Sequential fault detection for sealed deep groove ball bearings of in-wheel motor in variable operating conditions

2.1. Feature extraction of bearing faults

It is generally known that the localized defects of each bearing-element generate a series of shock vibrations every time a running roller passes over the surfaces of the defects, and these vibrations occur at bearing characteristic frequencies, which are estimated based on the geometry of the bearing, its rotational speed, and the location of the defect. Moreover, the bearing characteristic frequencies capture in high-frequency domain [19]. By identifying the type of the occurring bearing characteristic frequency, envelope processing is used to extract the feature spectrum of bearing faults. Certainly, upper-envelope, lower envelope and full envelope are applied to detect faults. In this paper, absolute envelope processing is proposed, as shown in Fig. 1.

Here, ORD is used as example to introduce the absolute envelope processing. Firstly, band-pass filter is used to remove the noise signal outside the frequency domain captured by bearing faults. Where, $t_0$ expresses the pass-time of the defect, $f_1$ and $f_2$ are the cut-off frequencies of original spectrum. Secondly, the filtered signal is processed through absolute envelope, then to obtain an envelope signal which obviously presents shock characteristics. Finally, lower frequency is used to resample from envelope signal to a new wave, then the new wave is transformed into the frequency domain by fast Fourier transform (FFT). The pass-frequency $f_0$ corresponding to the pass-time of the defect $t_0$ and its harmonic pass-frequencies 2$f_0$, 3$f_0$, … will capture in the spectrum of the new wave. When a pure rolling motion is assumed, ball pass frequency outer (BPFO), ball pass frequency inner (BPFI) and ball spin frequency (BSF) can be estimated on the basis of the geometry of the bearing, its rotational speed, and the location of the defect [33-35].

Fig. 1Feature spectrums extraction using envelope processing

Ball pass frequency outer (BPFO) can be calculated as follows:

Ball pass frequency inner (BPFI) can be calculated as follows:

Ball spin frequency (BSF) can be calculated as follows:

where $n$ is the number of rolling elements, $f_r$ is the rotating frequency (Hz), $d$ is the diameter of rolling elements (mm), $D$ is the pitch diameter (mm), and $\theta$ is the contact angle of the rolling element (rad).

In practice, some sliding motion may occur on the working process of rolling bearing. Then slight deviation of the characteristic frequency locations can be caused. Therefore, these pass-frequencies calculated by above-mentioned equations should be regarded as theoretical value only, and the frequency region adjacent to theoretical value will be focused to identify the fault-type of rolling bearing.

2.2. Symptom parameters of fault feature

To perform intelligent diagnosis in the operating process, it is necessary that symptom parameters (SPs) are used to represent the fault features of machinery or parts. In the field of condition diagnosis, many SPs have been defined for fault diagnosis and fault-type identification [17, 19, 36]. Since bearing fault is a typical impact fault, and its characteristic frequencies generally capture in high-frequency domain, six dimensionless SPs in frequency domain are selected to represent the bearing fault features of in-wheel motor in this research, the signals processed by the absolute envelope method are used to calculate these SPs, as follows:

where, $N$ is the number of spectrum line, $f_i$ is the frequency, $F\left(f_i\right)$ is the power spectrum. ${\sigma }_f=\sqrt{\left({\sum }_{i=1}^N(f_i-\overline{f}{)}^2\cdot F(f_i)\right)/(N-1)}$ and $\overline{f}=\left({\sum }_{i=1}^N{f}_i\cdot F\left(f_i\right)\right)/{\sum }_{i=1}^{N}F\left(f_i\right)$.

2.3. Principal component analysis for Selecting sensitive SPs

Principal component analysis (PCA) is a multivariate statistical-analysis technique, in which a group of correlated variables are transformed into a new group of variables which are uncorrelated or orthogonal to each other [37, 38]. PCA can be done by eigenvalue decomposition of a data covariance matrix or the singular value decomposition of a data matrix. These decompositions are usually performed after mean centering the data for each attribute. The results of a PCA are usually discussed in terms of component scores and loadings. Recently, PCA has been applied to process fault diagnosis [39, 40].

To improve the rapidity of identifying targets, the PCA principle is illustrated for selecting two sensitive SPs from six SPs selected primitively on the basis of a training data set of $n$ samples (observations) and $m$ variables (SPs). While these data are standardized by subtracting from its mean and dividing its standard deviation, and the standardized data is denoted by a matrix $X(X\in R^{n\times m})$, it is easy to obtain the covariance matrix $A$ with the eigenvector ${\alpha }_i=({\alpha }_{i1},{\alpha }_{i2},\cdots ,{\alpha }_{im})$ and the eigenvalue ${\lambda }_i({\lambda }_1>{\lambda }_2>\cdots {>\lambda }_n>0)$. Then a new component $Y_i$ and its cumulative contribution rate ${\eta }_i$ are given as follows:

where $X_i=({SP}_{i1},{SP}_{i2},\cdots ,{SP}_{im})$ is each row vector of $X$ and $i=1,2,\cdots ,n$.

In general, when the values of ${\eta }_i$ are more than 85 %, the corresponding components $Y_i$ are called as principal components which contain most of the information and the discriminatory features. Here, the $i$ value corresponding to the minimum value of ${\eta }_i$ is labeled with $i_0$. Moreover, the principal components load can express the correlation degree between principal component $Y_i$ and original variable ${SP}_{ij}$. Therefore, the comprehensive load $C_j$ of ${SP}_j$ can be calculated as follows:

It has been proved that the larger the comprehensive load $C_j$, the higher the sensitivity of the corresponding ${SP}_j$ will be. In this study, the comprehensive load $C_j$ of principal components is used to select the sensitive SPs as the evaluation standard of SP’s sensitivity, and then select two sensitive SPs are selected to contain most of the vibration information each direction. Finally, two sensitive SPs from each direction are combined to build optimal composition of symptom parameters (SPOC).

3.1. Soft margin SVMs

SVM is a classifier which uses statistical learning theory to create an optimal classification hyper-plane between the two classes and ensure that the distance between the boundary and the nearest data point in each class is maximized [41-46]. SVM can efficiently perform not only a linear classification but a non-linear classification using kernel function. In general, a classification hyper-plane can be expressed as follows:

where, $x$ is the sample vector, $N$ is the dimensional number of $x$, $\omega$ is a vector represented the weight coefficients of $x$, $b$ is a classification threshold.

Recently, almost all of practical applications of SVMs have employed kernel functions to achieve linear classifications in high-dimensional feature spaces. However, it is difficult to choose an appropriate kernel function and determine the parameters for a given value of the regularization and kernel parameters. In order to increase the enchantment of SVM, SVM is modified to permit minimum error and relax the condition for the optimal classification hyper-plane, soft margin SVM is proposed for a fuzzy inference system [47, 48].

3.2. Sequential diagnosis system

Sequential diagnosis system is presented to monitor machine condition and identify fault types, as shown in Fig. 2. The first soft margin SVM is used to detect faults while the others are used to identify sequentially fault types. Moreover, SPOCs are employed as the object of follow-on process to put a SVM in each step, then obtain the optimal hyper-plane until whole diagnosis system is established.

Fig. 2Sequential diagnosis system based on SPs and SVMs

4.1. Experimental system

Fig. 3 shows the experimental equipment for the bearing fault test of in-wheel motor, and the place of accelerometer and diagnosis bearing of in-wheel motor. The parameters of the in-wheel motor is listed that rated power is 350 W, rated voltage is 48 V, rated speed is 300 rpm. 6202RS bearings were utilized, which information are listed in Table 1. In order to research whether the most commonly occurring bearing faults of in-wheel motor are detectable, the faults of diagnosis bearings were artificially made using a wire-cutting machine in the ORD, IRD or RED, and the defect sizes were the same: the width was 0.2 mm and the depth was 0.1 mm. Moreover, a 3-axis accelerometer (Type: CXL25GP3; Sensitivity: 80 mV/g; Range: ±25 g) was mounted on the holder which was fixed the stator axis of in-wheel motor. The vibration signals of normal state and three faults were measured with a sampling frequency of 204.8kHz, and the sampling time is 20 s. Moreover, when the operating speed was operated from 200 to 300 rpm, BPFO, BPFI and BSF are 6.1-9.1 Hz, 20.6-30.9 Hz, and 4.3-6.4 Hz, respectively.

Fig. 3Experimental equipment for fault diagnosis

a) Experimental bench of in-wheel motor

b) Sealed deep groove ball bearing

4.2. Data processing

The vibration signals collected from bearing experiment were divided into 60 data samples, respectively. Each data sample was actually a data series containing 65536 data points, and was processed by absolute envelope method. Here, band-pass filter (10 kHz was lower cut-off frequency and 100 kHz was upper cut-off frequency) was selected, and new data sample was measured by with a new resampling frequency of 2 kHz from the curve of absolute envelope. Finally, new data sample was used to calculate six SPs. Therefore, 60 data samples were obtained in each condition.

According to the test experiment and the sequential diagnosis method as shown in Fig. 2, the diagnostic system was designed four steps. The goal of the first step will focus whether the diagnosis condition is normal state (N). If the diagnosis condition is N, the system will stop. Otherwise, the system will come in the second step, and identify whether it is ORD. If not, diagnosis process will push forward the third step, and judge whether it is IRD. Similarly, RED will be distinguished in the fourth step. Certainly, there are more fault types in real plant, more steps can be continued in sequential diagnosis system. In order to fleetly detect fault and identify fault type, two SPs have been selected to aim toward the goal of each step by PCA from each direction, as shown in Table 2. Here, $S{P}_i$, $S{P}_i^{\text{'}}$ and $S{P}_i^{\text{'}\text{'}}$ ($i=$ 1, 2,…, 6) are used to reflect the corresponding SPs from vertical, horizontal and axial directions, respectively. Then the SPOCs are confirmed in each step.

4.3. Diagnosis system establishment and diagnosis results

To verify the effectiveness of the proposed method for diagnosing the faults of motor bearing, 30 data samples obtained from three directions were used to obtain 30 datasets of SPOCs in each condition, then to train the diagnosis system. Firstly, the establishment of the diagnosis system was described. In the first step, N was the diagnosis goal as one class with label 1. Then each $SPOC$$x_i=\left\{S{P}_{1i},S{P}_{4i},S{P}_{2i}^{\text{'}},S{P}_{3i}^{\text{'}},S{P}_{1i}^{\text{'}\text{'}},S{P}_{6i}^{\text{'}\text{'}}\right\}$ and its status label $y_i=$ 1 ($i=$1, 2, …, 30) were built to identify whether it is $N$. Another class was abnormal state (UN) which included three bearing faults of ORD, IRD and RED with label –1, each SPOC $x_j=\left\{S{P}_{1i},S{P}_{4i},S{P}_{2i}^{\text{'}},S{P}_{3i}^{\text{'}},S{P}_{1i}^{\text{'}\text{'}},S{P}_{6i}^{\text{'}\text{'}}\right\}$ and its status label $y_j=$ –1 ($j=$ 1, 2, …, 90) were built to affirm whether it is UN. Secondly, the SPOCs and its status labels of N and UN were combined to input into a SVM as the training data, shown as $X=\left[x_i;x_j\right]$ and $Y=\left[y_i;y_j\right]$. Since soft margin SVM was adopted, minimum error and relax coefficient were designed. In this research, it was proved to be optimal that minimum error was 5 % and relax coefficient was 0.05. Moreover, the weight coefficients $\omega$ of $X$ and classification threshold $b$ were determined, then the optimal classification hyper-plane between N and UN was obtained, as shown in Table 3. Similarly, the goal state of the second step was ORD to label with 1, other faults (UORD) were regarded as another class with label –1. In the third step, IRD was designed for the diagnosis goal with label 1, other faults (UIRD) were regarded as another class with label –1. In the fourth step, RED was designed for diagnosis goal with label 1, other faults (URED) were regarded as another class with label –1. Then the training data of each step were determined to obtain the corresponding optimal classification hyper-plane, respectively. The optimal classification hyper-plane of each step can be expressed as shown in Table 3. Then the diagnosis system was established.

Secondly, other 30 data samples of each condition were used to test the performance of the diagnosis system, then SPOCs of each condition in each step were selected according to Table 2. Here, the test data of ORD was used to introduce the proposed method. In the first step, the test data with SPOC $x_i=\left\{S{P}_{1i},S{P}_{4i},S{P}_{2i}^{\text{'}},S{P}_{3i}^{\text{'}},S{P}_{1i}^{\text{'}\text{'}},S{P}_{6i}^{\text{'}\text{'}}\right\}$ was input into the first SVM classifier, it was obtained that the probability which the diagnosing condition was N state was 12.5 %. Obviously, the diagnosing condition was judged to be $UN$ state, and then the diagnosis system turned into the second step. Therefore, SPOC $x_i=\left\{S{P}_{2i},S{P}_{5i},S{P}_{3i}^{\text{'}},S{P}_{4i}^{\text{'}},S{P}_{1i}^{\text{'}\text{'}},S{P}_{5i}^{\text{'}\text{'}}\right\}$ was used to identify whether it is ORD state. When inputting the new test data into the second SVM, the probability which the diagnosing condition was ORD state was 90.9 %, and the diagnosis system stopped. It is correct that the diagnosing condition was judged to be ORD state. Similarly, the test data of other diagnosing conditions were input to the diagnosis system respectively, the diagnosis results were shown in Table 4.

All the test results shown above verified that the sequential diagnosis method based on vibration information are available for diagnosing sealed deep groove ball bearings (SDGBBs) used in in-wheel motor.

4.4. Experimental comparison

To verify the effectiveness of absolute envelope processing (AEP), FT, WT, EMD, and STFT were used to extract the features of SDGBBs’ faults on the basis of the same experiment data. Similarly, each method was performed with appropriate parameter, respectively. In this paper, FT was executed with 10 kHz of lower cut-off frequency and 100 kHz of upper cut-off frequency. In the process of WT, Morlet wavelet was chosen as a mother wavelet function. For STFT, Gaussian window was used as a window function. Secondly, six SPs defined in Section 2.2 were calculated. Finally, PCA was employed to select the sensitive SPs and the comprehensive load was used to compare the SPs’ sensitivity. Moreover, the working time of each single processing was considered. Table 5 displays the results and correlative information from different directions in the first step. Sensitive SPs selected by each method and comprehensive loads have difference, however these differences except FT were smaller. But the working times of WT, EMD and STFT were three times more than AEP’s working time. Though the working time of FT was less, the SPs’ sensitivity was also lower. Moreover, the results and correlative information in other steps were similar to the first step. When all of these methods were considered together, it was not hard to know AEP’s advantage which AEP not only extracts effectively the features of SDGBBs’ faults, but also improves the progress of signal processing.

To verify the effectiveness of sequential fault diagnosis on the basis of soft margin SVMs, SVMs with different kernel functions and neural networks (NNs) were performed respectively. In SVMs’ work, polynomial and Gaussian RBF functions are used to obtain the optimal hyper-plane. When the degree of the polynomial with 1, 2, 3 and the width of the Gaussian RBF kernel parameter with 0.2, 0.4, 1.6 were selected respectively to train the sequential diagnosis system, it was verified that the training accuracies were more than 96.5 % when the polynomial degree was 2 in the second and third steps, and the Gaussian RBF kernel parameter was 0.4 in the first and fourth steps. Therefore, the first comparison system was built on the basis of SVMs with two polynomial functions and two Gaussian RBF functions. Then the diagnosis results of the same test data were shown in Table 6. Though each state was rightly judged, the diagnosis accuracy of each step was obviously lower than the diagnosis results of the proposed method as shown in Table 4.

In NNs’ application, the diagnosis system was still considered to divide four steps. The NN’s structure of each step was set as four layers. The first layer was set six neurons to input SPOC, the last layer was set two neurons to output the diagnosis the goal state and other state, and the second and third layers were middle layers. The neuron numbers of middle layers were selected respectively for 4, 5, 6,…, 10 to train the sequential diagnosis system, it was verified that the training accuracy was more than 95 % in each step when the neuron numbers of the second and third layers were 8 and 5. Therefore, the second comparison system was built on the basis of NNs. Then the diagnosis results of the same test data were shown in Table 7. It was obvious that each diagnosing state was rightly judged, but most part of the diagnosis accuracies were also lower than the diagnosis results using soft margin SVMs.

Moreover, the training time and test time of each method were focused in Table 8. Since there were 120, 90, 60, 60 data samples which were sequentially input to train the classifier of the diagnosis system in each step, the training time of each method in the first step was more than in the second step, and the training time of second step was more than in the third and fourth steps. For test data, 30 samples of each state were used to test the diagnosis accuracy of each step, so the test times of all steps were nearly same for the same method. However, there were obvious differences for different methods, and the training time and test time using soft margin SVMs were nearly half of spending times by other methods.

Throughout all the test results, it was verified that sequential diagnosis method based on soft margin SVMs is available and rapid for realizing the on-line application of SDGBBs used in in-wheel motor.

Hongtao Xue contributed to conceive and design the methodology, to write the manuscript. Man Wang mainly implemented the methodology, ran and analyzed the experiments. Zhongxing Li and Peng Chen analyzed the results and reviewed the paper.