## 5 Dynamic Simulation and Real Application
### Application to Vibration Signals of Aeroengine
After VMD and two-layer SVD, the Hilbert spectrum of 1\(\times\) and 3\(\times\) are shown in Figure 27a. Compared to Figure 27b-i, the fluctuating pattern of the IFs are clearly shown in Figure 27a, and the 1\(\times\) and 3\(\times\) can be recognized. The HHT was utilized to analyze the raw signal, which is not purified. The raw signal was decomposed into five IMFs. Only IMF\({}_{2}\), IMF\({}_{3}\), which are related to 1\(\times\) and 3\(\times\), are shown in the Hilbert spectrum. Additionally, the Hilbert spectrum of IMF\({}_{3}\) is shown in Figure 27b, IMF\({}_{3}\) is almost 1\(\times\), IF of IMF\({}_{3}\) fluctuates around 93.6 Hz most of the time. However, it suffers from mode mixing at 0.03 s, 0.21 s, 0.25 s, 0.42 s, 0.60 s and 0.64 s. Additionally, at these times the instantaneous frequency is incorrect. The Hilbert spectrum of IMF\({}_{2}\) is shown in Figure 27c; the instantaneous frequency of IMF\({}_{2}\) fluctuates irregularly, with a large fluctuation range from 0 s to 0.8 s. The minimum value of its fluctuation is even less than 0 Hz from 0.3 s to 0.5 s, which is obviously wrong. Figure 16.3 The Fourier spectrum of IFs. (**a**) Fourier spectrum of the IF(1\(\times\)); (**b**) Fourier spectrum of the IF(3\(\times\)). Figure 16.4 The singular values of SVD of IFs. (**a**) Singular values of SVD of IF of C\({}_{1}\)(t); (**b**) Singular values of SVD of IF of C\({}_{2}\)(t). Then, the median filtering was utilized to the IFs and instantaneous amplitudes (IAs); the result is shown in Figure 27d. IMF\({}_{3}\) is almost 1\(\times\), but it suffers from mode mixing, 3\(\times\) cannot be seen from IMF\({}_{2}\), and the FM feature of IMF\({}_{2}\), IMF\({}_{3}\) disappear. So, signal denoising before Hilbert transform is necessary. The short-time Fourier transform (STFT) was used for the original signal, when the length of the window is 256. Additionally, the result of STFT is shown in Figure 27e. By comparing Figure 27a-c,e, it is found that both Figure 27a,e can successfully show 1\(\times\) and 3\(\times\), whereas Figure 27b,c do not display 1\(\times\) and 3\(\times\) correctly due to noise. However, Figure 27e has a lower frequency resolution than Figure 27a. To make the frequency resolution of STFT higher, the length of the window was adjusted to 2560. By comparing Figure 27a,f, it is found that Figure 27f cannot show the characteristic of rapid change of instantaneous frequencies. The time-frequency diagram obtained by using STFT cannot guarantee good frequency resolution and the fast-changing instantaneous frequency can be displayed at the same time, no matter how we adjust the length of the window. In fact, the time-frequency diagram obtained by this method cannot guarantee good time resolution and frequency resolution [29]. Applying VNCMD to the signal, when using this method, the number of signal components needs to be set to K in advance [30; 31; 32], when K is set equal to 2, the decomposition result is shown in Figure 27g, 1\(\times\) is correctly decomposed, but 3\(\times\) cannot be seen. When K is set to 3, the VNCMD result is shown in Figure 27h. It can be seen that the instantaneous frequency of 1\(\times\) is wrong at 0.65 s, and 0.72 s, and the instantaneous frequency of 3\(\times\) is wrong from 0.62 s to 0.8 s. When K is set to 4, the VNCMD result is shown in Figure 27i. The first three modes almost overlap each other, and their center frequency is all 93.6 Hz, and the instantaneous frequency of 3\(\times\) is wrong from 0.4 s to 0.8 s.  
Next, we analyzed the second set of fault data, when the speed of the low-pressure turbine is about 6660 r/min (111.0 Hz). The waveform of the vibration signal is shown in Figure 28a, The FFT of the vibration signal is shown in Figure 28b. After comparing with the simulated vibration signal in Figure 11b, it is found that the Fourier spectra of the dynamic simulation signal and the aeroengine test signal are relatively similar, with higher 1\(\times\) amplitudes and lower 2\(\times\) and 3\(\times\) amplitudes.