## 4 Simulation Signal Analysis
As can be seen from Figure 5b, the fundamental frequency, fractional harmonics and high harmonics are all clearly presented. Then, the simulation signal is processed by EMD, SSD and ISSD respectively, and the results are shown in Figures 6-8, respectively. Figure 6a displays the decomposed sub-components using EMD and Figure 6b depicts the corresponding HHT TFS, i.e., the EMD-HT TFS. To clearly reveal the relationship between the frequency components exhibited in the TFS and the rotation frequency, the y-axis of the TFS in this paper was described using the frequency order, which is defined as the ratio of the actual frequency to the rotation frequency. Six sub-components were extracted after the EMD decomposition, but some of them are false components without physical meaning. The model aliasing problem appeared on the HHT TFS, which makes the HTT TFS cannot accurately reflect the fault information. As shown in Figure 7a, four sub-components were derived after the SSD decomposition. The corresponding SSD-HT TFS as depicted in Figure 7b reflects the characteristic frequencies of the simulated rubbing fault. Moreover, the \"intrawave frequency modulation\" phenomenon which is one of the characteristics of rubbing fault as reported in [35] was clearly extracted. However, the end effects can also be found from the SSD-HT TFS. The appearance of the end effect can cause the change of the energy-frequency-time distribution in the TFS to some extent. Consequently, the energy-frequency-time distribution of the TFS will lose its real regularity. To ensure a more accurate measure of the energy distribution characteristics of the TFS by using the CFBEE index, the end effect problem should be solved. Compared with Figure 7a, the sub-components obtained by using ISSD as shown in Figure 8a overcome the end effects. The corresponding ISSD-HT TFS displayed in Figure 8b is also away from the end effects. Hence, the superiority of ISSD method over the other two methods can be demonstrated from the simulated results. Figure 5: (**a**) The simulated rubbing fault signal; (**b**) its frequency spectrum. Finally, the CFBEE of the ISSD-HT time-frequency spectrum shown in Figure 8b was calculated as [0.0111, 0.2689, 0.1326 1.1420, 0.0544, 0.3804, 0.7641, 0.0169]. It is visible that the values of the energy entropy of different characteristic frequency bands are different. As the regularity and complexity of the normalized energy sequences of different characteristic frequency bands are different and the entropy index can measure the regularity and complexity of the signal [13; 14; 16; 30], the local information of the energy distribution around the characteristic frequency bands can be measured by the CFBEE indicator. Sometimes, the frequency components in the time-frequency spectrums of different rotor states are similar and difficult to distinguish by naked eyes, the CFBEE indicator can be used as an automatic fault feature extraction method to estimate the differences between different rotor states. The vibration signal of the rotor collected in the practical operating condition is frequently influenced by external noise [6]. To further verify the effectiveness of the developed method in  
Figure 6: (**a**) The decomposed sub-components using EMD; (**b**) the corresponding HHT time–frequency spectrum. Figure 7: (**a**) The decomposed sub-components using SSD; (**b**) the corresponding SSD-HT time–frequency spectrum. extracting the fault characteristics from the noisy vibration signals. Additive Gaussian random noise \(n(t)\) was simulated by employing the following equation:  
\[n(t)=0.3 randn(1,N), \tag{24}\]  
where \(N\) represent the length of \(n(t)\). Figure 9 shows the waveform of the additive Gaussian random noise. Then, the obtained noise was added to the simulated rubbing fault signal shown in Figure 5a to generate a mixed signal, which was shown in Figure 10a. Figure 10b describes the FFT spectrum of the mixed signal. Some harmonics of the rotation frequency disappeared compared to Figure 5b due to the influence of noise.