## 5 Dynamic Simulation and Real Application
### Application to Vibration Signals of Aeroengine
Then, the signal from the reconstruction matrix was recovered to obtain the purified signal. C\({}_{1}\)(t), C\({}_{2}\)(t) are the purified signals of u\({}_{1}\)(t), u\({}_{4}\)(t), and C\({}_{1}\)(t), C\({}_{2}\)(t)are shown in Figure 15.6 The envelope spectrums for C\({}_{1}\)(t) (i.e., 1\(\times\)) and C\({}_{2}\)(t) (i.e., 3\(\times\)) are revealed in Figure 15.7 The main fluctuating frequencies of the envelope of 1\(\times\) are 23.4 Hz and 93.6 Hz. Additionally, 23.4 Hz is 1/4-times the rotation frequency, 93.6 Hz is the rotation frequency. As for the 3\(\times\), the fluctuating frequencies of the envelope of 3\(\times\) are also 23.4 Hz and 93.6 Hz. After comparing with the simulated vibration signal in Figure 10b, it is found that the fluctuating frequencies of the envelope of 1\(\times\), which was obtained by processing the simulated signal, do not contain 1/4-times the rotation frequency. The fluctuating frequencies of the envelope of 1\(\times\), which was obtained by processing the signal of the real aeroengine test, do not contain two-times 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 15.8 The decomposed signals. (**a**) The waveform of u\({}_{1}\)(t); (**b**) The waveform of u\({}_{4}\)(t). Figure 15.9 Vibration signal of aeroengine when the low-pressure turbine speed is 5616 r/min. (**a**) The waveform of signal when the low-pressure turbine speed is 5616 r/min; (**b**) Fourier spectrum of signal when the low-pressure turbine speed is 5616 r/min. Second-layer SVD was applied to IFs of C\({}_{1}\)(t), C\({}_{2}\)(t), and the first 20 singular values obtained are shown in Figure 16.0 The first seven largest singular values for matrix reconstruction were selected, several component matrices were obtained, and the component matrices were superimposed to obtain the reconstruction matrix. Then, the signal was recovered from the reconstruction matrix. FFT was applied to the IFs of fundamental harmonic (1\(\times\)) and its three multiples (3\(\times\)); the FFT of the IFs are revealed in Figure 16.1 We can see that the main fluctuating frequencies of the IF of 1\(\times\) are 70.2 Hz, 93.6 Hz, 105.3 Hz. Additionally, 70.2 Hz is 3/4-times the rotation frequency, 93.6 Hz is the rotation frequency, 105.3 Hz is 9/8-times the rotation frequency. IF, which fluctuates around the rotation frequency, is a feature of rub-impact fault. Additionally, the main fluctuating frequencies of the IF of 3\(\times\) are 70.2 Hz, 93.6 Hz, 187.2 Hz. Additionally, 70.2 Hz is 3/4-times the rotation frequency, 105.3 Hz is 9/8-times the rotation frequency, 105.3 Hz is 9/8-times the rotation frequency, 105.  
frequency, 93.6 Hz is the rotation frequency, 187.2 Hz is two-times the rotation frequency. After comparing with the simulated vibration signal in Figure 10a, 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 3/4-times, and 9/8-times the rotation frequency. The fluctuating frequencies of IF of 1\(\times\), which were obtained by processing signal of the real aeroengine test, do not contain two-times the rotation frequency, three-times the rotation frequency, four-times the rotation frequency, and five-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. For comparison, the proposed method was used: HHT, STFT, and Nonlinear Chirp Mode Decomposition (VNCMD) on the signal, as shown in Figure 16.2 For the VMD method, \(K\) was set in advance to determine the number of modes in the data, and \(K\) was set to 4.