# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
## 5 Results and Discussion
### Experimental Analysis  
The research presented here was motivated by the need to conduct modal analysis on vibrating screens. Vibrating screens, depicted in Figure 5, are large machines used to separate materials by size using screens that are stimulated with an unbalanced rotating shaft. These machines are unique in vibration analysis as they are designed to vibrate at specific frequencies rather than suppress vibrations completely. The machines we examine weigh several metric tons, have different drive geometries, and are suspended on springs and/or dampers. Typically, these machines operate in the 10-20 Hz range, with free body modes in the 1-5 Hz range. We conducted hammer impact tests using four and eight accelerometers connected wirelessly to collect the data. We used a real dataset containing 40 recordings to further validate our findings. For statistical analyses, they were separated into 10 groups of four. The extracted damping ratios and the variances for each method are summarized in Table 3.  
Figure 5: Vibrating screens. The results in this table illustrate the variability in damping ratio estimates and variances across different methods and frequencies. Although the true damping ratio is unknown, simulations suggested that the LSRF and CTWT-2 methods would perform well; however, the real data indicate some discrepancies from those expectations. At 4.71 Hz, the CTWT-2 method provides a damping ratio of 0.0482 with a variance of \(1.58\times 10^{-6}\), indicating strong stability and reliability. The PP method gives a damping ratio close to that of CTWT-2 at 0.0521 but has the highest variance (\(1.09\times 10^{-3}\)), indicating greater uncertainty. Similarly, the LSRF method also provides a damping ratio close to CTWT-2 at 0.0411 but with a higher variance (\(2.09\times 10^{-4}\)). This suggests that, while these methods yield similar damping ratio estimates, CTWT-2 offers more consistent results at this frequency. Conversely, CTWT-1, pLSCF, and Yoshida show damping ratios much lower than CTWT-2 and LSRF, which performed best in simulations. For 17.81 Hz, the CTWT-2 method again stands out with the lowest variance (\(2.11\times 10^{-9}\)), suggesting high stability in its estimates. The LSRF method provides a damping ratio close to CTWT-2 at 0.0090, but with the highest variance (\(2.32\times 10^{-5}\)), indicating potential instability. CTWT-1 also offers a damping ratio estimate close to CTWT-2, reinforcing its reliability. The PP method similarly provides a close estimate, showing consistent performance. In contrast, the Yoshida method's estimate diverges more from the CTWT-2 and LSRF estimates, while the pLSCF method exhibits a greater difference. At 27.68 Hz, CTWT-2 continues to provide the lowest variance (\(1.20\times 10^{-9}\)), reinforcing its reliability. The CTWT-1 method also gives damping ratios close to CTWT-2 at this frequency, demonstrating consistent performance. The PP method performs well, offering estimates close to CTWT-2. However, the pLSCF and Yoshida methods produce estimates that are significantly different from the rest, indicating less alignment with the more reliable methods. Also, the identified damping ratios exhibit greater consistency at higher frequencies. At lower frequencies, even minor inaccuracies in frequency determination can lead to substantial relative errors, thereby significantly affecting the accuracy of the damping ratio estimates. Conversely, at higher frequencies, the influence of frequency errors is diminished, resulting in more stable and reliable outcomes. Overall, while simulations suggested that LSRF and CTWT-2 would be among the best performers, the actual data indicate that CTWT-2 consistently delivers the lowest variances across all frequencies, marking it as a reliable method for damping ratio estimation. CTWT-1 provides damping ratios close to CTWT-2 at higher frequencies but with higher variances, indicating less stability. The PP method performs well across frequencies, offering estimates close to CTWT-2.