Singular spectrum analysis and continuous hidden Markov model for rolling element bearing fault diagnosis

Abstract

Rolling element bearings are vital components in rotating machines, and it is important to diagnose bearing faults to avoid serious accidents in equipment. In this paper, singular spectrum analysis (SSA) is utilized to extract the bearing fault features. SSA is a non-parametric technique of time series analysis which decomposes the acquired vibration signals into an additive set of time series. Based on the selected singular features from SSA, a continuous hidden Markov model (CHMM) is introduced to diagnose the bearing fault. The detailed description and identification results of applying the proposed method to rolling element bearing fault diagnosis are shown in experiment 1. In experiment 2, a rolling element bearing accelerated life test is performed to simulate the performance variation of the bearing. The result demonstrates that the singular features and CHMM can reflect the performance degradation of the bearing from health to failure. A conclusion can be made that SSA and CHMM are feasible and effective in bearing fault diagnosis and performance assessment.

Keywords

Continuous hidden Markov model fault diagnosis performance degradation assessment rolling element bearing singular spectrum analysis

1. Introduction

Rolling element bearings are pivotal in most rotating machines, and faults in bearings are among the main causes of equipment breakdown. Hence, the detection and diagnosis of rolling element bearings are important to prevent malfunction and failure (Cong et al., 2013). When fault occurs in the bearing, no matter how light it is, it will cause a series of faults in connected components (Ming et al., 2011). Based on condition monitoring of the bearing, faults can be recognized by the changes in vibration signature using signal processing techniques. Therefore, vibration signals generated by faults have been widely studied, many powerful diagnostic techniques have been applied in this field (Dong et al., 2011; Randall and Antoni, 2011; Wang and Liang, 2011).

When a possible fault occurs, there are two assignments in fault diagnosis, namely feature extraction (FE) of the defective bearing and identification of the fault. The aim of signal processing is to extract the information related to bearing health for better understanding and interpretation of the bearing’s condition (Lei et al., 2007). Considerable research has been implemented in FE. The time domain analysis such as zero-crossing (William and Hoffman, 2011) and difference histograms (van Wyk et al., 2009) have been introduced to bearing fault diagnosis. Some statistical indicators such as root mean square (RMS), kurtosis, crest factor etc. have also been performed in bearing fault diagnosis (Cong et al., 2011). Based on the fast Fourier transform (FFT), envelope spectrum analysis (Feldman, 2011), spectral kurtosis (Antoni and Randall, 2006) and spectral entropy (Pan et al., 2009) etc. have also been applied to distinguish bearing faults. Furthermore, several time-frequency analysis methods have been employed in extracting the bearing features, which include, for example, wavelet analysis (Pan et al., 2010; Su et al., 2010), empirical mode decomposition (EMD) (Zhao et al., 2012). The singular spectrum analysis (SSA) is a non-parametric technique which incorporates the elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical system and signal processing (Hassani, 2010). Based on the singular value decomposition (SVD) method, the original time signal can be decomposed and reconstructed, and the principal components of the signal can be selected to identify the fault. The SSA provides a lower noise feature than the original signal (Cadzow, 1988), so it has been applied in hydrology forecasting (Marques et al., 2006), EMG on-set detection (Vaisman et al., 2010), tool wear monitoring (Kilundu et al., 2011) and headstock fault diagnosis (Zhao and Ye, 2011), etc. Because the SSA is effective in reducing the noise of the signal, it is adopted to extract the features of bearing, and the selected singular values can be utilized to diagnose and assess the performance of the rolling element bearing.

Another task in bearing fault diagnosis is how to build an intelligent assessment model after extracting feature vectors. Numerous algorithms have been proposed in bearing fault classification, which include the support vector machine (SVM) (Wang et al., 2012), artificial neural network (ANN) (de Moura et al., 2011), decision tree (DT) (Sun et al., 2007), etc. In this work, a continuous hidden Markov model (CHMM) is introduced for bearing fault classification. CHMM is a time series statistical model which provides a rich mathematical structure and a solid theoretical basis (Nair and Sreenivas, 2010). Therefore, it has been widely used in speech analysis, image recognition, electroencephalogram estimation and bearing fault identification (Lee et al., 2004; Liu et al., 2010; Park and Lee, 2011; Andersson et al., 2012). The selected singular features (SFs) are adopted as the input of CHMM, and the bearing fault can be classified by the proposed method. The following part of the paper will give the details of the SSA and CHMM for rolling element bearing fault diagnosis.

The paper is organized as follows: first, the theory of SSA and CHMM is introduced. Then the detailed process of SSA and CHMM is illustrated for the bearing fault diagnosis. In Section 3, data from a rolling element bearing experiment are employed to validate the effectiveness of SSA and CHMM in fault diagnosis. In Section 4, an accelerated bearing life test is performed to obtain the bearing performance variety in a whole lifetime. SSA and CHMM are utilized to estimate the change of bearing performance. Our conclusion is presented in Section 5.

2. Fault detection theory based on SSA and CHMM

2.1. Singular spectrum analysis

The aim of SSA is to make a decomposition of the original series into the sum of a small number of independent and interpretable components such as slowly varying trend, oscillatory components and structureless noise (Hassani, 2010). The SSA method builds a trajectory matrix from an original time series. The matrix is then processed by SVD and gets a sum of unit-rank matrix known as elementary matrices. Each of these matrices can be transformed into a reconstructed time series by taking the average of the diagonals. These time series are called principal components (Salgado and Alonso, 2006).

The vibration signal acquired from a rotating machine can always be expressed as a time series. And the details of SSA can be expressed as follows:

Embedding. Given a time series X, where $X = (x 1, x 2 \dots, x p)$ . Let L be the window length, with 1 < L < P. The one dimensional original time series X is transferred into a sequence with L dimensional vectors H_i by a succession of overlapping views of the series through a sliding L point window. Here, $H i = (x i, x i + 1 \dots, x i - L + 1) T$ , $i = 1, 2, \dots, K$ and $K = P - L + 1$ . The K vectors H_i will build an L × K trajectory matrix $H = [H 1, H 2, \dots, H K]$ and it is a Hankel matrix with equal elements of the diagonals, i.e. i + j = const. The form of the matrix is

H [x 1 x 2 \dots x P - L + 1 x 2 x 3 \dots x P - L + 2 x L - 1 x L - 2 \dots x P - 1 x L x L - 1 \dots x P]

(1)

Singular value decomposition. The trajectory matrix can be decomposed into a sum of unit rank, bi-orthogonal elementary matrices through SVD method. It can be proved that

H = H 1 + 2 + \dots + H d

(2)

where d is the number of non-zero eigenvalues of the matrix

S svd = HH T

. Let e₁, e₂ , … e_d be the non-zero eigenvalues of

S svd

(

d = L

if all eigenvalues are non-zero) in decreasing order. The elementary matrices are obtained by

H i = \sqrt{e i} U i W_{i}^{T}, i = 1, \dots, d

Here, U_i is the corresponding eigenvector and W_i can be calculated from

W i = H T U t / \sqrt{e i}, i = 1, \dots, d

. The obtained d singular values are the square root of the eigenvalues of matrix

S svd

. The plot of singular values arranged in decreasing order is known as singular spectrum (SS). And the singular spectrum reflects the different spectral components of the original signal.

Diagonal averaging (reconstruction). Each elementary matrix H_i can be transformed into a principal component by applying a linear transformation called diagonal averaging or Hankelization. The principal component is defined as $G i, G i = (g 1, g 2, \dots, g P) i$ and it is a new one dimensional time series with length P. The diagonal averaging algorithm is as follows (Salgado and Alonso, 2006): Let Y be a matrix of dimension $L \times K$ with elements $y ij, 1 \leq i \leq L, 1 \leq j \leq K$ . The diagonal averaging transforms the matrix Y into the one dimensional series $g 1, g 2, \dots, g P$ (principal component).

g k = {\frac{1}{k} \sum_{m = 1}^{k} y m, k - m + 1 for 1 \leq k \leq L * \frac{1}{L *} \sum_{m = 1}^{L *} y m, k - m + 1 for L * < k \leq K * \frac{1}{P - k + 1} \sum_{m = 1 = K * + 1}^{P - K * + 1} y m, k - m + 1 for K * + 1 < k \leq P + 1

(3)

where

L * = min (L, K), K * = max (L, K)

and

P = L + K - 1

. It is obvious that the initial series X is decomposed into the sum of d series

X = G 1 + G 2 + \dots + G P

(4)

where

G i = (g 1, g 2, \dots, g P) i

i = 1, 2, \dots, d

and

d = L

if no eigenvalue is zero in H_i. These singular values can be selected as SFs to detect the bearing fault.

Independence between principal components. The independence of two components can be measured by absolute value of correlation coefficient, which is given by

ρ ij (g i, g j) = | \frac{Cov (g i, g j)}{σ (g i) σ (g j)} | = | \frac{E [(g i - μ (g i)) (g j - μ (g j))]}{σ (g i) σ (g j)} |

(5)

where

Cov (g i, g j)

is the covariance between g_i and g_j, E is the expected value operator,

σ (g)

is the standard deviation of g. If the

ρ ij

is near zero, the components g_i and g_j are independent. If it is near 1, the two components are similar (Alonso and Salgado, 2008).

2.2. Continuous hidden Markov model

CHMM is a time series statistical model with a doubly embedded stochastic process, where the underlying stochastic process is not observable but can only be seen through another set of stochastic processes that produce the sequence of observations (Rabiner, 1989). The CHMM technique consists of two processes: training and estimating. And the definition of CHMM has been described in Lee et al. (2004) and Rabiner (1989).

2.2.1. Definition of CHMM

Before describing the CHMM, the components of the hidden Markov model (HMM) are defined as follows (Rabiner, 1989):

States. The number of states N in the model is associated with a specific condition of bearing such as health, inner race fault etc. The individual hidden states are denoted as $S = {S 1, S 2, \dots, S N}$ , and the state in time t is q_t, $q t \in S$ .

Observation symbols. The number of distinct observation symbols per state is M. Let $V = {V 1, V 2, \dots, V m}$ be the individual observation symbols. They correspond to the physical output of the system being modeled. An observation sequence can be defined as $O = {o 1, o 2, \dots, o T}$ , and it is obvious that the observation at time t can be defined as o_t. Each observation o_i is one of the symbols from V, and T is the number of observations in the sequence. The observation sequence can be looked on as different feature vectors of the bearing.

The state transition probability matrix A = {a_ij}, in which

a ij = P (q t + 1 = S j | q t = S i), 1 \leq i, j \leq N

(6)

with the property below:

\sum_{j = i}^{N} a ij = 1, \forall i

(7)

where i and j denote the state indices.

The observation probability matrix in state j, and $B = {b j (k)}$ , where

b j (k) = P (o t = v k | q t = S j), 1 \leq j \leq N, 1 \leq k \leq M

(8)

Similar to a_ij, b_j(k) has the following property

\sum_{k = i}^{M} b ij (k) = 1, 1 \leq j \leq N

(9)

The initial state distribution $π = {π i}$ , which is given by

π i = P (q 1 = S i), 1 \leq i \leq N

(10)

and

π i \geq 0

with the property

\sum_{i = 1}^{N} π i = 1

Generally, an HMM can be defined by the above five parameters: N, M, A, B and π. For convenience, the model is expressed with a compact notation

λ = (A, B, π)

(11)

CHMM is an extension of HMM in continuous signal. Therefore, it is advantageous to utilize HMM with continuous observation densities. Usually, a Gaussian mixture model (GMM) is adopted to ensure the observation probability of the model can be re-estimated in a consistent way (Rabiner, 1989; Zhu et al., 2009).

b j (o t) = \sum_{m = 1}^{M j} c jm N (o t, μ jm, Cov jm), 1 \leq j \leq N

(12)

where M_j is the number of Gaussian mixtures in state S_j, c_jm is the mixture weight of the mth Gaussian mixture in state S_j.

N (\cdot)

defines a Gaussian density with mean vector

μ jm

and covariance matrix Cov_jm for the mth Gaussian mixture in state S_j. In summary, a CHMM can be presented as

λ = (π, A, C, μ, Cov)

(13)

where

C = {c jm}

μ = {μ jm}

Cov = {Cov jm}

. And the mixture weight c_jm satisfies the stochastic constraint

{\sum_{m = 1}^{M} c jm = 1, 1 \leq j \leq N c jm \geq 0, 1 \leq j \leq N, 1 \leq m \leq M

(14)

2.2.2. CHMM model estimating

In order to get accurate and reliable results in diagnosis, the CHMM needs to be trained to generate the model parameters. The aim of training a CHMM is to adjust the parameters $λ = (π, A, C, μ, Cov)$ to maximize the probability of observation sequence O. That is, the object is to find $λ *$ , where

λ * = \arg max P (O | λ)

(15)

Figure 1 shows the training procedure of CHMM, and the details are as follows:

Take the training samples. Here, the training samples can be the various features extracted from original data. And it can be viewed as the observation vectors in CHMM.

Initialize the parameters of CHMM. The state transition probability distribution and initial state distribution are assigned arbitrarily subjected to equations (6) and (10) respectively. There are N observations and M Gaussian mixture centers, hence the total training samples can be clustered into $N \times M$ using k-means method. And the parameters of C, μ, Cov can be estimated by (Rabiner, 1989):

\overset{\land}{c} jm = \frac{numberofobsevationvectorsclassifiedinclusterm}{numberofobservationvectorsinS j}

(16)

\overset{\land}{μ} jm = samplemeanofobservationvectorsclassified inclustermofS j

(17)

\overset{\land}{C} ov jm = samplecovariancematrixofvectorsclassified inclustermofS j

(18)

Calculate the optimum state sequence. The state sequence can be obtained by a path backtracking method (i.e. Viterbi algorithm) based on the given model parameters λ. The aim of the algorithm is to find the best state sequence {q₁, q₂ , … , q_T} which can best explain the given observation sequence {o₁, o₂ , … , o_T}. Define the quantity

δ t (i) = max q 1, q 2, \dots q 1 P (q 1, q 2, \dots, q t = i, o 1, o 2, \dots, o t | λ)

(19)

as the maximal probability along a single path at time t. It accounts for the first t observations and ends in state i. By induction,

δ t + 1 (j) = max 1 \leq i \leq N [δ t (i) a ij] b j (o t + 1), 1 \leq t \leq T, 1 \leq j \leq N

(20)

Re-estimate the parameters of CHMM. The new model parameters can be re-estimated by the Baum-Welch method. The aim of the algorithm is to adjust the model parameters λ to maximize the output of $P (O | λ)$ , which can best explain the real state by the training data. When λ and O are given, the probability of the model in state S_i at time t can be defined as

γ t (i) = P (q t = S i | o, λ)

(21)

Similarly, the probability of being in state S_j at time t with the mth component Z_jm is

γ t (j, m) = P (q t = S j, z jt = Z jm | o, λ)

(22)

And

ξ (i, j)

is the probability of being in state S_i at time t, and state S_j at time

t + 1

, that is

ξ t (i, j) = P (q t = S i, q t + 1 = S j | o, λ)

(23)

The new model parameters can be calculated by

\bar{π} = γ 1 (i)

(24)

\bar{a} ij \frac{\sum_{t = 1}^{T - 1} ξ t (i, j)}{\sum_{t = 1}^{T - 1} γ t (i)}

(25)

And the new parameters of observation probability distribution are

\bar{c} jm \frac{\sum_{t = 1}^{T - 1} γ t (j, m)}{\sum_{t = 1}^{T} \sum_{m = 1}^{M j} γ t (j, m)}

(26)

\bar{μ} jm = \frac{\sum_{t = 1}^{T} γ t (j, m) o t}{\sum_{t = 1}^{T} γ t (j, m)}

(27)

\bar{C} ov jm = \frac{\sum_{t = 1}^{T} γ t (j, m) (o t - μ jm) (o t - μ jm) T}{\sum_{t = 1}^{T} γ t (j, m)}

(28)

where M_j is the number of the mth Gaussian mixture component in state S_j, o_t is the observation vectors in time t.

\bar{C} jm

is the re-estimated mixture coefficient of the mth mixture in state S_j.

\bar{μ} jm

and

\bar{C} ov jm

are the re-estimated mean vector and covariance matrix respectively.

Iterate steps (3) and (4) until the convergence is reached. In this work, the aim of iteration is to improve the probability of observation in each step. And the convergence threshold is set as the increment of two successive re-estimated probabilities is less than 10⁻⁵. The final result of re-estimated model parameters will be saved on the CHMM database.

2.3. Bearing fault diagnosis based on SSA and CHMM

When CHMM is obtained by a training process, fault diagnosis can be performed using the pre-trained model. The testing samples are input into the model to calculate the probabilities $P (O | λ)$ . In order to prevent data underflow, the logarithm of probability is adopted, and the log-likelihood is employed as an index to estimate the bearing conditions (Rabiner, 1989). There are two experiments in this work. One is the bearing fault diagnosis using SSA and CHMM; another is the bearing performance assessment based on SSA and CHMM. There are some differences in these two applications.

Figure 2 reveals the scheme of bearing fault diagnosis based on SSA and CHMM. At first, both the training data and testing data are processed through SSA respectively, and some SFs are extracted as the diagnosis features. It should be noted that the length of data and the dimension of the SFs should be the same for both training and testing data. The second step is training the CHMMs. Similar to other pattern recognition techniques, the training should be performed at different bearing conditions. If there are $L b$ bearing conditions, different CHMM models should be trained respectively and saved to the CHMM database. Therefore, it should be $L b$ CHMMs and can be denoted as ${λ i}_{i = 1}^{L b}$ . At last, the testing data can be input into the CHMM database, and the probabilities under different λ are calculated respectively. The presence with the maximal probability is the condition of testing bearing, i.e.

condition = \arg max 1 \leq j \leq L b \log (P (O | λ j))

(29)

In bearing performance assessment, the training procedure is somewhat different to the fault diagnosis. It only uses the data at normal condition as training data, and the scheme is shown in Figure 3. Because the condition of bearing is always from health to failure, the aim of bearing performance assessment is to evaluate bearing status to prevent the unexpected breakdown. The SSA is applied at whole lifetime bearing and the SFs are extracted. Then the features from the normal condition are employed to train the CHMM. Finally, the testing SFs are input pre-trained into CHMM to calculate the log-likelihood of the testing samples. The result can be utilized to estimate the deviated degree of the bearing from the normal condition. In order to weaken the influence of the observation vectors’ length, the output of CHMM should be processed by

PI = \frac{\log P (O | λ)}{T}

(3)

where T is the length of observation vectors. The processed log-likelihood is utilized as performance index (PI) to assess the bearing performance quantitatively.

In the following section, two experiments will be introduced to verify the feasibility and validity of SSA and CHMM in bearing fault diagnosis and performance assessment.

3. Experiment 1 of the artificial rolling element bearing fault detection

3.1. Experiment introduction

In the first experiment, test data of normal and fault bearings were from Ocak and Loparo (2004). Four bearing conditions are analyzed in this study, which include normal condition (NC), inner race fault (IRF), outer race fault (ORF) and rolling body fault (RBF). The test bearing is 6205-2RS JEM SKF, deep groove ball bearing. The parameters are shown in Table 1. Faults in the test bearing components were processed respectively by the electrical discharge machining (EDM) method. And the fault severity levels were 0.007, 0.014, 0.021 and 0.028 inch. The test rig consisted of a 2 HP Reliance Electric IQPreAlert motor, a torque transducer, accelerometer and a dynamometer. Because the analyzed object was bearing, the data acquired from the accelerator in the driven end were used in this study. The data were collected under different fault sizes and loads. That is, in testing each fault size, the rotor speed decreased from 1797 rpm to 1730 rpm, which corresponded to change of the loads from 0 HP to 3 HP respectively. The sampling rate of data is 12 kHz. Figure 4 reveals the spectrum of bearing at normal conditions and another three fault conditions. It is obvious that the frequency components of different bearing conditions are different.

Table 1.

Parameters of bearing 6205-2RS.

Type	Ball number	Ball diameter (mm)	Pitch diameter (mm)	Contact angle
6205-2RS	9	7.94	39.04	0

3.2. Bearing fault diagnosis based on SSA and CHMM

The process of bearing fault diagnosis has been displayed in Figure 2. The first step is to extract the SFs and select the features.

3.2.1 Selection of features based on SSA

It is suitable to apply the singular spectrum in bearing fault diagnosis, because each principal component represents the different frequency spectrum of the signal (Alonso and Salgado, 2008). Figure 5(a) reveals a singular spectrum of outer race fault bearing where $L = 10$ . The largest singular value represents the main amplitude component after SVD. The graphical representation of correlation coefficients between SFs is in Figure 5(b), where the gray scale from white ( $ρ ij = 1$ ) to black $ρ ij = 0$ reflects the correlation between components i and j. It is clear that singular features 1 and 2 are highly similar ( $ρ 12 = 0.993$ ), so we can select only one of them in fault diagnosis. The original time wave and spectrum are shown in Figure 6(a) and (b) respectively. The component of each SF is displayed in Figure 6(c). The amplitudes of the first few components are higher than the rest. The spectrum of each component is shown in Figure 6(d). It is clear that the spectral energy of the first four SFs is distributed from 2000 Hz to 4000 Hz. And the spectral range of SF5, SF6 and SF9 mainly focus on 4000 Hz to 6000 Hz. The other three SFs correspond to the low frequency range from 0 to 2000 Hz. Therefore, SFs represent different spectral components of the original signal and can be utilized in bearing fault diagnosis.

The details of extracting the singular values based on SSA have been illustrated in Section 2.1. There are two steps in generating the SFs. The first step is selecting the window length L. It can be deduced from embedding and SVD that L determines the number of singular values (eigenvalue number). Figure 7 shows the singular spectrum of various bearing conditions under different window lengths, where every time series is 1000 points. The identification ability of SFs under different window length is different. In Figure 7(a), the first two singular features (SF1 and SF2) can distinguish the various bearing conditions, but in the other three features it is difficult to classify the bearing conditions. And it is hard to discover the frequency variation of the original signal at different frequency ranges using only SF1 and SF2. It can be seen from Figure 7(b) to (d) that more SFs are generated with the increase of L. All these three window lengths can be utilized to extract the singular features. But the increase of window length will cause the increase of time in computing. And the singular value of SFs decreases abruptly after SF4. It is difficult to distinguish the bearing conditions using the last five singular features when $L = 15$ or $L = 20$ . Therefore, $L = 10$ is selected in this work (Muruganatham et al., 2013). The second step in singular feature extraction is selecting the appropriate SFs which best identify the bearing conditions. It can be seen from Figure 7(b) that SF1, SF2, SF4, SF5, SF6 and SF7 can classify the bearing conditions clearly. Figure 6 gives an example of the spectrum of ORF, and it can be observed from Figure 6(d) that these SFs represent different frequency components of the original signal. Considering that the correlation coefficients between component SF1 and SF2 in all bearing conditions are higher than 0.98, the SF2 is neglected, and the features SF1, SF4, SF5, SF6 and SF7 are selected.

Because the classification ability of SFs are not sensitive to the sample size (Muruganatham et al., 2013), the sample size 1000 is selected to extract the SFs for each bearing condition respectively. All SFs are normalized and used as input of CHMM to train and test the fault type of bearing.

3.2.2. Bearing fault diagnosis by CHMM

The selected SFs are utilized as input of CHMM to identify the bearing faults. A total of 400 datasets of five normalized singular features (SF1, SF4, SF5, SF6 and SF7) are extracted from different bearing conditions (NC, IRF, ORF and RBF). Here 30 datasets of each bearing condition are selected randomly as training sets, and the other 280 datasets are employed as testing set. Ten-fold crossed validation is utilized to evaluate the classification accuracy.

Because there are four bearing conditions of testing data, the CHMM with four states, each state with two Gaussian mixtures, are used to model the vibration signals. The CHMM parameters can be initialized and estimated by the Baum-Welch algorithm presented in Section 2.2. The model of each bearing condition should be trained respectively. So a CHMM database with four models: $λ NC$ , $λ IRF$ , $λ ORF$ , $λ RBF$ can be obtained which corresponds to different bearing conditions. Then the testing data will be inputinto the pre-trained CHMM database to calculate the log-likelihood.

The results are shown graphically in Figure 8. For convenience, only 120 datasets (30 testing samples for each bearing condition) were displayed. The actual conditions of these testing datasets have been indicated on the top of each plot. For example, in Figure 8(a), the real condition of the first 30 samples (test number: 1 to 30) are normal condition, each testing sample is input into the CHMM database and calculate the log-likelihood under four models. It is obvious that the normal condition always obtains the maximum log-likelihood over the four models. Therefore, these samples can be classified into the normal condition which is coincident to the actual condition of the testing data. The rest of the datasets can be analyzed similarly. Figure 8(b) is the local enlargement of 8(a). It can be observed that most of the testing samples can be correctly identified (only one misclassification in IRF and three in RBF).

Table 2 shows the diagnosis results of total 280 testing datasets. As a comparison, different features are selected as an input of CHMM. At first, the results show that in all cases, the normal condition can be recognized. It can be observed from Figure 7(b) that the selected SFs can clearly distinguish the normal condition from other bearing faults. Secondly, when input is only the SF1, the diagnosis accuracy of IRF, ORF and RBF are much lower than the other two cases. Because the singular features reflect the variation of different spectral ranges, SF1 alone is not sufficient to present the spectral variation of the original signal. Thirdly, the average accuracy of the other two cases increase more than 10% compared to SF1. This validates the fact that the selected SFs can effectively reflect the variation of different bearing conditions. Furthermore, the average accuracy using six singular features improves by only 0.14% compared to five features. This may be because the SF1 and SF2 are highly correlated (

ρ 12 > 0.98

), and it is feasible to select only one of them. Therefore, a conclusion can be made that the selected singular features and CHMM are efficient and accurate in bearing fault diagnosis.

Table 2.

Diagnosis results by selecting different singular features.

No. of singular features	Accuracy of testing data (%)				Average testing accuracy (%)
No. of singular features	NC	IRF	ORF	RBF	Average testing accuracy (%)
1	100	88.29	79.14	77.43	86.22
1,4,5,6,7	100	96.30	99.14	96.86	98.08
1,2,4,5,6,7	100	96.30	99.43	97.14	98.22

NC: normal condition; IRF: inner race fault; ORF: outer race fault; RBF: rolling body fault

4. Experiment 2 of rolling bearing fault in the accelerated life test

4.1. Experiment introduction

The experiment introduced in Section 3 is a traditional research way of fault diagnosis using an artificial defect machining method. It cannot simulate the natural fault state in practice, especially early faults. In modern fault diagnosis, more attention is focused on proactive maintenance, which requires assessing the bearing performance in a whole lifetime (Jardine et al., 2006). Hence, an accelerated life test is performed to acquire the vibration data over a whole lifetime.

As shown in Figure 9, the experiment system is made up of the test bench and data acquisition system. Accelerated bearing life tester (ABLT-1A) was provided by Hangzhou Bearing & Research Center (HBRC), which contributed an AC motor, a transmission system, a lubrication system, a loading system and a data acquisition system. It simultaneously hosted four test bearings on the shaft driven by an AC motor and coupled by rub belts. And a load of 12.744 kN was appended on the bearings in order to accelerate the bearing into failure. The data acquisition system mainly included three PCB348A acceleration sensors, a SCXI-1531 signal conditional model and an NI PCI-6023E data acquisition card. Data acquisition software was programmed by National Instruments LabVIEW. The locations of acceleration sensors are displayed in Figure 9(b). The sketch of the acquisition system can be seen in Figure 10. The data sampling rate was 25.6 kHz and acquired 20,480 points per minute (one group). Oil lubrication was adopted. The type of tested bearing was 6307, whose parameters and operation conditions were shown in Table 3. Figure 11 reveals the time wave of bearing under three conditions: normal, degradation and failure. With the increase of defect, the impact characteristics caused by defects become clearer. Figure 12 reveals the fault surface of a final failure bearing. It is obvious that the variation of bearing performance is an intricate process.

Table 3.

Parameters and operating conditions of bearing 6307.

Type	Ball number	Ball diameter (mm)	Pitch diameter (mm)	Contact angle	Motor speed (rpm)	Load (kN)
6307	8	13.494	58.5	0	3000	12.744

4.2. Bearing performance assessment based on SSA and CHMM

The whole lifetime vibration data were collected from normal to degradation and to final failure. And the total acquisition duration was 2469 min. The details of bearing performance assessment based on SSA and CHMM are illustrated in Section 2.3. As shown in Figure 3, the first step is generating the SFs of bearing. Similar to experiment 1, the window length $L = 10$ is utilized to calculate the singular features in a whole lifetime. Figure 13 gives an example of SFs extracted from different bearing states (normal, degradation and failure). It is obvious that the failure state can be easily separated from normal and degradation states by the first five SFs (SF1 to SF5), but the singular values under normal and degradation states are approximate especially the SF5. In the last five SFs (SF6 to SF10), the degradation states can be identified from the other two states, but the normal and failure states are hard to distinguish. Furthermore, the maximum correlation coefficient between the SFs is 0.974, so the SF1, SF2, SF3, SF4, SF6 and SF8 may be used as features.

In order to validate the efficiency of SFs in bearing performance assessment, the RMS index is also employed as a comparison. It can be seen from Figure 14 that an increase occurs at about 1295 min and keeps stable for a long time. Then the RMS increases at 2304 min and fluctuates for a short time and increases abruptly at 2337 min. The first tiny increase in 1295 min discovers the occurrence of the incipient fault. With the occurrence of a single point fault in bearing, the impact between rolling elements and races will excite the bearing’s natural frequencies (Berry, 1991). It may cause an increase of vibration amplitude, but the fault is very weak and difficult to be seen. With the increase in fault level, the value of RMS will increase too. The second transit appearing at 2304 min indicates that the bearing falls into failure. RMS can be used to detect bearing faults, but the variation from normal to degradation is too weak to distinguish the early fault that occurred in the bearing, so the SFs are utilized to assess the bearing performance.

Figure 15 reveals the singular values of 10 SFs in bearing's whole lifetime. It is observed that the singular value of the first five SFs (SF1 to SF5) increases abruptly at about 2304 min and 2337 min, which is consistent with the RMS. Similarly, with these features it is difficult to detect early faults. However, it can be seen from SF6 to SF10 that there are two transformation processes during the whole lifetime. The first one is that the singular value rises at 1295 min and keeps stable till 2304 min in which the second transit occurs. In the second transformation process, the singular value decreases abruptly. It is obvious that the last five SFs can detect the early fault in the bearing, which is beneficial in equipment maintenance. But singular values of these SFs increase at 1295 min and decrease at 2304 min. Hence, the consistency is worse than the first five. Therefore, considering the detecting ability of SFs to the different bearing conditions, the SF1, SF2, SF3, SF4 and SF6 are utilized to assess the bearing performance.

From the 2469 min whole lifetime datasets, five SFs of each group (one min) are extracted respectively. Then these SFs are employed as observation sequences and input into CHMM to train and assess the bearing performance. The SFs in the first 300 min (normal condition) are used as training data, and the SFs over the whole lifetime are testing data. A CHMM with three states, each state with two Gaussian mixtures are utilized to train the model parameters. At last, the testing data are input into the pre-trained CHMM to calculate the performance index illustrated in equation (30).

As a comparison, SF1, SF6 and SF8 are also used to assess the bearing performance. The results are revealed in Figure 16(a). It can be seen from Figure 16(b) that there a transition occurs at 1295 min when the training and testing feature is SF6 or SF8, but the PI value of SF1 is almost invariable. It demonstrates that SF1 is not sensitive to the incipient fault, however both SF6 and SF8 can detect that the fault occurs at 1295 min. Moreover, Figure 16(c) shows that the PI value of SF1 decreases at 2304 min and 2337 min when using SF1 or SF8. But with SF6 it is difficult to recognize the bearing failure. The PI of SF8 increases at 2304 min but it fluctuates significantly between 2350 and 2384 min. The consistency of SF8 is worse than SF1 and SF6. Therefore, it is not appropriate to use SF8 in bearing performance assessment.

Figure 17 shows the assessment results by all SF1, SF2, SF3, SF4 and SF6. Similar to RMS, there are also two transitional processes in a whole lifetime. The time of the first transit occurs at 1295 min, and the PI value decreases abruptly from 7.93 (1294 min) to 5.98 (1295 min) and then keeps stable till the second transit occurs. Unlike RMS, the degradation state can be clearly recognized by the SFs and CHMM. The second transit appears at 2304 min where the PI declines from 7.03 (2303 min) to 6.27 (2304 min) and fluctuates for a short time. Then in 2337 min, the PI value rapidly decreased until final failure. The results show that these SFs are more accurate in estimating the bearing performance than RMS and single SF. Furthermore, the PI reflects the variation of bearing performance quantitatively. It is observed that the selected SFs and CHMM can clearly identify the bearing states from normal to degradation to failure. In particular, the first transit discovers the occurrence of the incipient bearing fault. If the bearing can be replaced before final failure, it may prevent serious breakdown of equipment, which is significant in guiding the equipment maintenance.

5. Conclusion

In this study, singular spectrum and CHMM are applied to detect the bearing fault. In order to validate the effectiveness of the proposed method, the artificial rolling element bearing fault and accelerated life test are utilized to collect the test data. The selection of SFs in bearing fault diagnosis after SVD is discussed, and the results demonstrate that the SFs can effectively recognize different bearing conditions. Furthermore, the maximal log-likelihood calculated from CHMM can be employed as a performance index to quantitatively assess the degree of degradation of the bearing. The incipient fault and different running stages can be detected by using SFs and CHMM, which is beneficial in equipment maintenance and preventing a severe accident.

Footnotes

Funding

This work was supported by the National Natural Science Foundations of China (grant numbers 51035007, 51175329 and 51105243).

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