## 3 Simulation Examples
### Mode Shape Extraction Using Displacement Approximation of Contact Points
#### 3.3.3 Mode Shape Extraction with CEEMDAN-IWT  
Figure 21 illustrates the CEEMDAN of residual accelerations (\(\Delta u_{1,2}\)) and residual displacements (\(\Delta u_{1,2}\)) between Vehicle 1 and Vehicle 2 after wavelet denoising using the threshold function determined by the improved wavelet threshold at the noise level of 10%, in which the amplitude of the high-frequency IMF component and the low-frequency IMF component fluctuated slightly. According to SNR and MSE in Table 5, there is no significant change in acceleration or displacement before and after noise reduction, and the denoising effect of IWT is quite limited. At a noise level of 10%, for \(\Delta\dot{u}_{1,2}\) the acceleration and its SNR increase approximately 29.26%, and the corresponding MSE is reduced by approximately 73.77%. For \(\Delta u_{1,2}\) the displacement and the SNR increase by approximately 36.15%, and the MSE decreases by approximately 35.75%. In CEEMDAN-IWT, an improved wavelet threshold function is used to determine the optimal threshold to properly retain these small-amplitude, high-frequency contents. The comparison of Figures 15 and 22 reveals that all the wavelet-denoised high-frequency IMF components could still maintain concentrated  
Figure 20: The first three beam mode shapes from the approximate residual displacement STFT-based extraction after being denoised by CEEMDAN-NSPCA under noise levels of 1%, 5%, and 10%, compared to the analytical first mode shape (**a**), the second mode shape (**b**), and the third mode shape (**c**). frequency contents and effectively remove noise, particularly for IMF-1 to IMF-4 and IMF-1 to IMF-5 for both acceleration and displacement. Consequently, CEEMDAN-IWT could provide a better denoised signal than IWT, as shown in Figure 23 and Table 5. In Figure 24, under the 10% noise level, the first mode, which is well extracted and has a MAC value close to one, and the second and third modes, which have MAC values of 0.634 and 0.529, show little change. Subsequently, the modal shapes extracted using CEEMDAN-IWT are closer to the analytical solutions compared to IWT. For the first mode, both are close to one. The second and third modes are improved by approximately 15.11% and 22.87%, respectively. This indicates that while the IWT effectively filters noise from the high-frequency IMF components, it inevitably removes some low-frequency contents. Figure 21: The wavelet-denoised IMF components (**a**) and spectrograms (**b**) of residual acceleration \(\Delta\dot{n}_{1,2}\) and the wavelet-denoised IMF components (**c**) and spectrograms (**d**) of residual displacement \(\Delta u_{1,2}\) by IWT. Figure 22: The wavelet-denoised IMF components (**a**) and spectrograms (**b**) of residual acceleration \(\Delta\dot{n}_{1,2}\) and the wavelet-denoised IMF components (**c**) and spectrograms (**d**) of residual displacement \(\Delta u_{1,2}\) by CEEMDAN-IWT. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Index** & \multicolumn{3}{c}{**SNR**} & \multicolumn{3}{c}{**MSE**} \\ \hline
**Signals** & \(\Delta\dot{n}_{1,2}\) & \(\Delta\dot{n}_{2,3}\) & \(\Delta\dot{n}_{1,2}\) & \(\Delta\dot{n}_{2,3}\) & \(\Delta\dot{n}_{1,2}\) & \(\Delta\dot{n}_{2,3}\) & \(\Delta\dot{n}_{1,2}\) & \(\Delta\dot{n}_{2,3}\) \\  & & & & & (\(\times 10^{-5}\)) & (\(\times 10^{-6}\)) & (\(\times 10^{-9}\)) & (\(\times 10^{-9}\)) \\ \hline Noise & 19.719 & 19.900 & 19.789 & 20.219 & 3.548 & 4.586 & 2.749 & 7.237 \\ contaminated & 19.877 & 19.48 & 19.860 & 20.249 & 3.422 & 4.499 & 2.663 & 7.029 \\ IWT & 25.695 & 26.666 & 27.566 & 28.821 & 0.897 & 0.958 & 4.516 & 9.766 \\  & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The SNR and MSE of denoised residual accelerations and displacements by IWT and CEEMDAN-IWT under the 10% noise level.