Evaluation of rolling bearing performance degradation using autoregressive model energy ratio and support vector data description

Fa-ling WANG, Jian-min ZHOU, Chen-chen ZHANG, Wen-hao YIN, Long ZHANG

(Key Laboratory of Conveyance and Equipment of Ministry of Education, East China Jiaotong University, Nanchang 330013, China)

Abstract: Rolling bearings have different degrees of degradation in performance during long-term work. If the degraded state of the rolling bearing can be identified, maintenance measures can be taken. Aiming at the performance degradation evaluation of rolling bearings, a method for evaluating the degradation of rolling bearing performance is proposed, which combines the vibration signal autoregressive model (AR) energy ratio and the support vector data description (SVDD). Firstly, the residual components of the vibration signal in the whole life cycle of the bearing are obtained by AR filtering, and the energy ratio is calculated as the feature vector of the bearing state. Then, SVDD is trained by using the feature vector of the bearing under normal state, and the hypersphere under normal state is obtained. The relative distance between the feature vector of bearing life cycle sample and the hypersphere is used as the quantitative evaluation index of bearing performance degradation. Finally, the early fault alarm threshold is set to determine the early fault point. The results show that compared with the performance degradation assessment methods of common monitoring indicators, the early fault detection capability of the proposed method is stronger, and the description of each stage of bearing performance degradation is more accurate.

Key words: Rolling bearing, Performance degradation assessment, AR model, Support vector data description,Boxplot

1 Introduction

Rolling bearings, as important components of rotating machinery, not only support the load but also allow relative motion [1-2]. They are also common failure units due to their complex running conditions. The performances of bearings directly affects the operation reliability of the whole equipment [3- 4], therefore, fault diagnosis and degradation assessment based on condition monitoring have been a key means to ensure the reliable operation of equipment, reduce the production downtime and save maintenance costs, etc.[5].

The process of fault diagnosis and performance degradation evaluation of bearing generally includes: data acquisition, data processing and data analysis. It is a common method to obtain vibration signal of bearing by a sensor. Vibration analysis of vibration signal is a common means of fault diagnosis and degradation evaluation [6-8]. The performance of rolling bearings is influenced by many factors, such as rotation speed, temperature, and lubrication conditions [9]. The running state is the final results of those factors. The vibrations reflect synthetically the present running state of the bearing. Many researches have been developed about the feature extraction from the vibration signals to assess the performance degradation. Time-domain features and frequency-domain features are the first choice because of the easy calculation and the definite physical meaning. Root mean square (RMS), Kurtosis Factor, the average amplitude of the defective frequency and its first six harmonics were successively used in [10-11], but the original features have different advantages and limitations as well. For example, RMS is a good stability feature which steadily grows with the fault development of the bearing. However, it is difficult to discover the incipient defects by RMS. On the contrary, Kurtosis Factor is sensitive to impulse faults. It distinctly appears at the initial fault stage while it decreases with the fault development. Kurtosis Factor has a high sensitivity for incipient faults, but the poor stability for the serious damage. Few original features satisfy the conditions of sensitivity and stability simultaneously. A composite index is necessary which is both sensitive to the initial defect and rises stably as the damage grows.

In recent decades, many advanced signal processing techniques have been proposed for extracting bearing fault features from measured vibration signals, such as envelope analysis, wavelet [12], multiwavelet [13], dual-tree complex wavelet [14], empirical mode decomposition [15], cyclostationarity [16], spectral kurtosis [17], ensemble empirical mode decomposition [18], variational mode decomposition[19,20] and stochastic resonance [21]. Although the proposed methods have their own advantages, they still have some deficiency. For example, a wavelet method is not sensitive to nonlinear signal. Empirical mode decomposition is prone to modal aliasing. Although the variational mode decomposition can solve the empirical mode decomposition mode aliasing problem, it needs to use the optimization algorithm to select the preset scale and penalty factor, which increases the complexity and difficulty of the operation.

Therefore, scholars turn their attention to how to structure an intelligent assessment model based on the original features. Several scholars have proposed some comprehensive indexes and obtained impressive results. Hong et al. used self-organizing map neural network (SOM) to evaluate the health status of rolling bearings, train SOM with data in the nonfault state, and obtain health monitoring values from the data under test [22]. Liu et al. used the firefly optimization algorithm to obtain the optimal initial weight and threshold of BP neural network, and used the optimized BP neural network to evaluate the degree of bearing performance degradation [23]. However, BP neural network has problems such as local minimization and different network structure selection. The SOM algorithm converges slowly and the network size is difficult to determine. In addition to the artificial neural network model, there are methods such as support vector machine [24], hidden Markov model [25], and Gaussian mixture model [26]. But there are some problems in these methods. For example, the evaluation model established by SVM generally requires a large number of fault samples to participate in the training, but it is difficult to obtain a large number of data samples with different verification levels of equipment failure in practical applications. HMM uses exponential distribution to describe the interval length of each health state, which is usually not quite consistent with the actual situation. Support vector data description (SVDD) is a single value classification method based on the boundary idea proposed by Tax et al. [27], which has good robustness and superior computing performance. In addition, this method only needs normal samples for model training, which provides a solution to the problem of abnormal data shortage in fault diagnosis. Therefore, SVDD has been widely used in degradation assessment of bearings [28-29].

Autoregressive (AR) model is a time sequence analysis method whose parameters comprise important information of the system condition. An accurate AR model can reflect the characteristics of the dynamic system. Additionally, it is indicated that the parameters of AR model are very sensitive to the condition variation [30]. In many cases, the AR parameters are estimated by using the least squares method or the Yule-Walker equation method based on correlation function for the stationary signals. When the AR model is applied to the non-stationary signals, its parameters are difficult to estimate by these approaches. However, bearing fault vibration signals composed of a series of impulse signals and intermixed various noise are typical non-stationary and non-Gaussian signals. Aiming at this problem, Ref. [31] proposed an approach combining empirical mode decomposition (EMD) method and AR model. The EMD method was used as a pretreatment to decompose the nonstationary vibration signal of a roller bearing into a number of intrinsic mode function (IMF) components which are stationary, then the AR model of each IMF component was established, and the means of AR parameters of the IMF components were regarded as the feature vectors. Nevertheless, due to the defects of the algorithm, mode mixing is an inevitable problem in EMD. When mode mixing arises, single IMF contains widely different characteristic time scales or a similar time scale appears in adjacent IMFs. The problem would cause each IMF to not be able to reflect the real physical meaning. In this paper, an autoregressive model (AR) is applied to separate the original vibration signal of the bearing into the random parts and the deterministic parts and then the energy ratio between the random parts and the original signal is calculated as the fault feature. The energy ratio characteristics are not only sensitive to fault characteristics, but also can fully reflect the degradation trend of bearing performance.

Based on SVDD model, this paper takes the AR model energy ratio of rolling bearing vibration signal as the characteristic sample to establish the performance degradation evaluation model of bearing, so as to obtain the performance degradation evaluation index of bearing, and set the early failure threshold. Finally, the validity and feasibility of the method are verified by the experimental data of bearing life cycle, and the correctness of the evaluation results is verified.

2 Theoretical background

2.1 Autoregressive model method

The vibration signal collected by the bearing failure test can be composed of the deterministic periodic signal and the random components, which is caused by the imbalance and misalignment of the shaft and random components [32]. The random components consist of the impulses generated by the fault bearing, and the random noise. It can be expressed by Eq.1.

Sk=(gk+nk+dk)*hk

(1)

Wheregkis the fault impulse,nkis noise,dkis the periodic part andhkis the transmission path effect.

When a fault develops, the periodic signal generated by the shaft will remain the same, but then the energy of the component will increase. So the energy ratio between the random components and the original signal will increase. This characteristic enables the energy ratio to fully reflect the degradation trend of bearing performance. AR model is used to remove the periodic part of the vibration signal and the AR model residual signal only consists of the random parts of the signal. If the bearing remains in a healthy condition, the AR residual signal will only represent the random prediction error of the AR modal [39].

The use of model can be described as

(2)

The vibration signal is filtered by an AR filter and the residual signal is obtained by subtract the original signal. And then the energy ratio between the residual signal and the filtering signal is calculated. The energy ratio is used as the feature of the rolling element bearing. The energy of a discrete signal series (s) of length N is defined as Eq. 3 and the energy ratio is defined as Eq.4:

(4)

Where,siis the discrete signal series consisting of N elements,Ykis the periodic parts of the discrete signal consisting of N elements andεiis the random parts of the discrete signal consisting ofNelements.

Fig.1 shows the process of extracting the energy ratio as a characteristic from the vibration signal. Where (a) is the scheme of the AR prediction filter and (b) is the process diagram from vibration signal to feature formation.

Fig.1 The process of extracting the energy ratio as a characteristic from the vibration signal

The choice of the order of the AR model is very important for the accuracy of the fault diagnosis results. There are many related literatures at home and abroad that have conducted in-depth research on this problem. Among them, the most widely used is the information quantity criterion discriminant method, which mainly includes AIC standard, FPE standard, MDL standard and CAT standard. In this paper, the AIC criterion is used to select the order of the AR model. According to the AIC criterion, the order of the AR model is 14th order.

2.2 Support vector data description

Support vector data description (SVDD) proposed by Tax and Duin is inspired by the theory of support vector machine (SVM) proposed by Vapnik [33]. The main idea of SVDD is to find an optimal hypersphere with minimal volume containing all or most targets, as shown in Fig.2.

Fig.2 Schematic of the two-dimensional SVDD

Consider a training set {xi,i=1,2,…,n},nis the total number of samples. The optimal supersphere containing all or most of the normal samples is sought. This hypersphere is described by centercand radiusR, and satisfies the following optimizational function

(5)

whereCis a penalty parameter which controls the tradeoff between the volume of hypersphere and errors, andξiis slack variables which permit a few training data to be outside the hypersphere.

Generally speaking, Equation (5) is solved by introducing Lagrange multipliers and it can be transformed into the following maximizing functionLwith respect to the Lagrange multipliersαi

(6)

Since the data in the input space are not always linearly predicted, a kernel functionK(xi,yj)=(Φ(xi)·Φ(yj)) is introduced to replace the inner product (xi,yj), whereKis a Mercer kernel. The kernel functionK(xi,yj) can map the data into a high-dimensional feature space and transform the nonlinear problem to a linear model. Any function meeting the Mercer’s theorem can be employed as kernel function, but not all of them are useful for SVDD. Gaussian kernel is the most commonly used function. It is defined as follow

(7)

Whereσis the width parameter. Since Gaussian kernel can restrain the growing distances for large feature spaces, it is regarded asK(xi,yj). Then Equation (6) becomes

(8)

Allαiare got by solving Equation (8) and only a few of them are nonzero. The samples withαi>0 are called support vectors. Then the radius R is obtained by any support vectorxsv

(9)

For a new samplexw, its distance to the center c can be described as follow

(10)

In this paper, the theory of SVDD is introduced into the evaluation of bearing performance degradation. Then the relative distance between the new samplexwand the hypersphere boundary can be used as the degradation value (DV) ofxw. It is defined as follow

DV=(Rw-R)/R

(11)

IfDV≤0,xwis accepted as a target which indicates that the bearing runs in a normal state. Otherwise, it is an outlier which indicates that the bearing runs in a degradation state.

2.3 Adaptive threshold setting

Box-plot consists of five statistics in the data: upper limit (Maximum value in non-abnormal range), upper quartile, median, lower quartile and lower bound (Minimum value in non-abnormal range). It is a method for describing data, which can be used to identify data outliers, compare the shapes of several batches of data, and so on. As shown in Fig. 3. This paper uses it as an adaptive threshold to determine the data outliers. The abnormality data in the bearing life is identified by the data of the performance degradation index, and the fault diagnosis is performed [34].

Fig.3 Box plot

The basic principle of box plot analysis is to arrange the data from small to large and calculate the quartile of the data. Calculate the anomaly of the data by quartiles:

K≤L1-1.5(L3-L1)

(12)

K≥L3+1.5(L3-L1)

(13)

WhereKis an outlier, that is, the data is in an abnormal state;L1is the upper quartile;L3is the lower quartile.

The drawing of the box plot is based on the actual data to calculate the quartiles. The drawing process does not require the form of the data, so the actual appearance of the data can be expressed more intuitively. Up to a quarter of the data can be moved away from normal data without affecting the quartile to a large extent, so outliers do not affect the drawing criteria of the box plot, so the effect of identifying outliers is more objective. In summary, the box plot has certain advantages in judging data outliers [35]. Therefore, this paper selects the box plot as the abnormal data identification method, and selects the maximum value of the box plot as the adaptive alarm threshold for performance degradation assessment.

The box-plot diagram is introduced as a method for setting the adaptive alarm threshold for performance degradation evaluation of rolling bearings. If there are consecutive multiple evaluation index values exceeding the abnormality of the box plot defined by the evaluation index value, it indicates that the performance degradation state of the bearing has undergone a large change. At the same time, an adaptive alarm line that changes with time can be obtained according to the constant change of the DI value.

3 Establishment of performance degradation assessment model

In this paper, a performance degradation assessment method of bearings based on AR energy ratio and SVDD is proposed. The steps of the proposed method are illustrated as follows.

Step1: The full life cycle signal of the rolling bearing are Collected. The AR model order is determined by the AIC criterion. The bearing life cycle signal is filtered using an AR model.

Step2: The ratio of filtered residual component energy to filtered signal energy is calculated, which is taken as the characteristic.

Step3: The features obtained in step 2 are divided into training samples and test samples to establish an SVDD model. Where the test sample is. Then, the radiusRcan be obtained by (10).

Step4: As for the test samplexw, the generalized distance is calculated using (11) which is related to the model established in step 3.

Step5: The degradation value (DV) ofxwis calculated using Equation (12). Then the degradation index with a series of DVs of testing signals can be obtained. Here set all the values of DV to 0 whenDV≤0.

Step6: From the degradation index that the bearing runs in a normal state whileDV≤0. Otherwise, it runs in a degradation state. Moreover, DV also reflects the degree of fault severity of a bearing, namely the larger DV means the larger degree of fault severity.

The SVDD performance degradation assessment model is shown in Fig.4.

Fig.4 Performance degradation assessment model

4 Experiment and result analysis

4.1 Description of the experiment

The experimental data is based on the bearing fatigue life test bench of the Intelligent Maintenance System Center of the University of Cincinnati, USA [36]. The test bench is shown in Fig.5:

Fig.5 Rolling bearing accelerated fatigue life test bench

The test bench spindle is equipped with four double row roller bearings and the bearing type Rexnord ZA-2115. The radial load of the bearing is approximately 6 000 pounds. The rotation speed is maintained at 2 000 r/min. The data is collected once every 10 minutes with the NI DAQ-6062E data acquisition card. The sampling frequency is 20 kHz. The data is collected for 1 s each time, and the sampling length is 20 480 points. Install the acceleration sensor PCB353B33 in the horizontal and vertical positions. The oil circulation system is used to adjust the flow rate and temperature parameters of the lubricating oil. The magnetic plug is mounted inside the conduit of the feedback oil. Three sets of data were obtained in the experiment. In this paper, a total of 984 samples of the second set of data were used for performance degradation evaluation. Since the vibration signals of the last two samples of the bearing were abnormal, the total number of samples was 982.

4.2 SVDD model evaluation results

First, the first 200 groups of normal data were selected as training samples.AR filtering was performed on each group of data to obtain the energy ratio, and the eigenvector was constructed to establish SVDD model under normal state.

When the SVDD model is established, the penalty factor C and the kernel function parameter σ have a certain influence on the evaluation result. The largerC, the more anomalous samples that are excluded from the model, and the model imposes stricter constraints on the SVDD hypersphere, increasing the number of support vectors. WhenCis constant, the value ofσwithin a certain range can make the hypersphere relatively stable, that is, the number of stable support vectors is generated. As a training SVDD model, it is desirable that the SVDD hypersphere can represent a range of data samples, and it is not expected to have too many constraints on the range. Therefore, the selection ofCshould avoid excessive support vectors, and ensure that the value ofσcan make the hypersphere relatively stable. Based on this, this paper setsC=0.1,σ=1, so that the established SVDD model is relatively loose and stable [37].

After obtaining the SVDD hypersphere, 982 sets of data within the whole life cycle of the bearing were taken as samples to be tested, and the eigenvector was obtained according to the above method, so as to obtain the DI value within the whole life cycle of the bearing. In order to eliminate the influence of burrs caused by interference signals, five-point sliding average method was used to smooth the curve, and the performance degradation evaluation authority was obtained as shown in Fig.6.The solid line is the performance degradation evaluation curve and the red dotted line is the adaptive alarm line of the early failure threshold.

Fig.6 Performance degradation evaluation results based on AR energy ratio and SVDD model

As can be seen from the Fig.6, at the 533rd sample, the bearing began to have an early failure, and then the DI value increased significantly, indicating that the failure began to deepen. From the 697 sample, the curve has been rising and started to oscillate repeatedly, indicating that the bearing failure has become serious and began to deteriorate. After the 965th sample, the curve showed a slight increase and then a sharp decline, indicating that the serious failure deteriorated rapidly after the minor wear, and the bearing was also close to failure.

From the above analysis, the performance degradation process of the bearing in the experiment can be divided into four stages, which are the normal bearing stage (1～532), the early failure stage (533～696), the severe failure stage (697～964), and the deterioration to the Failure phase (965～982). If effective maintenance measures can be taken in the early failure phase, equipment is shut down in the near-failure phase and bearings are replaced in time to avoid unnecessary economic losses.

The RMS and kurtosis indicators are commonly used monitoring indicators for monitoring equipment operating conditions [38]. RMS is a stability indicator whose value generally increases as the degree of failure increases. The kurtosis indicator is a sensitivity indicator that is generally sensitive to early failures. The RMS value change over the life cycle is shown in Fig.7.

Fig.7 RMS value over the life cycle

It can be seen from Fig.7 that the earliest fault point that can be detected by the RMS value is the 535th time, which lags behind the evaluation results based on the wavelet packet energy entropy and the RBF neural network by two moments (that is, lags by 20 minutes). The RMS value can only be judged when the bearing fault has sharply deepened. In the sharp deterioration phase (701 to 964 moments), the root mean square value also appears to increase and decrease repeatedly. This shows that the bearing fault at this stage does have repeated deepening and smoothing, and the severity of the change in the RMS value is far less strong than the evaluation method proposed in this paper.

Then analyze the change of the kurtosis index in the whole life cycle, as shown in Fig.8.

Fig.8 Kurtosis index over the life cycle

It can be seen from Fig.8 that the earliest fault point that can be detected by the kurtosis index is the 648th time, but it lags 115 times (that is, 1150 minutes later) than the performance degradation evaluation method based on AR energy ratio and SVDD.

It can find from the article of Hu et al that in the performance degradation evaluation of the same set of data, the detection of early failure point is the 541st sample, which is 80 minutes later than the method proposed in this paper [39].

From the above analysis, the performance degradation evaluation results of AR energy ratio and SVDD model proposed in this paper can judge the early failure point earlier, and the performance degradation evaluation curve is consistent with the failure degradation trend of the rolling bearing.

5 Conclusion

According to the vibration characteristics of the bearing after damage, the energy ratio of the residual component and the filtered signal is used as the characteristic after the AR model is filtered, which is applied to the evaluation of bearing performance degradation. In view of the advantages of SVDD, a method for evaluating the degradation of rolling bearing performance based on AR energy ratio and SVDD is proposed. Calculate and obtain the performance degradation evaluation index of the bearing to solve the problem of setting the early fault threshold. Through the analysis of the bearing life cycle test data, the results show that the method is more effective, can find early faults more timely and track the development trend of the fault well, and the consistency with the fault degree change is better, and can Set early failure thresholds based on actual conditions to provide a stronger basis for the development of equipment maintenance plans.