## 5 Experimental Validation
### Result and Discussion  
As analyzed above, the presented approach is applicable to processing multi-component signals and is more effective than the existing methods (i.e., OVMD, LMD, EMD, and WT) in extracting time-frequency contents and fault features. The advantage of merging several procedures (i.e., waveform matching extension, GWO, VMD, and TEO) is that it makes the resulting algorithm more adaptive to the practical situation and that it can directly extract time-varying characteristics. Besides, the presented approach can improve the precision of signal decomposition. However, the above results are geared to qualitative analysis, which indicates that a quantitative analysis should also be conducted to further show the performance of the proposed method. Here, three evaluation indexes (i.e., the energy error \(\theta\), orthogonal index \(OI\), and correlation coefficient shown in Equation (11)) are adopted to compare, on a quantificational level, the decomposability of different methods. Figure 20: The analyzed results of the rotor oil-whip signal: (**a**) Temporal waveform, (**b**) FFT spectrum, time-frequency representations obtained by (**c**) IAVMD, (**d**) OVMD, (**e**) LMD, (**f**) EMD, and (**g**) WT. The energy error \(\theta\) of one signal \(x(t)\) can be given by [58]:  
\[\theta=\frac{\left|\sqrt{\sum\limits_{i=1}^{n+1}RMS_{i}^{2}}-RMS_{x}\right|}{RMS _{x}},\ RMS_{x}=\sqrt{\frac{\sum\limits_{i=1}^{N}x^{2}(i)}{N}} \tag{20}\]  
where \(RMS_{x}\) represents the root mean square of \(x(t)\), \(RMS_{i}\) represents the root mean square of the \(i\)-th modes, and \(n+1\) represents the mode number. According to Ref. [58], the smaller \(\theta\) value represents a higher decomposition accuracy and a lesser end effect. The orthogonal index of one signal \(x(t)\) can be given by [59]:  
\[OI=\frac{\sum_{i=1}^{N_{C}}\sum_{j=1}^{j<i}\left|\sum_{k=1}^{N}C_{ik}\times C_{ jk}\right|}{\sum_{k=1}^{N}\left(x_{k}-r_{k}\right)^{2}} \tag{21}\]  
where \(N_{C}\) represents the mode number, \(N\) means the data length of modes, \(C_{ik}(t)\) and \(C_{jk}(t)\) respectively denote the \(i\)-th and \(j\)-th modes at the sifting step \(k\), and \(x_{k}\) and \(r_{k}\) denote the raw signal and residual term, respectively. Similarly, according to Ref. [59], the smaller the value of the orthogonal index is, the better the decomposition performance is. Furthermore, it should be explained that we evaluate their performance through calculating the correlation coefficient between the raw signal with the signals reconstructed by different methods. Tables 2-4 list the energy error \(\theta\), orthogonal index \(OI\), and correlation coefficient of the decomposition results obtained by different approaches for the simulation data and experimental signal, respectively. It can be known from Tables 2 and 3 that two evaluation indexes (\(\theta\) and \(OI\)) of our proposed IAVMD approach are lower than those in four similar approaches (i.e., OVMD, LMD, EMD, and WT), which implies that our approach is equipped with satisfactory a decomposition performance in analyzing complicated compound signals compared with other similar methods (i.e., OVMD, LMD, EMD, and WT). However, it should be pointed out that the computational load of our approach is greater than those in other methods (i.e., OVMD, LMD, EMD, and WT) because of the waveform matching and parameter optimization procedure. Besides, OVMD also has a high computation time including the time cost of two stages (i.e., parameter optimization and signal decomposition). Hence, with the help of the high-end computing platform and computer technology, the improvement of the calculation efficiency of the proposed approach is regarded as our following work and research direction.