## 5 Dynamic Simulation and Real Application
### Application to Vibration Signals of Aeroengine
The raw signal is decomposed into two modes using VMD. D\({}_{1}\)(t) is 1\(\times\). Additionally, D\({}_{1}\)(t) is shown in Figure 16.5  
The first layer SVD was performed to denoise D\({}_{1}\)(t) and retain the first 250 singular value components; the first 250 singular values obtained are shown in Figure 16.6 The signal after denoising is E\({}_{1}\)(t), E\({}_{1}\)(t) is shown in Figure 31 The envelope spectrum of the signal after signal denoising is shown in Figure 16.7 It can be seen that the main fluctuating frequencies of 1\(\times\) on the envelope spectrum are 97.1 Hz and 111.0 Hz. Additionally, 97.1 Hz is 7/8-times the rotation frequency and 111.0 Hz is equal to the rotation frequency. After comparing with the simulated vibration signal in Figure 13b, it is found that the fluctuating frequencies of the envelope of 1\(\times\), which is obtained by processing the simulated signal, do not contain 7/8-times the rotation frequency. The fluctuating frequencies of the envelope of 1\(\times\), which were obtained by the processing signal of the real aeroengine test, do not contain two-times the rotation frequency. During the engine test, the rub-impact occurred more complicated than in the dynamic simulation, and the parameters \(k_{c}\), \(\delta_{0}\), etc. in the model were not completely consistent with the parameters in the real aeroengine test. Figure 16.8 Time–frequency representations of the vibration signal obtained utilizing several methods. (**a**) Two-Layer SVD of signal after VMD (K = 4); (**b**) HHT (IM\({}_{3}\)); (**c**) HHT of signal (IM\({}_{2}\)); (**d**) HHT of signal after median filtering (IMF\({}_{2}\) and IMF\({}_{3}\)); (**e**) The STFT of signal (The length of the window is 256); (**f**) The STFT of signal (The length of the window is 2560); (**g**) VNCMD of signal (K = 2); (**h**) VNCMD of signal (K = 3); (**i**) VNCMD of signal (K = 4). Figure 16.9 Vibration signal of aeroengine when rotation speed of the low-pressure turbine is 6660 r/min. (**a**) The waveform of the vibration signal when rotation speed of the low pressure turbine is 6660 r/min; (**b**) FFT of the vibration signal when rotation speed of the low pressure turbine is 6660 r/min. Figure 17.0 Singular values of SVD of \(\rm D_{1}(t)\). Figure 17.1 The waveform of \(\rm D_{1}(t)\). The second layer SVD was performed on the instantaneous frequency. The FFT of the instantaneous frequency is shown in Figure 17.2 It can be seen that the fluctuating frequencies of the instantaneous frequency are mainly 97.1 Hz, 111.0 Hz, and 208.1 Hz, which are, respectively, equal to 7/8-times, one-time, and 15/8-times the rotation frequency. Moreover, the fluctuating frequencies of the instantaneous frequency of 97.1 Hz and 111.0 Hz are consistent with the fluctuating frequencies on the envelope spectrum. If fluctuating around the rotation frequency is a feature of rub-impact fault, and it can identify rub-impact fault more clearly than the lower amplitude 2\(\times\), 3\(\times\) in Fourier spectrum in Figure 28b. After comparing with the simulated vibration signal in Figure 13a, it is found that the fluctuating frequencies of IF of 1\(\times\), which was obtained by processing the simulated signal in dynamic simulation, do not contain 7/8-times, and 15/8-times the rotation frequency. The fluctuating frequencies of IF of 1\(\times\), which was obtained by the processing signal of the real aeroengine test, do not contain two-times the rotation frequency, and three-times the rotation frequency. During the engine test, the rub-impact occurred more complicated than in the dynamic simulation, and the parameters \(k_{c}\), \(\delta_{0}\), etc. in the model were not completely consistent with the parameters in the real aeroengine test. Figure 17.3 Purified 1\(\times\) signal. Figure 17.4 Envelope spectrum of 1\(\times\). For comparison, the proposed method was used; HHT, STFT, VNCMD, on the signal, as shown in Figure 17.5 Using the proposed method, the instantaneous frequency result obtained is very good, as shown in Figure 34a.