Enhancement and extraction of periodic transient based on hybrid intelligent fusion algorithm and its application in rolling bearing fault detection

Abstract

Weak fault detection of rolling bearing presents difficult, because the periodic transient signature produced via localized incipient damage is easily submerged by various interference components and background noise. Hybrid intelligent fusion method is a breakthrough strategy for revealing feature frequency of rolling bearing fault by comprehensively using a variety of intelligent signal processing technologies, possessing the advantage of each technology. Considering the rolling bearing often construct a transmission device combination with gear and shaft, its vibration signal is often vulnerable to other multi-morphology components, such as harmonic modulation, noise. Thus, how to identify the fault frequency in repetitive transients is crucial to accurately identify rolling bearing fault detection. To address this issue, a novel hybrid intelligent method is proposed to effectively apply on periodic transients extraction, enhancement and rolling bearing fault diagnosis. The innovation of this method is to solve three problems, namely, the separation of multi-morphology components, noise reduction without periodic transients distortion, weak fault frequency enhancement. The proposed method is tested and validated on simulated signal, rolling bearing fault signal from accelerated rolling bearing degradation rig. In addition, comparisons with other classical rolling bearing fault detection methods have been conducted to highlight the superiority of the proposed methodology.

Introduction

Rolling bearing is a precise mechanical component of transmission device, which is often affected by axial load, radial load, impact load, and various external excitations under complex operating condition, as a result, the structural fatigue cracks can be easily induced in internal components. If appropriate maintenance strategies are not implemented in time, it is convenient to cause the breakdown of the transmission device, which may result in considerable economic losses. Therefore, weak fault detection of the rolling bearing is vital for ensuring the safety and reliability of transmission device.

In reality, rolling bearing fault vibration signal often exhibits similar transient translation phenomenon commonly, as a result, repetitive transients can be regarded naturally as the translation result of first transient. On the other hand, the rolling bearing fault vibration signal also contains strong interference components. Considering repetitive transients in vibration signal often reflect rolling bearing incipient fault, using proper signal processing methods to extract and enhance periodic transients from vibration signal, and furthermore identify fault frequency is primary for rolling bearing condition maintenance.

The typical methods for rolling bearing fault detection can be roughly considered as joint time-frequency,^1,2 filter technology^3,4 and time domain adaptive decomposition,^5,6 such as variational mode decomposition (VMD), empirical mode decomposition. These methods prominently promote safety working level. Compared with time-frequency, morphological component analysis (MCA)^7,8 uses parameterized waveform which can self-adaptive conform to actual signal structure characteristic and directly extract fault frequency consequently. Recently, some new approaches, for example, dynamic model theory, ⁹ based on blind deconvolution method, a series of improvement algorithms, such as minimum entropy deconvolution, ¹⁰ maximum correlated kurtosis deconvolution, ¹¹ multipoint optimal minimum entropy deconvolution adjusted ¹² also have been developed and applied for detecting fault. By utilizing filter technology, a series of application methods are utilized to enhance the repetitive transients induced by the localized defect in rolling bearing or determine the frequency band containing diagnostic information, such as spectral kurtosis, ¹³ Autogram, ¹⁴ and average Inforgram. ¹⁵ Besides, the sparse decomposition represented by resonance sparse signal decomposition (RSSD) ¹⁶ is also widely used, it can separate signal into different sub-signals according to resonant property. However, its separation effect is compactly related with parameter values in RSSD, so how to determine the reasonable values is an unavoidable problem, especially for the extraction of periodic transients. Furthermore, in the context of incipient rolling bearing fault, repetitive transients are often submerged by strong background noise, resulting in weak fault phenomenon, so it is necessary to develop a specific noise reduction method which can remove as many noise as possible while maintaining effectively periodic transients. As wavelet transform can effectively separate useful component and noise in time domain, threshold method based on wavelet transform is widely used consequently, such as NSDWT, ¹⁷ ACWT, ¹⁸ RSGWT ¹⁹ and so on. However, these methods present signal distortion deficiency when dealing with periodic transients. Due to down-sampled operation and sub-division operation, the existing wavelet transform presents translation-varying defect, if these methods are applied directly on periodic transients, the decomposed coefficient energy has large fluctuation. As a result, applying those methods to achieve noise reduction will inevitably miss some weak periodic transients. Concentrating on detecting rolling bearing incipient fault, a novel hybrid intelligent fusion diagnosis method need to address following issues, concentrating on how accurately extracting periodic transients from fault vibration signal, achieving noise reduction without periodic transients distortion, identifying and enhancing weak fault frequency from spectrum respectively.

Hence, a novel hybrid intelligence fusion method by making full use of RSSD, DTCWT and IES is proposed and can effectively make up for the deficiency of a single signal processing algorithm. In the new method, firstly, an optimal Q-factor resonance sparse decomposition is designed for achieving periodic transients extraction by the superior matching to sparse basis function. Furthermore, we also give a no distortion noise reduction based on dual tree complex wavelet adjacent coefficient contraction, it can not only effectively extrude transient component energy, but also has amplitude fidelity. Finally, we put forward an improved envelope spectrum and use it to identify and enhance rolling bearing weak fault frequency. Simulation signal and rolling bearing fault signal shows that the proposed hybrid intelligence fusion method can effectively be applied on rolling bearing incipient fault diagnosis.

Proposed method in this paper

The remainder of this paper is organized as follows. In Section 2, we give an optimal Q-factor RSSD to extract repetitive transients, and a neighcoeff threshold based on DTCWT is proposed to achieve noise reduction without distortion, finally, a weak fault frequency identification and enhancement based on IES is developed. As for rolling bearing incipient fault, we divide it into three dominating problems, as for each problem, we use targeted algorithm to solve it and achieve algorithm fusion which can effectively improve the accuracy of diagnosis. In Section 3, the performance of proposed method is validated using the simulated signal and compared with the reported methods. Section 4 presents the result of proposed method and reported methods on the experimental rolling bearing fault signal. The main conclusions are summarized in Section 5.

Solving solution of first problem: An optimal Q-factor RSSD

Assuming that rolling bearing fault vibration signal x , considering the rolling bearing fault vibration reflects oscillating and multi-component property, in the parameter of classical RSSD, Q_H-factor and Q_L-factor describes quantitatively the resonant property of wavelet bases S ₁ and S ₂, therefore, we should give them an appropriate values, which induces that the wavelet bases can self-adaptive be conformed to impact morphology for realizing the optimal matching between RSSD and periodic transients, they can be extracted ultimately. Q-factor is defined as the ratio of the center frequency f_c and frequency bandwidth BW, as expressed in formula (1), RSSD uses MCA as shown in formula (2) to non-linearly decompose fault vibration signal into two sparsely represented high-resonance component $\overline{X_H}$ , low-resonance component $\overline{X_L}$ which has sparsest representation according to sparsely coefficient matrixes W ₁ and W ₂ respectively as shown in formula (3).

$$Q=f_c/\mathrm{BW}$$

(1)

$$\begin{array}{c}{\mathit{W}}_1,{\mathit{W}}_2=\mathrm{argmi}{n}_{{\mathit{W}}_1,{\mathit{W}}_2}\hfill \\ \{{\Vert \mathit{x}-{\mathit{S}}_1{\mathit{W}}_1-{\mathit{S}}_2{\mathit{W}}_2\Vert }_2^2+{\lambda }_1{\Vert {\mathit{W}}_1\Vert }_1+{\lambda }_2{\Vert {\mathit{W}}_2\Vert }_1\}\hfill \end{array}$$

(2)

$$\overline{X_H}={\mathit{S}}_1{\mathit{W}}_1,\overline{X_L}={\mathit{S}}_2{\mathit{W}}_2$$

(3)

In order to accurately extracting repetitive transients from fault vibration signal x , we should achieve the seriously multi-morphology decoupling effect between high resonance component and low resonance component and the latter also reflect periodic transients. For above purpose, we propose an optimal Q-factor RSSD based on formula (2). It uses the kurtosis indicator to assess the degree of periodic transient attribute in the low-resonance component $\overline{X_L}$ and applies stepwise iterative to obtain maximum value of $\mathrm{Kurtosis}\left(\overline{X_L}\right)$ . Finally, its correspondingly optimal Q-factor ${Q_L}^{*}$ and ${Q_H}^{*}$ constructs wavelet basis ${{\mathit{S}}_2}^{*}$ and ${{\mathit{S}}_1}^{*}$ for separating low resonance component ${\overline{X_L}}^{*}$ acting as estimated value of $\overline{X_L}$ in formula (4). Finally, the separation flowchart of periodic transients is shown in Figure 1 and the detailed procedures are also given.

$${\overline{X_L}}^{*}={{\mathit{S}}_2}^{*}{\mathit{W}}_2^{*}$$

(4)

• Step1: Initialization. The correspondingly Q_H and Q_L initial value is 4.0 and 1.2 respectively, and the range of Q_H is from 1 to 40, the range of Q_L is from 1 to 10, as for redundancy factor r, its constant value is 3.

• Step2: Optimizing Q_L value. We need to fix Q_H value at initial value. In every interval ΔQ, we take an assigned value to Q_L, and using RSSD to decompose vibration signal into high and low resonance component, calculating kurtosis value of low resonance component. Finally, through fitting algorithm, an optimal Q_L value can be obtained corresponding to maximum kurtosis value, which is symbolized as ${Q_L}^{*}$ .

• Step3: Optimizing Q_H value. We fix Q_L value at $Q_L={Q_L}^{*}$ , and obtain the optimal Q_H value at $Q_H={Q_H}^{*}$ according to step2.

• Step4: Iteration. As long as Q_H or Q_L does not meet the convergence condition, corresponding optimization process is continued, and replacing existing value, repeating step2-step3.

• Step5: Termination. When both Q_H and Q_L converge to corresponding optimal values, and they do not exceed the specified number of iterations, the whole optimization process is finished.

Figure 1.

An optimal Q-factor based RSSD based on stepwise iterative optimization.

Solving solution of second problem: A neighcoeff threshold based on DTCWT

Although the low resonance component ${\overline{X_L}}^{*}$ presents periodic transients, it will inevitably contain some noise, we introduce neighcoeff threshold based on DTCWT to improve SNR and obtain remarkable periodic transients ${\overline{X_L}}^e$ . This algorithm can takes full account of the wavelet adjacent coefficient correlation of periodic transients and maintain the periodicity as shown in Figure 2. As a result, it has a advantage on the signal fidelity. Finally, the specific steps are shown as follows.

• Step1: Transformation. The low resonance component ${\overline{X_L}}^{*}$ is transformed by DTCWT $\tilde{\mathrm{\Theta }}=W·{\overline{X_L}}^{*}$ , ${\tilde{\theta }}_{j,k}$ symbolizes the k-th coefficient on j-th scale.

$$\tilde{\mathrm{\Theta }}={\left[{\tilde{\theta }}_{j,k}\right]}_{j,k\in Z}$$

(5)

• Step2: Grouping. Grouping wavelet coefficients at each scale, such as J scale (the number of members is defined as 1 in each group $b_i^j$ ), and then expand central element of each group to left and right into a large group $B_i^j$ (the number of left and right extensions is also defined as 1).

• Step3: Adjacent threshold reduces noise. Adopting NeighCoff contraction rule to process each wavelet coefficient ${\tilde{\theta }}_{j,k}$ , which is based on calculating threshold factor ${\lambda }_j$ and contraction factor ${\beta }_i^j$ as in formula (6)–(8).

$${\lambda }_j=\frac{\mathrm{median}\left(\left|{\tilde{\mathrm{\theta }}}_{j,k}\right|\right)\sqrt{2\ln (n_j)}}{0.6745}$$

(6)

$$S_{j,i}^2=\sum _{(j,k)\in B_i^j}{\tilde{\theta }}_{j,k}^2$$

(7)

$${\beta }_i^j=\max (1-\frac{{\lambda }_j^2}{S_{j,i}^2},0)$$

(8)

Figure 2.

A noise reduction algorithm based on DTCWT neighcoeff threshold.

Finally, the wavelet coefficients by noise reduction are calculated in formula (9).

$${\tilde{\theta }}_{j,k}={\beta }_i^j{\tilde{\theta }}_{j,k}$$

(9)

• Step4: Inverse transformation. The remarkable periodic transients ${\overline{X_L}}^e$ is obtained in formula (10).

$${\overline{X_L}}^e=W^{-1}·\hat{\mathrm{\Theta }},\hat{\mathrm{\Theta }}={\left[{\tilde{\theta }}_{j,k}\right]}_{j,k\in Z}$$

(10)

Solving solution of second problem: a fault frequency identification and enhancement based on improved envelope spectrum

When we possess the remarkable periodic transients ${\overline{X_L}}^e$ , we need to excavate the fault information contained in it for identifying rolling bearing fault state, this goal depends on the fault frequency recognition in the signal ${\overline{X_L}}^e$ . As rolling bearing occurs fault, the signal ${\overline{X_L}}^e$ often presents modulation phenomenon, in this regard, the natural frequency or rotating frequency often act as carrier frequency f and the fault frequency is modulation frequency α, as a result, envelope spectrum (ES) analysis is a better tool. In especial, enhanced envelope spectrum (EES) ²⁰ is an effective ES-based method, it quantitatively reveals the correlation between frequency f and frequency f + α in a two-dimensional plane which comprises spectral frequency f and cyclic frequency α by analyzing the second-order cyclostationary property. As EES is generated by integrating spectral coherence (SCoh) along the full spectral frequency band, this “fully integrating” principle makes it susceptible by noise when it is directly used to deal with rolling bearing incipient fault, because the interference from other cyclic frequencies is unquestionable to dominate the entire EES.

To address this issue, a Candidate Fault Frequencies Optimization-gram (CFFOgram) is proposed to locate the informative spectral frequency band from SCoh as shown in Figure 3. It can automatically identify potential fault frequency based on local peak values of SCoh and converts the integral range from full frequency band to informative band which suppress the effect of interference. Furthermore, an improved ES (IES) based on CFFOgram is given to identify the fault-related cyclic frequencies. It is certain that this advantage can reveal the fault information hidden in the SCoh plane and is suitable for weak fault frequency enhancement of rolling bearing.

Figure 3.

The flow chart of IES based on CFFOgram.

Given the remarkable periodic transients ${\overline{X_L}}^e$ , F_s is the sampling frequency, we can obtain the instantaneous auto-correlation function of ${\overline{X_L}}^e$ expressed in formula (11). E symbolizes the ensemble average operator, * represents the complex conjugate, and ${\tau }_m=m/F_s$ . T symbolizes the period of transient.

$$R_{{\overline{X_L}}^e}(t_n,{\tau }_m)=E\left\{{\overline{X_L}}^e\left(t_n\right){\overline{X_L}}^e{(t_n-{\tau }_m)}^{*}\right\}=R_{{\overline{X_L}}^e}(t_n+T,{\tau }_m)$$

(11)

Furthermore, we derive the double discrete Fourier transform of $R_{{\overline{X_L}}^e}(t_n,{\tau }_m)$ represented by SC in formula (12) and corresponding cyclic spectra $S_{{\overline{X_L}}^e}^k\left(f\right)$ in formula (13), α symbolizes discrete cyclic frequency into $\alpha =k/T$ , f also presents continuous Fourier frequency.

$$\begin{array}{c}{S}_{{\overline{X_L}}^e}(\alpha ,f)\hfill \\ =\underset{N\to \infty }{lim}\frac{1}{(2N+1)F_s}\sum _{n=-N}^N\sum _{m=-\infty }^{\infty }{R}_X(t_n,{\tau }_m)e^{-j2\pi n\frac{\alpha }{F_s}}{e}^{-j2\pi m\frac{f}{F_s}}\hfill \end{array}$$

(12)

$$S_{{\overline{X_L}}^e}(\alpha ,f)=\{\begin{array}{c}{S}_{{\overline{X_L}}^e}^k\left(f\right),\alpha =k/T\hfill \\ \multicolumn{1}{c}{0,\mathrm{elsewhere}}\end{array}$$

(13)

In order to compute IES conveniently, we give a normalized expression of $S_{{\overline{X_L}}^e}^k\left(f\right)$ represented by ${\gamma }_{{\overline{X_L}}^e}(\alpha ,f)$ in formula (14), $S_{{\overline{X_L}}^e}(0,f)$ symbolizes the classic power spectral density.

$${\gamma }_{{\overline{X_L}}^e}(\alpha ,f)=\frac{S_{{\overline{X_L}}^e}(\alpha ,f)}{\sqrt{S_{{\overline{X_L}}^e}(0,f)S_{{\overline{X_L}}^e}(0,f-\alpha )}}$$

(14)

As ${\gamma }_{{\overline{X_L}}^e}(\alpha ,f)$ presents the correlation between f and f + α quantitatively in ${\overline{X_L}}^e$ , if the spectral frequency spectrum slice (SFSS) ${\gamma }_{{\overline{X_L}}^e}(\alpha ,·)$ presents non-zero values which is parallel to the spectral frequency axis in the (α,f) plane, and the cyclic frequency spectrum slices (CFSSs) also reveal a series of non-zero values parallel to cyclic frequency axis, we define them as CFFs, it infers that the cyclic frequency α is generated by rolling bearing fault. As a result, if we can effectively locate the non-zero or extreme value on SCoh, the informative narrowband can be separated for enhancing weak fault frequency, and the detailed steps are as follows.

Firstly, we discretize ${\gamma }_{{\overline{X_L}}^e}(\alpha ,f)$ as ${\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}(\alpha ,f)$ which has both an equidistant discrete spectral frequency $f_g=g·F_s/N_w,g=0,\dots ,N_W-1$ and equidistant discrete cyclic frequency ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N={\alpha }_{\max .}$ If we give a spectral frequency $f_g$ , the ${\chi }_{{\overline{X_L}}^e}(n,g),n=1,\dots ,N$ is defined as indicator and it is determined by the local maxima distribution in g-th CFSS ${\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}(·,f_g)$ . Its expression is shown in formula (15), parameter L is used to control the sparsity of local maxima on CFSSs, and usually recommended to have a value of 3–5. Finally, a series of elements ${\chi }_{{\overline{X_L}}^e}(n,g)$ form an matrix $\chi ={\left(\left({\chi }_{{\overline{X_L}}^e}(n,g)\right)\right)}_{N\times N_W}$ .

$$\begin{array}{c}{\chi }_{{\overline{X_L}}^e}(n,g)=\hfill \\ \{\begin{array}{c}1,\mathrm{if}L\le n\le N-L\mathrm{and}{\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}({\alpha }_n,f_g)\ge {\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}({\alpha }_{n+l},f_g),l=\pm 1,...,\pm L\hfill \\ \multicolumn{1}{c}{0,\mathrm{elsewhere}}\end{array}\hfill \end{array}$$

(15)

Secondly, we need to focus on the non-zero elements in the matrix $\chi ={\left(\left({\chi }_{{\overline{X_L}}^e}(n,g)\right)\right)}_{N\times N_W}$ , because they correspond to local extremum of CFSSs in entire SCoh plane, these extremum are basically induced by a certain fault-related cycle frequency ${\alpha }_n$ . Compared with other cyclic frequencies, the CFSSs have more obvious amplitudes at this cyclic frequency ${\alpha }_n$ , which indicates that the majority of non-zero extremum in the matrix $\chi ={\left(\left({\chi }_{{\overline{X_L}}^e}(n,g)\right)\right)}_{N\times N_W}$ are concentrated in n-th row. To capture this critical information, we derive a vector $\eta =[\eta \left(1\right),\eta \left(2\right),..\eta (N){]}^T$ , its element corresponds to the sum of n-th row elements in matrix $\chi$ .

$$\eta \left(n\right)={\sum }_{m=1}^M{\chi }_{{\overline{X_L}}^e}(n,m),n=1,2,..,N$$

(16)

The n-th element $\eta \left(n\right)$ quantifies the number of local maxima contained in the n-th SFSS ${\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}({\alpha }_n,·)$ , we rearrange its elements into $\tilde{\eta }=[{\tilde{\eta }}_{\left(1\right)},{\tilde{\eta }}_{\left(2\right)},\dots ,{\tilde{\eta }}_{(N)}{]}^T$ according to the following principles.

1. For any $i<j$ , ${\tilde{\eta }}_{(i)}\ge {\tilde{\eta }}_{(j)}$ should conform.

2. If ${\tilde{\eta }}_{(i)}={\tilde{\eta }}_{(i+1)}$ , the $k_i$ and $k_{i+1}$ should retain so that ${\tilde{\eta }}_{\left(i\right)=}\eta \left(k_i\right)$ and ${\tilde{\eta }}_{(i+1)=}\eta \left(k_{i+1}\right)$ , giving a indicator $\mathrm{SFSSAmp}\left(n\right)=\sum _{m=1}^M{\chi }_{{\overline{X_L}}^e}(n,m)·{\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}(a_n,f_m)$ which need to satisfy $\mathrm{SFSSAmp}\left(k_i\right)>\mathrm{SFSSAmp}\left(k_{i+1}\right)$ .

The advantage of indicator $\mathrm{SFSSAmp}\left(n\right)$ is that it reveal the sum of local maxima amplitude contained in the n-th SFSS ${\gamma }_{{\overline{X_L}}^e}^{\mathrm{est}}({\alpha }_n,·)$ , furthermore, we obtain the CFFs $\hat{\alpha }\left(d\right)$ shown in formula (17), they contains fault-related frequency and the interference frequency which distribute on each CFSS and are the basis for identifying informative spectral band in CFFOgram.

$$\hat{\alpha }\left(d\right)=\{{\alpha }_n|\eta \left(n\right)={\tilde{\eta }}_{\left(d\right)},n=1,2,..,N\},d=1,2,..,D.$$

(17)

It can be seen that the CFFs $\hat{\alpha }\left(d\right)$ is identical to cyclic frequency ${\alpha }_n$ which present the most local maxima of SFSS. Furthermore, we divide the full spectral frequency band into each narrowband by using 1/3-binary tree filter banks to seek the informative band along the spectral frequency which contains the fault information, and compute its corresponding IES. Given an approved i-th narrowband, its $\mathrm{IE}{S}_{l,i}\left(\alpha \right)$ at the l-th level is shown in formula (18).

$$\mathrm{IE}{S}_{l,i}\left(\alpha \right)=\frac{1}{F_s/2^{l+1}}{\int }_{F_s·(i-1)/2^{l+1}}^{F_s·i/2^{l+1}}|{\gamma }_{{\overline{X_L}}^e}(\alpha ,f)|\mathrm{df}$$

(18)

Besides, we use the ratio of the energy of all CFFs to the energy of $\mathrm{IE}{S}_{l,i}\left(\alpha \right)$ to quantify the fault information level in each narrowband, a diagnostic index DI to quantify the fault frequency information hidden level in $\mathrm{IE}{S}_{l,i}\left(\alpha \right)$ shown in formula (19) and formula (20).

$$D{I}_{l,i}=\frac{{\sum }_{n=1}^N{I}_{\{\hat{\alpha }\left(d\right),d=1,2,..,D\}}({\alpha }_n)·|\mathrm{IE}{S}_{l,i}\left({\alpha }_n\right){|}^2}{{\sum }_{n=1}^N|\mathrm{IE}{S}_{l,i}\left({\alpha }_n\right){|}^2}$$

(19)

$$I_{\{\hat{\alpha }\left(d\right),d=1,2,..,D\}}\left({\alpha }_n\right)=\{\begin{array}{c}1,\mathrm{if}{\alpha }_n\in \{\hat{\alpha }\left(d\right),d=1,2,..,D\}\hfill \\ \multicolumn{1}{c}{0,\mathrm{if}{\alpha }_n\notin \{\hat{\alpha }\left(d\right),d=1,2,..,D\}}\end{array}$$

(20)

As for $I_{\{·\}}({\alpha }_n)$ , it is a binary function with a value of 0 or 1. Because $D{I}_{l,i}$ is defined as the ratio of all CFFs energy to the $\mathrm{IE}{S}_{l,i}\left(\mathrm{\alpha }\right)$ energy, if $D{I}_{l,i}$ has a larger value, corresponding $\mathrm{IE}{S}_{l,i}\left(\alpha \right)$ contains more CFFs information naturally. Therefore, we calculate the $D{I}_{l,i}$ of each spectral frequency narrowband and display the result on CFFOgram. Ultimately, the optimal spectral frequency narrowband presenting maximum diagnostic index DI is used to improve envelop spectrum for detecting rolling bearing fault, and we call it IES based on CFFOgram, the detailed implementation procedure of IESCFFOgram is shown as follows.

• Step1: Giving an appropriate window length N_w and maximum cyclic frequency $\alpha$ to establish SCoh by using fast SC algorithm. ²⁰

• Step2: The full spectral frequency band in SCoh is divided into a series of narrowbands by using the 1/3-binary tree structure strategy, and calculating its corresponding $\mathrm{IE}{S}_{l,i}\left(\alpha \right)$ according to formula (18).

• Step3: Identifying the CFFs on the entire SCoh plane and calculating the diagnostic index DI of corresponding narrow band $\mathrm{IE}{S}_{l,i}\left(\alpha \right)$ according to formula (19).

• Step 4: Selecting the spectral frequency narrowband presenting the maximum DI value and its IES acts as the optimal envelope object to perform weak rolling bearing fault detection.

Finally, considering that the IESCFFOgram algorithm involves some parameters, their settings must be careful. For example, the window length of STFT N_w in fast SC algorithm determines the spectral frequency resolution, the maximum cyclic frequency ${\alpha }_{\max }$ determines the Maximum value of fault frequency and its harmonics. As for variable D in formula (20), it dominates the number of CFFs along the cyclic frequency axis. For parameter $p_1$ , it determines the proportion of CFFs in the entire cycle frequencies.

$$D=p_1·{\alpha }_{\max }$$

(21)

Simulation rolling bearing fault vibration signal analysis

Rolling bearing fault vibration signal model

In this section, we apply a simulation signal to demonstrate the feasibility of proposed method and corresponding flowchart is shown in Figure 4.

Figure 4.

The demonstration on proposed method by simulation signal.

We give a rolling bearing fault vibration simulation signal as shown in formula (22).

$$\mathrm{mvs}\left(t\right)=b\left(t\right)+r\left(t\right)+h\left(t\right)+n(t)$$

(22)

In above formula, $b\left(t\right)$ represents the periodic transients generated by rolling bearing fault, as following in formula (23).

$$b\left(t\right)=\sum _i{A}_i·e^{-{\beta }_1(t-i/f_m-\omega )}\sin (2\pi f_1(t-i/f_m-\mathrm{\Delta }/f_m))$$

(23)

Where $f_m=\mathrm{Rs}·f_r=100\mathrm{Hz}$ is the fault feature frequency and $f_r=26\mathrm{Hz}$ symbolizes the nominal shaft rotating frequency, $\mathrm{Rs}$ is set to $50/13$ for simulating outer race fault. $A_i$ represents the i-th shock amplitude in periodic transients, which also specified as a uniform random number in $[0.9,1]$ for outer race fault. $f_1$ and ${\beta }_1$ symbolizes resonant frequency and decay coefficient, respectively. $\mathrm{\Delta }$ represents a random variable which simulates the rolling bearing jitter effect caused by slight speed fluctuation, and its value conforms to uniform distribution $U[-{\mathrm{\Delta }}_{max},{\mathrm{\Delta }}_{max}],{\mathrm{\Delta }}_{max}=0.5\%$ . As for $r\left(t\right)$ , it represents random impulses and is simulated in the form of oscillating attenuation function as following in formula (24).

$$r\left(t\right)={\sum }_j^J{D}_j·e^{-{\beta }_2(t-{\epsilon }_j)}\sin (2\pi f_2(t-{\epsilon }_j))$$

(24)

$J$ and $D_j$ symbolizes the number and amplitude of random impulse respective, they are specified as 2. ${\epsilon }_j$ denotes the occurrence time of j-th random impulse, ${\beta }_2=700$ and $f_2=4500\mathrm{Hz}$ symbolizes decay coefficient and resonant frequency in $r\left(t\right)$ .

As for $h\left(t\right)$ , it simulates the harmonics which are produced by the shaft rotation and consist of three sinusoidal components, and its expression is shown in formula (25). $P_i$ , $f_i$ and ${\theta }_i$ symbolizes the amplitude, frequency, and phase in the i-th harmonic, respectively, as listed in Table 1.

$$\begin{array}{c}h\left(t\right)=P_1·\sin (2\pi f_{1}t+{\theta }_1)+P_2·\sin (2\pi f_{2}t+{\theta }_2)\hfill \\ +P_3·(0.5+0.3+\sin (2\pi f_{3}t+{\theta }_3))\hfill \end{array}$$

(25)

Table 1.

Parameters of the discrete harmonic signal.

Parameters	$P_1$	$P_2$	$P_3$	$f_1$ (Hz)	$f_2$ (Hz)	$f_3$ (Hz)	${\theta }_1$	${\theta }_2$	${\theta }_3$
Values	0.3	0.3	0.6	73	170	7.5	$\pi /6$	− $\pi /3$	0

As for $n(t)$ , it represents the Gaussian noises with a standard deviation $\sigma =0.9333$ . The sampling rate and duration time for $\mathrm{mvs}\left(t\right)$ are assumed as 12.8 kHz and 1 s, respectively, corresponding time-domain waveform of simulation signal $\mathrm{mvs}\left(t\right)$ is shown in following Figure 5.

Figure 5.

The simulated signal.

Furthermore, the spectrum in Figure 6 presents strong harmonic phenomenon in low frequency range, so that the fault-related frequency is barely identified in the resonance frequency band centered at 2.45 kHz.

Figure 6.

The spectrum of simulated signal.

Adopting optimal Q-factor RSSD to separate signal $\mathrm{mvs}\left(t\right)$ , after 16 iterations, the optimal Q-factor ${Q_L}^{*}=1.4$ and ${Q_H}^{*}=3.2$ is obtained, and corresponding kurtosis value is 3.0359. Finally, the varying rule between Q-factor is shown as follows in Figure 7.

Figure 7.

The varying low of Q-factor on the process of iteration.

According to the above procedures, the decomposition result is shown as follows. Figure 8 on top is high resonant component, reflecting obvious modulation characteristic, Figure 8 in the middle is low resonance component ${\overline{X_L}}^{*}$ , presenting periodic transients accompanied with noise, inducing that the adjacent impact interval $T^{\prime }=0.01s$ is difficult to detect directly. Figure 8 in low is the residual component, which is the error between simulation signal and high resonance component, low resonance component, it can be seen that its amplitude is weak. Through above processing, we finish the initial processing of simulation signal and obtain periodic transients accompanied with noise.

Figure 8.

Signal decomposition based on optimal Q-factor RSSD.

Furthermore, applying neighcoeff threshold based DTCWT on low resonance component ${\overline{X_L}}^{*}$ to reduce noise, the decomposition level is 4 and corresponding five sub-signals is shown in Figure 9 which can be seen that the transients play a dominant role, This phenomenon shows that the proposed method in this paper can effectively enhance the weak periodic transients.

Figure 9.

Sub-signals obtained by neighcoeff threshold based DTCWT.

Finally, we combine those sub-signals into a remarkable periodic transients ${\overline{X_L}}^e$ as shown in Figure 10, which presents the impact period $T=0.01s$ . In order to quantify noise reduction effect, the corresponding kurtosis values $\mathrm{Kurtosis}\left({\overline{X_L}}^{*}\right)=3.003$ , $\mathrm{Kurtosis}\left({\overline{X_L}}^e\right)=78.64$ are calculated respectively by using following formula $\mathrm{Kurtosis}\left(x\right)=E{(x-\mu )}^4/{\sigma }^4$ , which can prove the effectiveness.

Figure 10.

Remarkable periodic transients ${\overline{X_L}}^e$ .

Now, we give the SCoh of remarkable periodic transients ${\overline{X_L}}^e$ in Figure 11. Firstly, we can intuitively conclude that the amplitude value is more significant at $f=2000\mathrm{Hz}$ along the spectral frequency axis and it also reveals the fault frequency $\alpha =f_m=100\mathrm{Hz}$ and its harmonics hidden periodicity. This phenomenon indicates that the ${\overline{X_L}}^e$ contains a signal component whose carrier frequency is $2000\mathrm{Hz}$ and modulation frequency is $100\mathrm{Hz}$ . However, we also can find obvious interference spectral lines in the SCoh, and its corresponding EES is displayed in Figure 12. Therefore, if we can effectively focus on the frequency range near $f=2000\mathrm{Hz}$ , analyze its corresponding envelope spectrum, the fault frequency can be effectively enhanced.

Figure 11.

SCoh of enhancement low resonance component.

Figure 12.

Envelop spectrum of EES.

For solving above problem, we use IESCFFOgram to enhance fault frequency by identifying the informative band of SCoh with $p_1=5\%$ , and its corresponding result is shown in Figures 13 and 14. Obviously, the selected informative spectral frequency band is $f_c=1866.6667\mathrm{Hz},B_W=533\mathrm{Hz}$ which conforms to our judgment. This result infers that the proposed method can effectively extract informative frequency band. Finally, the spectral peaks of fault frequency and its harmonics in IES are outstanding compared with Figure 12.

Figure 13.

IESCFFOgram of enhancement low resonance component.

Figure 14.

IES of simulation signal.

Anti-interference performance evaluation

For quantitatively analyzing the diagnosis effect of proposed method on incipient fault, an adjusted diagnostic feature (ADF) is utilized to quantify the performances of anti-interference in formula (26), the ADF is defined to be the sum of the H-harmonics of fault frequency f_m under the given noise level, in this paper, the value of H is 5, where the $A\left(f\right)$ presents the spectral amplitude determined by frequency f

$$\mathrm{ADF}=\frac{1}{H}{\sum }_{h=1}^{H}A\left(h{f}_m\right)$$

(26)

In this section, we also use the simulation signal in section 3.1, and the SNR which is changed by varying the standard deviation of the white noise from 0.29 to 15.29 with equal spacing of 1.5 is investigated to compare the result of proposed method on ADF. Figure 15 displays the ADF values determined by the proposed method and EES which is used in place of the IES.

Figure 15.

ADF values for processing the simulated fault signal under the verity of different standard deviations.

It can be seen that the proposed method does have a slight advantage under the interference of strong background noise, as the noise becomes strengthening, indicator ADF decreases slowly, but this phenomenon is not very significant. Considering the dominating function of the IESCFFOgram is to enhance the weak fault frequency, in order to present its superiority, we also give the ADF values corresponding to EES instead of IESCFFOgram, it can be seen that the values is obviously lower compared with IES, and they are sensitive to strong noise, especially if the standard deviation exceeds 10, they fluctuate within.^10,16 On the contrary, the proposed method demonstrated good resistance to the fluctuation. The above analysis fully illustrates that, in the presence of strong interference, the proposed method is a better choice for revealing the hidden fault information.

Verification with experimental rolling bearing outer ring fault

The simulation results have verified the effectiveness of proposed method on extracting periodic transients and identifying and enhancing feature frequency. In this section, the diagnostic performance is demonstrated on XJTU-SY early rolling bearing outer ring fault data from school of mechanical engineering in Xi’an Jiao Tong University. Figure 16 displays the rolling bearing accelerated life test bench used in this study comprising a digital display, motor speed controller, rotating shaft, AC motor, support bearing, hydraulic loading system, vertical acceleration sensor, horizontal acceleration sensor and test rolling bearing.

Figure 16.

Rolling bearing accelerated life test bench.

The rolling bearing test bench produces variable working condition by adjusting radial force and speed, and style of fault rolling bearing is LDK UER204 as shown in Figure 17, its parameters is shown in following Table 2.

Figure 17.

Outer race wear.

Table 2.

LDK UER204 rolling bearing parameters.

Parameter	Value	Parameter	Value
Outer ring speed $n_a/\min$	2100	Ball diameter /mm	7.92
Inner ring speed $n_b/\min$	2400	Bearing pitch diameter $d_{\mathrm{v}}$ /mm	34.55
Outer race diameter/mm	39.80	Number of balls $Z$	8
Inner race diameter/mm	29.30	Contact angle $\alpha$ / (°)	0

According to the theoretical calculation formula of rolling bearing fault frequency, corresponding outer ring fault frequency $f_a=107.91\mathrm{Hz}$ by using formula (27).

$$f_a=\frac{n_a}{2\times 60}(1-\frac{d}{d_v}\cos \alpha )Z$$

(27)

Finally, the corresponding rolling bearing outer ring fault vibration signal is shown in Figure 18 in the full life cycle, its sampling frequency is 25.6 kHz.

Figure 18.

Fault vibration signal in full life cycle.

Due to the large noise amount in the fault vibration signal during the acquisition process on test bench, it can be seen from spectrum in Figure 19 that the spectral amplitude below 200 Hz is obvious, the fault frequency can not be identified directly. As a result, it is necessary to extract periodic transients and identify, enhance failure frequency in it. In order to validate the application value of the proposed method in rolling bearing early fault diagnosis, we intercept sub fault signal in full life cycle for analyzing it. Here we use $t=1,N=2^{14}$ , because at this time, the rolling bearing gradually varies from normal state to early stage. By using optimal Q-factor RSSD, in order to highlight impact property in low resonance component, we need to majorly analyze sub-band near the natural frequency connected with rolling bearing fault, it can be roughly determined that the resonance band of natural frequency exists near the frequency band of 0–500 Hz, and the initial value of RSSD is set as shown in 2.1 section. Finally, optimal low resonance component is dominated in Figure 20.

Figure 19.

The spectrum of fault vibration signal.

Figure 20.

Optimal low resonance component of outer ring fault signal.

Furthermore, we use neighcoeff threshold based on DTCWT to reduce noise in low resonance component by decomposing it into five sub-signals at four levels, and establish a remarkable periodic transients which has been enhanced impact property and shown in Figure 21. From the quantitative view, above conclusion can also be proven, $\mathrm{Kurtosis}\left(\mathrm{Figure}19\right)=2.14$ , $\mathrm{Kurtosis}\left(\mathrm{Figure}20\right)=2.98$ .

Figure 21.

Remarkable periodic transients in outer ring fault signal.

Finally, we apply the IESCFFOgram to explore and enhance fault frequency by combination spectral frequency and cyclic frequency, and the results are displayed in Figures 22 and 23.

Figure 22.

IESCFFOgram of outer ring fault signal.

Figure 23.

IES of outer ring fault signal.

For comparison, the classical SCoh and its corresponding EES is also shown in Figures 24 and 25.

Figure 24.

SCoh of outer ring fault signal.

Figure 25.

EES of outer ring fault signal.

In actual, just by observing the cyclic axis in Figure 24, as the spectral frequency at $f=12000\mathrm{Hz}$ has certain correlation with almost all the cycle frequencies, we cannot see the candidate fault frequency with human vision in Figure 25. However, by adopting IESCFFOgram, we can effectively locate the informative fault frequency band and enhanced weak fault feature frequency and its harmonics. On the other hand, we also can not ignore that the IES also contains abundant other frequencies, as the sub-signal we use is corresponding to early operation stage in the rolling bearing whole life cycle, in this time period, the rolling bearing basically has no failure, or the failure is very weak.

Furthermore, we use another fault sub-signal whose $t=3000000,N=2^{14}$ to verify the diagnostic effectiveness of proposed method. In this phase, the rolling bearing has a definite failure, but the degree of failure is still not very obvious. Figure 26 presents the SCoh, it can be that the cycle frequency associated with fault characteristic frequency can be seen intuitively, but there is still a strong noise component. Its EES can also confirm above conclusion which shows in Figure 27.

Figure 26.

SCoh of fault sub-signal.

Figure 27.

EES of fault sub-signal.

Applying the IESCFFOgram to obtain corresponding IES which is seen in Figure 28, there is no doubt that the proposed method can enhance fault feature frequency quantitatively.

Figure 28.

IES of fault sub-signal.

Conclusion

This paper proposes a hybrid intelligent fusion algorithm for extracting periodic transients and identifying, enhancing fault frequency for rolling bearing fault detection. The core and key innovation of this method is that it can periodic transients extraction without distortion and weak fault frequency enhancement. Firstly, we uses an optimal Q-factor RSSD to extract transient sub-signal from fault vibration signal. Then, a neighcoeff threshold based on DTCWT is applied to reduce noise. Finally, IES based on CFFOgram is to identify weak fault frequency. Simulation and experimental data from School of Mechanical Engineering, Xi’an Jiao tong University are applied to verify the effectiveness of proposed method. The following conclusions can be drawn from this study:

1. Q-factor is a core parameter directly related to low resonance component, and its selection plays a decisive role in extracting periodic transients. A strategy is given to obtain an optimal Q-factor for self-adaptive matching with periodic transients in fault vibration signal by using stepwise iterative algorithm which possesses good search and optimization advantage.

2. On account that the adjacent wavelet coefficients of periodic transients in low resonance component have a good correlation, the neighcoeff threshold in this paper can effectively reduce noise, improving SNR and enhance periodic impact property in low resonance component.

3. The IES based on IESCFFOgram can adaptively excavate the fault information hidden in the SCoh, and enhance weak fault frequency compared with EES. Therefore, it is suitable for the fault identification of rolling bearing.

In future research, we need to develop a proper health indicator to judge the failure stage of rolling bearing, if the indicator exceeds the threshold, or presents aberration, we can apply the proposed method in this paper as soon as possible to achieve early fault diagnosis. In addition, we should consider the feasibility of extending this method to fault diagnosis under variable speed condition. Finally, we also should solve the problem of identifying multiple informative spectral frequency bands of SCoh in future work.