## 4 Simulation Validation  
### Case 1: Sinusoidal Superposition Signal  
Considering that the real rotor vibration signal is generally expressed as multiple harmonic superposition signals (e.g., the linear superposition of fundamental frequency, fractional frequency, and ultraharmonics), to confirm the availability of the presented approach we here employ a numerical signal \(y(t)\) consisting of four sine waves (their frequencies are, respectively, 30 Hz, 60 Hz, 100 Hz, and 150 Hz) to simulate the rotor fault signal, which is considered as follows:  
\[ \begin{array}{l}y(t)=x_{1}(t)+x_{2}(t)+x_{3}(t)+x_{4}(t)\\ \quad\quad=\sin(60\pi t)+\sin(120\pi t)+\sin(200\pi t)+\sin(300\pi t)\end{array}  \tag{18}\]  
where \(x_{1}(t)\) is used to simulate the fundamental frequency component of the rotor fault signal, \(x_{2}(t)\) is used to simulate the double frequency component of the rotor fault signal, \(x_{3}(t)\) is used to simulate the fractional frequency component of the rotor fault signal, and \(x_{4}(t)\) is used to simulate the higher harmonic component of the rotor fault signal. For the simulation signal \(y(t)\), its sampling frequency and sampling number are set as 3000 Hz and 1500, respectively. Figure 8 draws the numerical signal \(x(t)\) and its FFT spectrum. In the IAVMD method, the numerical signal \(y(t)\) is first preprocessed by waveform matching extension, after which the optimal combined parameters (\(\hat{\alpha}\), \(K\)) of VMD obtained using GWO are (1986, 4). That is, VMD with \(\hat{\alpha}=1986\) and \(K=4\) is applied to analyze the numerical signal \(y(t)\), and the  
Figure 7: The flowchart of the proposed IAVMD method. ultimate decomposition results are shown in Figure 9. Note that, except for the input signal and two key parameters (\(\hat{\alpha}\), \(K\)), another four parameters (e.g., time-step of the dual ascent \(\tau\), DC part, uniform initialization of omegas, and tolerance of the convergence criterion \(\varepsilon\)) of VMD have a fixed value, set as 0, 0, 0, and \(10^{-6}\), respectively. Seen from Figure 9, four frequency components of the numerical signal \(y(t)\) are separated efficiently. Figure 10a,b plots the two-dimensional and three-dimensional TFR based on IAVMD, respectively. One can clearly see that the time-frequency trajectory obtained by IAVMD is very clear in Figure 10, which indicates that the proposed IAVMD can exactly decompose the sinusoidal superposition signal containing multiple components. As a comparison, particle swarm optimization-based VMD (OVMD) and three other techniques (i.e., LMD, EMD, and WT) are devoted to disposing the same signal \(y(t)\). The optimization range of OVMD is the same as for the proposed method. Here, the optimized parameters of OVMD are respectively determined as \(\hat{\alpha}=1120\) and \(K=4\) by using the PSO algorithm. The decomposition results acquired by OVMD, LMD, EMD, and WT are shown in Figure 11a-d, respectively. Besides, the acquired TFR using four techniques (i.e., OVMD, LMD, EMD, and WT) are shown in Figure 12a-d, respectively. Note that, in the comparison process, to ensure a fair comparison, the TFR of four methods (i.e., OVMD,  
Figure 11: The decomposition results obtained by different methods for simulation 1: (**a**) VMD, (**b**) LMD, (**c**) EMD, and (**d**) WT. Figure 10: The time-frequency representations obtained by IAVMD for simulation 1: (**a**) Two-dimensional and (**b**) three-dimensional. LMD, EMD, and WT) are obtained by TEO, where WT adopts the 'db1' wavelet function containing a good orthogonality and where three decomposition layers are performed (i.e., four reconstructed wavelet coefficients can be obtained for implementing the time-frequency analysis). As shown in Figure 12a, four sub-components are extracted effectively, but they suffer from certain boundary fluctuations (see the red dotted line). From Figure 12b,c, one can observe that the first two IMFs suffer from serious mode aliasing and that the signal representation in the time-frequency plane is highly dispersed.