# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 5 Results and Discussion
### Numerical Simulation
Errors in this estimate can greatly reduce the accuracy of these methods. Figure 3 shows how the accuracy of various damping ratio extraction methods degrades as the number of samples shifted from the actual impact time increases. The two subfigures display results for different SNR levels: Figure 2(a) 10 dB SNR and Figure 2(b) 0 dB SNR. As shown in this figure, even a small error in the impact time estimation can lead to significant variations in the damping ratio, with FFT-based methods showing high levels of sensitivity to these errors. One approach to estimating the time of impact is through wavelet analysis, which identifies the point of maximum energy. This technique allows us to estimate the impact time even when the exact moment is unknown. By estimating the time of impact and computing the FFT of the impulse input, we can perform frequency normalization to assess the system's frequency response. Figure 4 illustrates the outcomes achieved through wavelet analysis to determine the time of impact. Figure 3(a) displays the extracted distributions of the damping ratios for various methods across different SNR levels. Meanwhile, Figure 3(b) presents the MSE of the damping ratio estimates for the same methods. These figures emphasize the performance and reliability of the distributions under varying SNR conditions, demonstrating their effectiveness in scenarios where the precise time of impact is not clearly defined. Figure 2: Analysis of damping ratio extraction methods with random but known times of impact, frequency normalized. The figures show how each method performs when the impact timing is not fixed, presenting a more challenging but realistic scenario. (**a**) Damping Ratio vs. SNR. (**b**) Mean Squared Error vs. SNR. Analyzing the graphs, it is evident that different methods perform variably under non-aligned conditions. The damping ratio distribution in Figure 3(a) shows that methods like LSRF and CTWT-2 tend to have lower damping ratio variations, indicating higher accuracy. In contrast, the PP method shows higher variations, especially at lower SNR levels, suggesting lower robustness to noise. In terms of mean squared error, as shown in Figure 3(b), the methods LSRF and CTWT-2 again demonstrate superior performance with lower MSE values across all SNR levels. This indicates the reliability and precision of these methods in generating accurate damping ratio estimates under non-aligned conditions. The pLSCF method also performs well at higher SNR levels but displays higher sensitivity to increased noise levels, leading to a dramatic decrease in performance as the SNR decreases. The PP method exhibits the highest MSE values even in lower levels of noise, indicating it is less effective in estimating damping ratios when the impact time is not precisely known. The CTWT-1 method performs comparably to LSRF, CTWT-2, and pLSCF at high SNR levels, demonstrating its reliability under less noisy conditions. However, as the SNR decreases, CTWT-1 shows more robustness compared to pLSCF but is less effective compared to LSRF and CTWT-2. The Yoshida method performs better than CTWT-1 at low SNR levels but is worse than CTWT-2 and LSRF. Additionally, it has higher MSE values at high SNR levels compared to other methods. In conclusion, the LSRF and CTWT-2 methods emerge as the most robust and accurate for damping ratio estimation under non-aligned conditions, due to their lower variation in damping ratios and lower mean squared error values across different SNR levels. These findings demonstrate the importance of choosing reliable methods for practical applications where the exact impact time is uncertain. While examining the performance of damping ratio extraction methods under various SNR conditions provides valuable insights, it is also crucial to understand the impact of the number of recordings on these estimates. In practical applications, the number of recordings can vary, affecting the reliability and accuracy of the methods.