# Diagnosis Method for Mechanical Faults Based on Rotation Synchrotexting Chirplet Transform
## 3 Numerical Experiments  
In this section, a simulated signal is constructed to test the performance of the RSECT method in addressing the nonstationary signals with multiple components. Some advanced TF analysis methods, such as SET, GLCT [22], are also taken for comparison. An AM-FM signal is defined as  
\[\left\{ \begin{array}{l}f_{1}(t)=\left[\sin(2\pi(25t_{1}+10\sin(t_{1})))\sin (2\pi(34\cdot 2t_{2}))\right]\\ f_{2}(t)=\left[\sin(2\pi((25+8)\,t_{1}+10\sin(t_{1})))\sin(2\pi(34\cdot 2+8)t_{ 2})\right]\\ f_{3}(t)=\sin(2\pi(8t+3 arc\tan((t-5)^{2})))\\ t_{1}\in(0\;6)\;\;\;t_{2}\in(6\;7)\;\;\;t=[t_{1}t_{2}]\end{array} \right. \tag{13}\]  
The sampling frequency is 100 Hz, the sampling time is 14 s, and the window width length is selected as 100. The time-domain waveform of the simulated signal is shown in Figure 1. To verify the effectiveness of the RSECT, here, it is compared with the SET and GLCT [22]. Figures 2-4 show the time-frequency distribution obtained from the GLCT, SET, and RSECT methods, respectively. From Figures 2-4, the SET and RSECT have better TF energy aggregation in processing the non-stationary signals. While the GLCT can effectively extract the frequency characteristics of the signals, its TF aggregation is poor due to the limitation of Heisenberg uncertainty principle. Comparing Figures 3 and 4, it can be found that the RSECT has more advantages in processing the FM signal components, while the SET has more divergence in signal energy because the window function can only be translated in the horizontal direction. That is, the fixed window width leads to a larger bandwidth in the SET for FM signal processing. Despite the construction of the SEO operator, the energy around the non-time-frequency ridges is inevitably processed, resulting in a redundancy signal, which is not conducive to the aggregation of TF energy. Therefore, the SET is not able to obtain better TF results than the RSECT. The RSECT reduces the energy bandwidth when processing the FM signal by enabling the window direction of the window function to rotate adaptively according to the frequency ridge. Therefore, the RSECT can aggregate the TF energy of the processed signal more. To evaluate the anti-noise performance and the TF aggregation performance in the noisy environment of the RSECT, a gaussian white noise of 10.4 dB is added to the simulation signal, i.e., a noisy simulation signal with a signal-to-noise ratio (SNR) of 10 is constructed to simulate the actual fault vibration signal with noise. The time-domain waveform of the simulated signal with noise is shown in Figure 5, and its TF representations generated by the three different methods are shown in Figures 6-8. Figure 4: TF distribution of the simulated signal based on RSECT. Figure 3: TF distribution of the simulated signal based on SET. Figure 2: TF distribution of the simulated signal based on GLCT. Figure 5: Time-domain waveform of the simulated signals with noise. Figure 6: TF distribution of the simulated signal with noise based on GLCT. Figure 7: TF distribution of the simulated signal with noise based on SET. Comparing Figures 6-8, the TF results obtained by all three methods can reflect the constant frequency characteristics and FM characteristics of the noisy simulation signal and they all achieve very good anti-noise effect. Specifically, the TF accuracy of the GLCT is significantly inferior to that of the SET and RSECT. Comparing Figures 7 and 8, it can be found that the RSECT improves the anti-noise effect while ensuring that the TF results is not degraded. The SET also has a fatal flaw, that is, the instantaneous frequency difference of adjacent modes in processing signals with multi-frequency is greater than two times the frequency support range of the selected window function. However, the chosen frequency support range of the window function can only rely on manual trial and error, lacking basis. The RSECT proposed in this paper is adaptive and overcomes the shortcomings of the SET well.