## 3 Simulation Examples
### Mode Shape Extraction Using Displacement Approximation of Contact Points
#### 3.3.2 Mode Shape Extraction with CEEMDAN-NSPCA  
At a noise level of 10%, the residual accelerations (\(\Delta u_{1,2}\)) between Vehicle 1 and Vehicle 2 are decomposed by CEEMDAN with a standard deviation of 0.0577. Comparing the original IMF components from CEEMDAN in Figure 15 with the orthogonalized IMF components by PCA in Figure 16, the residual displacement and residual acceleration amplitude after PCA fluctuate greatly. The spectrum of all components is altered vastly for both acceleration and displacement through PCA, and the amplitude increases. Moreover, according to the cumulative contribution rate of 0.85, the residual acceleration of IMF-1 to IMF-7 and the residual displacement of IMF-1 to IMF-6 will be discarded to lead to the loss of high frequency contents. In Figure 16, IMF-1 to IMF-4 are high-frequency components, and IMF-5 to IMF-10 are low-frequency components as seen in the CEEMDAN-NSPCA flow charts. As shown in Figure 2, the high-frequency component was selected for PCA. Meanwhile, the high-frequency and low-frequency IMF components from CEEMDAN and the original residual displacement (\(\Delta u_{1,2}\)) are IMF-1-\(\cdot\)IMF-5 and IMF-6-IMF-10, and the standard deviation is 5.119 \(\times\) 10\({}^{-4}\). For both residual accelerations and displacements, it is evident that their IMF components (IMF-1 to IMF-4 and IMF-1 to IMF-5) with a high frequency contain contents with the higher frequency range, which cover the second (16.0850 Hz) and third modes (36.1911 Hz). Instead of involving all components, only high-frequency IMF components are selected for PCA in the CEEMDAN-NSPCA process, including IMF-1 to IMF-4 for acceleration and IMF-1 to IMF-5 for displacement. All these orthogonalized IMF components by PCA are marked in Figure 17. It is easy to find that the method effectively suppresses the noise of the high-frequency components of acceleration and displacement and retains the effective information. With the same cumulative contribution rate of 0.85, the orthogonalized components are picked for the final signal's construction, including HF-IMF-1 to HF-IMF-3 for acceleration and HF-IMF-1 to HF-IMF-3 for displacement, which contain the higher-frequency contents as mentioned above. Figure 14: The STFT of the approximate displacement residual for \(\Delta u_{12}\) between \(u_{c1}\) and \(u_{c2}\). Figure 15: The IMF components (**a**) and spectrograms (**b**) of residual acceleration \(\Delta\dot{u}_{1,2}\) and the IMF components (**c**) and spectrograms (**d**) of residual displacement \(\Delta u_{1,2}\) by CEEMDAN. Figure 16: The orthogonalized IMF components (**a**) and spectrograms (**b**) of residual acceleration \(\Delta ti_{1,2}\) and the IMF components (**c**) and spectrograms (**d**) of residual displacement \(\Delta ti_{1,2}\) by CEEMDAN-PCA. Figure 17: The IMF components (**a**) and spectrograms (**b**) of residual acceleration \(\Delta i_{1,2}\), the IMF components (**c**) and spectrograms (**d**) of residual displacement \(\Delta i_{1,2}\) by CEEMDAN-NSPCA, and the high frequency component is HF-IMF and the low frequency component is LF-IMF. The denoised signals by CEEMDAN-PCA and CEEMDAN-NSPCA are plotted in Figure 18. The SNR and MSE of noise-contaminated and denoised signals are provided in Table 4, wherein \(\Delta i_{1,2}\) and \(\Delta i_{2,3}\) are the residual accelerations between consecutive contact points. The smoother denoised signals by CEEMDAN-PCA show remarkable deviations away from the original signals rather than the denoised ones by CEEMDAN-NSPCA, which introduces more noise, resulting in the decline of signal smoothness. For \(\Delta i_{1,2}\) the acceleration and the signal-to-noise ratio (SNR) increase by approximately 180.85%, while the corresponding mean square error (MSE) reduces by approximately 99.99%. For \(\Delta u_{1,2}\) the displacement and the SNR increase by approximately 132.33%, and the corresponding MSE decreases by approximately 99.99%.