GEAR FAULT DIAGNOSIS BASED ON ADAPTIVE TIME-FREQUENCY FEATURE EXTRACTION AND BSA-SVM METHOD

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001 Vol. 10, Issue 3, Jun 2020, 3111-3124 © TJPR Pvt. Ltd.

GEAR FAULT DIAGNOSIS BASED ON ADAPTIVE TIME-FREQUENCY FEATURE EXTRACTION AND BSA-SVM METHOD LONG HOANG1, VANTRONG THAI2 & TUANLINH NGUYEN3 1

School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam 2,3

Faculty of Mechanical Engineering, Hanoi University of Industry, Hanoi, Vietnam

ABSTRACT There are many types of approaches to the detection of gear faults. The method of a novel adaptive feature ex-traction and optimized machine learning is applied to diagnose a gear faults. Firstly, the vibration signal produced by local faults is decomposed into intrinsic scale components (ISCs) by using local characteristic-scale decomposition (LCD) method and initial feature vector matrices are obtained. Then, singular value decomposition (SVD) techniques to obtain singular values extract fault feature values, while avoiding the selection of reconstruction parameters. Secondly, the Backtracking Search Optimization Algorithm (BSA), an evolutionary algorithm, is pro-posed and demonstrated to be effective though various benchmark problems. The paper proposes an optimization method for the SVM parameters based on BSA, being so called BSA-SVM. This is a new approach method applied in diagnosing gear faults. The experimental results prove this method, the results will be more accurately and shorter time cost.

KEYWORDS: Signal Processing, Fault Detection, Gears, Artificial Neural Networks & Backtracking Search Optimization Algorithm

Received: May 15, 2020; Accepted: Jun 05, 2020; Published: Jul 11, 2020; Paper Id.: IJMPERDJUN2020295

Original Article

that the proposed method operates highly effective and mostly feasible for identifying gear faults in practice. By applying

1. INTRODUCTION Gears are widely applied in mechanical machines as the most important parts. Recently, the gear vibration monitoring for fault detection and diagnosis has been unceasingly investigated due to its useful application. Many researchers investigated the dynamic modeling of gearbox vibration aiming at quantifying the effect of different types of gear train damages on the resultant gear case vibration [1]. In the field of gear fault diagnosis, the method of vibration analysis plays as a crucial part of the diagnosis technologies [2]. The vibration analysis could be summed up in three steps: acquisition and preprocessing of a vibration signal; fault feature extraction [3]; and fault pattern recognition. If a pair of gears runs in faults, then the dynamic behaviour will appear to be non-stationary and nonlinear, and the vibration signals will reveal the same features. Accordingly, a method of signal decomposition is investigated to handle signals before extracting feature parameters. The original vibration signals are disintegrated on some components[3]. Among them there is alternative feature information of the original vibration signal with different bands. By doing so, the interference or coupling of the signal characteristic information can be reduced.

More interestingly, the effective separations of fault features of the signal can also reflect the essence of fault information better.

Fault diagnosis of rotating machinery mainly includes two key steps. One key step is feature extraction.

Fault features could be generally extracted by applying a suitable vibration signal processing approach[4]. Recently,

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the singular value decomposition (SVD) technique has been applied to fault diagnosis of rotating machinery [5]. However, the SVD technique would suffer a complex selection of reconstruction parameters. The empirical mode decomposition (EMD) method [6] is an excellent time-frequency analysis approach to deal with non-stationary vibration signals produced by machinery fault diagnosis. However, the EMD method still has some drawbacks, such as end effect and mode mixing [7]. We have developed a new time-frequency analysis approach, namely, local characteristic-scale decomposition (LCD) method [5].

Similar to EMD method, LCD method can decompose a complex signal into several intrinsic scale components (ISCs). It is most important point that LCD is superior to EMD in running time, restraining the end effect and relieving mode mixing [8]. Therefore, LCD method is used as vibration signal processing approach in this paper. Since each ISC contains special time-scale information of original signal, the ISCs of vibration signal, which reflect nature characteristics of original signal, can be employed to construct initial fault feature matrix for SVD technique avoiding difficulty of choosing reconstruction parameters. Hence, SVD technique combining with LCD method, namely, LCDSVD, is introduced to extract the fault feature of roller bearing in this paper.

The content is structured as follows: Section 2 presents the LCD and LCD-SVD energy entropy methods. Section 3 addresses the algorithm of optimizing parameters relied on the BSA. Subsequently, the method of diagnosing faults relied on the BSA-SVM is presented in Section 4. The main content of the paper, Section 5, will bothpresent the optimization algorithm related to BSA-SVM and LCD-SVD and demonstrate its strength empirically. Finally, the conclusion is highlighted in Section 6.

2. SINGULAR VALUE DECOMPOSITION TECHNIQUE BASED ON LCD METHOD For the nonlinear time series of 𝑁 data points {(𝑖), 𝑖 =1, 2, . . .,}, a trajectory matrix X𝐿×𝐾 can be formed based on phase space reconstruction theory. Let window length 𝐿 be an integer and 𝐾 the number of lagged vectors X𝑖= (𝑥𝑖, . . .,𝑖+𝐾−1), where 1<𝐿<𝑁, 𝐾=𝑁−𝐿 + 1, 1≤ 𝑖≤ 𝑙.

The trajectory matrix X𝐿×𝐾 can be expressed as

𝑋𝐿×𝐾

𝑥1 𝑥2 𝑥2 𝑥3 =[⋮ ⋮ 𝑥𝐿 𝑥𝐿+1

⋯ 𝑥𝑘 ⋯ 𝑥𝑘+1 … ⋮ ] … 𝑥𝑁

(1)

Based on singular value decomposition (SVD) theory, a trajectory matrix 𝑋𝐿×𝐾 can be decomposed into a series of mutually orthogonal, unit-rank, and elementary matrices by using SVD, that is, 𝑋𝐿×𝐾 = 𝑈𝑆𝑉 𝑇 where U∈𝑅𝐿×𝐿and V∈𝑅𝐾×𝐾 and S is a 𝐿 × 𝐾diagonal matrix; let 𝜆1, 𝜆2, 𝜆3, . . ., 𝜆𝑛be nonzero diagonal elements arranged in decreasing order. They are called singular values of matrix 𝑋𝐿×𝐾, namely, singular spectrum, where 𝑛 =rand(𝑋𝐿×𝐾), 𝜆1 ≥ 𝜆2 ≥ 𝜆3 ≥ ⋅⋅⋅ ≥ 𝜆𝑛> 0. The number 𝑛 of nonzero diagonal elements reflects the complexity of frequency components in the nonlinear time series[9]. When 𝑛 is bigger, the frequency components are more complex. Therefore, the singular spectrum of matrix 𝑋𝐿×𝐾 describes the characteristics of the nonlinear time series correspondingly. However, reconstruction parameters, such as lag time and embedding dimension, would have effect on the result of SVD method.

However, there is no mature theory for selecting the reconstruction parameters [10]. Targeting the nonlinear and Impact Factor (JCC): 8.8746

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nonstationary characteristics of vibration signals from bearing exhibiting local faults and the limitations of SVD techniques, LCD-SVD technique is presented to extract the fault feature vector in this investigation. Firstly, the nonlinear and nonstationary vibration signal is decomposed into a number of ISCs. Then, after selecting several ISCs to form a trajectory matrix, SVD is applied to obtain singular values as fault feature vector. Therefore, LCD-SVD technique should be able to extract the nonlinear and nonstationary characteristics of vibration. At the same time, this kind of method could easily avoid the difficulty of determining reconstruction parameters of SVD. Moreover, in this signal processing, because only several special ISCs are selected after analysis, the noise would be eliminated and the LCD-SVD method could improve the accuracy of feature extraction.

Figure 1: Time Domain Wave form of ISCs of Vibration Signal of Gear Fault.

3. BACKTRACKING SEARCH OPTIMIZATION ALGORITHM The main content of BSA method is to generate a trial population including two new crossover and mutation operators.

The BSA strategy is considered as powerful exploration because this method both generates trial populations and controls the amplitude of the search-direction matrix and search-space boundaries[11]. Particularly, the BSA stores a population from a randomly chosen previous generation for use in generating the search-direction matrix. The BSA strategy is very strong in terms of both a global exploration and local exploitation with a good feature of local minima avoidance[12]. The main five procedures of the BSA method are presented as follows in figure 2: Problem and Algorithm Parameter:An overview of five processes is provided as follows:

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Figure 2:Flow Chart for Backtracking Search Optimization Algorithm[12].

3.1. Initialization The procedures of BSA begin from initializing the population P as follows: Pi,j = lowerj + (upperj − lowerj ) × random, i = 1,2, … , NP

(2)

j = (1,2, … , DP) In which: NP is the size of population (PopSize). DP is the dimensional number of the problem. random is a real value which is uniformly allocated between 0 and 1. lowerj stands forthe lower bound for the jth factor of the ith individual.upperjis the upper bound for the jth factor of the ith individual.

3.2. Selection-I The generation of old P is the historical population, in the procedure of the Selection-I, and is used to make a calculation for the search direction. The formulation is as follows: Pi,j = lowerj + (upperj − lowerj ) × random, i = 1,2, … , NP

(3)

j = (1,2, … , DP) In each iteration, oldPis defined as follows: 𝑖𝑓𝑎 < 𝑏 𝑡ℎ𝑒𝑛 𝑜𝑙𝑑𝑃 ≔ 𝑃|𝑎, 𝑏 ∈ [0,1]

(4)

In which:= means the update operation. Two random numbers, a andb, in the range [0, 1]. The above equation assures that the BSA algorithm’s population might be randomly chosen from a historical population. The algorithm memorizes the historical population till it is changed via a random permutation.

3.3. Mutation Initially, the trial population is created via mutation operation as per the bellow formulation: T = P + (oldP- P) × F Impact Factor (JCC): 8.8746

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whereF is a scale factor which controls the amplitude of mutation (oldP- P).

In this paper, F = 3 • random,where random is a random real number with uniform distribution in the range (0,1).

By involving the historical population in calculating oldP -P, BSA learns from its memory of previous generations to obtain a trial population. [12] 3.4. Crossover The final trial population T is generated by crossover. The guidance of the trial individuals characterized by improved fitness values supports the search direction in optimizing the problem. BSA’s crossover works as the following procedure.

A binary integer-valued matrix (map) of size NP× DPis computed in the first step.

The individuals of T are generated by utilizing relevant individuals of P. If mapi,j= 1, T is updated with Ti,j:=Pi,jPi,j.

3.5. Selection-II In the stage of Selection-II, Ti makes the corresponding Pi better in the manner of fitness value which is utilized for updating Pi[12]. As the BSA finds the best optimal value (Pbest) which is dominant comparing to all previous ones, Pbest takes the place of all optimal solutions, and these values are updated as the most fitting value of Pbest.

4. SVM PARAMETER OPTIMIZATION BASED ON BSA 4.1. Support Vector Machine (SVM) The SVM is a type of machine learning technique. The SVM relies on the theory of statistical learning. The SVM is handling the training samples as the input to a higher-dimensional characteristic space through the use of a mapping function  7 . Assuming that there is a given set of the training samples G  xi , yi , i  1,2,..., l in which each sample

xi  R d belongs to a class by y   1,1 , and the training data is not linearly separable in the space of feature, then the target function can be expressed as follows [13]: Minimize  ( ) 

1 .  C 2

l



i

i 1

subject to yi ( . ( xi )  b)  1  i ,  i  0 i={1,2,…,l} in which

(6) is the normal vector of the hyperplane, C is thepenalty parameter, b is the bias, ξi is non-negative slack

variables, and ϕ(x) is the mapping function.

By introducing a set of Lagrange multipliers αi ≥ 0, the optimization problem could be rewritten as: Maximize

L( , b, ) 

l

 i 1

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i 

l



1  i j yi y j K ( xi x j ) 2 i , j 1

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subject to 0   i  C ,

 y i

i

0

i 1

The function of making decision can be achieved as:

 l  f ( x)  sgn  i yi K ( xi x)  b  i 1 



(8)

The SVM method used the radial basis kernel function which is the most common kernel function as it indicates in the bellow equation: 2

K ( xxi )  exp(  x  xi / 2 2

(9)

Where σ is the kernel parameter.

4.2. SVM’s Parameter Optimization Relied on BSA The performance of SVM is significantly impacted by its parameters. It needs to be selected as a penalty factor C and kernel parameter σin the function of Gaussian kernel. It is not easy to select these parameters. As a whole, C and σ are chosen empirically. Therefore, this paper applied the BSA as a technique for optimizing the parameters of SVM. As a consequence, C and σplayed as the variables[14]. SVM’s error testing became the fitness function for the sake of optimization. The SVM’s error testing is given as: fitness(x) = Test ErrorSVM(x);

(10)

wherex = (C; σ) and the test error of SVM was defined as: 𝑦

𝑓 𝑇𝑒𝑠𝑡_𝐸𝑟𝑟𝑜𝑟𝑆𝑉𝑀 = (1 − 𝑦 +𝑦 ) 100% 𝑡

(11)

𝑓

Where 𝑦𝑡and 𝑦𝑓 write down the number of true and false classified samples, respectively. The desirable value is small to classification accuracy is high.Fig. 3 shows the flow chart of BSA-SVM

Figure 3: Parameter Optimization Flowchart of SVM based on BSA.

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4.3. Experimental Results The paper used Thyroid, Seed and Escherichia coli(E. coli) as three common sets of benchmark data provided by the University of California, Irvine to quantify the performance of the suggested BSA-SVMmethod.

Table 1 showed the training and test sets. Every sampling set is separated into two subsets. In which, one subset is used for SVM training, and the other is for the testing of the obtained model. The training set accounted for 70%, and the test set took 30% of the total samples. Based on the trial and error detection practice, this proportion was selected in order to evaluate the performance of the achieved SVM which was optimal for the available samples.

Three data sets were used to make classification among the BSA-SVM, GA-SVM, PSO-SVM, and CMAES-SVM methods. The values selection is the same in all four above methods in order to make comparison fairly (e.g., iteration = 30 andPopSize = 30). For the PSO, the parameters were fixed withthe values given in the literature[15] (i.e., W = 0:9, c1 = 0:5,and c2 = 1:25). For CMAES, the parameters were fixedwith the values given in the literature[16] (i.e., σ0= 0:25 andµ = 4 + 3 log(N)). Each method’s testing result was considered as the average value of 30 runs.

The training data andtest data were both mixed and randomly divided, as shown in table 1.

Lin et al. argued that [17] the lower and upper bounds of Cwere given in [0:01; 35000] and σ in [0:01; 32] for the BSA-SVM, GA-SVM, PSO-SVM, and CMAES-SVM classifiers.Each search method gave the values of C and σ in order togive the smallest value of the classification error. Table 2 presented experimental results.

The effectiveness of the proposed method was illustrated in Tables from 2 to 4 on the base of the detailed classification results of each data set. The Thyroid and Seeddata sets encompassed three classes, thus, the authors needed two SVM classifiers. The E. Colidata set embraced five classes; therefore the authors neededfour classifiers.These tables show the optimal parameters (C and σ), average test error, and average cost timedone by different algorithms.

Table 1: Properties of Problems Name

Data Train Test Validation Data Input Class

Thyroid

225

150

52

150

5

3

Seed

220

145

55

160

7

3

E. coli

320

225

60

180

7

5

The results of the classification of the E.Coli, Seed and Thyroid were given in table1. The next Table indicates that the cost timing and testing error of BSA-SVM was a bit lower in comparison with those of other methods as GASVM, PSO-SVM, and CMAES-SVM. Civicioglu said that[18] the BSA utilized a mutation mechanism in which there was acomplex crossover and one individual. Moreover, by using its memory, the BSA took an advantage of the experiences achieved from previousgenerations.Tables 2-4 demonstrate that the BSA-SVM classifier achieved higher classification accuracy in the way of a shorter time as comparing to othermethods. Next, the BSA-SVM method was used to diagnose a gear fault.

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Table 2: Results of Thyroid Data Set Training Samples

Test Samples

Average Cost Time (s)

BSA-SVM

150

52

15.83

0.000

[19]

PSO-SVM

150

52

36.31

2.358

[20]

GA-SVM

150

52

31.12

1.442

[21]

CMAES-SVM

150

52

32.56

2.622

[22]

Method

Average Test Refs.

Errors (%)

Table 3: Results of Seed Data Set Training Samples

Test Samples

Average Cost Time (s)

Average Test Errors (%)

Refs.

BSA-SVM

160

55

17.65

0.910

[19]

PSO-SVM

160

55

40.69

3.993

[20]

GA-SVM

160

55

36.38

3.552

[21]

CMAESSVM

160

55

37.56

2.298

[22]

Method

Table 4: Results of E. Coli Data Set Training Samples

Test Samples

Average Cost Time (s)

Average Test Errors (%)

Refs.

BSA-SVM

180

60

26.73

4.641

[19]

PSO-SVM

180

60

59.43

5.364

[20]

GA-SVM

180

60

44.78

7.787

[21]

CMAESSVM

180

60

42.52

8.582

[22]

Method

5. GEAR FAULT DIAGNOSIS BASED LCD-SVD AND BSA-SVM The gear vibration signal is analyzed by the LCDSVD energy entropies with different work statuses and obviously different fault patterns which make the energy of every ISC changed when the gear works incorrectly. The BSA-SVM input vector has energy features of every ISC component. By doing so, the work status and the gear fault patterns can be identified. figure 4 shows the flow chart of LCDSVD and BSA-SVM method for the gear fault diagnosis. The fault diagnosing method is described as follows: 

Choose several signals as samples in three cases: gear normal, gear chipped and gear broken conditions.



Decompose the original vibration signals by the LCDSVD method into several ISCs. The first m ISCs including the most dominant information of the fault will be selected to extract the characteristics.



The total energy Ejof the first m ISCs was calculated by:

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

Ej

 C (t ) dt , (j=1, 2,...,m)

(12)

j



(4) Build up the feature vector T with the energy levelsas the following element:

T  [ E1 , E2 ,..., Em ]

(13)

Sometimes the energy is the biggest, thenT isadjusted by the normalization of the feature advantage of the subsequent analysis and processing. Herethe paper lets

 E   

2

m

 j 1

Ej

   

1

2

(14)

Then T’=[E1/E, E2/E, Em/E] and T is a normalized vector

Figure 4: Flowchart of Gear Fault Diagnosis Relied on LCDSVD and BSA-SVM. 

Run the BSA-SVM, with the parameters of SVM, named as N, which are optimized by the BSA.



Input the vector T into the BSA-SVM classifier and initiate training. Equation (11) gives the fitness function. The net results of N are inputted to the BSA-SVM classifier.



The output of BSA-SVM is defined with the work conditions of the gear of three classes: class 1-normal [-1 -1 +1]; class 2-chipped tooth [-1 +1 -1]; class 3-broken tooth [+1 -1 -1].

Data collected in this paper comes from the public datasets distributed by the Prognostics and Health Management

(PHM) Society[23], and the data sets were sampled synchronously from two accelerometers mounted on each side of the gearbox housing shown in figure 5. Data collection was implemented at five different shaft speeds of 30, 35, 40, 45 and50

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Hz under low and high loads from the brake with the sample frequency fs = 200/3 kHz. The low and high load shave resulted in a small difference in shaft speed; each load case is repeated four times. There are total 560 different cases to be diagnosed with six cases using helical gears and the other eight spur gears. On the input shaft, the tachometer generates10 pulses per revolution, and the data from the tachometer is very accurate. Here we chose a helical gear to research with three states: normal, chipped tooth and broken tooth, and the sampling frequency can be taken as 1024Hz.The vibration signals of helical gear with three conditions (normal, chipped tooth and broken tooth) were tested, and 40 vibration signals from the helical gear in each condition were taken from 6 groups collected at random as the test data.

Firstly, the vibration signals of each group with three conditions, namely normal, chipped tooth and broken tooth, were decomposed by the LCDSVD into a number of ISCs. The first eight ISCs embracing the most dominant information which is selected and arranged in accordance with the frequency components as c1(t), c2(t),…,c8(t)from high to low. Then, the fault features of vector T were obtained following to Eqs. (3), (6) and (24). Finally, the BSA-SVM was used to identify the various patterns. In order to have a fair comparison, the original vibration signals are chosen similarly. These signals are decomposed by the LCD into ISCs. Of which, the first eight ISCs are selected and arranged in the manner of high to low values based on frequency components as c1(t), c2(t),…,c8(t). The T vector including fault feature can be achieved according to Eqs. (6) and (24).And then, the feature vector is taken into the input of BSA-SVM. The identification results for the test samples based on the LCDSVD and compared with the LCD method are shown in table 5.

Figure 5: Common Test Rig. The input vector of the BSA-SVM method is used with the LCDSVD or LCD analysis as preprocessor for the energy extraction in each frequency band in identifying the fault gears. These input vectors affect the identification of the fault gears. The frequency components after decomposition by LCDSVD had better results than the LCD method, therefore, the method BSA-SVM based on the LCDSVD is better than that based on the LCD. The experimental results were demonstrated in table 5. As a result, the accuracy of identification is good, and the time is shorter. In practical cases, they may be used for a fault diagnosis.

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Table 5: Results of Diagnosing Fault Gear Relied on LCD or LCDSVD Signal Method

E6

E7

E8

Test Error

E1

E2

E3

E4

E5

LCD

0.5235

0.4849

0.5299

0.4010

0.1811

0.1152 0.0369 0.0128

Normal (2.18%)

LCDS VD

0.6528

0.5133

0.3636

0.2725

0.2441

0.0749 0.0327 0.0079

Normal (0%)

LCD

0.6946

0.4619

0.4561

0.3967

0.1846

0.0946 0.0856 0.0057

Chipped tooth (2.18%)

LCDS VD

0.7357

0.5721

0.4812

0.3519

0.2183

0.0612 0.0319 0.0158

Chipped tooth (0%)

LCD

0.6552

0.4925

0.4521

0.3876

0.1732

0.0629 0.0773 0.0036

Broken tooth (2.18%)

LCDS VD

0.6832

0.4719

0.4128

0.4101

0.1396

0.0652 0.0376 0.0051

Broken tooth (0%)

S1

S2

S3

6. CONCLUSIONS Obviously, the fact of fault diagnosis is required in order to maintain and ensure the working condition of gears in a good state. This paper suggests a method of optimizing SVM parameters relied on BSA. A new approach method, a gear fault diagnosis relied on LCDSVD and BSA-SVM, is presented and demonstrated. All the results are shown above in the main content section. These experimental results proved the effectiveness of BSA-SVM. More concretely, the testing results of some real-world benchmark data sets showed that the BSA-SVM classifier has a high accuracy with low computational cost time in comparison with some other methods as the GA-SVM, PSO-SVM and CMAES-SVM. Moreover, the experimental results proved the test errors and cost timing will be considerably reduced as combining the BSA-SVM classifier and the EEMD method in diagnosing the gear faults.

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Gear Fault Diagnosis Based on Adaptive Time-Frequency Feature Extraction and Bsa-Svm Method

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25. Yiming Huang, Di Wu, Zhifen Zhang, Huabin Chen, and Shanben Chen, EMD-based pulsed TIG welding process porosity defect detection and defect diagnosis using GA-SVM. Journal of Materials Processing Technology, 2017. 239: p. 92-102.

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AUTHROR’S PROFILE

Long Hoang, Ph.D. in the School of Mechanical Engineering, Hanoi University of Science and Technology, Vietnam, His research interests are CAD/CAM/CAE, reverse engineering Computation Geometry, cutting tool profiling by enwrapping, gear machining.

VanTrong Thai, Ph.D. in the Department of Mechanical Engineering, Hanoi University of Industry, Vietnam. His research interest are fault diagnosis, mechanical engineering, cutting tool design, PVD-CVD.

Nguyen Tuan Linh, Ph.D. in the Department of Mechanical Engineering, Hanoi University of Industry, Vietnam. His research interest are mechanical engineering, manufacturing technology, machine dynamics, and modeling and simulation.

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