# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 5 Results and Discussion
### Numerical Simulation  
The central signal guiding our investigation is \(\text{Signal}^{k}(t)=B^{k}x(t-\tau_{0}^{k})+\epsilon^{k}(t)\), which is articulated in Equation (4). This equation encapsulates various factors, such as amplitude variation \(B_{k}\), time-shift \(\tau_{0}^{k}\), and noise \(\epsilon_{k}(t)\), all of which affect the behavior of the signal. The influence of these parameters, together with the total number of recordings \(K\), the correlation between the sampling frequency and the modal frequency, and the overall number of samples in each recording \(L\), collectively shape the results of our algorithmic evaluations. To ensure a satisfactory resolution for capturing system dynamics, we selected a damping ratio of \(\zeta=0.01\), a sampling frequency of \(\text{Fs}=800\) Hz, a signal length of \(L=4000\), and damped frequency of \(\omega_{d}=15.516\) Hz. To ensure reliability and conduct statistical analysis, each simulation was repeated 1000 times. To determine the accuracy of damping ratio extraction by different algorithms, we used analysis of variance (ANOVA) testing in R with the null hypothesis that the number of samples has no effect on the output. Table 1 summarizes the outcomes of this test for aligned recordings, in which the impact takes place at the start of the recordings (\(\tau_{0}^{k}=0\)). The total number of recordings \(K\) varies from 10 to 100 with intervals of 10. * The CTWT-2 method shows a significant effect of the number of samples on damping ratio estimates, with a relatively low \(p\)-value (0.015), indicating a notable influence of the number of recordings on its performance. * The CTWT-1 method shows a moderate effect with a \(p\)-value of 0.079, suggesting some influence of the number of recordings, though less pronounced compared to CTWT-2. * The LSRF method demonstrates a limited impact with a \(p\)-value of 0.377, suggesting lower sensitivity to the number of recordings compared to CTWT methods. * The pLSCF method demonstrates a lower relative effect with a higher \(p\)-value of 0.831, indicating the least influence of the number of recordings on its performance. * The PP and Yoshida methods also show limited effectiveness with \(p\)-values of 0.429 and 0.493, similar to the LSRF method. These results suggest that, while the CTWT-2 method is significantly affected by the number of recordings, the CTWT-1 method shows a moderate effect, and the LSRF, pLSCF, PP, and Yoshida methods are less sensitive to variations in the number of samples. In addition to examining the impact of the number of recordings, this article also explores the influence of noise on the performance of these damping ratio extraction methods. As illustrated in Figure 1, we evaluate how varying signal-to-noise ratios (SNRs) affect these methods. The analysis assumes that the time of impact is precisely known and occurs at the beginning of the recordings. The dashed line in the figure denotes the true value of the damping ratio. Our investigation contains a comprehensive range of SNR conditions, extending from \(-5\) dB to 30 dB, thereby facilitating a thorough comparison of the efficacy of the different methodologies employed. As depicted in this figure, all methods perform well at higher SNR levels. At an SNR of \(-5\)dB, the PP method shows the best performance, exhibiting the lowest error. Additionally, the CTWT-1 method performs commendably at this level of noise, surpassing the LSRF but not achieving the same level of performance as the PP method. At an SNR of 0 dB, LSRF shows the best performance with the lowest error, indicating high accuracy and precision. When comparing the two CTWT methods, the CTWT-1 method consistently demonstrates strong performance across various SNR levels, resulting in lower errors. The box plots for CTWT-1 closely align with the true values. In contrast, CTWT-2 exhibits slightly higher error levels. This suggests that the differences in the averaging step between the two methods affect CTWT-2's accuracy.