal parameter identification using reconstructed displacements at different periods. It is clear that the accuracy of modal frequency identification was high, and the first three modes were accurately identified.