Fault diagnosis for rotating machinery based on Local Mean Decomposition morphology filtering and Least Square Support Vector Machine

Abstract

In order to improve the accuracy of fault diagnosis, a fault diagnosis method for rotating machinery based on Local Mean Decomposition (LMD) morphological filtering and Least Square Support Vector Machine(LS-SVM) is proposed. Firstly, the collected vibration signals are processed by LMD which decomposes them into a series of product function (PF) components, and PF components are carried out by method of the morphological filtering and signal recombination. Then, the feature vector is obtained from the new PF components by LMD method. Secondly, a new kernel function is proposed to improve LS-SVM, which can solve the problem of kernel function parameters selection. Further, Lagrange parameters are processed by feature weighting method, and the weighted average value of the Lagrange parameters is calculated as the threshold value of Pruning algorithm, which can solve sparsity problem in LS-SVM. Thirdly, Energy features are inputted into the new LS-SVM to recognize machinery faults according to LS-SVM parameters. Finally, the performance of the proposed method was verified by a fault diagnosis case in a rolling element bearing. The results indicated that this new method could judge and classify the multi-fault of rotating machinery quickly and effectively.

Keywords

Fault diagnosis for rotating machinery Local Mean Decomposition morphological filtering Least Square Support Vector Machine

1 Nomenclature

Variables

Explanation

a function of structural risk minimization

x _i

inputted quantity

y _i

target value

ξ _i

error variable

φ (•)

kernel function

weighted vector

adjustable parameter

deviation quantity

number of inputted value

α_i

Lagrange multiplier

the number of data types to be classified

the number of feature vectors of this type

1 Introduction

Rotating machinery covers a broad range of industrial applications and plays an important role in many crucial areas. So the failure of rotating machinery can result in damage of mechanical structure and huge economic loss. As a result, developing effective fault diagnosis techniques for rotating machinery to avoid accidents and increase machine reliability is very important. Substantially, fault diagnosis of rotating machinery is a problem of pattern recognition, in which fault feature extraction and intelligent fault recognition are two the most important aspects. Therefore, the accuracy of fault diagnosis can be improved by developing more effective feature extraction methods and fault classification methods [1].

The raw vibration signals are the source of the fault information for rotating machinery. However, it is usually very difficult to extract the fault features of the raw vibration signals which contains much noise and complicated multiple faults with different types and severity [2]. So the method of Local Mean Decomposition (LMD) morphological filtering is provided. LMD is a advanced self-adaptive time–frequency domain signal processing method proposed, which processes non-linear and non-stationary original signals effectively. The collected vibration signals are processed by LMD which decomposes them into a series of PF components that have a real physical meaning. Next, the PF components are processed by method of morphological filtering, noise in components can be eliminated effectively by this method. Consequently, the fault features extracted from the PF components are more distinct than those extracted from the raw vibration signals.

After fault features extraction, an effective fault classification method is necessary to accurately recognize faults for rotating machinery. In 1999, Least Square Support Vector Machine (LS-SVM) was proposed as a new type of SVM by Suykens [3]. LS-SVM is improved SVM which introduces the least square linear system, and converts the inequality constraints into equality constraints. So the training process of LS-SVM is transformed into solving linear equations, which greatly improves the efficiency of the SVM, reduces the difficulty of learning, and improves the fault classification effect of the traditional SVM. However, in the process of calculation, LS-SVM reduces the sparsity of traditional SVM and keeps the problem of kernel parameters selection, which hinders its application [4].

For the above introduction, the method of LMD morphology filtering and LS-SVM is proposed, which is applied to the fault diagnosis for rotating machinery. Firstly, the raw vibration signals are preprocessed by LMD which decomposes them into a series of PF components. Then, the PF components are processed by morphological filtering method. Next, the false components are removed by using the correlation analysis method, and the obtained signal is reorganized by the method of signal recombination. Besides, the signal is decomposed by the LMD method again, and the feature vector is extracted and normalized. Secondly, a new Kernel function is applied to adapt to the characteristics of the data and a self-adaptive threshold value determination method of pruning algorithm is applied to solve kernel parameters selection problem and poor sparsity problem in LS-SVM. In the end, the normalized energy features are inputted to the new Least Square Support Vector Machine and the features of signal are trained and predicted, which can judge and classify the fault information for rotating machinery quickly and effectively.

2 LMD morphological filtering and feature extraction

LMD method is an self-adaptive signal decomposition method, which can decompose the signal into a series of PF components [5, 6]. Then, the signals are processed by a combination of open and close operations of morphological filter [7]. Using the same structure element information and the combination of open and close operation of Morphological filter, the open-close(OC) and close-open(CO) morphological filter are defined. Because there is statistical deviation phenomenon in the process of filtering, the morphological filter is used alone that can’t get a good effect. So the combined morphological filter y (n) = [OC (f (n)) + CO (f (n))]/2 is used to reduce the noise of the signal [8]. Signal processing and features extraction methods using LMD morphological filtering are as follows:

Analyzing Structure features of measured object.

When the signals are decomposed by LMD method, several PF_1i components are obtained. Then, PF_1i components are removed false components by analyzing the correlation, new PF_2i components are obtained.

New PF_3i components are obtained from PF_2i components by using the combined morphological filtering method which can reduce the noise of signal.

A new signal x′ (t) is obtained by reconstructing the PF_3i components. The signal x′ (t) is decomposed by LMD again, which obtains the new PF_4i components.

According to the analysis of the structure in step (1), the PF_4i components are classified: high frequency components ${PF}_{4 i}^{'}$ containing main information and low frequency components ${PF}_{4 i}^{″}$ containing secondary information. Next, energy features in ${PF}_{4 i}^{'}$ are extracted by formula E = [e₁, e₂, …, e_m], and Energy features are normalized by formula $E^{'} = E / \sum_{i = 1}^{m} e_{i} = [e_{1}^{'}, e_{2}^{'}, \dots, e_{m}^{'}]$ . Finally, E′ is taken as the inputted feature vector of LS-SVM.

In this paper, the method of LMD–correlationanalysis–morphological filtering–signal recombi-nation–LMD is used as a new signal processing method, which can improve the effect of noise reduction and the accuracy of feature extraction. In step (5),the features extraction method is to reduce the dimension of the inputted LS-SVM from the source, reduces the sparsity and guarantees the reliability of the data.

3 Least square support vector machine

LS-SVM is improved from the basis of SVM. In the objective function, the slack variable is replaced by the sum of square errors and the quadratic programming problem is transformed into a problem of solving linear equations, which can analyze parameters availably [9 –11]. Meanwhile, LS-SVM method leads to the reduction of signal sparsity and the problem that the inputted signal dimension is too large, and keeps the limitation of traditional SVM in the selection of kernel function and parameter.

3.1 Least square support vector machine

The function modeling method in LS-SVM can be described as follows [12 –15]: $min J (ω, ξ_{i}) = \frac{1}{2} | | ω | |^{2} + \frac{1}{2} γ \sum_{i = 1}^{l} ξ_{i}^{2}$ (1) $y_{i} = ω^{T} φ (x_{i}) + b + ξ_{i}$ (2)

Where J is a function of structural risk minimization, x_i is inputted quantity; y_i is target value, ξ_i is error variable, φ (•) is kernel function, ω is weighted vector; γ is adjustable parameter, b is deviation quantity, and l is number of inputted value.

The Lagrange function is introduced:

$\begin{matrix} L & = & \frac{1}{2} | | ω | |^{2} + \frac{1}{2} γ \sum_{i = 1}^{l} ξ_{i}^{2} \\ - \sum_{i = 1}^{l} α_{i} (ω^{T} φ (x_{i}) + b + ξ_{i} - y_{i}) \end{matrix}$ (3) where α_i is Lagrange multiplier.

According to the solution conditions of Lagrange, derivative of each variable can be calculated separately as follow: ${\begin{matrix} \frac{\partial L}{\partial ω} = ω - \sum_{i = 1}^{l} α_{i} φ (x_{i}) = 0 \\ \frac{\partial L}{\partial b} = \sum_{i = 1}^{l} α_{i} = 0 \\ \frac{\partial L}{\partial ξ_{i}} = α_{i} - γ ξ_{i} = 0 \\ \frac{\partial L}{\partial α} = y_{i} - (ω^{T} φ (x_{i}) + b + ξ_{i}) = 0 \end{matrix}$ (4)

Eliminating ω and ξ, the results of derivative can be obtained. According to Mercer conditions, classification models of LS-SVM can be expressed by using Kernel function: $K (x, x_{i}) : f (x) = \sum_{i = 1}^{l} α_{i} K (x, x_{i}) + b$ (5)

Where α_i and b can be obtained by using Equation (4), and the kernel function is an arbitrary symmetric function satisfied the Mercer condition.

According to LS-SVM, its main parameters are adjustable parameter γ and kernel function parameter σ. Parameter γ indicates the punishment degree for sample error. The greater the value of the parameter γ is, the greater the punishment degree to the wrong sample is. When the parameter σ is certain and parameter γ increases to a certain degree, the classification effect of LS-SVM gradually tends to be stable.

3.2 Selection of kernel function and kernel parameters

The classification accuracy of different inputted values is influenced by different kernel functions studied. For example, the linear kernel function is the best for the classification of linear data. In order to get the kernel function which is suitable for the data type of this paper, the following research is done. Figure 1 shows the data trend obtained in the way that feature vector is sorted by energy. From Fig. 1, The distributions of feature vectors are irregular and nonlinear, but in the overall trend, they are expressed as the inverse proportion function in the first range.

The translation invariant kernel function is provided to fit characteristics of this feature vector, as shown in Equation (6): $K (x, y) = \frac{k}{e^{| | x - y | |^{2} / - z} + 1}$ (6)

Where k is the number of data types to be classified, z = ∑||x - y||/ - n, n is the number of feature vectors of this type. The parameters in the kernel function are determined self-adaptively by the data, which can solve the problem of determining the parameters of the kernel function.

From Equation (6), it can be seen that the kernel function uses the structure of inverse proportion function and satisfies translation invariant kernel function theorem [16 –18]. The proof is as follows:

In order to make the Equation (6) to meet the translation invariant kernel condition, Its Fourier transform results must be greater than or equal to 0.

$\begin{matrix} F [K (w)] \\ = (2 π)^{- n / - 2} \int_{R^{n}} e^{- j ω x} K (x) dx \\ = (2 π)^{- n / - 2} \int_{R^{n}} e^{- j ω x} • \frac{k}{e^{| | x | |^{2} / - z} + 1} dx \geq 0 \end{matrix}$ (7)

3.3 Weighted LS-SVM algorithm

For the sparsity problem in LS-SVM, the pruning algorithm is used to reduce the sparsity in this paper [16]. The effect of pruning algorithm is influenced by threshold value determination. So a self-adaptive threshold value determination method is provided to replace the subjective threshold value determination. The method is shown as follows:

The weight of the a_i is calculated by formula $P_{i} = \frac{| a_{i} |}{\sum | a_{i} |}$ ;

Weighted arithmetic mean value of the a_i is solved according to formula $η = \sum_{i = 1}^{N} P_{i} | a_{i} | / - N$ where η is used as threshold value of pruning algorithm.

4 Method of LMD morphology filtering and LS - SVM

In this paper, the LMD algorithm is combined with the morphology filtering method. So the signal pretreatment and feature extraction are finished together. What’s more,this method is combined with the improved LS-SVM algorithm. So algorithm of LMD morphology filtering and LS-SVM is provided. This method has simple structure and simple calculation process. The flow chart is shown in Fig. 2:

5 Experimental study

In this paper, the drive end bearing data of the Case Western Reserve University are used as the study objects. The type of bearing is 6205-2RS JEM SKF deep groove ball bearing and the sampling frequency is 12000 Hz [19]. Time domain diagram of outer-race fault signal of bearing is shown in Fig. 3 (SNR = 2.8369):

The signal in Fig. 3 is processed by the method of LMD morphological filtering. The PF₁ component of the first LMD decomposition is carried out by correlation analysis method. Correlation coefficients are shown in Fig. 4.

From Fig. 4, it is shown that the correlation coefficient between the PF 5–12 components and the source signal is too low. So they don’t need to be processed and can be removed directly. The first 4 PF components are processed by morphological filtering method. The noise reduction effect is shown in Fig. 5. From Fig. 5(a), it is shown that the PF components are relatively disordered and irregular. In addition, the PF components become relatively ordered and regular after morphological filtering method in Fig. 5(b), which shows that the irregular noise is effectively eliminated and that the feature information of the original signal is more obvious.

The signal x′ (t) is derived from the reorganization of the signal in Fig. 5(b). The time domain diagram is shown in Fig. 6. Thus the noise reduction effect and signal-noise ratio (SNR = 5.4352)of the signal x′ (t) can be obtained. Compared with the signal-noise ratio (SNR = 2.8369) of the original signal. It can be seen that the method can effectively reduce the noise of the vibration signal and that it is simple andpractical.

The PF₂ components are obtained by the way that the signal x′ (t) is decomposed by LMD. Because of the particularity of the bearing structure, the fault information is mainly concentrated in the first 4 high frequency components decomposed by the LMD method. So the first 4 high frequency components are extracted as the original data of the feature vector. According to the type of bearing fault, the measured data are divided into 4 categories, each category contains 30 groups of data, each group contains 6000 points. 10 sets of data in each category are randomly selected as the test data, and the other data of 5 groups are selected as training data. The energy data of the outer-race fault selected randomly are shown in Table 1. From Table 1,

The extracted train data are inputted into the improved LS-SVM for Classified training. In order to compare the classification effect of the new proposed kernel function, its classification effect is compared with the classification effect of LS-SVM using radial basis kernel function. In the process, Parameter γ of two LS-SVM methods is equal to 10, Radial basis kernel parameter δ is equal to 0.4. Classification results of the two LS-SVM methods are shown in Fig. 7.

According to Table 2 and a new method for the calculation of threshold value, the weighted average value of the Lagrange parameters of four kinds of fault data can be calculated. They are 0.02017, 0.00803, 0.01907 and 0.02336. So data a₅ and a₉ of the normal state of bearing and data a₄ of the inner-race fault are removed, but no data of the outer-race fault and the rolling element fault are deleted. Data obtained newly are brought into improved LS-SVM,which is trained circularly. Data can meet the requirements after second training, which indicates that the new threshold value determination method can solve the sparsity problem of the self-adaptive LS-SVM algorithm. The results of the second training are shown in Fig. 8.

From Fig. 9, it can be seen that the new method of determining the threshold value is different from the traditional threshold value determination method in training times. The values of the traditional threshold values are 0.01, 0.05 and 0.1.

In Fig. 9, the number 1–4 in the transverse coordinates are respectively representative of the normal state of bearing, outer-race fault of the bearing, inner-race fault of the bearing and the rolling element fault. As can be seen, when the traditional threshold values are 0.05 and 0.1, many fault data of outer-race, inner-race and rolling element are deleted in the process, especially more than half fault data of outer-race of bearing are deleted; When the traditional threshold value is 0.01, the results of the processing are the same as that of the new threshold values. When the threshold values are different values, the number of cyclic training are less, which is 2, 2, 3, 2, respectively. It can be known that randomness of traditional threshold values are too large, Which cause the deletion of a large of data and credibility reduction of the classification. However, there are no such problems in new threshold values because the threshold value is determined by the data themselves. In the meantime, in order to verify the accuracy of the improved LS-SVM method, the receiver operating characteristic (ROC) curve is introduced. Figure 10 shows the ROC curve for the LS-SVM and the modified LS-SVM method based on the radial basis kernel function. It can be seen that the improved LS-SVM method is optimized for the LS-SVM method, and the effect is obvious.

The Table 3 shows the results of training LS-SVM test data. Normal rolling bearing in Sample 1 is trained, only the first numerical value is positive and the outputted result is the target output in Table 3. So the bearing is judged to be normal. Similarly, for the other 3 outputted test sample results, only the second, the third, the fourth value is positive, So this 3 kinds of bearing fault is the outer-race fault of the bearing, inner-race fault of the bearing and the fault of rolling element respectively. The outputted results are in accord with the reality. So the improved LS-SVM method can be applied to the actual bearing fault diagnosis, and the output results are credible.

With the same sample data, the improved LS-SVM algorithm is compared with the traditional algorithm of bearing fault diagnosis, which can get different diagnostic results. Diagnostic results are shown in the Table 4. From the Table 4, compared with other algorithms, the improved LS-SVM algorithm has a further improvement on the bearing fault diagnosis effect.

6 Conclusions

In order to improve the accuracy of fault diagnosis, a fault diagnosis method for rotating machinery based on Local Mean Decomposition morphological filtering and Least Square Support Vector Machine is proposed. The method has the following advantages:

Firstly, LMD–correlation analysis-morphology filtering–signal recombination–LMD is used as a new Signal processing method, which can effectively remove noise and improve the accuracy of the feature vector.

Secondly, the new kernel function is proposed to meet the data features by studying the trend of the feature vector. So the determination of parameters is self-adaptively realized and the problem of LS-SVM parameter selection is solved. By analyzing the structural features of the target, the selection of PF components are decided, the dimension of the feature vector is reduced, the computation quantity is reduced and the sparsity of LS-SVM is reduced to a certain extent. Adopting the pruning algorithm, a new threshold value determination method is provided, which also solves the problem of negativesparsity.

Finally, the method of LMD morphology filtering and LS-SVM is compact and simple, which can effectively solve the problem of fault diagnosis for rotating machinery.

Footnotes

Acknowledgments

Project supported by the 2013 National Natural Science Foundation of Shandong province (no. ZR2013FM005) and higher school science and technology plan projects in Shandong province (no. J10LG22).

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