A novel feature extraction method of compound faults of bearing based on ITD and information fusion

Abstract

A method combining signal decomposition and information fusion is proposed to solve the difficulty in extracting the compound fault features of rolling bearing. Firstly, the method decomposes the homologous signals into the sum of a series of proper rotational components, which are obtained by the sensors in horizontal and vertical directions with intrinsic time scale decomposition (ITD). Secondly, autocorrelation noise reduction and normalization processing are performed for the component signals obtained; weight coefficients of signals are adaptively determined according to normalized autocorrelation function (AF) of components. Thirdly, autocorrelation functions of all component signals are weighted and blended according to weight coefficient obtained. Finally, fault feature frequency of rolling bearing is extracted with power spectrum of signals blended. Equally, the failure types of bearings are judged. To verify the effectiveness of proposed method, the data which corresponds to different failure types is analyzed and verified; meanwhile, other methods is compared with presented method. The result indicates that the feature information of compound failure can be extracted precisely, and the compound failure type of bearings can be judged accurately with presented method.

Keywords

Intrinsic time-scale decomposition fault identification rolling bearing multiple faults

Introduction

The running state of rolling bearing as the core component of rotating machine can directly affect the stability and service life.¹ As the running state of rotating machine is harsh and rolling bearing operates in high-temperature and high-speed environment, the probability of failure is high extremely.¹ Equally, as time goes on, the working performance of bearing degenerates, which will put the stability of equipment and safety of whole system in higher risk. Consequently, it is crucial for the safety and stability of equipment to precisely determine the running state and the failure type of rolling bearing.² Bearing fault often exists in the form of being compound as found from the analysis of engineering practice; due to intercoupling of fault information and the influence of noise and component signals irrelevant to faults, compound faults of bearings are far more difficult to be recognized than single.³

Vibration signal is of strong non-stationarity and nonlinearity when rolling bearing has a fault.⁴ Signals can be decomposed into multiple component signals of certain physical significance with different ways and approaches by signal decomposition algorithm, which is beneficial to the precise extraction of failure feature information.⁵ Accordingly, in the identification of rolling bearing failure, signal decomposition algorithms have been studied extensively. Conventional signal decomposition algorithms include wavelet transform (WT),⁶ empirical mode decomposition (EMD),⁷intrinsic time scale decomposition (ITD),⁸ variational mode decomposition (VMD) and son on. To solve the problem arising from end effect and modal aliasing, other algorithms have been developed, including ensemble EMD, complementary ensemble EMD and complete ensemble EMD. VMD takes advantage of, establishes and solves constrained variation to decompose signals and the option of number of decomposition layers and penalty factor will affect the decomposition effect of VMD. ITD is characterized by excellent time-frequency aggregation, small end effect and fast calculation and applicable to the processing of nonlinear and non-stationary signals.⁹ Zhang, et al., took advantage of ITD algorithm to decompose signals. They judged fault of bearing according to the optimal proper rotation component chosen by maximum kurtosis and verified the advantage of ITD in fault diagnosis through comparison with EMD.¹⁰ To solve the problem of insufficient or excessive decomposition caused by the number of decomposition layers of ITD, Yu et al., took advantage of correlation coefficient to adaptively determine the number of decomposition layers of ITD.¹¹ It can be found that signal evaluation index is required in the adaptive determination of parameters and option of component signals after decomposition. As information entropy, kurtosis and correlation coefficient is rather sensitive to fault features of vibration signal, they often serve as the basis for parameter determination in signal decomposition and option of sensitive fault components after decomposition. Among them, kurtosis is more sensitive to impact feature of fault signals and often integrated with signal decomposition algorithm to optimize and determine parameters¹² and choose optimal sensitive component signals¹³ during recognition of bearing faults.

In case of a fault occurs, the fault feature which comes from vibration signal is more evident in some direction or position. Due to uncertainty of fault, these directions or positions cannot be predicted. Meanwhile, when fault is identified according to the signal from single sensor, as only single source of information is applied, it is very likely to get incomplete information resulting in missed diagnosis and further aggravation of fault. Therefore, information fusion is an important item in fault identification. Multi-sensor information fusion system makes use of multiple (or kinds of) sensor resources to combine and optimize criterions, reduce and complement information to create new results. Feature layer fusion of information fusion reduces the handling capacity of original data, increases processing speed and instantaneity of system and can reduce the influence of noise and redundant information on system processing by extracting representative features.¹⁴ Jiang et al.,¹⁵ proposed a multi-sensors feature fusion method, extracted various kinds of entropies of vibration signals with information entropy theory and classified the faults according to the established feature fusion model. Homologous signal fusion, signals collected by sensors in orthogonal directions on the same section of rotor, can enhance identification ability of fault and improve the detection accuracy of system. Yu et al.,¹⁶ chose cross-correlation function of homologous signal and actualized the information fusion of vibration acceleration signals from engine casing and combined ITD algorithm and complexity parameter of Hjorth parameter to precisely identify the faults of intershaft bearing.

Fault information can be enhanced and diagnosis accuracy of fault identification can be improved by giving larger weight coefficient to signals with distinct fault features. Therefore, to precisely determine weight coefficient is an important study for fault identification according to information fusion algorithm. Liu, et al.,¹⁷ constructed weight fusion rule based on fuzzy entropy and blended different frequency components from multiple sensors. Tong, et al.,¹⁸ merged together the decision score of each sensor with fuzzy rank and minimized the difference between actual and predicted values. Huang, et al.,¹⁹ determined the weighting coefficient of band-limited eigenmode function (BLIMF) through defined energy ratio, weighted the envelope of each BLIMF to rebuild new envelope line and successfully diagnosed a bearing fault.

Based on the above studies, the paper has proposed a new fault recognition method for bearing according to signal decomposition and information fusion algorithms. The method breaks up homologous vibration signals with ITD algorithm; the component signals obtained are subjected to autocorrelation noise reduction and normalization processing to reduce the noise and further highlight the periodic features of signal; kurtosis of normalized AF of each component signal is calculated to adaptively determine weight coefficient of component signal; features of component signals are enhanced and signals are rebuilt with the weight coefficient obtained; compound failures of bearings are determined exactly with the power spectrum of rebuilt signals.

Main contributions are as follow:

(1) ITD is combined with information fusion to fully represent fault information and reduce the influence of irrelevant components on the extraction of compound fault features of rolling bearings.

(2) Homological signals from sensors in horizontal and vertical directions are taken to capture the fault features in different directions, represent vibration information more comprehensively and improve the reliability of fault identification.

(3) Weight coefficients of signals are adaptively determined according to the kurtosis of normalized autocorrelation function of components. Fault features are enhanced and the signal components with obvious fault features can be highlighted more precisely.

(4) Compound faults of bearings are exactly extracted. Through analysis of power spectrum of reconstructed signals, fault features of bearings can be extracted, noise and influence of irrelevant components on fault features reduced, and compound faults of bearings exactly judged.

Methodology

ITD of homologous signal

Intrinsic time scale decomposition is a time sequence analysis technology for non-stationary signal. With ITD algorithm, vibration signal can be decomposed into several independent proper rotation components (PRCs) and a residual trend component (R).

Assuming the vibration signal from some sensor is $X_{t}$ and operator of baseline signal is L (varying with the local extremum of original signal), then, the calculation process of ITD as is shown in formulas (1)–(4). Specific steps are as follow:

Step 1: vibration signal is decomposed into the sum of baseline signal L and rotation component H:

X_{t} = L X_{t} + (1 - L) X_{t} = L_{x t} + H_{x t}

(1)

In formula (1),

L_{x t}

and

H_{x t}

is the baseline signal and PRC obtained from one decomposition.

Step 2: extreme point of sequence $X_{t}$ and its moment ( $τ_{k}$ , k = 1,2,...,M) is extracted and $τ_{0} = 0$ . Baseline factor of piecewise linearity L is defined within the interval ( $τ_{k}$ , $τ_{k + 2}$ ] and the extraction of piecewise baseline signals is as follow:

{L X}_{t} = L_{k} + \frac{L_{k + 1} - L_{k}}{X_{k + 1} - X_{k}} (X_{t} - X_{k}), t \in (τ_{(k)}, τ_{(k + 1)}]

(2)

In the formula (2),

X_{t}

is the extreme point of original signal;

τ_{k}

is the time of extreme point (k = 1, 2,...,M); M is the number of extreme points;

L_{k + 1}

can be expressed as:

L_{k + 1} = α [X_{k} + (\frac{τ_{k + 1 -} τ_{k}}{τ_{k + 2 -} τ_{k}}) (X_{k + 2} - X_{k})] + (1 - α) X_{k + 1}

(3)

The range of linear gain parameter $α$ is 0< $α$ <1. Usually, $α$ = 0.5.²⁰

Step 3: baseline signal is regarded as new original signal and step 1–2 is repeated.

The result of decomposition of signal $X_{t}$ is expressed as:

X_{t} = L_{x t} + H_{x t} = H_{x t} + (H + L) L_{x t} = H \sum_{k = 0}^{p - 1} L_{x t}^{p} + L_{x t}^{p}

(4)

In the formula (4), H is the extraction operator of proper rotation component;

L_{x t}^{p}

is residual trend component.

According reference 20, the number of decomposition layers of ITD is set as 4, then there are 8 component signals obtained after ITD of signals acquired by sensor in horizontal $(x (t)$ ) and vertical direction $(y (t)$ . The component signals corresponding to $x (t)$ and $y (t)$ are labelled as $H_{x t i} (t)$ and $H_{y t i} (t)$ ( $i = 1, 2, 3, 4)$ .

To reduce the influence of noise, component signals acquired after decomposition are self-correlated and normalized. Given autocorrelation function of some component signal $H_{x t i} (t)$ is $R_{x, x i} (τ)$ , then $R_{x, x i} (τ)$ is expressed as formula (5):

R_{x, x i} (τ) = \int_{- \infty}^{+ \infty} H_{x t i} (t) H_{x t i} (t + τ) d t

(5)

After autocorrelation treatment, the signal $R_{x, x i} (τ)$ and $R_{y, y i} (τ)$ $(i = 1, 2, 3, 4)$ corresponds to the autocorrelation functions of $H_{x t i} (t)$ and $H_{y t i} (t) (i = 1, 2, 3, 4)$ which is normalized. With $R_{x, x i} (τ)$ $(i = 1, 2, 3, 4)$ as an example, the formula (6) is as shown:

{P R C}_{x i} (t) = \frac{R_{x, x i} (τ)}{a}, (i = 1, 2, 3, 4)

(6)

In the formula,

a

is the maximum of

R_{x, x i} (τ)

(i = 1, 2, 3, 4)

. Through formulas (5) and (6), self-correlated and normalized signals

{P R C}_{x i} (t)

and

{P R C}_{y i} (t)

(i = 1, 2, 3, 4)

can be obtained.

To self-adaptively determine weight coefficient of component signals

Fault information can be strengthened by giving greater weight coefficient to the signals with evident fault features. Kurtosis is one of parameters of statistics designed to depict sharp degree. As it is sensitive to impact feature of bearing fault and unrelated to rotate speed, dimension and load of bearing, it is often used to detect an early injury of rolling bearing. When bearing is fault-free and due to the influence of various uncertain factors, the amplitude distribution of vibration signal is close to normal distribution. With the presence and development of fault, the probability density of high amplitude increases in vibration signal, distribution of signal amplitude deviates from normal distribution, normal curve deflects or disperses, and kurtosis increases with it. The larger kurtosis of signal is, the more likely it is to contain more evident fault information and the easier to extract the fault features.²¹ Based on that, weight coefficient is established with kurtosis as the basis when homologous signal is fused together to enhance fault information.

Calculation formula of kurtosis is as shown:

K u r t o s i s (x) = \frac{{\frac{1}{n} \sum_{i = 1}^{n} (x - \bar{x})}^{4}}{{({\frac{1}{n} \sum_{i = 1}^{n} (x - \bar{x})}^{2})}^{2}}

(7)

In the formula (7),

x

is the instantaneous amplitude value of some component signal and

\bar{x}

is the average value of amplitude.

${P R C}_{x i} (t), (i = 1, 2, 3, 4)$ and ${P R C}_{y i} (t), (i = 1, 2, 3, 4)$ represents the AF normalized of component signals $H_{x t i} (t)$ and $H_{y t i} (t)$ of horizontal and vertical signals after ITD. Features of component signals are enhanced in the following steps:

Step 1: the kurtosis $K_{x} (i)$ and $K_{y} (i), (i = 1, 2, 3, 4)$ of $PR C_{x i} (t), (i = 1, 2, 3, 4)$ and $PR C_{y i} (t), (i = 1, 2, 3, 4)$ is calculated.

Step 2: the weight coefficients $W_{n} (i)$ of component signals are calculated according to the kurtosis of component signals and formula (8).

W_{n} (i) = \frac{K_{n} (i)}{\sum_{i = 1}^{4} K_{x} (i) + \sum_{i = 1}^{4} K_{y} (i)}, (n = x, y)

(8)

Step 3: weight coefficients are multiplied by normalized AF of each component signal and the result is the weighted component signal ${PR C^{'}}_{n i} (t)$ :

{PR C^{'}}_{n i} (t) = W_{n} (i) {P R C}_{n i} (t), (n = x, y)

(9)

Step 4: the weighted component signals are blended and the new blended signal $f$ of feature enhanced is obtained:

f = \sum_{i = 1}^{4} PR C_{n i}^{'} (t), (n = x, y)

(10)

Overall plan of proposed method

The paper includes the following into consideration:

(1) The vibration signals of rolling bearing are polydirectional and it is very likely that faults show different vibration features in horizontal and vertical directions.

(2) Single sensor can only capture the vibration information from one direction and the representation of fault features is one-sided.

(3) When different faults occur, the ability of sensors in different directions to represent faults varies. As it is impossible to forecast the type of faults, it will be more beneficial for fault judgment to choose the signals from sensors in 2 directions for fusion analysis.

(4) On the same section, the signals from sensors in horizontal and vertical directions are homological information which contain the fault information of vibration signals from 2 orthogonal channels through the complementarity of orthogonal information in the space and can reflect fault features more comprehensively and accurately.

Therefore, the paper blends and analyzes the homologous signals obtained by the sensors installed in horizontal and vertical direction. The block diagram of thinking of proposed method is shown in Figure 1.

Figure 1.

Block diagram of thinking of proposed method. Note: AF, Autocorrelation function; PRCs, proper rotation components.

Specific steps of proposed method:

Step 1: according to formulas (1)–(4), signals from horizontal and vertical sensors are decomposed by ITD algorithm and 8 component signals in total can be acquired (each sensor is related with 4 components).

Step 2: according to formulas (5) and (6), autocorrelation function of component signal is normalized.

Step 3: according to formulas (7)-(10), features of component signals are enhanced and information is fused.

Step 4: according to the power spectrum of blended signal $f$ , failure features are extracted and failure types are judged.

To verify the effectiveness and superiority of proposed method, a comparison is made between ITD-Kurtosis-Enhance-Fusion and other comparison schemes which are as follow:

(1) Comparison scheme 1, ITD-Max-Kurtosis: signals obtained by sensors are decomposed by ITD, sensitive component signals of fault are chosen according to maximum kurtosis, and bearing faults are judged according to the analysis of sensitive fault component signals.

(2) Comparison scheme 2, ITD-Max-Kurtosis-Fusion: signals from 2 orthogonal sensors are decomposed by ITD and component signals sensitive to faults are chosen and blended with maximum kurtosis as criterion. Bearing faults are evaluated according to the analysis of blended signals. This scheme is different from the method of paper in: (1) weight coefficient of component signal is not adaptively determined and feature is not enhanced; (2) for the component signals from 2 sensors, one component signal each is chosen for fusion.

Verification and analysis

Experiment of compound faults of bearing and fault data

The data is from rotor-rolling bearing testbed shown in Figure 2(a). Main components of testbed include rolling bearing, shaft, bearing block, adjustable-speed motor, gear box and rotor disc. Speed is measured by an eddy-current sensor. 3 acceleration sensors are installed in horizontal (left and right end of bearing block) and vertical (the top of bearing block) positions. The installation position of sensors corresponds to CH1-CH4 shown in Figure 2(a). Bearing is damaged by wire cut electrical discharge (cutting depth is 0.2 mm).²² The diameter of ball is 9.36 mm, pitch diameter of bearing 36 mm and number of rolling elements 7. 3 classical compound failures of rolling bearing are displayed in Figure 2(b)–(d), including outer race (OR) and rolling element (RE) combined failure, inner race (IR) and RE combined failure, and OR, IR and RE combined failure.

Figure 2.

Rotor-rolling bearing experimental rig and failure bearings. Note. RE, Rolling element; IR, Inner race; OR, Outer race.

For simple description, the fault feature frequency of outer race, inner race, rolling element and retainer is represented by $F_{o}$ , $F_{i}$ , $F_{b}$ and $F_{c}$ and rotate frequency by $F_{r}$ .

The presented method is analyzed and verified with the classical data of 3 types of compound faults randomly chosen. According to the calculation formula of feature frequency of bearing,¹¹ geometrical parameter and rotate speed, the feature frequency of bearing corresponding to each data is shown in Table 1. In Table 1, the bold represent the fault characteristic frequencies which match with the failure types of the bearing.

Table 1.

Information of typical failure data.

Cases	Sensors	Fault types of bearings	Rotational speed (r/min)	Feature frequency of bearings (Hz)
Cases	Sensors	Fault types of bearings	Rotational speed (r/min)	$F_{r}$	$F_{c}$	$F_{b}$	$F_{o}$	$F_{i}$
Case 1	CH2 + CH3	OR + RE failure	1554	25.91	9.50	45.12	66.49	114.85
Case 2	CH2 + CH3	IR + RE failure	1546	25.77	9.45	44.89	66.15	114.26
Case 3	CH2 + CH3	OR + IR + RE failure	1542	25.71	9.43	44.77	65.98	113.97

Failure identification of rolling bearing with proposed method

The effectiveness of proposed method is verified and analyzed. Firstly, Case 1 (compound failure of OR and RE) in Table 1 is randomly taken as an example. This state corresponds to rotate speed 1542 r/min. According to dimension of bearing, rotate speed and formula of feature frequency,¹¹ the feature frequency as is shown in Table 1, $F_{r} = 25.91 Hz$ , $F_{b} = 45.12 Hz$ , $F_{i} = 114.85 Hz$ , $F_{o} = 66.49 Hz$ and $F_{c} = 9.50 Hz$ .

The time domains of homologous signals which corresponds to the horizontal sensor (CH2) and vertical (CH3) are displayed in Figure 3(a) and (b); the frequency spectrum and the power spectrum of Figure 3(a) and (b) is displayed in Figure 3(c)–(f) respectively. According to reference 20, the number of decomposition layers of ITD proves 4. Figure 3(g) and (h) are the component signals of Figure 3(a) and (b) after ITD; normalized autocorrelation function of each component signal is Figure 3(i) and (j). The kurtosis of signals in Figure 3(i) and (j) is calculated and shown in Table 2. According to these kurtosis values, the weight coefficients of kurtosis index of component signals are shown in Figure 3(m). The features of component signals are strengthened according to these weight coefficients and the feature-enhanced component signals are shown in Figure 3(k) and (l). Signals are blended according to Figure 3(k) and (l) and blended signal is shown in Figure 3(n), and its power spectrum is shown in Figure 3(o).

Figure 3.

Failure identification results of the proposed method.

Table 2.

The kurtosis values of each component signal.

$P R C s$	$PR C_{x} (1)$	$PR C_{x} (2)$	$PR C_{x} (3)$	$PR C_{x} (4)$
Kurtosis	28.84	7.18	18.55	29.80
$P R C s$	$PR C_{y} (1)$	$PR C_{y} (2)$	$PR C_{y} (3)$	$PR C_{y} (4)$
Kurtosis	75.76	26.07	13.87	25.76

It can be found from the analysis of frequency spectrum and power spectrum of signals collected by horizontal and vertical directions in Figure 3(c)–(f) that:

Though some characteristic frequencies accompany with bearing failure can be found from frequency and power spectrums, there are too many noises which are averse to extraction of fault feature information.

It can be found from the signal power spectrum obtained by the proposed method in Figure 3(n) and (o) that noise component is greatly reduced and there are obvious and prominent frequency components shown as follow:

(1) 600.6 Hz (600.6/9 = 66.73), 1250 Hz (1250/19 = 65.79), 1521 Hz (1521/23 = 66.13), 1731 Hz (1731/26 = 66.58) and 1868 Hz (1868/28 = 66.71); these frequencies correspond to 9×, 19×, 23×, 26× and 28× of $F_{o}$ ; it can be judged from these feature frequencies that there is a fault in outer race.

(2) 151.4 Hz ((151.4 − 2 $F_{c}) /$ 3 = 44.13), 297.9 Hz ((297.9 + 2 $F_{c}) /$ 7 = 45.27), 356.4 Hz (356.4/8 = 44.55) and 710.4 Hz((710.4 + $F_{c}$ )/16 = 44.99); these frequencies correspond to 3×, 7×, 8× and 16× of $F_{b}$ ; it can be judged from these feature frequencies that there is a fault in rolling element.

From the analysis of (1) to (2), it can be concluded that a combined failure involving the outer race and rolling element has occurred, which aligns perfectly with the actual failure type of the bearing. This has verified that the proposed ITD-Kurtosis-fusion method not only can control the noise effectively but also can enhance failure information. Meanwhile, the extraction of failure feature information is very comprehensive and the type of compound failure can be judged precisely.

Comparative method - case 1

To further verify the effectiveness and advantage of proposed method, this section has compared the proposed method with other methods. The specific description of each comparison scheme is shown in section “Comparative method - case 1”, Comparison scheme 1 and Comparison scheme 2. For comparison and verification, this section chooses the same data with section “Failure identification of rolling bearing with proposed method” and the number of decomposition layers of ITD is 4. The kurtosis of each component signal is still shown as Figure 3(m). It can be known from Figure 3(m) that the component signal $PR C_{x} (4)$ corresponding to horizontal sensor CH2 has the maximum kurtosis; the component signal $PR C_{y} (1)$ corresponding to vertical sensor CH3 has the maximum kurtosis.

Comparative scheme 1

From Figure 3(m), according to comparative scheme 1, $PR C_{x} (4)$ and $PR C_{y} (1)$ is chosen as sensitive component signal of fault. The time domain and power spectrum of $PR C_{x} (4)$ and $PR C_{y} (1)$ is as shown in Figure 4(a)–(d).

(1) From the analysis of Figure 4(c), power spectrum of horizontal sensor, no outstanding feature frequency matched with bearing failure can be found; it cannot be judged that a bearing fault occurs.

(2) From the analysis of Figure 4(d), power spectrum of vertical sensor, frequency components 1250 Hz (1250/19 = 65.79), 1521 Hz (1521/23 = 66.13), 1731 Hz (1731/26 = 66.58) and 1868 Hz (1868/28 = 66.71) can be found. These frequencies correspond to 19×, 23×, 26× and 28× of $F_{o}$ ; no remarkable feature frequency component matched with rolling element failure can be found. Therefore, the judgement of bearing fault type is not correct.

Figure 4.

Results of Comparison scheme 1-case 1.

Comparative scheme 2

Bearing fault is judged according to comparative scheme 2. The sensitive component signal chosen with maximum kurtosis as criterion is still $PR C_{x} (4)$ and $PR C_{y} (1)$ . These 2 component signals are blended in the way of weight. The time domain of blended signals is displayed in Figure 5(a) and Figure 5(b) represents the power spectrum of Figure 5(a).

Figure 5.

Results of comparison scheme 2.

From the analysis of Figure 5(b), it can be inferred that Comparative scheme 2 is also capable of effectively controlling noise and exhibits the following classical frequency components:

1250 Hz (1250/19 = 65.79), 1521 Hz (1521/23 = 66.13), 1731 Hz (1731/26 = 66.58) and 1868 Hz (1868/28 = 66.71); these frequencies are 19×, 23×, 26× and 28× of $F_{o}$ . These frequencies still correspond to outer race failure of bearing and no remarkable frequency component corresponding to the fault of rolling element can be found. Therefore, the judgment of fault type is incorrect.

Meanwhile, the result of Figure 5(b) is similar to Figure 4(d). This is mainly because the amplitude of $PR C_{x} (4)$ is far from that of $PR C_{y} (1)$ . Therefore, the new blended signal acquired by blending of $PR C_{x} (4)$ and $PR C_{y} (1)$ in the way of equal weight has the similar result to $PR C_{y} (1)$ . The blended signal still and only has the frequency multiplication of feature frequency of outer race failure $F_{o}$ ; there is no classical frequency component found corresponding to the fault of rolling element; therefore, the type of compound fault of bearing cannot be precisely judged.

Compared with other comparison methods, the proposed method does not need to choose component, but only relies on adaptively-determined weight coefficient to enhance and blend the features of component signals of homologous signals, and extracts the feature frequency matched with the type of compound faults of bearing. The extraction of feature information is the most comprehensive and the compound fault of bearing can be precisely judged.

Effectiveness analysis under different combined failure types

Next, the proposed method is verified under different combined failure types. An analysis is given to the vibration signals which correspond to typical compound failure of outer race and rolling element (Case 2) and outer race, inner race and rolling element (Case 3).

The rotate speed of Case 2 is 1546 r/min. After calculation, failure feature frequencies of bearing are $F_{r} = 25.77 H z$ , $F_{b} = 44.89 H z$ , $F_{i} = 114.26 H z$ , $F_{o} = 66.15 H z$ and $F_{c} = 9.45 H z$ .

The rotate speed of Case 3 is 1542r/min and the failure feature frequencies of bearing are $F_{r} = 25.71 H z$ , $F_{b} = 44.77 H z$ , $F_{i} = 113.97 H z$ , $F_{o} = 65.98 H z$ and $F_{c} = 9.43 H z$ .

Figure 5(a) and (b) shows the fault signal of CH2 and CH3 related with Case 2. Figure 5(c) and (d) shows the fault signal of CH2 and CH3 related with Case 3. After ITD of fault signals of Case 2–3, the diagrams of component signals are Figure 5(e)–(h). Figure 5(i)–(l) is the diagram of each component signal after normalized autocorrelation treatment of Case 2 and Case 3. Table 3 shows the kurtosis of component signals. With kurtosis, the weight coefficient of kurtosis index of component signal is calculated, and the result is shown in Figure 5(m) and (n). Feature of autocorrelation functions of component signals are enhanced according to weight coefficients. After enhancement, the signal is as shown in Figure 5(o)–(r). Figure 5(s) and (t) shows the fused signal of Case 2 and Case 3 obtained by the proposed method. Figure 3(u) and (v) shows the power spectrum of Figure 5(s) and (t).

Table 3.

The kurtosis values of each component signal.

$P R C s$	$PR C_{x} (1)$	$PR C_{x} (2)$	$PR C_{x} (3)$	$PR C_{x} (4)$
Kurtosis(Case2)	57.03	12.04	15.44	14.76
Kurtosis(Case3)	19.30	8.33	17.21	19.35
$P R C s$	$PR C_{y} (1)$	$PR C_{y} (2)$	$PR C_{y} (3)$	$PR C_{y} (4)$
Kurtosis(Case2)	18.58	17.02	17.07	10.62
Kurtosis(Case3)	16.96	15.80	17.80	16.64

With the analysis of Figure 6(u), when the compound fault of bearing is IR and RE (Case 2), from the power spectrum of signal obtained by proposed method, noise is effectively controlled. Meanwhile, the following obvious and prominent frequency components can be found:

(1) 139.2 Hz (139.2/3 = 46.4), 705.6 Hz ((705.6 + $F_{c}$ )/16 = 44.69), 1221 Hz ((1221 $- F_{c}$ )/27 = 44.87) and 1794 Hz (1794/40 = 44.85). These frequencies correspond to 3×, 16×, 27× and 40× of $F_{b}$ . According to these frequencies, what can be estimate is that a failure occurs in RE.

(2) 114.7 Hz, 212.4 Hz ((212.4 + $F_{r} - F_{c}$ )/2 = 114.35), 290.5 Hz ((290.5 + 2 $F_{r}$ )/3 = 113.97), 1272 Hz ((1272 $- F_{r} + F_{c}$ )/11 = 114.15) and 1812 Hz (1812 + $F_{r} - F_{c}$ )/16 = 114.27). These frequencies correspond to 1×, 2×, 3×, 11× and 16× of $F_{i}$ . According to these frequencies, what can be judged is that a failure occurs in IR.

Figure 6.

Results of Case2 and Case3 with proposed method.

According to (1) and (2), what can be concluded is that a compound fault of RE and IR happens, which is consistent with the failure type of the bearing.

With the analysis of Figure 6(v), when the type of compound fault of bearing is IR, OR and RE (Case 3), the signal power spectrum obtained by the proposed method can still help to effectively control the noise and reveal the following classical and prominent frequency components:

(1) 139.2 Hz (139.2/3 = 46.4), 163.6 Hz ((163.6 + 2 $F_{c}$ )/4 = 45.65), 293 Hz ((293 − $F_{r}$ )/6 = 44.55) and 708 Hz ((708 + $F_{c}$ )/16 = 44.84). These frequencies correspond to 3×, 4×, 6× and 16× of $F_{b}$ . According to these frequencies, what can be estimate is that a bearing has a failure of RE.

(2) 856.9 Hz (856.9/13 = 65.92), 922.9 Hz (922.9/14 = 65.92) and 991.2 Hz (991.2/15 = 66.08). These frequencies correspond to 13x, 14x and 15x of $F_{o}$ ; according to these frequencies, it can be judged that a bearing has a failure of OR.

(3) 908.2 Hz (908.2/8 = 113.53), 2861 Hz (2861/25 = 114.44); these frequencies correspond to 8×, 25× of $F_{i}$ . According to these frequencies, it can be judged that a bearing has a failure of IR.

According to (1)–(3), what can be concluded is that a bearing has the compound failure of rolling element, outer race and inner race. This is consistent with the failure type of this bearing.

It can be concluded that when a bearing has different compound failure types, the proposed method can still extract typical and prominent feature information matched with bearing fault, and precisely determine the fault type of bearing.

The paper has proposed a compound fault identification method which combines ITD, information fusion and feature enhancement. Homological signals are decomposed by ITD. The kurtosis of autocorrelation function of each component signal is normalized and the weight coefficients are adaptively determined after feature enhancement. Homological information is fused in the way of weight addition after feature enhancement. The following conclusion can be drawn:

(1) Fault information can be represented more accurately through the fusion of component signals after decomposition;

(2) By the strategy of combining component signals of homological signals from the sensors in vertical and horizontal directions, the fault information of vibration signals in 2 directions can be complementarily fused and fault features extracted more completely. The accuracy and reliability of fault identification is improved. The misdiagnosis caused by incomplete information captured by single sensor can be solved.

(3) Through the kurtosis of normalized autocorrelation function of component signals, the weight coefficients can be adaptively determined in feature enhancement. Signals can be fused and fault features enhanced according to weight coefficients. Compared with the contrast methods in which optimal components are directly chosen by kurtosis and optimal components are directly fused, the proposed method of paper can obtain more complete fault information and the judgment of the type of compound faults of bearings is more accurate.

(4) The proposed method is not sensitive to the type of compound faults. For different types of compound faults, the proposed method can extract the classic feature frequencies corresponding to the types of compound faults and accurately judge a compound fault of bearing.

The proposed method is verified according to tester signals. Future studies will continue to verify the engineering applicability of proposed method. In the selection of ITD of decomposition layers, the experimental method in reference is referred. The adaptive determination of ITD will be studied further in the future.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China [Grant No: 51605309], Department of Education of Liaoning Province [Grant No: LJKMZ20220529], Aeronautical Science Foundation of China [Grant No: 20230033054001], Natural Science Foundation of Liaoning Province [Grant No: 2022-MS-299], Steady fundamental supporting project phase II for scientific research institute of military industry [Grant No: 03020051] and the fundamental research funds for the universities of Liaoning province [LJ212410143047, LJ232410143072].

ORCID iDs

Mingyue Yu

Yunbo Wang

Data Availability Statement

The datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.*

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