## 5 Dynamic Simulation and Real Application
### Application to Vibration Signals of Aeroengine
It can be clearly seen that the instantaneous frequency is the sum of several sinusoids. After using HHT, 1\(\times\) is decomposed, as shown in Figure 34b. However, the instantaneous frequency fluctuates irregularly, with a large fluctuation range around 0 s, and a small fluctuation range around 0.8 s. The STFT result of the signal is shown in Figure 34c. 1\(\times\) is shown in the time-frequency diagram. The time-frequency diagram obtained by this method cannot guarantee good time resolution and frequency resolution [29]. Directly using VNCMD on the signal, the number of signal components was set to K equal to 2, and 1\(\times\) was almost decomposed, as shown in Figure 34d, but the result at 0.5 s on the time-frequency diagram is wrong. Figure 17.6 Time–frequency representations of the vibration signal obtained utilizing 4 methods. (**a**) Two-Layer SVD of signal after VMD (K = 2); (**b**) HHT of signal; (**c**) STFT of signal; (**d**) VNCMD of signal (K = 2). Figure 17.7 FFT of the instantaneous frequency (1\(\times\)). Next, let us analyze the third set of data. At this time, the speed of the low-pressure turbine is 5220 r/min (87.0 Hz), and the waveform of the vibration signal is shown in Figure 35a. The FFT of the vibration signal is shown in Figure 35b, the amplitude of 1\(\times\) is high, amplitudes of 2\(\times\) to 6\(\times\) are low. The raw signal is decomposed into two modes using VMD. F\({}_{1}\)(t) is 1\(\times\), the waveform of the F\({}_{1}\)(t) is shown in Figure 17.8  
The first layer of SVD was performed on F\({}_{1}\)(t) for signal denoising, and the first 250 singular value components retained, and the first 250 singular values obtained are shown in Figure 17.9 The signal after denoising was G\({}_{1}\)(t), and G\({}_{1}\)(t) is shown in Figure 14:0 The envelope spectrum of the signal after denoising is shown in Figure 14:1 It can be seen that the main fluctuating frequencies of 1\(\times\) on the envelope spectrum are 87.0 Hz and 108.8 Hz. Additionally, 87.0 Hz is equal to the rotation frequency, 108.8 Hz is 5/4-times the rotation frequency. Figure 14:2 Vibration signal when rotation speed of the low-pressure turbine is 5220 r/min. (**a**) The waveform of the vibration signal when rotation speed of the low-pressure turbine is 5220 r/min; (**b**) FFT of the vibration signal when rotation speed of the low-pressure turbine is 5220 r/min. Figure 14:3 The waveform of F\({}_{1}\)(t). The second layer SVD is performed on the instantaneous frequency. The FFT of the instantaneous frequency is shown in Figure 14:4 It can be seen that the fluctuating frequencies of the instantaneous frequency are mainly 87.0 Hz, 108.8 Hz, and 174 Hz, which are, respectively, one-time, 5/4-times, and two-times the rotation frequency. IF fluctuating around the rotation frequency is a feature of the rub-impact fault. Moreover, the fluctuating frequencies of the instantaneous frequency are 87.0 Hz and 108.8 Hz, which are consistent with the fluctuating frequencies on the envelope spectrum. Figure 14:5 Singular values of SVD of F\({}_{1}\)(t). Figure 14:6 Purified 1\(\times\) signal. Figure 14:7 Envelope spectrum of 1\(\times\). For comparison, the proposed method was used; HHT, STFT, VNCMD, on the signal, as shown in Figure 14:8 Using the proposed method, the instantaneous frequency result obtained was very good, as shown in Figure 41a. It can be clearly seen that the instantaneous frequency is the sum of several sinusoids. After using HHT, 1\(\times\) was decomposed, as shown in Figure 41b. However, the instantaneous frequency fluctuates irregularly. The STFT result of the signal is shown in Figure 41c. You can see 1\(\times\) in the time-frequency diagram. The time-frequency diagram obtained by this method cannot guarantee good time resolution and frequency resolution [29]. Using VNCMD on the signal, the number of signal components was set K equal to 2, and 1\(\times\) was also decomposed, as shown in Figure 41d. Figure 14:9 Time–frequency representations of the vibration signal obtained utilizing 4 methods.