A simulation model based fault diagnosis method for bearings

Abstract

Bearings are essential parts in mechanical transmission systems, and their running states directly affect the reliability and stability of the systems. Therefore, an efficient diagnosis method is necessary to detect faults in bearings. In the present, a simulation model based fault diagnosis method for bears is proposed by combination of finite element method (FEM), wavelet packet transform (WPT) and support vector machine (SVM). In this method, firstly, the agreeable finite element models to simulate faulty bearings are presented to obtain the vibration response signals. Secondly, the vibration signals are decomposed into eight signal components using WPT. Ten time-domain feature parameters of all the signal components are calculated to generate the training samples to train the SVM. Finally, the eight signal components decomposed by WPT from the measured vibration signal in a bear, which are serve as a test sample into the trained SVM, and the work condition of the bearing can be determined. Experimental investigations are performed to verify the effectiveness of the present method. The classification accuracy rates for four type faults, i.e., inner race fault, rolling body fault, outer race fault and the combination of rolling body and outer race faults, are 79%, 81%, 71% and 76%, respectively.

Keywords

Fault detection bearing numerical simulation wavelet packet transform support vector machine

1 Introduction

The running state of rotating machinery directly affects the performance of the entire machine, including reliability and stability. Thus, fault diagnosis in rotating machinery is a hot topic in the past several decades, and numerous fault detection methods [1 –6] and criteria [7 –9] were proposed. For the fault detection of bearings, various researchers have reported many methods. Wang et al. [10] gave a good review to the development of a spectral kurtosis method to detect faults in rotating machines. Masmoudi et al. [11] developed a single point bearing fault diagnosis method using simplified frequency model. Do and Nguyen [12] presented an adaptive empirical mode decomposition (EMD) method for bearing fault detection. Tian et al. [13] proposed a motor bearing fault detection method using spectral kurtosis-based feature extraction coupled with K-nearest neighbor distance analysis. Xiang et al. [14] developed a novel personalized diagnosis methodology using numerical simulation and an intelligent method to detect faults in a shaft. Guo and Deng [15] presented an improved EMD method using the multi-objective optimization to extract fault features in faulty rolling bearing. Liu et al. [16] proposed an adaptive stochastic resonance in a new nonlinear system to improve the bearing fault diagnosis efficiency. Shao et al. [17] presented a novel deep autoencoder feature learning method for rotating machinery fault diagnosis. Li et al. [18] developed a Bayesian approach to consequent parameter estimation in probabilistic fuzzy systems to classify bearing faults.

Because the vibration signals carry dynamic information corresponding to different faults about the machine state, mechanical fault diagnosis depends largely on the features extraction methods to extract fault features [19]. As a signal processing tool, WPT [20, 21] decomposes the original signal into multi-layers, and the same frequency bandwidth is offered in each layer. Then, time-domain feature parameters, such as standard deviation, peak, skewness, kurtosis, root mean square, peak factor, clearance factor, shape factor, impulse factor and kurtosis factor, which can be calculated and served as training samples for classification methods, such support vector machine (SVM), deep neural network (DNN), Genetic algorithms (GAs), etc. SVM is a pattern recognition approach, which has been widely used in fault classification [22]. However, it has greatly limited the fault classification using SVM in real-world applications because of the lack of faulty training samples collected from the real-world running machines. Numerical simulations provide a possible way to simply obtain the fault samples for every type of faults under the complex running conditions. Obviously, it is necessary to build up agreeable numerical simulation models to simulate real-world running machines.

For the above reasons, a simulation model based fault diagnosis method is proposed using FEM and SVR to detect faults in bearings.

2 The simulation model based fault diagnosis method

The basic idea for the simulation model based fault diagnosis method is given. Figure 1 shows the flowchart of the proposed fault diagnosis method.

Fig.1

The flowchart of the proposed fault diagnosis method.

Firstly, the agreeable finite element models to simulate faulty bearings are presented to obtain the vibration response signals. The explicit sub-steps are:

Built up FEM model of bearings without faults.

Insert fault types in the components of bearings.

Apply FEM updating technique if necessary.

Obtain FEM model of bearing with faults.

Obtain simulation faulty signals.

Secondly, the vibration signals are decomposed into eight signal components using WPT. Ten time-domain feature parameters of all the signal components are calculated to generate the training samples to train the SVM. The explicit sub-steps are:

Decompose the simulation faulty signals using WPT at level j = 3 to get eight signal components.

Ten time-domain feature parameters, as shown in Table 1 are calculated (eight signal components) to generate the training samples to train SVM.

Obtain the trained SVM.

Table 1

Ten time-domain feature parameters

Feature	Equation
Standard deviation x_std	$x_{std} = \sqrt{\frac{\sum_{i = 1}^{N} {(x (i) - x_{m})}^{2}}{N}}$
Peak x_p	x_p = max \|x (i) \|
Skewness x_ske	$x_{ske} = \frac{1}{N} \sum_{i = 1}^{N} (x (i) - x_{m})^{3}$
Kurtosis x_kur	$x_{kur} = \frac{\sum_{i = 1}^{N} {(x (i) - x_{m})}^{4}}{{Nx}_{std}^{4}}$
Root mean square x_rms	$x_{rms} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} x_{i}^{2}}$
Peak factor PF	$PF = \frac{x_{p}}{x_{rms}}$
Clearance factor CLF	$CLF = \frac{x_{p}}{{(\frac{1}{N} \sum_{i = 1}^{N} \sqrt{\| x (i) \|})}^{2}}$
Shape factor SF	$SF = \frac{x_{rms}}{\frac{1}{N} \sum_{i = 1}^{N} \| x (i) \|}$
Impulse factor IF	$IF = \frac{x_{p}}{\frac{1}{N} \sum_{i = 1}^{N} \| x (i) \|}$
Kurtosis factor KF	$KF = \frac{x_{kur}}{x_{rms}^{4}}$

x is the data; N is the number of data points; x_m is the mean value of x.

Finally, the eight signal components decomposed by WPT from the measured vibration signal in a bear, which are serve as a test sample into the trained SVM, and the work condition of the bearing can be determined. The explicit sub-steps are:

Obtain the measured signals of faulty bearings in real-world running machines.

The similar procedures (dotted arrow, as shown in Fig. 1) are performed to the measured signals to generate test samples.

Submit the test samples into the trained SVM, diagnosis results will be finally determined.

3 Simulation procedures

FEM models of the bearings (N205) are constructed using commercial finite element analysis (FEA) software ANSYS to get enough faulty training samples. Figure 2 shows finite element model and geometrical dimensions of the bearing. Suppose the rotation speed is kept constant at 1500 rpm, Solid 186 element in software ANSYS are employed to conduct simulation of bearings (material parameters are: Young’s modulus E = 210GPa, Poisson’s ratio μ = 0.3, density ρ = 7860 kg/m³). The FEM model updating technique (mean absolute percentage error, MAPE) [23] is used to make the models of bearings represent the real-world ones precisely. Furthermore, the faulty models (shown in Fig. 3) will be constructed to simulate the dynamic response of the bearings with faults. Figure 3(a-d) show the inner race fault, the rolling body fault, the outer race fault, and the combination of rolling body and outer race faults, respectively. The simulation signals of bearings with faults are obtained, as shown in Fig. 4(a-d), respectively.

Fig.2

The finite element model and geometrical dimensions of the bearing. (a) finite element model, (b) the geometrical dimensions.

Fig.3

The finite element model of the bearing with faults. (a) the inner race fault, (b) the rolling body fault, (c) the outer race fault, (d) the combination of rolling body and outer race faults.

Fig.4

The simulation signals with four kinds of faults. (a) the fault in inner race, (b) the rolling body fault, (c) the outer race fault, (d) the combination of rolling body and outer race faults.

The simulation signals of bearings with four kinds of faults are obtained from FEM models. The signals are decomposed by WPT using Daubechies wavelet (Db6) into three layers (level j = 3). Therefore, eight frequency bands (signal components) are obtained, as shown in Fig. 5.

Fig.5

Frequency bands reconstructed by WPT of inner race fault.

In the present, only the decomposed signals of inner race fault are given, and the corresponding ten time-domain feature parameters are calculated using the equations listed in Table 1, the calculation results are recorded in Table 2.

Table 2

The calculation results in each frequency band of each fault in simulations

The fault in inner race
	1	2	3	4	5	6	7	8
SD	0.000993	0.001299	0.006024	0.00204	0.002165	0.011845	0.006628	0.008692
P	0.003253	0.003833	0.025215	0.00518	0.006278	0.047057	0.02187	0.030054
Ske	0.043702	0.014676	– 0.00195	0.010556	0.001611	– 0.00251	0.001805	0.293678
Kur	2.645857	3.265865	5.473104	2.532214	3.202765	4.024322	3.920326	4.53399
PF	2.376294	249.6374	218.9686	48.23902	60.25186	281.11	190.4562	2495.594
RMS	0.001369	1.54×10^–5	0.000115	0.000107	0.000104	0.000167	0.000115	1.20×10^–5
CLF	4.568345	4.56952	7.011213	3.694837	4.412477	6.205045	5.339617	5.927259
SF	1.671518	0.015228	0.026162	0.065217	0.061555	0.018435	0.023072	0.001906
IF	3.972018	3.801599	5.72865	3.145998	3.708789	5.182152	4.394283	4.757349
KF	7.53×10¹¹	5.87×10¹⁹	3.11×10¹⁶	1.90×10¹⁶	2.72×10¹⁶	5.13×10¹⁵	2.25×10¹⁶	2.16×10²⁰
The rolling body fault
SD	0.00174	0.001737	0.008997	0.003879	0.004352	0.008286	0.007148	0.011923
P	0.018146	0.008169	0.033346	0.040906	0.009818	0.026755	0.022107	0.090189
Ske	– 4.90093	– 0.05547	– 0.02418	– 1.91177	0.002094	– 0.00841	0.000639	– 0.60181
Kur	47.53274	5.568967	5.672053	38.92367	2.33585	3.233324	3.326969	16.33202
PF	27.92489	185.1346	143.6414	74.75231	26.09063	31.61192	454.5075	175.8144
RMS	0.00065	4.41×10^–5	0.000232	0.000547	0.000376	0.000846	4.86×10^–5	0.000513
CLF	24.18984	8.525899	6.75942	22.39637	3.218623	4.991958	5.02465	15.13115
SF	0.667889	0.035843	0.037259	0.233289	0.105652	0.131306	0.00894	0.066866
IF	18.65073	6.635852	5.351962	17.4389	2.756522	4.150844	4.063418	11.75602
KF	2.67×10¹⁴	1.47×10¹⁸	1.95×10¹⁵	4.34×10¹⁴	1.17×10¹⁴	6.3×10¹²	5.94×10¹⁷	2.36×10¹⁴
The outer race fault
SD	0.00102	0.00127	0.005551	0.002719	0.007906	0.018507	0.008216	0.012309
P	0.003617	0.003871	0.017843	0.007882	0.026097	0.068951	0.039895	0.068693
Ske	0.320086	– 0.05615	0.002697	0.028613	0.003633	– 0.00413	0.001355	– 0.10558
Kur	3.767353	2.959094	3.581219	2.805354	4.729002	5.466631	6.326811	11.42975
PF	2.721551	65.99579	144.5031	83.77905	98.33147	131.7289	626.3762	247.6666
RMS	0.001329	5.87×10^–5	0.000123	9.41×10^–5	0.000265	0.000523	6.37×10^–5	0.000277
CLF	5.685581	4.437985	5.097386	4.21132	6.144333	6.282372	7.97306	11.37818
SF	1.712054	0.057473	0.029182	0.04297	0.048383	0.038967	0.010506	0.035296
IF	4.659442	3.792951	4.2169	3.599966	4.757526	5.133107	6.580823	8.741621
KF	1.21×10¹²	2.50×10¹⁷	1.54×10¹⁶	3.58×10¹⁶	9.53×10¹⁴	7.28×10¹³	3.84×10¹⁷	1.93×10¹⁵
The combination of rolling body and outer race faults
SD	0.011044	0.024568	0.039809	0.036926	0.03326	0.094922	0.052986	0.090095
P	0.033839	0.07982	0.126022	0.109921	0.085823	0.25154	0.259717	0.252378
Ske	– 0.39144	0.00324	0.002749	– 0.0441	– 0.00392	0.003198	– 0.00234	– 0.0647
Kur	2.892417	3.4491	2.960959	2.669367	2.575741	2.834888	5.667568	2.551054
PF	2.80593	24.9208	96.56177	57.29365	76.03666	123.6716	177.535	197.5363
RMS	0.01206	0.003203	0.001305	0.001919	0.001129	0.002034	0.001463	0.001278
CLF	4.675108	4.980093	4.698164	4.364965	3.603785	3.929606	7.721923	3.97166
SF	1.390826	0.1676	0.041155	0.064622	0.041226	0.026842	0.036398	0.017319
IF	3.902561	4.176715	3.974005	3.70245	3.134676	3.319591	6.461873	3.421133
KF	1.37×10⁸	3.28×10¹⁰	1.02×10¹²	1.97×10¹¹	1.59×10¹²	1.66×10¹¹	1.24×10¹²	9.57×10¹¹

4 Experimental investigation

To verify the feasibility of the simulation model based fault diagnosis method, experimental investigations using the same bearings as used in FEM simulation, are applied in this section. The experimental setup is shown in Fig. 6. Four different faulty signals are collected including inner race faults, rolling body faults, outer race fault and the combination of rolling body and outer race faults, as shown in Fig. 7(a-d), respectively. In the experiments, the rotation speed is kept constant at 1500 rpm, as well as in simulation. An accelerometer mounted at the top of bearing housing. The sampling frequency fs = 6000 Hz, and the signal length is N = 8000 points. Signals are acquisited by the accelerometer and sent to a computer through a signal conditioner (AVANT MI-7016). The measured vibration signals of four kinds of faults in bearings are shown in Fig. 8(a-d), respectively.

Fig.6

The experimental setup.

Fig.7

The faults in bearings. (a) the fault in inner race (b) the rolling body fault, (c) the outer race fault, (d) the combination of rolling body and outer race faults.

Fig.8

The measured signal with four kinds of faults. (a) the measured signal with inner race faults. (b) the measured signal with rolling body faults, (c) the outer race fault, (d) the combination of rolling body and outer race faults.

Then, using the procedures mentioned Section 2, eight frequency bands of measured vibration signals are obtained. Figure 9 shows the eight frequency bands (signal components) decomposed by WPT for the inner race fault. The calculation results of corresponding ten time-domain feature parameters are also listed in Table 3.

Fig.9

Eight frequency bands (signal components) decomposed by WPT for the inner race fault.

Table 3

The calculation results in each frequency band for the four types of faults in the experimental setup

The fault in inner race
	1	2	3	4	5	6	7	8
SD	0.000982	0.001377	0.007641	0.002222	0.004297	0.012713	0.006241	0.008242
P	0.002442	0.00433	0.02882	0.006017	0.014689	0.047283	0.02509	0.022751
Ske	– 0.30396	0.080851	0.000204	– 0.06345	– 0.00321	0.002316	– 0.00313	0.049053
Kur	2.499496	3.145308	4.586853	2.693243	4.313011	5.670531	5.72323	2.848798
PF	7.080766	128.4414	190.183	86.63463	143.2935	353.357	176.8988	1473.443
RMS	0.000345	3.37×10^–5	0.000152	6.95×10^–5	0.000103	0.000134	0.000142	1.54×10^–5
CLF	3.580408	4.859228	6.525532	3.86576	5.7165	6.745275	6.969139	4.05755
SF	0.43304	0.031389	0.027311	0.038429	0.032292	0.01513	0.031899	0.002341
IF	3.066258	4.031618	5.194177	3.329258	4.627265	5.346292	5.642903	3.449676
KF	1.77×10¹⁴	2.44×10¹⁸	8.70×10¹⁵	1.16×10¹⁷	3.91×10¹⁶	1.77×10¹⁶	1.41×10¹⁶	5.01×10¹⁹
The rolling body fault
SD	0.001115	0.001581	0.011423	0.00198	0.002872	0.008103	0.013139	0.007848
P	0.005787	0.006867	0.039703	0.006499	0.007351	0.028352	0.068297	0.021972
Ske	– 0.59297	– 0.02146	0.010138	0.049512	– 0.00197	0.001594	0.001895	– 0.03206
Kur	4.87443	5.473273	4.48035	3.072007	2.738286	3.308884	11.09794	2.780425
PF	3.38172	65.30763	245.6014	81.41226	193.7938	491.5534	239.505	281.4001
RMS	0.001711	0.000105	0.000162	7.98×10^–5	3.79×10^–5	5.77×10^–5	0.000285	7.81×10^–5
CLF	8.119675	7.471865	6.24619	4.832865	4.039407	5.121127	10.78893	4.177653
SF	2.001327	0.092108	0.019933	0.050434	0.016995	0.008891	0.034659	0.012491
IF	6.767928	6.015351	4.895692	4.10594	3.293482	4.370191	8.300918	3.515092
KF	5.68×10¹¹	4.48×10¹⁶	6.56×10¹⁵	7.56×10¹⁶	1.32×10¹⁸	2.99×10¹⁷	1.68×10¹⁵	7.48×10¹⁶
The outer race fault
SD	0.000927	0.001537	0.006885	0.004087	0.008736	0.014059	0.008766	0.017768
P	0.002581	0.004178	0.028965	0.010729	0.03879	0.063291	0.029099	0.08696
Ske	0.215043	0.025429	0.000401	– 0.0065	0.000989	– 0.00094	– 0.00262	0.066377
Kur	2.928928	2.732971	5.467655	2.862381	8.043841	5.751471	3.369821	8.479827
PF	3.05495	46.61843	354.7595	70.42248	669.7228	910.4376	164.2945	643.3004
RMS	0.000845	8.96×10^–5	8.16×10^–5	0.000152	5.79×10^–5	6.95×10^–5	0.000177	0.000135
CLF	4.214914	3.997833	7.62562	4.212566	8.139355	7.372027	5.067044	10.19701
SF	1.155774	0.072692	0.016808	0.04843	0.009738	0.006659	0.025877	0.011966
IF	3.530833	3.388767	5.962706	3.410594	6.522029	6.06277	4.251391	7.697922
KF	5.74×10¹²	4.24×10¹⁶	1.23×10¹⁷	5.31×10¹⁵	7.15×10¹⁷	2.46×10¹⁷	3.42×10¹⁵	2.54×10¹⁶
The combination of rolling body and outer race faults
SD	0.080517	0.164504	0.572863	0.341805	0.179674	0.366851	0.364994	0.552108
P	0.510846	1.030153	4.877952	2.676613	0.907112	2.21461	2.341712	3.882709
Ske	– 1.70158	0.455364	– 0.00158	0.031504	– 0.00032	0.002889	0.000229	– 0.088
Kur	21.62376	21.78764	44.3939	31.72826	13.4228	19.60964	22.68718	28.54686
PF	46.53941	252.2603	6516.269	963.4559	2325.119	2950.467	800.7769	1439.836
RMS	0.010977	0.004084	0.000749	0.002778	0.00039	0.000751	0.002924	0.002697
CLF	30.2898	26.36053	74.10235	45.98174	25.65204	21.95604	30.62978	29.29691
SF	0.348056	0.061321	0.004769	0.024177	0.005309	0.004611	0.020838	0.012385
IF	16.19832	15.46888	31.07882	23.2935	12.34306	13.60558	16.68631	17.83181
KF	1.49×10⁹	7.83×10¹⁰	1.41×10¹⁴	5.33×10¹¹	5.79×10¹⁴	6.18×10¹³	3.1×10¹¹	5.4×10¹¹

The method of SVM is applied to classify the fault type of the bearing. Table 4 presented the final classification results. The classification accuracy ratios for the inner race fault, rolling body fault, outer race fault, and the combination of rolling body and outer race faults are 79%, 81%, 71% and 76%, respectively.

Table 4

The recognition results for the four types of faults

Different Type Faults	Classification Accuracy
Inner race fault	79%
Rolling body fault	81%
Outer race fault	71%
Combination of rolling body and outer race fault	76%

5 Conclusions

This paper suggests a faults detection method for bearing based on numerical simulation and support vector machine. The results show that the classification accuracy rates of inner race fault, rolling body fault, outer race fault and combination of rolling body and outer race fault are 79%, 81%, 71% and 76%, respectively. The results verified that the proposed method has the ability to detect the fault types of bearing. This method is possible to be extended to detect faults in more complex mechanical systems.

Footnotes

Acknowledgments

The authors are grateful to the support from the National Science Foundation of China (Nos. 51575400, 51505339, 51405346), the Zhejiang Provincial Natural Science Foundation of China (LQ16E050005).

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