Study on rolling bearing fault diagnosis approach based on improved generalized fractal box-counting dimension and adaptive gray relation algorithm

Abstract

Aiming at the nonlinear and nonstationary characteristics of bearing vibration signals as well as the complexity of condition-indicating information distribution in the signals, a novel rolling element bearing fault diagnosis approach based on improved generalized fractal box-counting dimension and adaptive gray relation algorithm was proposed in this article. First, an improved generalized fractal box-counting dimension algorithm was developed to extract the characteristic vectors of fault features from the bearing vibration signals, to offer more useful and distinguishing information imaging different bearing health status in comparison with traditional fractal box-counting dimension. After feature extraction by improved generalized fractal box-counting dimension algorithm, an adaptive gray relation algorithm, in which the concept of weight coefficient and adaptive distinguishing coefficient was introduced into the calculation of the relation degree, was employed to fulfill an intelligent bearing fault diagnosis. The experimental results demonstrate that the proposed approach can more effectively and accurately identify different bearing fault types as well as severities compared with the existing intelligent methods, and it can solve the learning problem with an extremely small number of samples.

Keywords

Rolling bearing fault diagnosis fractal box-counting dimension gray relation theory vibration signal

Introduction

Rolling bearings as a main component have been widespread used in various types of rotating machines. Bearing failure is one of the foremost causes of failure and breakdowns in rotating machinery, resulting in significant economic loss.^1–3 In order to keep machinery running with high reliability and reduce economic losses, it is essential to develop a reliable and effective bearing fault diagnosis method. Among different bearing fault diagnosis methods, vibration-based methods have attracted extensive attention during the last few decades.⁴

Bearing vibration signals contain abundant information about machine health status; therefore, it is possible to obtain dominant characteristics from vibration signals by signal processing techniques.⁵ A great number of signal processing methods have been used for rolling bearing fault detection and diagnosis. However, due to the nonlinear factors, for example, stiffness, friction and clearance, bearing vibration signals in particular under faulty conditions, will behave nonlinear and nonstationary characteristics.⁶ Besides, the measured vibration signals contain not only the working condition information related to the rolling bearing itself, but also a wealth of information of other moving parts and structures of machinery and equipment, which belongs to the background noise compared with the former.⁷ Since background noise is often relatively large, the slight rolling bearing fault information may easily be buried in the background noise and become difficult to extract. As a result, the conventional time and frequency domain methods aiming mainly at linear vibration signals, and even advanced signal processing methods (e.g., wavelet transform) may not make an accurate assessment of the rolling bearing health status.⁸

With the development of nonlinear dynamics theory, a great deal of nonlinear analysis techniques has been proposed aiming at recognizing and predicting the complex nonlinear dynamic behavior of bearings.^4,9 One of the most typical ways is to extract and refine the fault features from vibration signals over the combination of some advanced signal processing methods, for example, wavelet package transform (WPT),^8,10 Hilbert transform (HT),^10,11 empirical mode decomposition (EMD),¹¹ and higher order spectra (HOS),¹² to recognize the fault frequency and compare to the theoretical value with involvement of expert’s empirical judgment. With the development of artificial intelligence, the procedure of rolling bearing fault diagnosis is more and more treated as a procedure of pattern recognition,¹³ and its effectiveness and reliability mainly depend on the selection of dominant characteristic vector of the fault features.¹³ Recently, some entropy-based methods, for example, approximate entropy (ApEn),^14,15 sample entropy (SampEn),¹⁶ fuzzy entropy (FuzzyEn),^16,17 hierarchical entropy (HE),^13,18 and hierarchical fuzzy entropy (HFE),¹³ were developed to extract dominant characteristic vector of the fault features from the bearing vibration signals and obtained evident effect. Here, we exploit a fractal theory-based method, that is, an improved generalized fractal box-counting dimension algorithm, to extract dominant characteristic vector of the fault features from the bearing vibration signals. Fractal theory is one of the most important branches for the contemporary nonlinear science, and it is in particular suitable for processing all kinds of complex nonlinear and nonstationary phenomenon¹⁹ and thus may also suitable for fault feature extraction from bearing vibration signals.

Normally, after feature extraction, a pattern recognition technique is needed to achieve the rolling element bearing fault diagnosis automatically.¹³ Various pattern recognition methods have been developed for mechanical fault diagnosis nowadays, among which the most commonly used ones are artificial neural networks (ANNs)^20–22 and support vector machines (SVMs).²³ The training of ANNs needs a large number of samples which is hard or even impossible to achieve in practical applications, in particular the fault ones. SVMs based on statistical learning theory, which is of specialties for a smaller number of samples, have better generalization than ANNs and ensure that the local and global optimal solution are exactly the same.²⁴ However, the accuracy of a SVM classifier is decided by the selection of optimal parameters for SVMs.^24,25 In order to ensure the diagnostic accuracy, an optimization algorithm^24,25 or/and complex multi-class concept^13,26 have to be used complementally to improve the effectiveness of SVMs. In this article, in order to solve the issue of generality versus accuracy, an adaptive gray relation algorithm (AGRA), in which the concept of weight coefficient and adaptive distinguishing coefficient was introduced into the calculation of the relation degree, was developed to achieve accurate pattern recognition based on a small number of samples.

In summary, a novel approach for rolling element bearing fault diagnosis is proposed based on improved fractal box-counting dimension and AGRA. At first, the fault features from the bearing vibration signals, which offer more useful and distinguishing information imaging different bearing health status, are extracted by improved generalized fractal box-counting dimension algorithm. And then, the fault types of rolling element bearings as well as various levels of severity are recognized by the outputs of the AGRA.

The rest of this article is organized as follows. A review of fractal box-counting dimension and its improved version are introduced in sections “Traditional fractal box-counting dimension” and “Improved generalized fractal box-counting dimension,” respectively. The gray relation theory for fault classification is given briefly in section “Gray relation algorithm,” and the AGRA is illustrated in section “Adaptive gray relation algorithm,” followed by section “Proposed approach” which presents the proposed rolling bearing fault diagnosis approach. Next, experimental validation of the proposed approach is given in section “Experimental validation.” Finally, conclusions are presented in section “Conclusion.”

Fractal box-counting dimension

Fractal theory is one of the most important branches for the contemporary nonlinear science, which is in particular suitable for processing all kinds of complex nonlinear and nonstationary phenomenon, and it may be also suitable for fault feature extraction from bearing vibration signals.

Traditional fractal box-counting dimension

Suppose A is a nonempty bounded subset of Euclidean space $R^{n}$ to be calculated, and $N (A, ε)$ is the least number of boxes with the side length of $ε$ covering A, and thus, the box-counting dimension can be defined as follows

D = lim_{ε \to 0} \frac{\log N (A, ε)}{\log (1 / ε)}

(1)

For the practical vibration discrete signal, due to the existence of the sampling frequency, the sampling interval $σ$ is the highest resolution for signal. Therefore, it is no meaning to calculate the box-counting dimension when $ε \to 0$ . It is often made the minimum length of the box $ε = σ$ .

Considering a sequence of vibration discrete signal $x (i)$ as the closed set of Euclidean space $R^{n}$ , the calculation process of box-counting dimension is described as in the following.

Use approximate method to make the minimum side length of the boxes covering the sequence of vibration discrete signal $x (i)$ equal to the sampling interval $σ$ , and calculate the least number of boxes $N_{k σ}$ with side length of $k σ$ covering the sequence of vibration discrete signal $x (i)$ , thus

p_{1} = max {x_{k (i - 1) + 1}, x_{k (i - 1) + 2}, \dots, x_{k (i - 1) + k + 1}}

(2)

p_{2} = min {x_{k (i - 1) + 1}, x_{k (i - 1) + 2}, \dots, x_{k (i - 1) + k + 1}}

(3)

p (k ε) = \sum_{i = 1}^{N_{0} / k} | p_{1} - p_{2} |

(4)

where $i = 1, 2, \dots, N_{0} / k, k = 1, 2, \dots, K$ . $N_{0}$ is the number of sampling points, $K < N_{0}$ . $p (k ε)$ is the longitudinal coordinate scale of the sequence of vibration discrete signal $x (i)$ . Thus, $N_{k ε}$ can be expressed as follows

N_{k ε} = \frac{p (k ε)}{k ε + 1}

(5)

Select a good linearity of the fitting curve $\log k ε ~ \log N_{k ε}$ as the scale-free area, thus

\log N_{k ε} = d_{B} \log k ε + b

(6)

where $k_{1} \leq k \leq k_{2}$ and $k_{1}$ and $k_{2}$ are the beginning and the end of the scale-free area. respectively.

As usual, the least square method is used to calculate the slope of the current fitting curve which is the fractal box dimension D of the sequence of vibration discrete signals $x (i)$

D = - \frac{(k_{2} - k_{1} + 1) \sum (\log k) \cdot \log N_{k ε} - \sum (\log k) \cdot \sum \log N_{k ε}}{(k_{2} - k_{1} + 1) \sum \log^{2} k - {(\sum \log k)}^{2}}

(7)

Improved generalized fractal box-counting dimension

The traditional fractal box-counting dimension algorithm has been introduced in section “Traditional fractal box-counting dimension.” It has been widely used in strictly self-similar signals such as the signals in the electromagnetic fault diagnosis, image analysis, and biological medicine. However, for the common bearing vibration signals, they do not satisfy the self-similar structure of fractal theory to some degree. Therefore, when using the traditional fractal box-counting dimension algorithm to calculate box-counting dimension of the vibration signals, the curve fitting often does not have good linear structure. Aiming at this issue, here an improved generalized fractal box-counting dimension algorithm was developed to overcome the defect of the traditional fractal box-counting dimension.

First, resample the vibration discrete signals and then properly add the sampling points, to reduce the value of $ε$ in order to improve the calculating precision of the box-counting dimension of a sequence of vibration discrete signal $x (i)$ . Reconstruct the phase space of the sampling signals and determine the number of iterated dimension of the reconstructed phase space according to the sampling points at last. The specific calculation process is described in the following.

Consider the vibration discrete signal $x (i)$ which is composed of $2^{n}$ sampling points and resample the vibration discrete signal $x (i)$ in order to improve the calculating precision. Suppose the number of sampling points is $N = 2^{K} (K > n)$ , and thus the number of reconstructed dimension of phase space of the vibration discrete signal $x (i)$ is set, respectively, as $R = K + 1 = 2, 3, 4, \dots, \log 2^{K} + 1$ .

The derivative process of the number of boxes covering the sequence of vibration discrete signal $x (i)$ can be described as follows:

When $k = 1, P_{1} = max {x_{i}, x_{i + 1}}, P_{2} = min {x_{i}, x_{i + 1}}, i = 1, 2, \dots, N_{0} / k$ , the reconstructed space dimension is 2.

When $k = 2, P_{1} = max {x_{2 i - 1}, x_{2 i}, x_{2 i + 1}}, P_{2} = min {x_{2 i - 1}, x_{2 i}, x_{2 i + 1}}, i = 1, 2, \dots, N_{0} / k$ , the reconstructed space dimension is 3.

When $k = 3, P_{1} = max {x_{3 i - 2}, x_{3 i - 1}, x_{3 i}, x_{3 i + 1}}, P_{2} = min {x_{3 i - 2}, x_{3 i - 1}, x_{3 i}, x_{3 i + 1}}, i = 1, 2, \dots, N_{0} / k$ , the reconstructed space dimension is 4.

When $k = K, P_{1} = max {x_{Ki - K + 1}, x_{Ki - K + 2}, \dots, x_{Ki + 1}}, P_{2} = min {x_{Ki - K + 1}, x_{Ki - K + 2}, \dots, x_{Ki + 1}}, i = 1, 2, \dots, N_{0} / k$ , the reconstructed space dimension is $R = K + 1$ .

After reconstruct the phase space of the sequence of vibration discrete signal $x (i)$ K times, the corresponding $\log N_{k ε}$ can be obtained at each time, and then the relation curve of $\log N_{k ε} ~ \log k ε$ can be drawn. Due to that the fitting curve does not have strict linear relationship, taking the derivation of the relation curve on K points over the improved generalized discrete signal box-counting dimension algorithm. The slopes $D_{1}, D_{2}, D_{3}, \dots, D_{K}$ at different points from the relation curve are the box-counting dimensions of the different reconstructed phase space. Take the slopes $D_{1}, D_{2}, D_{3} \dots, D_{K}$ obtained as the K number of characteristic parameters in the characteristic vector of fault features extracted from the sequence of vibration discrete signals $x (i)$ , which is the prerequisite for rolling element bearing fault pattern recognition.

Gray relation theory

Normally, after feature extraction, a pattern recognition technique is needed to complete the rolling element bearing fault diagnosis automatically. The research of gray relation theory is the foundation of gray system theory, mainly based on the basic theory of space mathematics, to calculate relation coefficient and relation degree between reference characteristic vector and each comparative characteristic vector.²⁷ Gray relation theory has a good potential to be used in fault classification with four reasons:^28–31

It has good tolerance to measurement noise;

It has the ability to assist the selection of characteristic parameters for classification;

It can solve the learning problem with a small number of samples;

Its algorithm is simple and can solve the issue of generality versus accuracy.

Gray relation algorithm

The dominant characteristic vectors (i.e. the improved generalized fractal box-counting dimension) of the fault features extracted from bearing vibration signals are classified as follows

B_{1} = [\begin{matrix} D_{1, 1} \\ D_{2, 1} \\ \dots \\ D_{K, 1} \end{matrix}], B_{2} = [\begin{matrix} D_{1, 2} \\ D_{2, 2} \\ \dots \\ D_{K, 2} \end{matrix}], \dots, B_{i} = [\begin{matrix} D_{1, i} \\ D_{2, i} \\ \dots \\ D_{K, i} \end{matrix}], \dots

(8)

where $B_{i} (i = 1, 2, \dots)$ is the fault pattern to be identified; $D_{k, i} (i = 1, 2, \dots)$ is each characteristic parameter; K is the total number of characteristic parameters chosen as characteristic vector.

The knowledge base from a small number of samples between fault signatures (i.e. the characteristic vectors) and fault patterns (i.e. the fault types of rolling element bearings as well as various levels of severity) is as follows

C_{1} = [\begin{array}{l} c_{1} (1) \\ c_{1} (2) \\ \dots \\ c_{1} (K) \end{array}], C_{2} = [\begin{array}{l} c_{2} (1) \\ c_{2} (2) \\ \dots \\ c_{2} (K) \end{array}], \dots, C_{j} = [\begin{array}{l} c_{j} (1) \\ c_{j} (2) \\ \dots \\ c_{j} (K) \end{array}], \dots, C_{m} = [\begin{array}{l} c_{m} (1) \\ c_{m} (2) \\ \dots \\ c_{m} (K) \end{array}]

(9)

where $C_{j} (j = 1, 2, \dots, m)$ is known as fault pattern; $C_{j} (j = 1, 2, \dots, m)$ is each characteristic parameter; and m is the total number of fault patterns.

For $ρ \in (0, 1)$

ξ (D_{k, i}, c_{j} (k)) = \frac{\min_{j} \min_{k} | D_{k, i} - c_{j} (k) | + ρ \cdot \max_{j} \max_{k} | D_{k, i} - c_{j} (k) |}{| D_{k, i} - c_{j} (k) | + ρ \cdot \max_{j} \max_{k} | D_{k, i} - c_{j} (k) |}

(10)

ξ (B_{i}, C_{j}) = \frac{1}{K} \sum_{k = 1}^{K} ξ (D_{k, i}, c_{j} (k)), j = 1, 2, \dots

(11)

where $ρ$ is the distinguishing coefficient, and its value is usually set 0.5; $ξ (D_{k, i}, c_{j} (k))$ is the gray relation coefficient for kth characteristic parameter of $B_{i}$ and $C_{j}$ ; and $ξ (B_{i}, C_{j})$ is the gray relation degree of $B_{i}$ and $C_{j}$ .

And then $B_{i}$ is classified to the fault pattern in which the maximum $ξ (B_{i}, C_{j}) (j = 1, 2, \dots)$ is obtained.

Adaptive gray relation algorithm

In order to enhance its tolerance to measurement noise and the ability to assist the selection of characteristic values for fault classification, the information theory was introduced into the calculation of the relation degree.

First, process the distance of characteristic parameter $| Δ d_{ij} (k) | = | D_{k, i} - c_{j} (k) |$ and then calculate the probability as follows

l_{ij} (k) = \frac{| Δ d_{ij} (k) |}{\sum_{j = 1}^{M} | Δ d_{ij} (k) |}

(12)

where M is the total number of the known fault patterns in the rule base.

Define the entropy value as follows

E_{ij} (k) = - \sum_{j = 1}^{M} l_{ij} (k) \ln l_{ij} (k)

(13)

And the maximum entropy value is as follows

E_{max} = - \sum_{j = 1}^{M} l_{ij} (k) \ln l_{ij} (k) = - \sum_{i = 1}^{M} \frac{1}{M} \ln \frac{1}{M} = \ln M

(14)

Then, the relative entropy value is obtained as follows

e_{ij} (k) = \frac{E_{ij} (k)}{E_{max}}

(15)

Referred to the concept of surplus degree in information theory, the definition of surplus degree for kth characteristic parameter is as follows

H_{ij} (k) = 1 - e_{ij} (k)

(16)

The essential meaning of surplus degree is to remove the difference between the entropy value of the kth characteristic parameter and the optimal entropy value of characteristic parameter. The bigger $H_{ij} (k)$ is, the more important the kth characteristic parameter is, and then the $H_{ij} (k)$ should be granted greater weight.

Finally, calculate the weight coefficient $a_{ij} (k)$ for kth characteristic parameter as follows

a_{ij} (k) = \frac{H_{ij} (k)}{\sum_{k = 1}^{K} H_{ij} (k)}

(17)

where $\sum_{k = 1}^{K} a_{ij} (k) = 1, a_{ij} (k) \geq 0$ .

Then, obtain relation degree by weight coefficient multiplying with the corresponding relation coefficient, as follows

ξ (B_{i}, C_{j}) = \frac{1}{K} \sum_{k = 1}^{K} a_{ij} (k) \cdot ξ (D_{k, i}, c_{j} (k))

(18)

And finally $B_{i}$ is classified to the fault pattern in which the maximum $ξ (B_{i}, C_{j}) (j = 1, 2, \dots, M)$ is obtained, that is, the fault types of rolling element bearings as well as various levels of severity are recognized.

As the selection of distinguishing coefficient value $ρ$ can also have a big effect on the calculation of the relation degree and thus influence accuracy for fault classification. In order to further increase the adaptive capability for gray relation algorithm (GRA), here the concept of adaptive distinguishing coefficient is introduced.

Process distinguishing coefficient in equation (10) as follows: Denote $Δ_{ν}$ is a mean value for all the distances of characteristic parameter $| Δ d_{ij} (k) | = | D_{k, i} - c_{j} (k) |$ , that is

Δ_{ν} = \frac{1}{m \cdot K} \sum_{j = 1}^{m} \sum_{k = 1}^{K} | D_{k, i} - c_{j} (k) |

(19)

Δ_{max} = max | D_{k, i} - c_{j} (k) |

(20)

Denote $ε_{Δ} = (Δ_{ν} / Δ_{max})$ , that is

ε_{Δ} = \frac{Δ_{ν}}{Δ_{max}} = \frac{\frac{1}{m \cdot K} \sum_{j = 1}^{m} \sum_{k = 1}^{K} | D_{k, i} - c_{j} (k) |}{max | D_{k, i} - c_{j} (k) |}

(21)

Then, the distinguishing coefficient $ρ$ can be determined as follows:

if $Δ_{max} > 3 Δ_{ν}$ , then $ε_{Δ} \leq ρ < 1.5 ε_{Δ}$ ;

else if $Δ_{max} \leq 3 Δ_{ν}$ , then $1.5 ε_{Δ} \leq ρ \leq 2 ε_{Δ}$ .

And the value of distinguishing coefficient $ρ$ can be set as a mean value as follows:

if $Δ_{max} > 3 Δ_{ν}$ , then $ρ = 1.25 ε_{Δ}$ ;

else if $Δ_{max} \leq 3 Δ_{ν}$ , then $ρ = 1.75 ε_{Δ}$ .

When the adaptive distinguishing coefficient $ρ$ is obtained, a new relation degree in equation (18) can be calculated.

Proposed approach

Based on the integration of the advantages of improved generalized fractal box-counting dimension algorithm and AGRA, a novel fault diagnosis method for rolling element bearings is put forward as follows:

The bearing vibration signals under different working conditions are acquired as samples which are divided into two subsets, the samples for knowledge base and the samples for testing.

The dominant characteristic vectors of fault features from the bearing vibration signals, which can provide more useful information reflecting bearing health status, were extracted by improved generalized fractal box-counting dimension algorithm.

Establish knowledge base according to the relationship between fault signatures (i.e. the characteristic vectors) and fault patterns (i.e. the fault types of rolling element bearings as well as different levels of severity) from the base samples for AGRA model.

The feature vectors of the testing samples are input into the AGRA model, and various bearing working conditions can be recognized by the output of the model.

Experimental validation

Experimental rig

All the rolling element bearing vibration signals analyzed in the article were downloaded from the Case Western Reserve University Bearing Data Center.³² The whole experiment system is made up of a torque transducer, a two horsepower, a dynamometer and a three-phase induction motor as shown in Figure 1. The horsepower and speed data are collected over the transducer. The desired torque load levels could be achieved by controlling the dynamometer. The motor shaft at the drive end is supported by the test bearing, and single-point faults were introduced into the test bearing using electro-discharge machining, with fault diameters of 7, 14, 21, and 28 mils. Bearing faults under consideration include inner race fault, ball fault and outer race fault, and an accelerometer with a bandwidth up to 5000 Hz was mounted on the motor housing at the drive end of the motor, and then bearing vibration data under various working conditions were collected using a recorder with a sampling frequency of 12 kHz. The deep groove balling bearing 6205-2RS JEM SKF was used in the experiment.

Figure 1.

Experimental setup.

Application and analysis

The experimental vibration data used for the analyses are from the tests which were conducted at the load of 0 horsepower and at the motor speed of 1797 r/min. Normal and three fault types of vibration data as well as those with various severities for each fault type are analyzed, and the detailed description of the data is shown in Table 1. Considering various fault categories and severities, the rolling element bearing fault diagnosis turns out to be an 11-class classification problem. The data set consisted of totally 550 data samples, where each data sample is truncated into a 2048-point time series from the original vibration data and there is no overlap between any two of them. Among these 550 data samples, 110 samples are selected at random as samples for knowledge base, and the rest 440 samples are automatically treated as testing data. Here, the testing data are four times the amount of the training data, since in the practical applications, the bearing vibration data under faulty conditions are hard to achieve and are usually of small sample size.

Table 1.

Description of experimental data set.

Bearing condition	Fault size (mils)	The number of base samples	The number of testing samples	Label of classification
Normal	0	10	40	1
Inner race fault	7	10	40	2
	14	10	40	3
	21	10	40	4
	28	10	40	5
Ball fault	7	10	40	6
	14	10	40	7
	28	10	40	8
Outer race fault	7	10	40	9
	14	10	40	10
	21	10	40	11

The fault features extracted from bearing normal condition and different fault conditions with fault size of 7 mils by traditional box-counting dimension are shown in Table 2 and by improved generalized box-counting dimension are shown in Figure 2. And the fault features extracted from bearing inner race fault condition with different levels of severity by traditional box-counting dimension are shown in Table 3 and by improved generalized box-counting dimension are shown in Figure 3. Here, due to that each data sample is truncated into a 2048-point time series from the original vibration data, which conforms to the calculating rule of box-counting dimension; thus, there is no need to do a resampling again. The side length of box sequentially increases from 2⁰, 2¹, 2², until the side length reaches 2¹⁰, and calculate the total number of boxes covering each sample. Then, the fitting curves can be obtained according to the method introduced in section “Improved generalized fractal box-counting dimension,” and the improved generalized box-counting dimension can also be obtained, to be used as dominant characteristic vectors of fault features to achieve the purpose of pattern recognition.

Table 2.

Traditional box-counting dimension of a random chosen sample from bearing normal condition and different fault conditions with fault size of 7 mils.

Signals	Normal	Inner race fault	Ball fault	Out race fault
Traditional box-counting dimension	1.5718	1.6173	1.7511	1.6000

Figure 2.

Improved generalized box-counting dimension of a random chosen sample from bearing normal condition and different fault conditions with fault size of 7 mils.

Table 3.

Traditional box-counting dimension of a random chosen sample from bearing inner race fault condition with different levels of severity.

Signals	7 mils	14 mils	21 mils	28 mils
Traditional box-counting dimension	1.6173	1.5795	1.6356	1.6491

Figure 3.

Improved generalized box-counting dimension of a random chosen sample from bearing inner race fault condition with different levels of severity.

Fault features extracted from rolling bearing vibration signals have great effect on the successful diagnosis. Evident differences of fault features among different rolling bearing health status are beneficial for classification by a pattern recognition technique. From Tables 2 and 3, it can be seen that the fault features extracted by traditional box-counting dimension method are limited, and the distances between different fault types of rolling element bearings as well as different levels of severity are close, which are not easy to be classified by a pattern recognition technique. From Figures 2 and 3, it is interesting to see that the dominant characteristic vectors of fault features extracted from the rolling element bearing vibration signals with different fault types as well as different levels of severity by improved generalized fractal box-counting dimension algorithm show apparent differences.

After establishing knowledge base according to the relationship between fault signatures (i.e. the characteristic vectors) and fault patterns (i.e. the fault types of rolling element bearings as well as different levels of severity) from the base samples, the feature vectors of the testing samples are input into the GRA model, and various bearing working conditions were recognized by the output of the GRA model shown in Table 4.

Table 4.

The identification result with traditional fractal box-counting dimension and improved generalized fractal box-counting dimension by GRA.

Label of classification	The number of testing samples	The number of misclassified samples		Testing accuracy (%)
Label of classification	The number of testing samples	Traditional	Improved	Traditional	Improved
1	40	18	0	55	100
2	40	8	0	80	100
3	40	20	4	50	90
4	40	18	0	55	100
5	40	14	0	65	100
6	40	22	4	45	90
7	40	31	0	22.5	100
8	40	32	4	20	90
9	40	22	0	45	100
10	40	31	0	22.5	100
11	40	16	4	60	90
Total	440	232	16	47.2727	96.3636

GRA: gray relation algorithm.

From Table 4, the identification result shows that the recognition effect with traditional fractal box-counting dimension is poor and misleading due to the limited fault features extracted, while the recognition effect with improved generalized fractal box-counting dimension appears much better, and the total success rate can reach more than 96% and also shows that the proposed fault feature extraction algorithm can be applied to the practical rolling element bearing fault diagnosis.

The feature vectors of the testing samples are also input into the AGRA model, and various bearing health status were recognized by the output of the AGRA model as shown in Table 5. One of the merits of AGRA algorithm introduced in section “Adaptive Gray relation algorithm” is no need to design complex computational framework and set concrete parameters, which can well solve the issue of generality versus accuracy.

Table 5.

The identification result with improved generalized fractal box-counting dimension by GRA and AGRA.

Label of classification	The number of testing samples	The number of misclassified samples		Testing accuracy (%)
Label of classification	The number of testing samples	GRA	AGRA	GRA	AGRA
1	40	0	0	100	100
2	40	0	0	100	100
3	40	4	3	90	92.5
4	40	0	0	100	100
5	40	0	0	100	100
6	40	4	2	90	95
7	40	0	0	100	100
8	40	4	4	90	90
9	40	0	0	100	100
10	40	0	0	100	100
11	40	4	3	90	92.5
Total	440	16	12	96.3636	97.2727

AGRA: adaptive gray relation algorithm; GRA: gray relation algorithm.

From Table 5, the identification result shows that the recognition effect with AGRA shows certain improvement, and the total success rate can reach more than 97%. In order to illustrate the effectiveness of the proposed approach, two other most commonly used intelligent methods (e.g. back-propagation (BP) and SVM²⁴) were used as comparative algorithm in the article, as shown in Table 6.

Table 6.

The identification result with improved generalized fractal box-counting dimension by AGRA, BP, and SVM.

Label of classification	The number of testing samples	The number of misclassified samples			Testing accuracy (%)
Label of classification	The number of testing samples	BP	SVM	AGRA	BP	SVM	AGRA
1	40	0	0	0	100	100	100
2	40	0	0	0	100	100	100
3	40	40	0	3	0	100	92.5
4	40	0	0	0	100	100	100
5	40	0	0	0	100	100	100
6	40	40	0	2	0	100	95
7	40	40	0	0	0	100	100
8	40	40	7	4	0	82.5	90
9	40	40	7	0	0	82.5	100
10	40	0	0	0	100	100	100
11	40	0	13	3	100	67.5	92.5
Total	440	200	27	12	54.545	93.860	97.273

AGRA: adaptive gray relation algorithm; SVM: support vector machine; BP: back-propagation.

From Table 6, we can see all these three methods can accurately recognize the rolling bearing health status between healthy and faulty conditions. However, due to the small number of training samples, the extrapolation and interpolation performance of ANN (BP) are assessed to be poor and its recognition effect is unacceptable. It is also shown that the proposed approach in the article has superior diagnostic performance to the existing feed-forward BP neural network and SVM.

Further discussion

In order to further analyze the effectiveness of the proposed approach, an extreme experimental validation, in which only one set of random base sample for these 11 classifications was used for establishing knowledge base for AGRA model, was carried out. The feature vectors of the testing samples are input into the AGRA model, and various bearing working conditions were recognized by the output of the model shown in Table 7.

Table 7.

The identification result with improved generalized fractal box-counting dimension by GRA and AGRA.

Label of classification	The number of testing samples	The number of misclassified samples		Testing accuracy (%)
Label of classification	The number of testing samples	GRA	AGRA	GRA	AGRA
1	40	0	0	100	100
2	40	0	0	100	100
3	40	11	18	55	72.5
4	40	0	0	100	100
5	40	0	0	100	100
6	40	2	2	95	95
7	40	3	3	92.5	92.5
8	40	5	6	85	87.5
9	40	1	1	97.5	97.5
10	40	2	3	92.5	95
11	40	11	13	67.5	72.5
Total	440	35	46	89.545	92.045

AGRA: adaptive gray relation algorithm; GRA: gray relation algorithm.

From Table 7, it is encouraging to see that the proposed approach shows good robust in rolling bearing fault diagnosis and the total success rate can still reach more than 92% by AGRA. And the related comparison with other two existing intelligent methods (i.e. BP and SVM²⁴) is shown in Table 8.

Table 8.

The identification result with improved generalized fractal box-counting dimension by AGRA, BP, and SVM.

Label of classification	The number of testing samples	The number of misclassified samples			Testing accuracy (%)
Label of classification	The number of testing samples	BP	SVM	AGRA	BP	SVM	AGRA
1	40	36	0	0	10	100	100
2	40	12	0	0	70	100	100
3	40	32	28	18	20	30	72.5
4	40	8	0	0	80	100	100
5	40	16	0	0	60	100	100
6	40	40	20	2	0	50	95
7	40	40	4	3	0	90	92.5
8	40	16	0	6	60	100	87.5
9	40	16	16	1	60	60	97.5
10	40	12	24	3	70	40	95
11	40	28	16	13	30	60	72.5
Total	440	228	108	46	41.818	75.454	92.045

AGRA: adaptive gray relation algorithm; SVM: support vector machine; BP: back-propagation.

From Table 8, we can see that compared with existing fault intelligent diagnostic methods (i.e. BP and SVM) of a rolling bearing, the proposed approach can effectively solve the learning problem with an extremely small number of samples for identifying different bearing fault types as well as severities.

Conclusion

In this article, a novel approach for rolling element bearing fault diagnosis is proposed based on an improved generalized fractal box-counting dimension and AGRA. First, the fault features from the bearing vibration signals, which can provide more useful information reflecting bearing health status, were extracted by improved generalized fractal box-counting dimension algorithm. And then the fault types of rolling element bearings as well as various levels of severity were recognized by the outputs of the AGRA. The experimental results demonstrate that the proposed approach can effectively and accurately identify the different bearing fault types as well as severities. And some other meaningful conclusions can be obtained as follows:

The proposed improved generalized fractal box-counting dimension algorithm is very suitable for rolling bearing fault feature extraction, to offer more useful and distinguishing information imaging different bearing health status.

GRA as a pattern recognition technique is very suitable for achieving the rolling element bearing fault diagnosis automatically, in particular for condition of a small number of samples.

With the introduction of concept of weight coefficient and adaptive distinguishing coefficient to calculate the relation degree, the proposed AGRA appears a certain improvement.

In the future research, in order to continually improve the diagnostic accuracy for the proposed approach, some advanced signal processing methods, for example, WPT, HT, EMD, and HOS, may be explored to be integrated into the improved generalized fractal box-counting dimension to more effectively extract dominant characteristic vectors.

Footnotes

Appendix 1 Acknowledgements

The authors are grateful to the Case Western Reserve University Bearing Data Center for kindly providing the experimental data.

Academic Editor: Jianqiao Ye

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Fundamental Research Funds for the Central Universities (HEUCFZ1005) and also supported by the National Natural Science Foundation of China (no. 61603239).

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