# Research on rolling bearing fault diagnosis based on support vector machine.

1. IntroductionThe rolling bearing is one of the most widely used and easily damaged parts in the machine. According to statistics in the use of rolling bearing in mechanical equipment, about 30% of the mechanical fault is related to the bearing fault and bearing fault not easy to be found, this will of machinery and equipment time damage, rolling bearing condition and directly affects the running state of the whole machinery and equipment and the entire production line, thereby affecting the efficiency of the plant operations and profit. Therefore, it is of practical significance to carry on working condition monitoring and fault diagnosis of rolling bearings (Abhijit, 2005; Delgado, 2015).

Rolling bearing is one of the most important parts of mechanical equipment, and its failure rate is very high. The working condition of the bearing affects the operation of the whole equipment and even the entire production line (Farshid, 2007; Rai, 2007). Therefore, it is very necessary and important to study the fault diagnosis of rolling bearing. In this paper, the rolling bearing is studied, the basic structure and the failure mode of the rolling bearing are analyzed, and the fault mechanism and vibration characteristics of the bearing are studied (Yunyun, 2011). According to the characteristics of the rolling bearing fault signal, the acceleration sensor to detect the vibration signal, of rolling bearings of four kinds of working state: normal operation, inner ring fault, outer fault and rolling element fault were vibration signal acquisition. Because the traditional fault diagnosis needs a large number of data samples, but the real test data samples are not easy to get, which brings great difficulty to the fault diagnosis. So in this paper, we introduce the support vector machine to fault diagnosis, and provide a new research method for fault intelligent diagnosis (Sugumaran, 2007; Xiaodong, 2011). In this paper, we use a combination of theoretical research and computer simulation research methods, first of all with the vibration signal of the sensor bearing. Then the threshold method of wavelet transform wavelet de-noising of rolling bearing vibration signal processing, signal interference signal is removed. Then use the wavelet packet technology was used to extract the noise signal frequency band energy. At last, the wavelet packet analysis get the frequency band energy feature sets as input vector for support vector machine using support vector machine intelligent classification judges the working state of the bearing.

2. Vibration mechanism of rolling bearing

The basic structure of the rolling bearing is shown in figure 1. It is composed of four parts: inner ring, outer ring, cage and rolling element. The general inner ring is fixed on the shaft neck, and the inner ring and the shaft are rotated together, and the outer ring is used for assembling the bearing seat hole.

[FIGURE 1 OMITTED]

1. Pitting corrosion: In the normal operation of the bearing, rolling by the load size different, resulting in rolling contact surfaces of the body and the inner and outer contact force changes, when the contact stress is beyond the limit value, the surface fatigue crack is generated and continuous expansion to the bearing surface formation of pitting corrosion. Pitting can cause noise, shock and vibration of the bearing to reduce the rotation precision of the rolling bearings, and the pitting corrosion is the main failure mode of the rolling bearing.

2. Erosion: It is a kind of plastic deformation, which is caused by the heavy load during the operation of the impact load or the stop.

3. Friction and corrosion of bearing: In fact, it is the corrosive wear of the bearing. It is caused by the vibration of the transport link before the bearing assembly and application installation. Vibration load caused by vibration load can also cause this situation.

4. Corrosion caused by the current through the bearing: When the motor is used, the bearing is not insulated, and the current will pass through the bearing and form a series of micro pits on the surface of the roller.

5. Indentation caused by hard particles: Hard particles into the lubricating oil and bearings can also lead to the destruction of the surface of the rolling contact surface.

Bearing at a certain load and speed, the bearing, the bearing seat of the vibration system will produce vibration; show the mechanism of its vibration. The vibration, including internal factors and external factors, internal is determined by bearing structure characteristics, the rotation failure and other factors; external factors for bearings where the drive shaft and the shaft on the other parts of the integrated effect. Rolling bearing signal acquisition is the vibration signal obtained by installing sensors in the bearing seat, which is collected by the internal and external factors together with the integrated vibration signal of the bearing system. Therefore, the vibration signal generated by the rolling bearing fault is separated from the comprehensive vibration, which is the key part of the bearing fault diagnosis, which has a great influence on the accuracy of the diagnosis. The natural frequency is only related to the characteristics of the parts, and has nothing to do with the rotational speed of the rolling bearing. We calculated the natural frequency when considering the internal factors of its material, size, structure etc.. The natural frequency is much higher than the frequency of the fault feature. The calculation formulas of the natural frequency of the rolling bearing components are as follows:

[f.sub.n] = 9.4 x [10.sup.5] hn([n.sup.2] - 1)/[b.sup.2][square root of [n.sup.2] + 1] (1)

In the formula, n represents the order of the natural frequency, where n starts from 2: 2,3,4,... H, said the ring thickness in millimeters, B, said the ring width in millimeters. From the above formula can be known, the bearing's width is smaller, the thickness is bigger, the inherent frequency of the inner and outer ring is bigger. Natural frequency of rolling elements:

[f.sub.b] = 0.424/[gamma] [square root of E/2[rho]] (2)

E means the density of the material elastic modulus, p means rolling radius. [gamma] means the larger the elastic modulus, the greater the material density and the radius, the greater the inherent frequency of the rolling element. When the inner ring, outer ring and the rolling body wear fault, with rotation of the bearing, its surface will produce the composition of the low frequency vibration, which is generated by surface of repeated impact, the periodic pulse vibration time is short, steep shape, called for "by vibration, the vibration frequency, also known for rolling bearing fault characteristic frequency can obtained through the bearing size and speed. This type of fault signal is also the main object of this thesis. After the extraction of the actual bearing signals, the classification and recognition ability of the support vector machine is to determine which kind of fault. According to four different damage locations of rolling bearings: rolling bearing inner and outer ring, rolling body and cage, the formula for calculating the fault characteristic frequency is given in table 1.

In table 1, fa is the turning frequency, [alpha] for the contact angle, d for the rolling body diameter, z for the number of rolling elements, D for the bearing diameter. But in practical applications, the fault characteristic frequency and the theoretical value obtained after the signal analysis are not exactly the same.

3. Noise reduction and wavelet packet energy extraction of rolling bearing

3.1. Wavelet analysis

If [PSI](t)[member of][L.sup.2](R) to meet the permit conditions:

[C.sub.[psi]] = [[integral].sup.[infinity].sub.-[infinity]] [absolute value of [??]([omega])]/[omega] d[omega] < [infinity] (3)

Then [PSI](t) called base wavelet or admissible wavelet. Based on the basis of wavelet function, we derive the formula of wavelet function, which can be expressed as:

[[psi].sub.a,b](t) = [[absolute value of [alpha]].sup.-1/2] [psi] (t - b/a) (4)

Through this function to decompose the signal processing, the decomposition process is the continuous wavelet transform.

([W.sub.[psi]] f) (t) = [[integral].sup.[infinity].sub.-[infinity]]f (t)[bar.[[psi].sub.q,b](t)]dt (5)

We again without loss of information under the premise, the parameters a and b or at the same time discretization, this conversion for the discrete wavelet transform, discrete wavelet function can be expressed as:

[[psi].sub.j,k] (t) = [a.sup.1j/2.sub.0][psi]([a.sup.-j.sup.0] t - kb) (6)

The coefficients of discrete wavelet transform can be expressed as:

[W.sub.j,k](t) =[[integral].sup.[infinity].sub.-[infinity]]f(t)[bar.[[psi].sub.j,k](t)]dt (7)

Signal reconstruction formula:

f(t) = C [[infinity].summation over (-[infinity]])] [[infinity].summation over (-[infinity]])[W.sub.f,k][[psi].sub.j,k] (8)

The signal was decomposed using a wavelet. In this paper, the DB4 wavelet three layer wavelet decomposition of noise signal with, obtained after the decomposition of the first layer to the third layer detail coefficient selected an appropriate threshold and threshold processing is divided into hard threshold and soft threshold method, the hard threshold method is the absolute value is less than the threshold of detail coefficients become zero, but in the process will appear between the breakpoint. Soft threshold method for the improvement of the shortcomings of the hard threshold method, the boundary is not set to 0 points, so that the signal can be reconstructed without a break point, and smoother. Hard threshold formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Soft threshold formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Wavelet denoising is an important part of how to select the threshold, if the threshold is too small, the effect of noise reduction will be affected, to reduce the follow-up signal fault recognition accuracy rate. If the threshold selection is too large, although some of doping doping the noise signal can be removed, but effective vibration signal will be mistaken for a signal to noise and be disposed of, so that the de noised signal distortion, cannot really shows the signal characteristics, misleading fault type diagnosis.

3.2. Experimental simulation of wavelet noise reduction

The quality of internal control not only has a direct impact on investment efficiency, but also has an indirect impact. The indirect impact affects that internal control quality through the financing constraints, agency costs and the quality of accounting information produce a certain impact and then affect the efficiency of investment. Wavelet denoising simulation using Daubechies wavelet DB4 wavelet family, it is better to extend the wavelet, suitable for wavelet denoising. Due to the threshold selection and quantification of has a direct influence on the effect of wavelet denoising, the threshold is too small will make noise cannot be removed and lost the purpose of noise reduction; if the threshold is too large, may will make the loss of the components of the original signal, so we choose the default threshold de-noising. Matlab simulation using wdencmp global threshold command, the decomposition coefficients by parameters THR, SORH specified threshold by means of wavelet function name with a specified wavelet. Figure 2-5 in order to normal bearing, inner ring fault, and outer ring fault, the original fault of the rolling body signal contrast after signal and noise reduction.

[FIGURE 2 OMITTED]

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4. Fault recognition based on support vector machine

4.1. Support vector machine

SVM is a method based on statistical learning theory and structural risk minimization theory, which provides a new way to solve the problem of small sample data classification and nonlinear problem. SVM is peacekeeping linear rise of, through nonlinear mapping; the sample space is mapped to a higher dimensional space within, the samples in high dimensional space correctly classified, so that you can through linear learning machine method to solve the problem of nonlinear classification in the sample space. Ascending dimension is mapping the samples to higher dimensional space, which can be linearized by linear hyper plane in high dimensional space, which can solve the problem that the sample is not easy to be classified in the low dimension space. For example, figure 6 shows a typical example: randomly distributed points in the two-dimensional space to linear programming, but by mapping to 3D space to achieve linear programming.

[FIGURE 6 OMITTED]

Here we only consider a loss function of SVM; the original problem is converted as:

[phi](w,[xi]) = 1/2[parallel]w[[parallel].sup.2] + C [m.summation over (i=1)][[xi].sub.i] (11)

The problem is to minimize the constraints, That is:

min [parallel]w[[parallel].sup.2]/2 + C [m.summation over (i=1)][[xi].sub.i] (12)

[y.sub.i] (w * x)+b) - 1 [greater than or equal to] 0 (13)

Then carries on the derivation of the dual problem, need to introduce the original problem of Lagrange's function:

L(w,b,a) = [1/2] [parallel]w[[parallel].sup.2] + C[m.summation over (i=1)][[xi].sub.i] - [m.summation over (i=1)] [[alpha].sub.i]([y.sub.i](w * x) + b) -1 + [[xi].sub.i]) (14)

Calculated:

[m.summation over (i=1)][[alpha].sub.i][y.sub.i][x.sub.i] = 0 (15)

W = [m.summation over (i=1)][[alpha].sub.i][y.sub.i][x.sub.i] (16)

C - [[alpha].sub.i] - [[beta].sub.i] = 0 (17)

Take into the plane equation, get the decision function:

f (x) = sgn ([m.summation over (i=1)][a.sub.i][y.sub.i] (x * [x.sub.i]) + b*) (11)

4.2. Experiment simulation

Intelligent diagnosis of rolling bearing can determine whether the normal work state of the bearing, but also can quickly and accurately the bearing fault classification and recognition, intelligent diagnosis model process first acquisition of bearing vibration signal, and then by wavelet decomposition and wavelet packet energy extraction, signal energy characteristics are obtained. Then, the optimal feature vector is obtained by using the data preprocessing with the extreme value normalization formula. Use to extract the feature vectors to form a set of sample and the sample set is divided into two parts: training and testing, with the training sample set to train the SVM, the SVM training and testing samples are tested on the set, test in the diagnosis of whether arriving at an accurate rate. Finally, using SVM to reach the requirements of the bearing can be intelligent diagnosis.

[FIGURE 7 OMITTED]

First, the training set is used to train the SVM, and then use the trained SVM model to predict the test set, in which the SVM program uses the libsvm toolbox. Figure 8 is the fractal dimension of data, which indicates that the 320 sets of feature vectors extracted by wavelet packet are distributed in each frequency band. Figure 9 is the final classification result. Table 4 is the statistical table for diagnostic results.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Diagnosis results show that the normal bearing 4 false positives, the inner ring fault does not appear false positives, outer ring fault occurs 3 miscarriages of justice, the rolling body fault occurs 2 false positives. Its diagnostic rate was 90% (36/40), 100% (40/40), 90.25% (37/40X90.5% (38/40).The final diagnosis accuracy rate is 94.375% (151/160), which is consistent with the accuracy of bearing fault diagnosis.

5. Conclusion

This paper analyzes the failure forms of rolling bearing, fault characteristic frequency and vibration mechanism and application of wavelet de-noising, wavelet packet energy extraction and support vector machine (SVM) and other technologies, using a rolling bearing fault intelligent diagnosis method based on support vector machine (SVM), verified by simulation experiments. The main results the following: due to the bearing signal is mixed with a lot of noise signal, the wavelet de-noising technique can effectively remove the original signal in the noise signal, effectively reduce the after fault diagnosis of error. Wavelet packet energy decomposition is a kind of effective of rolling bearing vibration signal feature extraction method, it can extract the signal energy in different frequency bands, the characteristic information of the signal in each frequency band is composed of feature vector can be a good characterization of rolling bearings. This article in view of the limited number of fault samples, the wavelet threshold denoising and support vector machine combination method to diagnose the fault of rolling bearing. Experimental simulation results show that the proposed method can identify the fault types of rolling bearings rapidly and accurately.

Recebido/Submission: 02/03/2016

Aceitacao/Acceptance: 4/07/2016

Acknowledgments

The work of this paper is supported by Project of Natural Science Foundation of Heilongjiang Province (E201215); The Ministry of education, the specialized research fund for the doctoral program of higher education of doctoral research (20120062110006).

References

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Delgado, A., Velthuis, M. (2015). Proposal for a continuous improvement IT governance framework at financial institutions. RISTI-Revista Iberica de Sistemas e Tecnologias de Informacao, (15), 51-67.

Farshid, T., Farshid, T. (2007). A ball bearing fault diagnosis method based on wavelet and EMD energy entropy mean. International Conference on Intelligent and Advanced Systems, 1210-1212.

Rai, V. (2007). Bearing fault diagnosis using FFT of intrinsic mode functions in HilbertHuang transform. Mechanical Systems and Signal Processing, 21, 2607-2615.

Sugumaran, V., Ramachandran, K. (2007). Automatic rule learning using decision tree for fuzzy classifierinfault diagnosis of roller bearing. Mechanical Systems and Signal Processing, 21, 2237-2247.

Xiaodong, W., Yanyang, Z. (2011). Multiwavelet denoising with improved neighboring oefficients for application on rolling bearing fault diagnosis. Mechanical Systems and Signal rocessing, 25, 285-304.

Yunyun, Y., Wei, T. (2011). Study of Remote Bearing Fault Diagnosis Based on BP Neural Network combination. Seventh International Conference on Natural Computation, 618-621.

Shusen Li (1) *, Qingchun Zhang (1), Xin Shang (2), Bin-bin zhang (2)

* 99483865@qq.com

(1) Northeast Forestry University, Harbin 150040, China

(2) Harbin Normal University, Harbin 150025, China

Table 1--Calculation formula of Characteristic frequency for rolling bearing Damage location characteristic frequency inner ring [f.sub.i] = z/2 [f.sub.a] (1+d/D cos[alpha])] outer ring [f.sub.o] = z/2 [f.sub.a] (1 - d/D cos[alpha]) Rolling body [f.sub.b] = D/2d [f.sub.a] (1 - [d.sup.2]/[D.sup.2] [cos.sup.2] [alpha] Holder [f.sub.c] = 1/2 [f.sub.[alpha]] (1 + d/D cos[alpha] /z (touch the inner ring) [f.sub.c] = 1/2 [f.sub.a] (1 - d/D cos[alpha]) /z (touch the outer ring) Table 2--The number distribution of number Training Test sample sample Bearing condition number number State mark Normal working conditions 40 40 1 Inner fault 40 40 2 Outer ring fault 40 40 3 Rolling element fault 40 40 Table 3--The part of training sample energy character vector serial Spectrum 1 Spectrum 2 Spectrum 3 Spectrum 4 1 72.6214 10.5231 4.2141 2.6753 2 80.2541 7.5102 3.4126 2.5218 3 75.5287 9.2643 2.6414 3.5624 4 16.2451 5.2928 10.1475 8.9336 5 14.6953 6.7852 8.8462 6.4568 6 16.9650 4.2568 6.5482 6.2564 7 21.5634 12.6354 10.9875 24.6845 8 20.5569 15.5462 9.5246 29.3654 9 18.6754 16.4548 6.6427 30.3445 10 19.0935 38.4625 3.7505 14.8023 11 16.5736 40.4254 4.5789 12.5473 12 17.3546 39.3648 4.6951 15.3648 serial Spectrum 5 Spectrum 6 Spectrum 7 Spectrum 8 1 3.8264 4.1152 0.5438 1.4807 2 2.1431 3.2164 0.5211 0.4207 3 1.7815 5.3124 0.8541 1.0552 4 4.1862 8.2793 40.1627 9.7528 5 6.7852 14.5486 36.4513 5.4314 6 5.3698 10.8965 42.5623 7.1453 7 3.6984 8.3546 12.2548 5.8214 8 2.3648 4.5862 10.3647 7.9012 9 6.3544 4.7841 6.3348 10.4093 10 14.5224 7.3201 0.6550 1.3936 11 15.3784 8.4524 0.9541 1.2564 12 13.3589 7.3548 0.6581 1.4694 Table 4--Result of fault diagnosis Correct Test sample identification Bearing condition number number Normal working 40 36 conditions Inner fault 40 40 Outer ring fault 40 37 Rolling element fault 40 38 accuracy Comprehensive Bearing condition rate accuracy Normal working 90% 94.375% conditions Inner fault 100% Outer ring fault 90.25% Rolling element fault 90.5%