The time domain waveform and frequency spectrum of vibration data measured when the sun wheel is cracked are shown in Figure 1. It can be seen from Figure 1 that there are some disordered peaks in the time domain waveform, but there is no obvious fault impact caused by local fault. In the frequency spectrum, there are several sidebands on both sides of the meshing frequency FM, and the interval of sidebands is the rotation frequency fc of the planet carrier, which belongs to the normal frequency component. Therefore, it is impossible to judge whether the sun wheel is faulty.

In order to extract the fault feature information, the proposed apoelmd method is used to decompose the fault vibration signal of the sun gear. The amplitude and integration times of the white noise are fixed at 0.01sd and 2 respectively, and the upper limit frequency of the white noise is determined by RRMSE. Figure 2 (a) shows different upper frequency limits

The relative root mean square error corresponding to the rate. It can be seen from Figure 2 (a) that the optimal upper limit frequency for adding white noise to the sun gear fault signal is 30.72 kHz (6 times of the sampling frequency). Figure 2 (b) shows the decomposition result of apoelmd, in which the instantaneous frequency of the first product function (Pf1 component) fluctuates around the meshing frequency 6 times, as shown in Figure 2 (c). According to the principle of sensitive component selection, the Pf1 component is further demodulated and analyzed. Figure 2 (d) shows the Fourier spectrum of the amplitude envelope of the Pf1 component. In the envelope spectrum, the amplitudes at all peak frequencies are greater than the normal signal. Among them, the fault characteristic frequency FS and its frequency doubling NF are dominant. Other peaks appear in the absolute rotation frequency FSR and its frequency doubling NFSR of the sun gear, the rotation frequency fc and its frequency doubling NFC of the planet carrier, and the combined frequency NFS ± MFC (m, n = 1, 2,…) ）And so on. Fig. 2 (E) shows the Fourier transform spectrum of instantaneous frequency of Pf1 component. In the figure, it can be clearly found that the characteristic frequency FS and its frequency doubling NF are dominant. Similarly, similar to the envelope spectrum, there are absolute rotation frequency FSR and its frequency doubling NFSR of solar gear, rotation frequency FC of planetary carrier and its frequency doubling NFC, and their combined frequency NF s ± MFC (m, n = 1, 2 ）And the amplitude is more prominent than the normal signal. These characteristics indicate that the sun gear has a local fault, which is consistent with the experimental simulation of the sun gear root crack fault state.

As a contrast, the traditional elmd and EEMD methods are used to decompose the fault signal of the sun gear, in which the amplitude of the added white noise is set to 0.20sd, and the number of integration is set to 200. The envelope spectrum and instantaneous frequency Fourier spectrum of the decomposed sensitive component are analyzed, and the results are shown in Figure 3. In the envelope spectrum of Figure 3 (a), the three methods all extract the fault characteristic frequency information of the sun wheel, but the processing results of the apoelmd method are in the fault characteristic frequency FS and its frequency doubling NF, and the combined frequency NF ± MFC (m, n = 1, 2,…) ）The peak value at the same position is significantly higher than that of traditional elmd and EEMD methods. In the instantaneous frequency Fourier spectrum in Fig. 3 (b), the fault characteristic frequency information extracted by apoelmd is clearer, and there are many noise frequency components with large amplitude in the processing results of elmd and EEMD methods, which brings great interference to the fault diagnosis of solar gear and is not conducive to making correct judgment. Through the above comparative analysis, it can be seen that the proposed apoelmd method is superior to the traditional elmd and EEMD methods in signal decomposition performance, weak fault feature extraction and noise suppression.