Vibration signal-based bearing fault diagnosis using optimized multi-scale entropy and ANFIS network

Vibration signal-based bearing fault diagnosis using optimized multi-scale entropy and ANFIS network

N. Fergani, N. Boutasseta, B. Oudjani, A. Deliou, I. Attoui Unité de Recherche en Technologie Industrielle (URTI/CSC)

Welding and NDT Research Centre B.P.64, Cheraga, Algeria.

[email protected]

Abstract—This paper presents an application of a multi-scale classification method to detect bearing-related faults in an experimental benchmark. Multi-scale analysis of the vibration signal allows the representation of nonlinear dynamics and coupling effects between different mechanical components of industrial equipment. An improved multi-scale entropy analysis is used as features extraction tool for the diagnosis procedure.

The classification of the state of health of the bearings is achieved using adaptive neuro-fuzzy inference system and neural networks for different faults scenarios with variable fault severity.

Experimental results show the importance of the choice of the features extraction method for the classification of faults and the determination of their severity.

Index Terms—Fault diagnosis, vibration analysis, multi-scale entropy (MSE), bearing faults.

I. INTRODUCTION

Conditional maintenance plays an important role in modern industry. Early detection of incipient faults allows the prevention of unnecessary shutdowns and unscheduled maintenance interventions, leading to a reliable, cost-effective operation of industrial equipment. Vibration signal analysis is an important tool for condition monitoring and fault diagnosis, where the raw vibration data issued from the dynamics of different mechanical components is used as a main source for feature extraction algorithms. The classification of the state of operation is then realized after a training of the classifier using the history of the monitored equipment.

Many feature extraction methods have been proposed to find the intrinsic characteristics hidden in the raw data that allow optimal decision on the state of operation of the monitored equipment. In [1], neural network based method was used as successful vibration-based damage detection procedure in rotating machinery. An intelligent method based on empirical mode decomposition (EMD), dimensionless parameters, fault decision table (FDT), MLEM2 rule induction algorithm and improved rule matching strategy (IRMS) was proposed in [2], the authors stated that these combined techniques work well together in fault diagnosis of rotating machinery. In [3], a multi-scale entropy based features

extraction method was combined with adaptive neuro-fuzzy inference to detect bearing faults in an experimental test bed. In this work, we propose a fault detection and diagnosis procedure based on an improved version of the method presented in [3]

by using temporal indicators for fault detection and optimized multi-scale entropy complexity measure for extracting features for the ANFIS-based classifier.

The paper is organized as follows: In the following section, we present the multi-scale entropy complexity measure.

Section III gives a description of the proposed fault diagnosis procedure. A description of the experimental test stand is presented in section VI. Results and discussions are given in section V. Conclusions are given at the end.

II. MULTI-SCALE ENTROPY A. Sample Entropy (SampEn)

The two main entropy measure statistics are: Approximate Entropy (ApEn) proposed by Pincus in [4], and Sample Entropy (SampEn) proposed by Richman and Moorman in [5].

SampEn is an improved version of ApEn, it solves the problems of bias caused by self-matching and data length restriction. SampEn gives a likelihood measure of the similarity between m consecutive data samples within a given tolerance threshold r. It is shown in [6] that the values of r and m have a direct effect on the complexity measure of the considered signal, and that the parameters should be optimized to give maximum entropy value.

B. Multi-Scale SampEn

Given the multi-scale nature of time series signals, the SampEn calculated at the first scale may not represent the true complexity measure signal. It has been shown in [6] that the entropy at the first scale for a white noise is higher than that of a 1/f noise, which is a misleading conclusion that indicates that the white noise is more complex than the 1/f noise. The calculation of entropy at various scales allow better representation of the complexity of a given time series [3].

The multi-scale SampEn is measured on the coarse-grained time series yτ of the original data samples {x₁, …, x_N} in order to extract the intrinsic vibration modes of the equipment. The

parameter τ represents the scale factor, it gives the level of decomposition of the original time series x_i as follows:

( )

∑

^τ

+

− τ

= τ =τ ^j

1 1 j i

i

j 1 x

y (1)

Where 1 ≤ j ≤ N/τ and 1 ≤ i ≤ N. SampEn is then calculated for each scale τ using the following steps for a time series with N points x(i), where 1 ≤ i ≤ N [5] :

1) We construct N-m+1 vectors e_m(i); such that : 1 ≤ i ≤ N- m+1, where e_m(i)={x(i+k): 1 ≤ k ≤ m-1} is composed of m data points; ^ei^m =

{

^x

( ) ( )

^i,xi+^1,...,x

(

ⁱ+^m−¹

) }

^.

2) The distance between two vectors:

d[e(i), e(j)]=max{|x(i+k)-x(j+k)|: 1≤k≤m-1}.

3) Define ^Bmi

( )

^r as (N-m-1)^-1 times the number of vectors e ^m_j within r of e . Where 1≤j≤N-m and j≠i to exclude self-^m_i matches.

4) Define:

∑

⁻

=

− −

= ^{N m}

1 i

m i 1

m(r) (N m) B (r)

B .

5) Define: A^m_i (r) as

(

^N−^m−¹

)

⁻¹ times the number of vectors e^m_j⁺¹ within r of e^m_i ⁺¹ with: 1≤j≤N-m (j≠i):

∑

⁻

=

− −

= ^{N m}

i m i

m r N m A r

A

1

1 ( )

) ( ) (

6) Estimate the Sample Entropy:

( ) ( )

( )

^







− 

= B r

r ln A N , r , m

SampEn _m

m

.

III. THE PROPOSED FAULT DETECTION AND CLASSIFICATION METHOD

The vibration signal issued from accelerometers is pre- processed an then passed to the fault detection procedure that checks for the presence of abrupt changes.

A. Fault detection

In this work, many temporal indicators have been tested to detect faults in the studied test stand. The RMS-based temporal indicator is found to be the most convenient approach to detect faults. In the case of N samples, the RMS indicator is given as follows:

(

²N

)

2 1 ... x N x

RMS= 1 + + (2)

A fault is detected if the value of the RMS indicator bypasses a pre-fixed threshold. The classification procedure is then executed to identify the occurred fault (Fig. 1).

B. Fault classification using ANFIS

The Adaptive Network Fuzzy Inference System (ANFIS) classification algorithm is based on the training of fuzzy if-then rules to model the input/output data [7]. In the case of vibration-based fault classification developed in this work, the state of the equipment is modeled using features extracted from the vibration signals (f1, f2,…). Figure 2 represents the architecture of the adaptive fuzzy inference system where f1 and f2 are the calculated features. A1, A2, B1, B2 are fuzzy sets. The nodes M, N, S are multiplication, normalization, and summation nodes respectively. The output function F is given as follows:

( ) (

^{2 1 2 2 2}

)

2 1 1 2 2 1 1 2 1 1

1 p f q f r

w w r w f q f w p w

F w + +

+ + + + +

= (3)

Where w1 and w2 are firing strengths, and W1 and W2 are the normalized versions or the firing strengths. The output function is then a linear combination of the design parameters p1, q1, p2, q2 that are determined during the training process. The output function can be rewritten as follows:

(

^{1 1 1 2 1}

)

²

(

^{2 1 2 2 2}

)

1p f q f r W p f q f r

W

F= + + + + + (4)

When the training process using a combination of a least- squares method and the back-propagation gradient descent method is finished, the resulting trained model will be used to classify faults.

IV. DESCRIPTION OF THE EXPERIMENTAL SETUP In order to validate the classification approach, we used the vibration data provided by Case Western Reserve University (CWRU) [8] as original signals for the training and testing procedures [6]. The vibration signal was collected using accelerometers on the experimental test stand shown in Figure 3 at a sampling frequency of 12 kHz. It is composed of a 2hp motor, a torque transducer, a dynamometer and control circuitry. The test bearings support the motor shaft. In this work we consider faults on the bearings’ ball, inner race and outer race with variable fault diameter and motor speed.

Fig 1. Proposed fault detection and classification method.

Figure 2. Architecture of the Adaptive Network Fuzzy Inference System (ANFIS).

V. RESULTS AND DISCUSSION A. Fault detection

The fault detection procedure was tested by simulating the occurrence of different types of faults on the test stand. The simulation was realized by the introduction on the vibration signature of the fault to the normal vibration signal. The results are given in Fig. 5. A fixed threshold allows the detection of an abrupt change in the value of the indicator which represents the occurrence of faults.

B. Fault classification

The fault classification procedure relies on the indicators used to train the classification network. In our work we use the multi-scale entropy measure for the extraction of efficient features. This type of complexity measure is characterized by two parameters: m and r.

In order to adapt the values of the parameters for the case of the vibration signal, we have simulated the SampEn entropy measure for different values of r and m. The result of the optimization process is given in Fig.6. The parameters r and m are bounded as follows: 0 ≤ r ≤ 1 and 2 ≤ m ≤ 7, according to the suggestions given in [3]. The optimal values of r and m to be used in the following experiments are 0.05 and 2 respectively.

Fig. 3. Experimental test stand [8].

0 2000 4000 6000 8000

-0.5 0 0.5

Healthy

0 2000 4000 6000 8000

-1 0 1

Ball

0 2000 4000 6000 8000

-2 0 2

Inner race

0 2000 4000 6000 8000

-5 0 5

Outer race

Fig. 4. Vibration signal under different bearings status.

0 0.5 1

-1 0 1

Ball fault

0 0.5 1

0 0.1 0.2

0 0.5 1

-2 0 2

Inner race fault

0 0.5 1

0 0.2 0.4

0 0.5 1

-5 0 5

Outer race fault

0 0.5 1

0 0.5 1

Fig. 5. Fault detection using the RMS indicator.

0 0.2

0.4 0.6

0.8 1 2

3 4

5 6

7 0

0.5 1 1.5 2 2.5

r m

SampEn

Fig. 5. Optimization of the parameters of SampEn (r , m).

The vibration data used in the training and testing of the classification network is given in Table 1. The dataset contains 520 samples divided into 120 samples as training samples and

400 samples for testing. The studied faults where classified into 10 classes depending on the fault and its severity.

Figure 7 presents the multi-scale SampEn calculated for different conditions of the bearing at different scales. The features are then extracted using the following measures:

Max: Maximum of the multi-scale SampEn.

Min : Minimum of the multi-scale SampEn.

Mean: Mean of the multi-scale SampEn.

Geo-Mean: Geometric mean of the multi-scale SampEn.

STD: Standard deviation of the multi-scale SampEn.

Sample values for these features for different states of the bearings are given in Fig.7. The extracted features are then used for the training process of the ANFIS network.

TABLE I. DESCRIPTION OF THE VIBRATION DATASET

Condition of the Bearings Fault diameter (inch) Number of training samples Number of testing samples Class

Healthy 0 12 40 1

Ball 0.007 12 40 2

0.028 12 40 3

Inner race 0.007 12 40 4

0.014 12 40 5

0.021 12 40 6

0.028 12 40 7

Outer race 0.007 12 40 8

0.014 12 40 9

0.021 12 40 10

120 400

520

0 2 4 6 8 10 12 14 16 18 20

0 0.5 1 1.5 2 2.5 3

τ

SampEn

Normal Outer_7 Outer_14 Outer_21 Ball_7 Ball_28 Inner_7 Inner_14 Inner_21 Inner_28

Fig. 6. Multiscale entropy for 20 scales of the original signal.

Max0 Min Mean Geo-Mean STD

0.5 1 1.5 2 2.5

Normal Outer_7 Outer_14 Outer_21 Ball_7 Ball_28 Inner_7 Inner_14 Inner_21 Inner_28

Fig. 7. Selected features for the training process.

Fig. 8. Decision surface for two inputs.

0 10 20 30 40 50 60 70 80 90 100

0.16 0.17 0.18 0.19 0.2 0.21 0.22

Performance evaluation

Epochs

RMSE value

Fig. 9. Performance of the ANFIS training.

In the training process, the extracted features are used to adapt the ANFIS network for each state of the rotating machinery. Figure 8 illustrates the decision surface in the case of two features, the performance of ANFIS during the training is given in Fig. 9. A classification accuracy of 93.33% was effectively realized using the proposed approach.

VI. CONCLUSION

In this work we have applied a fault diagnosis method based on using a complexity measure to detect and classify bearings related faults in an experimental test bed.

In the proposed supervision algorithm, fault detection is realized using temporal indicators which allow real-time monitoring of the equipment. The classification procedure is then executed to determine the state at which the system is operating using features extracted from the vibration signal.

The complexity measure using multi-scale SampEn along 20 scales was used to extract features that allow better representation of the different interactions between different mechanical parts. Its optimization for the case of the vibration signals used in this work allowed better evaluation of the state

at which the system is operating. The final step of identifying the type of fault was realized effectively using an ANFIS based approach.

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