## 5 Dynamic Simulation and Real Application
### Application to Vibration Signals of Aeroengine  
The structure sketch of the aeroengine is revealed in Figure 14. The rotation speed of the low-pressure turbine is approximately 5616 r / min (93.6 Hz), and the rotation speed of the high pressure turbine is approximately 10,445 r / min (174.1 Hz). A charge output accelerometer was used to measure the vibration acceleration, which was installed on the outer surface of aeroengine casing. There existed rubbing between the low-pressure turbine and the stator vane. The mounting position of the accelerometer is shown in Figure 15. During installation, according to the requirements on the \"installation and operating manual of the accelerometer\", the mounting torque is controlled within the range of 113 N.cm to 225 N.cm. A photo of the accelerometer is shown in Figure 16. The accelerometer was produced by PCB company. Its sensitivity is 1.02 pC/(m/s\({}^{2}\)), its frequency range is 9 kHz, its non-linearity \(\leq\)1%, its operating temperature range is (\(-\)71) \({}^{\circ}\)C to 260 \({}^{\circ}\)C, and its weight is 11 g. The accelerometer transfers the charge through the cable to the charge converter. A photo of the charge converter is shown in Figure 17. The charge converter was produced by DONGHUA company. It has 16 channels, its maximum measurement range is 100,000 pC, its maximum signal output voltage is \(\pm\)10 V, its current excitation is 2 mA, and its frequency range is 100 kHz. It transfers the voltage through the cable to a data acquisition device. Figure 14: Structure sketch of aeroengine. Figure 13: Fourier spectrum of the IF and Envelope spectrum when the rotation speed is 6660 r / min. (**a**) Fourier spectrum of the IF (1\(\times\)); (**b**) Envelope spectrum of 1\(\times\). A photo of the data acquisition device is shown in Figure 18. The data acquisition device was produced by DONGHUA company. Maximum sampling frequency of the data acquisition device is 1 MHz per channel. It has 16 data acquisition channels and its accuracy is \(\pm\)0.5%. It has multiple measurement ranges that can be switched; the minimum measurement range is \(\pm\)5 mV, and the maximum measurement range is \(\pm\)10 V. It transmits the signal to the computer through the network cable. Figure 16: Photo of the accelerometer. Figure 17: Photo of the charge converter. Figure 15: Position of accelerometer. During the aeroengine test, the root mean square (RMS) of vibration velocity almost reached the safety limit, as illustrated in Figure 15.0  
The waveform and Fourier spectrum of the vibration signal obtained from the aeroengine test is revealed in Figure 15.1 It is illustrated that there are two frequency peaks with high amplitude, i.e., the rotation frequency of the low-pressure turbine (X), and its three multiples (3\(\times\)). After comparing with the FFT of the simulated vibration signal in Figure 8b, it was found that the amplitude of 3\(\times\) of the vibration signal obtained from simulation is high is very low. 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 dynamic simulation were not completely consistent with the parameters in the real aeroengine test. Figure 15.2 RMS of vibration velocity and the safety limit (red: safety limit; blue: RMS of vibration velocity). Figure 15.3 Photo of the data acquisition device. The raw signal is decomposed into 4 modes using VMD [18]. u\({}_{1}\)(t) is 1\(\times\), u\({}_{4}\)(t) is 3\(\times\). Additionally, u\({}_{1}\)(t), u\({}_{4}\)(t) are shown in Figure 15.4  
Then, first-layer SVD was applied to u\({}_{1}\)(t), u\({}_{4}\)(t) for signal denoising. Firstly, the Hankel matrices were constructed. Then, SVD was performed on these two matrices; the first 250 singular values obtained are shown in Figure 15.5 The first 250 largest singular values for matrix reconstruction were then selected, several component matrices were obtained, and the component matrices were superimposed to obtain the reconstruction matrix.