1. Introduction
Rolling bearing is one of the most widely used parts in rotating machinery. The operational status of the rolling element bearing often directly affects the performance of the whole machine. Consequently, the fault identification and diagnosis of rolling bearings are of great significance to ensure the safe and reliable operation of mechanical equipment [
1,
2,
3]. Vibration signals caused by rolling bearing faults have been extensively studied and powerful diagnostic methods have been proposed. In recent years, the emerging artificial intelligence diagnosis technologies have been widely concerned by many scholars. A large number of artificial intelligent methods such as support vector machine [
4], fuzzy diagnosis method [
5], convolutional neutral network [
6], and recurrent neutral network [
7] were proposed to diagnose the mechanical faults. Furthermore, the sparse representation methods have also been widely used because of the signals sparsity. Adaptive impulse dictionary [
8], combined time-frequency dictionary [
9], matching pursuit [
10], and dictionary learning method [
11] were presented to diagnose faults of rotating machinery. In addition to the above mentioned two popular methods, many other research methods such as fault quantitative diagnosis [
12], fault mechanism research [
13,
14], fault diagnosis of low speed machinery [
15] and fault location diagnosis [
16] also gained attention. Some traditional methods such as Fourier transform, envelope analysis method, empirical mode decomposition, wavelet transform, spectral kurtosis, and morphological filtering have shown their advantages on single fault detection [
17,
18,
19]. However, these diagnosis methods all take the rolling bearing with single point of failure as the research object. In the actual conditions, the failure of rolling bearing usually manifests itself as a compound fault including the outer race defect, the inner race defect or the rollers defect. In general, due to the influence of operating environment, the interaction between multiple noise source and compound fault, the compound fault features are mutually coupled and interfere with each other, which causes major difficulty in compound faults detection [
20]. Therefore, the separation of the compound fault components from strong background noise is a difficult problem in the field of mechanical fault diagnosis [
21,
22].
In order to solve the problem issued above and improve the monitoring level of rolling bearings’ running state, some methods have been proposed for compound faults diagnosis, such as demodulation algorithm, variational mode decomposition, clustering algorithm, and blind source separation technique [
23,
24,
25,
26]. Reference [
27] proposed a method of combining wavelet analysis with blind source separation for roller bearing compound faults separation that needs multiple signal channels to analyze. However, composite signals analyzed by blind source separation are usually exported from different channels by multiple sensors. Therefore, multiple sensors need to be installed in the diagnosis process, which may bring a lot of inconvenience to engineering applications. McDonald et al. [
28] proposed a novel method named maximum correlated kurtosis deconvolution (MCKD), which is an effective tool for separating out the periodic impulse fault components from the vibration signal in circumstances of intense background noise. In recent years, MCKD has been widely used to extract periodic pulses for fault diagnosis [
29,
30,
31]. However, some shortcomings in practical application of MCKD limit its performance to extract the transient process in noisy vibration signal. One problem is that, like most of the existing technologies, the main concern of MCKD is the detection of a single fault located in the outer race, inner race or rolling element, i.e., MCKD is not suitable for extracting multiple fault-impact components simultaneously in compound fault diagnosis. However, the complex vibration signal is often mixed with a lot of background noise, which makes the diagnosis very difficult. Only by effectively extracting multiple fault shock signals from noise signals and enhancing fault characteristics [
32,
33], fault characteristic signals can be further separated accurately. For example, Reference [
34] proposed an improved matching pursuit algorithm, which successfully extracted the quantitative characteristics of bearing faults. Selesnick [
35] proposed a sparsity-enabled signal decomposition method based on resonance-based sparse signal decomposition (RSSD). According to the oscillatory behavior of the signal components, the analyzed signal can be decomposed into high and low resonance components by using the RSSD. Thus, high Q factor basis and low Q factor basis are constructed, and non-stationary signals are sparse decomposed. In the References [
36,
37], the RSSD method was introduced into the fault diagnosis of gearboxes. The results showed that the sparse decomposition effect was better. Therefore, the proposed RSSD method can be applied to extract multiple impact components in compound fault diagnosis. Another important problem is how to set the input parameters appropriately for the best performance, which contains the filter length
L, shift order
M and the deconvolution period
T. Among those parameters, which will affect the validation of MCKD,
L and
M should be set reasonably to highlight the superiority of MCKD on the base of satisfying the prime requirement of
T. If
L and
M are too large, the deconvoluted signal will be distorted. Therefore, the rigorous requirements for the parameters limit the application of MCKD. To solve this problem, the improved maximum correlation kurtosis deconvolution (IMCKD) is proposed in this paper. It can optimize the input parameters: The shift order
M and the filter length
L. In the case of different
M values, calculating the mean of Teager envelope spectrum kurtosis (MTEK) to select the optimal shift order
M; then, the Teager envelope spectral kurtosis (TEK) is used as the objective function to optimize the filter length
L by the iterative algorithm.
Based on the above discussion, a compound fault diagnosis method based on parallel dual-Q-factor and the improved MCKD is proposed in this paper. The method combines the compound fault characteristics of rolling bearings based on frequency characteristic analysis of composite signals. The parallel dual-Q-factor sparse decomposition algorithm is proposed for the compound faults of the inner and outer race of the bearing. The non-stationary signal is decomposed into the high resonance components and low resonance components (compound fault impact component and small amount of noise). Then, the multiple periodic failures of the compound fault signal can be extracted simultaneously. In order to further isolate and extract the compound faults, the low resonance component is processed by IMCKD, which is separated and extracted according to the different fault characteristics periods.
The remainder of this paper is organized as follows.
Section 2 briefly describes the theory of the parallel dual-Q-factors base sparse decomposition.
Section 3 introduces the improved maximum correlation kurtosis deconvolution (IMCKD) for selecting the input parameters appropriately.
Section 4 is devoted to descriptions of the proposed method.
Section 5 is dedicated to description of application of the proposed method with the simulation and experiment signals. The results show the effectiveness and reliability on the compound faults decoupling diagnosis for rolling bearings.
2. The Parallel Dual-Q-Factors Bases Sparse Decomposition
The resonance property of a signal is defined by the quality factor Q, the high Q-factor base represents the continuous oscillation component, and the low Q-factor base represents the transient oscillation component.
Figure 1 shows the concept of resonance properties of signals and the time-domain waveform of the signal. In
Figure 1, when Q = 1, it shows the single-period pulse signal, and they are defined as the low resonance signals because of the small quality factor Q. When Q = 3 it shows the multi-period pulse signals, and they are defined as the high resonance signals because of the large quality factor Q.
The RSSD method is actually a sparse representation using the high and the low tunable Q-factor wavelet transform (TQWT) [
35,
38]. The TQWT is fully discrete, has the perfect reconstruction property, is modestly overcomplete, and is based on the two-channel filter bank illustrated in
Figure 2 [
39].
In
Figure 2,
x(
n) is the original signal,
H0(
w),
H1(
w) and
H*0(
w),
H*1(
w) are the frequency response functions of the analysis and synthesis filters respectively;
v0(
n),
v1(
n) are the sub-band signals obtained after decomposition.
y0(
n),
y1(
n) are the synthetic signals. The scaling parameters are given by:
where,
β is the high-pass factor,
α is the low-pass factor, and
r represents the redundancy.
Suppose that the observed signal x can be represented as:
where
x1 and
x2 are components with different oscillation behavior.
The signal sparse decomposition method based on the compound Q-factor basis uses the morphological component analysis (MCA) method to separate the components nonlinearly in the signal according to the oscillation characteristics [
40,
41] and establishes the optimal sparse representation of the high resonance component and the low resonance component. The goal of MCA is to estimate
x1 and
x2 individually. Assuming that the signals
x1 and
x2 can be sparsely represented in base functions (or frames)
S1 and
S2 respectively, (obtained by TQWT), they can be estimated by minimizing the objective function [
42]:
where
W1—the transform coefficient of signal x1 under frame S1;
W2—the transform coefficient of signal x2 under frame S2;
S1—the filter banks of tunable-Q wavelet with the high quality factor;
S2—the filter banks of tunable-Q wavelet with the low quality factor;
λ1—weight parameter;
λ2—weight parameter.
In the sparse signal decomposition method based on the compound Q-factor bases, the objective function J is minimized through iterating and updating the coefficient W1 and W2 by using the split augmented Lagrangian searching algorithm (SALSA).
Eventually, it effectively separates the high resonance component and low resonance component, and extracts the transient impulse component. After
W1 and
W2 are obtained, the estimated high and low resonance components (solved by matching) are shown as follows: