## 3 Simulation Examples
### Mode Shape Extraction Using Displacement Approximation of Contact Points
#### 3.3.2 Mode Shape Extraction with CEEMDAN-NSPCA
After CEEMDAN-PCA denoising, the SNR of the residual acceleration and displacement polluted by noise decreases sharply, and the MSE increases sharply. It can be clearly found that standardizing a signal with a smaller standard deviation introduces more noise, and the more the SNR decreases, the more the MSE increases. The failure at this moving load's vibrated responses may be caused by putting all the IMF components into PCA indiscriminately. In Figure 15, the IMF components concentrated on higher frequencies have smaller amplitudes. So, the information carried by these IMF components is easily drowned after PCA. By contrast, the SNR and MSE of the denoised signal by CEEMDAN-NSPCA receive obvious improvement. The IMF components are distinguished before PCA. The noise signal is decomposed through CEEMDAN, minimizing the mode aliasing, extracting the high-frequency component for principal component analysis, and retaining the low-frequency component to retain its information to the greatest extent. As shown in Figure 19, the extracted mode shapes from the denoised signals using CEEMDAN-PCA are seriously distorted and even worse than those from the noise-contaminated signals in Figure 13. By applying CEEMDAN-NSPCA, the extracted mode shapes could remain in acceptable profiles under the 10% noise level, as shown in Figure 20. The MAC values are 0.998, 0.662, and 0.600 for these three modes at the 10% noise level, while the corresponding MAC values are 0.998, 0.600, and 0.553 without noise reduction. Subsequently, the modal shapes extracted using CEEMDAN-NSPCA are closer to the analytical solutions compared to CEEMDAN-PCA. For the first mode, both are close to one. The second and third mode are improved by approximately 10.33% and 8.49%, respectively. Figure 19: The first three beam mode shapes from the approximate residual displacement STFT-based extraction after being denoised by CEEMDAN-PCA 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**). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**SNR**} & \multicolumn{4}{c}{**MSE**} \\ \hline
**Signals** & \(\Delta\hat{u}_{\textbf{k},\textbf{2}}\) & \(\Delta\hat{u}_{\textbf{k},\textbf{3}}\) & \(\Delta\textbf{u}_{\textbf{k},\textbf{2}}\) & \(\Delta\textbf{u}_{\textbf{k},\textbf{3}}\) & \(\Delta\textbf{u}_{\textbf{k},\textbf{2}}\) & \(\Delta\textbf{u}_{\textbf{k},\textbf{3}}\) \\ \hline Noise & \(19.719\) & \(19.900\) & \(19.789\) & \(20.219\) & \(3.548\times 10^{-5}\) & \(45.86\times 10^{-6}\) & \(2.749\times 10^{-9}\) & \(7.237\times 10^{-9}\) \\ CEEMDAN-PCA & \(-28.269\) & \(-27.051\) & \(-68.644\) & \(-63.907\) & \(2.234\) & \(2.255\) & \(1.887\) & \(1.830\) \\ CEEMDAN- & \(24.361\) & \(28.842\) & \(22.201\) & \(27.014\) & \(1.219\times 10^{-5}\) & \(14.579\times 10^{-6}\) & \(1.553\times 10^{-9}\) & \(1.480\times 10^{-9}\) \\ NSPCA & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The SNR and MSE of denoised residual accelerations and displacements by CEEMDAN-PCA and CEEMDAN-NSPCA under the 10% noise level.