## 3 Experimentation and Analysis
### Result Analysis  
The result of Shannon entropy was then obtained using IVMD-SVD with the selected K-value. The process was applied to 30 measurements of each of the three rail conditions presented in Figure 6. The blue, red, and yellow bars represent healthy, 0.1 mm, and 0.75 mm depths of fault, respectively. In order to evaluate the effectiveness of IVMD-SVD it was compared to the energy ratio, which is a classic energy-based method [27]. The approach is based on the sum of the  
Figure 5: Optimal K-value selection from the acoustic impact response signal. Figure 6: Entropy results from IVMD-SVD. Figure 8 shows that, for all health conditions, the average distinction degree obtained from the entropy result was lower than that obtained using the energy ratio. This indicated that the entropy results processed using IVMD-SVD were more uniform than those obtained using the energy ratio method. The proposed method was, therefore, considered more robust against environmental interference, such as that caused by a change of impact location or force. In order to further validate the effectiveness of the IVMD-SVD algorithm, the evaluation process was repeated using vibration-based impact response signals. A vibration-based impact response signal is another impact response signal closely related to acoustic signals. The optimal K-value selection from the vibration-based impact response signals is shown as Figure 9. The products of the second derivatives at K-value 5 (\(-\)0.038) and K-value 6 (0.019) were negative. The absolute difference of these two second derivatives was 0.057 which was larger than the threshold of 0.05. This indicated that the optimal K-value was 6, which was the same as the conclusion drawn for the acoustic version of the impact response signal. The distinction degree results from IVMD-SVD entropy and energy ratio are presented in Figure 10.  
Figure 8: Comparison of distinction degrees between two methods for acoustic signals. Figure 9: Comparison of distinction degrees between two methods for vibration signals. The result for the vibration-based signal presented in Figure 10 was similar to that obtained from the acoustic signal. The distinction degree values calculated from IVMD-SVD were lower than those calculated from the energy ratio. Therefore, it was further demonstrated that IVMD-SVD produced more uniformly decomposed signals, and that there was consistency when applied to both acoustic and vibration-based signals. Hence, it was considered that the approach was suitable for reducing environmental effects in a range of recorded signals. To illustrate the effectiveness of appropriately selecting the K-value on fault classification, an index representing the capacity for identification (\(C_{k}\)) of different rail conditions was calculated. The index could be described as a subtraction of the maximum and minimum differences from the mean value of entropy [10], as given in Equation (19). \[ \begin{array}{l}c_{max\ i}=max\big{|}\overline{C_{i}}-E_{ij}\big{|}\\ c_{min\ i}=min\big{|}\overline{C_{i}}-E_{ij}\big{|}\\ C_{k}=\sum\limits_{i=1}^{K}|c_{max\ i}-c_{min\ i}|\end{array}  \tag{19}\]  
where the \(\overline{C_{i}}\) is the mean value of \(IMF_{i}(t)\). The identification capacity index results for each K-value, generated for the acoustic impact response signals, are shown in Figure 11.  
The values presented in Figure 11 represent the differences in signals obtained from different rail conditions, considered by K-value used in the processing. The effect of the signal decomposition was better when the value of \(C_{k}\) was higher. The figure shows that  
Figure 11: Capacity for identification. Figure 10: Comparison of distinction degrees between two methods for acoustic signals. the highest value of \(C_{k}\) corresponded to K = 6. This further demonstrated that the selected K-value was the most appropriate in differentiating the different rail conditions. This section demonstrated that the proposed method was suitable for the differentiation of rail conditions.