Diagnosis Method for Mechanical Faults Based on Rotation Synchroextracting Chirplet Transform

Abstract

The problems of the synchroextracting transform method being unable to handle FM signals and being prone to time–frequency feature discontinuity in a strong noise environment are addressed by the construction of a novel rotation synchroextracting chirplet transform under the framework of the synchroextracting transform. The method retains the advantage of the generalized linear chirplet transform that can fit the time–frequency characteristics of the original signal and retains the high precision time–frequency analysis ability of the synchroextracting transform. The simulation results show that the proposed method is obviously superior to the generalized chirplet transform and synchroextracting transform method. The method can obtain the time–frequency energy located at the time–frequency ridges of FM-AM signals and multicomponent signals with crossed-frequency components, and has high time–frequency analysis ability and anti-interference ability. Finally, the proposed method is applied to diagnose mechanical faults. The experimental results further verify the effectiveness of the proposed method, which can effectively extract the characteristic freque.ncy of fault signal.

1. Introduction

Synchroextracting transform (SET) is a novel time–frequency (TF) analysis technique, which was proposed by Yu et al. [1] under the inspiration of the synchrosqueezed transforms (SST) method [2]. SET constructs a synchroextracting operator (SEO) based on STFT to extract the TF coefficients at the TF ridge, thus removing the redundant signals around the TF ridge and greatly improving the accuracy of TF analysis. Due to the good TF aggregation performance and noise robustness performance, SET has been widely applied in petroleum exploration [3], fault diagnosis [4,5], power equipment testing [6], and bearing performance evaluation [7] once it was proposed. Yu [8] was the first to apply SET to the TF analysis of the vibroacoustic signals of excavators, and he obtained the conclusion that SET has better TF aggregation and better noise robust performance by comparing it with SST and STFT. Li et al. [9] verified that SET can greatly improve TF aggregation and can perfectly describe the local TF characteristics of FM signals through theoretical numerical simulation analysis and comparison with SST.

However, some shortcomings of SET are gradually exposed as the research progresses. For example, SET is limited by some difficulties in tackling complex non-stationary nonlinear signals. In addition, the results of SET for emphasized amplitude FM signals are pooled due to the assumption based on constant amplitude and linear FM signals. Influenced by STFT, the same frequency resolution is used for analyzing the high-frequency and low-frequency parts of the FM signals, which inevitably retains some disadvantages of STFT and leads to poor adaption. Therefore, SET is not able to reconstruct the original signal more perfectly. In order to overcome these problems in the traditional SET, some revised SET methods are proposed. The basic idea of these improved SET is to apply signal decomposition methods, such as: two-level adaptive FM modal decomposition [10], Empirical mode decomposition [11], primary kernel regression residual decomposition [12], nonlinear FM modal decomposition [13], and time–frequency coefficient modal great [14], decompose multi-component signals before SET processing. Liu et al. [14] proposed a method to identify the instantaneous frequency of time-varying structural response signals by combining SET with modal maxima of time–frequency coefficients. Kang [15] proposed a method combining SET and empirical wavelet transform method and applied this method to analyze the time–frequency of seismic signals at high-resolution. Liu et al. [16] proposed a method combining primary kernel regression residual decomposition with SET and applied it successfully to defect mechanical faults. Li et al. [17] proposed a method combining variational nonlinear frequency modulation modal decomposition with SET to diagnose the fault characteristic frequency of bearings. Although the above methods have solved the problem of frequency aliasing to some extent, the complexity and processing time increase. It is obvious that sacrificing the processing speed and computational complexity to eliminate frequency aliasing is a trade-off. In addition, this treatment often does not yield the ideal results because it does not fundamentally improve the shortcomings of SET.

Therefore, some scholars continue to explore new SET methods. Chen et al. [18] improved the results of SET for stressed FM signal by local frequency approximation and verified the noise robustness performance and signal reconstruction capability of the method. Kang et al. [19] proposed a method of synchroextracting generalized S transform considering the advantages of the generalized S transform and successfully applied it to process seismic signals. Zhou et al. [20] proposed the second-order SET and high-order SET methods to address the deficiencies of SET in processing emphasized FM-AM signals. The proposed second-order SET method has higher time–frequency accuracy, and the proposed high-order SET method has the clearest description of low-frequency components and high-frequency components. Chen [21] proposed a TF analysis method for recursive mapping demodulation high-order SET, which focuses on solving the demodulation of cross-aliasing signal components of external interference and the accurate estimation of local instantaneous frequencies in high-order.

However, although the above methods improve the TF accuracy and anti-interference performance of the SET to a certain level, they fail to improve the adaptive performance. To overcome the above problems, we replace the STFT in the SET framework with the generalized linear chirplet transform (GLCT) and propose a rotating synchroextracting chirplet transform (RSECT) method. The proposed method is characterized by retaining the advantages of the GLCT in fitting the TF ridge energy of signal, and also retaining the advantages of the SET in terms of high-precision TF. Therefore, the method has a strong TF resolution and can effectively characterize the time–frequency features of the fault signal. Simulation and experimental results verify the effectiveness of the proposed method.

The remainder of the paper is organized as follows. In Section 2, the basic concepts and principle of the RSECT method are introduced. This is followed by the numerical experiments of RSECT, presented in Section 3. In Section 4, the proposed method of RSECT is applied in the rubbing fault diagnosis. Before closing the paper, a brief conclusion is provided in Section 5.

2. The Basic Theory of RSECT

Generally, a nonstationary signal $f(t)$ can be defined as

$$f(t)=A(t)e^{i{\displaystyle \int \phi (t)dt}}$$

(1)

where $A(t)$ and $\phi (t)$ denote the instantaneous amplitude and the phase function of the signal, respectively. Before illustrating the core ideal of RSECT, we first express the linear chirplet transform (LCT) result of the signal, i.e.,

$$L(t,\omega ,c)={\displaystyle {\int }_{-\infty }^{+\infty }f(t)h(\tau -t)e^{-j\omega t}{e}^{-i\mathrm{c}{(\tau -t)}^2/2}}dt$$

(2)

where t is the time, ω is the frequency, c is an optional variable, $h(t)=e^{-(pi/{0.32}^2)t^2}$ is a gaussian window function, $e^{-ic{(\tau -t)}^2/2}$ is the discrete gaussian demodulation operator, and $L(t,\omega ,c)$ is a LCT coefficient. From Equation (2), LCT introduces the Gaussian demodulation operator on the basis of the STFT, the introduction of the Gaussian demodulation operator will make the kernel function of LCT with tilt characteristics. We can adjust the tilt angle of time–frequency ridge on time–frequency plane through optional variable c to find the best time–frequency energy. When the c value is appropriately selected, the time–frequency point in the neighborhood at a certain time will always make the Gaussian demodulation operator close to the demodulation part of the signal. At this time, the time–frequency energy located at the transient time–frequency has a high degree of aggregation, and the amplitude of the obtained coefficient $\left|L(t,\omega ,c)\right|$ of LCT will reach the maximum. The coefficient is calculated as follows.

$$c_m=\arg \max \left|L(t,\omega ,c)\right|\le \left|{\displaystyle {\int }_{-\infty }^{\infty }h(\tau -t)A(t)dt}\right|$$

(3)

Iterating each point in the TF plane, we can get $c_m=\arg \max \left|L(t,\omega ,c)\right|$ that maximizes $\left|L(t,\omega ,c)\right|$.

According to the calculation formula of the two-dimensional instantaneous frequency spectrum, the two-dimensional instantaneous TF distribution of the signal can be estimated as follow.

$${\phi }^{\prime }(t,\omega ,c)=-j\frac{{\partial }_{t}L(t,\omega ,c)}{2\pi L(t,\omega ,c)}$$

(4)

When $L(t,\omega ,c)$ is not 0, the synchroextracting operator (SEO) can be defined as follows

$$\delta (\omega -{\phi }^{\prime }(t,\omega ,c))=\{\begin{array}{ll}1& \omega ={\phi }^{\prime }(t,\omega ,c)\\ 0& \omega \ne {\phi }^{\prime }(t,\omega ,c)\end{array}$$

(5)

The SEO satisfies the properties of a Dirac delta function and is designed to extract the TF coefficients of $L(t,\omega ,c)$ along with the curves $\omega ={\phi }^{\prime }(t,\omega ,c)$, the synchroextracting LCT can be defined as

$$T_e(t,\omega ,c)=L_s(t,\omega ,c)\cdot \delta (\omega -{\phi }^{\prime }(t,\omega ,c))$$

(6)

We rotate the TF result by adding a gaussian demodulation operator. The angle of rotation can be calculated as $\arctan (-c)$. For a signal with unit sampling time $t\in [0,T_s]$ and sampling frequency $F_s$, its inclination angle on the TF ridge should satisfy

$$\alpha =\arctan (2\frac{T_s}{F_s}c)\hspace{1em}\hspace{1em}\hspace{1em}\alpha =(-\pi /2, \pi /2)$$

(7)

For the entire time domain, the c can be calculated

$$c=\tan (\alpha )\frac{F_s}{2T_s}$$

(8)

$$\alpha =-\frac{\pi }{2}+\frac{\pi }{N_c+1},-\frac{\pi }{2}+\frac{2\pi }{N_c+1}, \dots ,-\frac{\pi }{2}+\frac{N_c\pi }{N_c+1}$$

(9)

where $N_c$ is the number of discretization of the time–frequency interval. Then, the GLCT can be defined as

$$L_s(t,\omega ,c)={\displaystyle {\int }_{-\infty }^{+\infty }f(t)h(\tau -t)e^{-j\omega t}{e}^{-i\tan (\alpha ){(\tau -t)}^2/2}}dt$$

(10)

The parameter α is discretized into N values, that is, the original time–frequency plane is divided into N + 1 time–frequency planes, i.e.,

$$\alpha =-\pi /2+\pi /(N+1),-\pi /2+2\cdot \pi /(N+1)\dots -\pi /2+N\cdot \pi /(N+1)$$

(11)

Then the RSECT can be defined as

$$T_e(t,\omega ,N)=L_s(t,\omega ,N)\cdot \delta (\omega -{\phi }^{\prime }(t,\omega ,N))$$

(12)

When $N=1$, Equation (12) will degenerate into SET.

In the framework of the original SET, LCT is constructed by introducing the frequency shift rotation operator to the STFT. The SEO is constructed by using the two-dimensional transient distribution calculation formula. By performing N iterations for each energy point on the TF surface of the LCT, the RSECT with adaptive is obtained.

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

$$\{\begin{array}{l}{f}_1(t)=[\sin (2\mathsf{\pi }(25t_1+10\sin (t_1))) \sin \left(2\mathsf{\pi }\right(34.2t_2))]\\ f_2(t)=[\sin (2\mathsf{\pi }\left(\right(25+8)t_1+10\sin (t_1))) \sin \left(2\mathsf{\pi }\right(34.2+8)t_2)]\\ f_3(t)=\sin (2\mathsf{\pi }(8t+3arc\tan ({(t-5)}^2)))\\ t_1\in (0 6) t_2\in (6 7) t=[t_1{t}_2]\end{array}$$

(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]. Figure 2, Figure 3 and Figure 4 show the time–frequency distribution obtained from the GLCT, SET, and RSECT methods, respectively.

From Figure 2, Figure 3 and Figure 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 Figure 3 and Figure 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 Figure 6, Figure 7 and Figure 8.

Comparing Figure 6, Figure 7 and Figure 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 Figure 7 and Figure 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. Therefore, the RSECT gets better results than the GLCT and SET when dealing with multi-component complex non-stationary signals.

The more concentrated energy of the TF results indicates a stronger localization ability and better time–frequency characteristics. Rényi entropy is an important index to evaluate TF aggregation. A lower value of Rényi entropy indicates more concentrated energy. Therefore, to further verify the superiority of the proposed method numerically, white noise with SNR from 1 dB to 30 dB is added to the simulation signal, respectively. The Renyi entropy of TF representation generated by the three different methods is shown in Figure 9. It can be found that the noise increases the Rényi entropy, indicating that the noise can derive the energy aggregation of the above methods. In addition, the GLCT is susceptible to noise interference. The RSECT has the lowest Rényi entropy value, indicating that the method has the best TF aggregation and noise immunity. The Rényi entropy value of the SET is between the two methods.

To further test the effectiveness of the proposed method in practical applications, a more complex multi-component FM signal is constructed here, i.e.,

$$f(t)=\cos (400\pi t)+\cos (2\pi (80t+150t^2+\frac{3}{\pi }\cos (4\pi t)))$$

(14)

The signal is comprised of a constant-frequency signal and an FM signal, and the instantaneous frequency components of both of them are crossed. The sampling frequency is 1000 Hz, and the sampling time is 1 s. The time-domain waveform of the simulation signal is shown in Figure 10.

The simulated signal is processed by the RSECT, SET, and GLCT, respectively. Figure 11, Figure 12 and Figure 13 show the TF distributions obtained from the GLCT, SET, and RSECT, respectively. Comparing Figure 11, Figure 12 and Figure 13, the frequency characteristics of the constant-frequency signal and FM signals can be basically extracted regardless of the GLCT, the SET, or the RSECT. However, the three methods have obvious differences in the TF resolution of the signal. The GLCT is obviously inferior to the SET and RSECT. Comparing Figure 12 and Figure 13, the SET is significantly inferior to the RSECT when dealing with the instantaneous frequencies of cross-components. The RSECT combines the respective advantages of the GLCT and SET, which not only maintain the high-precision TF resolution but also have certain robustness to the processing of cross frequencies. From the local zoomed-in diagram of the TF distribution, it can be seen that the GLCT has a certain anti-aliasing property, but its TF accuracy is not high enough at the cross of frequency components. The SET meets a certain time–frequency accuracy, but the anti-aliasing property is not enough. Only the RSECT satisfies both the anti-frequency aliasing performance and high TF accuracy. Therefore, the RSECT gets better results than the GLCT and SET in processing cross-frequency signals.

4. Experiment Test

The data of the fault signal in this experiment comes from the fault data of the heavy oil machine set [1,12]. The device structure of collected mechanical fault signals is shown in Figure 14. In the figure, 1#, 2#, 3#, and 4# represent four sets of support bearings. This device consists of gas turbine, compressor, gearbox, motor, etc. The fault signal is collected at a speed of 5381 rpm, the sampling frequency is 2000 Hz. Therefore, the fault characteristic frequency is 90 Hz. The time-domain waveforms of the fault data are shown in Figure 15. Figure 16, Figure 17 and Figure 18 are the TF distribution obtained by the GLCT, SET, and RSECT.

From Figure 16, although the GLCT extracts the faulty feature frequency, the resolution of the features in the time and frequency domains is obviously unsatisfactory and needs further improvement. From Figure 17 and Figure 18 both the SET and RSECT can clearly extract the fault feature frequency of 90Hz with better resolution in both time and frequency domains, which has obvious advantages. Comparing Figure 17 and Figure 18, the RSECT retains the advantages of the SET, thus further improving the time–frequency resolution and the smoothness of the time–frequency features. The effectiveness of the proposed RSECT is further verified.

To further verify the effectiveness, the proposed method is applied to the analysis and processing of the rubbing test data of the double-disk rotor. The rotor experimental stage is illustrated in Figure 19a. The rotor is driven by an electric motor and the bearings are sliding bearings. The vertical and horizontal vibrations are measured by a non-contact eddy current sensor. The rubbing rotor test bench is shown in Figure 19b.

Radial rubbing of the rotor is artificially caused by a friction block, as shown in Figure 20.

The vibration signals in the horizontal and vertical directions are measured by setting the speed at 3000 r/min, 1024 sampling points and 1600 Hz sampling frequency. Figure 21 shows the time-domain diagrams of two sets of horizontal vibration signals. Figure 22 shows the frequency spectrograms corresponding to Figure 21. From the frequency spectrograms, the signals contain various frequency multiplier components, which are the typical spectral characteristics of rubbing faults. It can be roughly judged that the first group of signals corresponds to slight rubbing, while the second group is severe rubbing.

The 1600 Hz sampling frequency and 128 window width is taken to perform the GLCT, SET and RSECT for the signal of slight rubbing and serious rubbing, respectively. The corresponding time–frequency distribution of the three methods are shown in Figure 23, Figure 24 and Figure 25.

From the time–frequency results of the slight-rubbing-fault signal, the GLCT results only represent 1× and 2× component that are present throughout the signal, and the magnitude of the 2× component is less obvious. However, the other higher-order components are not clearly characterized. The SET and RSECT not only characterize the 1× and 2× components, but also the other high-order components such as 4×, 6×, 8×, and 10× components. In the characterization results of high-order components, the RSECT obtains higher amplitudes of the components, which is more advantageous than the SET. From the time–frequency results of the serious rubbing fault signal, the 1× component of the signal persists and has a relatively stable amplitude. The 2×, 3×, and 8× components also persist, but their magnitude varies slightly. The higher-order frequency components such as 5×, 8×, 9×, and 11× components appear intermittently in a more regular manner with small changes in amplitude. From the time–frequency results, the fault characteristics of rotor rubbing can be observed by the fact that only some low-order frequency components persist when slight rubbing occurs. The high-order frequency components with weaker amplitude are excited at intervals. When severe rubbing occurs, in addition to the low-order frequency components, more frequency components will persist, and the amplitudes of some of the frequency components will change slightly according to a certain rule, while the higher frequency components are still intermittently affected. When severe rubbing occurs, in addition to the low-order frequency components, more components persist and the amplitudes of some of these components change slightly according to a certain rule. While the higher-order frequency components are also excited at intervals.

5. Conclusions

(1) To improve the time–frequency analysis capability of the synchroextracting transform method for processing mechanical fault signals, this paper proposes a method called rotation synchroextracting chirplet transform. This method combines the advantages of the generalized linear chirplet transform and synchroextracting transform methods, which not only improves the adaptive performance of the synchroextracting transform method, but also has the characteristics of high time–frequency performance.

(2) To verify the effectiveness of the proposed method in processing AM-FM signals, simulation signals with strong AM and FM are constructed. The proposed method is compared with the synchroextracting transform and generalized linear chirplet transform methods. The simulation results show that the proposed method has obvious advantages over the other two methods. Finally, the proposed method is analyzed for experimental verification. The experimental results show that the proposed method can effectively extract the fault characteristic frequencies of mechanical fault signals with very high time–frequency resolution.(3) To further verify the capability of the proposed method, the method is introduced into the fault diagnosis of rubbing fault vibration signals and compared and analyzed with the synchroextracting transform and generalized linear chirplet transform methods. The proposed method can not only effectively extract the low-order components of the faults, but also outperforms the synchroextracting transform and generalized linear chirplet transform methods in characterizing the high-order components, thus showing that the proposed method can effectively extract the fault features of rotor rubbing faults.