Dynamic degradation observer for bearing fault by MTS–SOM system

Mechanical Systems and Signal Processing 36 (2013) 385–400

Contents lists available at SciVerse ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Dynamic degradation observer for bearing fault by MTS–SOM system Jinqiu Hu n, Laibin Zhang, Wei Liang College of Mechanical and Transportation Engineering, China University of Petroleum, Changping District, Beijing, China

a r t i c l e i n f o

abstract

Article history: Received 7 November 2011 Received in revised form 14 October 2012 Accepted 16 October 2012 Available online 30 November 2012

Rolling element bearings are used in a wide variety of rotating machinery from small hand-held devices to heavy duty industrial systems. Bearing fault or even failure is one of the foremost causes of breakdown in such rotating machines, resulting in costly system downtime. This paper presents a dynamic degradation observer for the identification and assessment of bearing degradation based on Mahalanobis–Taguchi system (MTS) and self-organization mapping (SOM) network called MTS–SOM system. It helps to differentiate especially the incipient fault stage and track the dynamic degradation trend of the running bearing by real-time vibration observations. The feature parameters from multifractal aspects are calculated first and further optimized by the MTS statistical method. Mappings of different degradation levels are then presented by SOM with optimal multifractal features, which help to differentiate each degradation stage and describe a degradation trajectory of the in-service bearing. The found results are validated by experiment, and a comparative study is carried out to verify the effectiveness of the proposed method. The contribution of the method considering both current and predictive perspectives on the fault degradation behavior is also showed for the elaborate maintenance management. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Dynamic degradation observer MTS Multifractal SOM Degradation trend

1. Introduction Rolling element bearings represent one of the most widely used industrial machine components, being the interface between the stationary and the rotating part of the machine. Condition monitoring of rotating machinery helps in early detection of faults and abnormity in time to prevent complete failure in advance, and also helps to reduce maintenance cost and loss through accidental events. The capability to detect the existence and severity of the degradation in a running bearing fast and accurately during operation is very important since an unexpected failure can lead to unacceptably long maintenance shutdown ([7,8]). Rolling bearing defects may be categorized as point or local defects and distributed defects. The vibrations are generated by geometrical imperfections on the individual bearing components. These imperfections are usually caused by irregularities during the manufacturing process or the occurrence of wear and tear during the operation process. The various distributed defects include but are not limited to surface roughness, waviness, misaligned races, and off-size rolling elements. The local defects include cracks, corrosion pitting and spalls on the rolling surfaces, where the corrosion pitting is a common failure mode.

n

Corresponding author. Tel.: þ86 13401021372. E-mail address: [email protected] (J. Hu).

0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.10.006

386

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

Thus, due to the importance of bearings and their complex fault degradation process, a variety of diagnosis approaches for bearing fault have been developed, among which vibration signal processing is the most frequently applied one, in order to keep machinery operating at its best and avoid catastrophic accidents. By applying certain processing techniques to bearing vibration signals, it is possible to obtain vital diagnostic information. In a condition monitoring and fault diagnosis system, extraction of feature information is normally followed by a fault mode classifier so that intelligent diagnosis is achieved. Various artificial intelligence (AI) techniques such as hidden Markov models (HMM) [15], artificial neural networks (ANN) [18] and support vector machines (SVM) ([20,17]) were widely used in the research of bearing fault diagnosis. Hong and Liang [3] proposed a new approach for bearing fault (single point) severity measure based on the continuous wavelet transform (CWT) results, and attempted to address the issues presented in the current version of the Lempel–Ziv complexity measure. Qiu and Lee [16] introduced a few enhanced and robust prognostic methods for rolling element bearings including a wavelet filter based method for weak signature enhancement for fault identification and a self organizing map (SOM) based method for performance degradation assessment. All of these feature extraction methods are effective for some certain kinds of faults and present satisfying results. However, the vibration features from different fault modes or at different degradation stages have different sort of contribution for diagnosis, so the research of feature optimization is necessary. Yu ([21,22]) proposed a new effective approach of dimension reduction and feature extraction (FE) based on locality preserving projections (LPP) for bearing degradation assessment, and further provided a comprehensive indication for quantifying the performance degradation of bearings by the integration of the exponential weighted moving average (EWMA) statistic and the negative log likelihood probability (NLLP) based on Gaussian mixture model (GMM). Pan [13] proposed a method using lifting wavelet packet decomposition and fuzzy c-means clustering to present degradation degrees of components. Although these optimal degradation indicators can manifest the severity levels especially during the stage near failure, they usually fail to be used to detect the incipient fault. The above two tasks as feature extraction and degradation assessment are difficult to be performed effectively due to the nonlinearity existing in the vibration signals of running bearings and the uncertainty inherent in the diagnostic information as well. Classification of signals from faulty or malfunctioning bearings requires expert knowledge and experience because the relationships between the causes and symptoms are very complicated [19]. The modeling of these relationships needs an effective pattern recognition (PR) procedure. The main shortcomings of the above conventional approaches which could be used for such purpose are summarized as follows: (1) Most of the above-mentioned existing approaches for the diagnosis and assessment of degradation of bearing faults can make intelligent identifications in certain limited cases; however, they usually fail to detect the incipient fault stage (early degradation) until severe degradation happens when it is very close to the complete failure. Hence the development of accurate indicators is an important consideration both for incipient fault diagnosis and degradation assessment. (2) Many features can be generated from vibration data; however, extracting the most useful information from these original features presents a big challenge. As different features have different sensitivity to the degree of degradation of the running bearing, a single feature usually fails to manifest the whole degradation process during the full cycle life of the in-service bearing. Therefore, a set of optimal degradation indicators consisting of a few effective features can provide the perfect information of the overall dynamic degradation process in a systematic way. (3) Supervised classification methods in the above literature need a great deal of samples corresponding to various fault modes and normal condition as well; however, there are few fault samples in the practice to train the model. In this situation, the calculated models usually diverge from the reality with obvious errors in the diagnostic results. In order to solve the above-mentioned problems and to overcome the shortcomings of the conventional approaches, it is necessary to develop an intelligent method that can identify different degradation levels especially the incipient fault stage. Therefore in this paper a new MTS–SOM system is proposed integrating of the Mahalanobis–Taguchi system (MTS) and self-organization mapping neural network (SOM). Based on multifractal theory first, the local conditions of fractal objects as incipient faults can be described more precisely, and the features of mechanical incipient faults can be successfully extracted by both the multifractal spectrum and the generalized dimensions. Degradation identification procedures are further developed by the statistical optimization of feature parameters and automatic classification of degradation levels based on the SOM network. Mappings of different degradation levels and their corresponding degradation trajectories are finally provided by MTS–SOM system for dynamic degradation assessment, which helps to make appropriate decision for proactive maintenance. The whole paper is organized as follows. In Section 2, the multifractal features of incipient faults are introduced and presented in detail. In Section 3, the proposed MTS–SOM system is described and the detailed scheme for degradation observing and predicting is given. In Section 4, a case study with experiment is presented to show the application results and, in Section 5 the conclusion of the paper is given. 2. Incipient fault identification by multifractal theory During the whole life cycle of bearings, there usually exist four stages which are normal condition, incipient fault, severe degradation and the last stage as complete failure. The incipient fault stage which follows the normal condition

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

387

stage is the early stage of degradation, during which the fault information is very weak, and hard to discover. Both microscopic defects and small localized damage can possibly occur at this stage. It usually has minor influence on the operation of the system with less damage loss and maintenance cost. However, if it is possible to capture the signs of incipient fault accurately in a timely fashion (the short and slight signs of an abnormal state that are produced before obvious faults develop), it would greatly help to perform proactive repair and make a great contribution to the risk control. Severe degradation is the third stage when obvious faults such as localized damage fully develop. In this stage, fault characteristics can be detected by many conventional methods mentioned in literatures. Severe degradation has severe influence on the operation of the system, resulting in the system efficiency reduced and unplanned outage happening. Logan and Mathew [11,12] were the first to pay attention to the occurrence and detection of chaotic behavior in time series vibration signals and detailed a method for quantifying a fractal dimension from the time series. It was demonstrated how the calculation of correlation dimension was performed experimentally on bearing vibration acceleration time series. In our early research, features in severe degradation of bearings were applied for diagnosis successfully [5], while this paper further focuses on the chaotic behavior of different degradation stages during the whole cycle life of bearings, and presents the multifractal characteristics of each stage, which are used for degradation identification and assessment. 2.1. Generalized dimensions Fractals refer to objects that are either self-similar or self-affine. The whole fractal structure can be regarded as the buildup of local fractals with different local fractal dimensions. This structure composed of different local fractal structures is called multifractal. So multifractal is defined as a set constituted by singular measures with multi-scaling exponents in fractal structure. For any fractal object, a set of generalized dimensions can reflect the spatial distribution information of the fractal structure. Based on Whitney embedding theorem and Packard phase space reconstruction theory, Grassberger and Procaccia [2] proposed an algorithm, using time series data, to calculate the correlation dimension of strange attractors in dynamical systems. Generalizing above theory, the correlation integral method to estimate the generalized dimensions of mechanical vibration signals is provided as follows: Set signal series as fxk ,k ¼ 1,2,. . .,Ng and embed them into m-dimensional Euclidean space Rm, then a point set J(m) can be obtained. The elements in the J(m) are denoted by Eq. (1), where t is delay time and Nm ¼N  (m 1)t is the number of vectors in the point set. X n ðm, tÞ ¼ ðxn ,xn þ t ,. . .,xn þ ðm1Þt Þ

n ¼ 1,2,. . .,N m

ð1Þ

The q-order correlation integral for a discrete fractal set is defined as Eq. (2), where rij denotes the distance of Xi and Xj in the point set J(m) with expression as Eq. (3). H(x) is Heaviside function shown in Eq. (4). 8 2 3q1 91=q1 Nm Nm = < 1 X   1 X 4 H rr ij 5 ð2Þ cðq,r Þ ¼ ; :N m Nm i¼1

2 r ij ¼ dðX i ,X j Þ ¼ 4

m 1 X

j¼1

2

31=2

ðxi þ lt xj þ lt Þ 5

ð3Þ

l¼0

( HðxÞ ¼

1,

x 40

0,

x r0

DðqÞ ¼ lim r-0

lnðcðq,r ÞÞ lnðrÞ

ð4Þ

ð5Þ

Then the generalized dimensions can be represented as Eq. (5), based on which the generalized dimensions D(q) can be obtained by the least square linear-fitting method [23]. Therefore the incipient fault information embedded in the nonstationary vibration signals of deteriorated bearings can be characterized by a set of fractal dimension values D(q), where q can be set as any integer. The selection and optimization of the parameter q will be further detailed in Section 3. 2.2. Multifractal spectrum Multifractal spectrum f(a) is also named as singularity spectrum, where a as singularity index indicates the local dimension of a certain small area of the fractal object. So the multifractal spectrum represents the fractal dimensions of particular subsets which have the same value of a, and describes the characteristics of the fractal object on different fractal levels. Usually a and f(a) are calculated from the following Legendre transforms:

tðqÞ ¼ ðq1ÞDðqÞ

ð6Þ

388

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

dtðqÞ dq

ð7Þ

f ðaðqÞÞ ¼ qaðqÞ þ tðqÞ

ð8Þ

aðqÞ ¼

Three important parameters of multifractal spectrum can be obtained from Eqs. (6)–(8), which are described as follows:

 The width of the multifractal spectrum which is Da ¼ amax amin . The value of Da represents the degree of 



nonhomogeneity of the distribution of probability measures on the overall fractal structure, which refers to the degree of fluctuation of the mechanical vibration signals. The difference of the fractal dimensions between the maximum probability and the minimum probability subsets, which can be calculated as Df ¼ f ðamax Þf ðamin Þ. The value of Df describes the proportion of the number of elements at the maximum and minimum in the subset constructed by related physical variables, which refers to the proportion of the large and small peaks of vibration signals. It is an important characteristic representing the degree of severity of mechanical vibrations. The maximum of the multifractal spectrum fmax. The peak of the multifractal spectrum fmax represents the changing speed of the number of units having the same probability as the scale e changes, which refers to the changing speed of the large and small peaks of vibration signals.

In a word the multifractal spectrum reflects the proportion and the degree of nonhomogeneity of the distribution of probability measures on the overall fractal structure. In this way above three parameters comprehensively describe the degree of fluctuation of mechanical vibration signals and the degree of severity of component’s degradation. Hence these parameters can be used for the degradation identification of the mechanical components especially when they are in incipient fault state. 2.3. Multifractal spectrum entropy Information entropy, which is typically used to describe the degree of information uncertainty based on the information theory, is a good measurement for assessing the degree of homogeneous of the probability distribution. According to the calculation method of entropy, the homogeneity of singularity distribution of fractal structures is described quantitatively, and the characteristics of mechanical vibration signals can be also presented in terms of multifractal spectrum entropy. Therefore, the multifractal spectrum entropy can describe the singular condition of the signal energy distribution and its probability of the geometric feature distribution, so it is conducive to identifying the characteristics of incipient faults. Set f(a) as the multifractal spectrum of a series of mechanical vibration signals, and based on the definition of entropy, the calculation of the multifractal spectrum entropy Hm is provided as follows: Hm ¼ 

k X

pi logpi

ð9Þ

i¼1

f ða Þ p i ¼ Pk i i ¼ 1 f ðai Þ

ð10Þ

where pi represents the proportion between f(ai) and the whole multifractal spectrum. Multifractal spectrum entropy Hm reflects the complexity of non-stationary signals caused by component’s degradation process. 3. MTS–SOM system 3.1. Fault feature optimization based on MTS Mahalanobis–Taguchi system (MTS) is a pattern information technology, which has been used in different applications to make quantitative decisions by constructing a multivariate measurement scale ([1,4,14]). The design model of MTS is shown in Fig. 1, which includes three key steps: (1) Construction of Mahalanobis distance (MD) scale; (2) Use of the signal-to-noise ratio (SNR) to evaluate the quality of measurement; (3) Optimization of the analytic information to improve the SNR with an orthogonal array. For the feature optimization based on MTS, mass samples of features with normal condition are utilized to develop the Mahalanobis distance space, whereas only a few fault samples are used for feature optimization.

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

389

Determination of the needed variables and the Mahalanobis reference space Acquisition of data samples with normal state as reference group

Construction of the Mahalanobis reference space

Calculation of the characteristics of the Mahalanobis reference space Calculation of the Mahalanobis distance for the reference group Analysis of the abnormal conditions as fault degradation with different levels

Validation of the reference space Calculation of the Mahalanobis distance for the abnormal conditions Design of the appropriate orthogonal array

Calculation of the SNR

Optimization of the Mahalanobis reference space

Selection of the variables which have contributions to the identification

Validation of the optimized reference space

Determination of the identification thresholds

Degradation identification based on the optimal reference space

Degradation identification and evaluation

Fig. 1. Design model of MTS.

3.1.1. Mahalanobis distance scale In the construction process of MTS, the Mahalanobis space (reference group) can be developed using the standardized variables of healthy or normal data. Let M the number of features obtained from original signals; N the number of signal samples; nij the standardized ith observation (sample) of the normal group for the jth variable (feature), i ¼ 1,. . .,n, j ¼ 1,. . .,m; Rm  m the correlation matrix for m standardized variables (features); R1 ½aij mn ; mm   Y ¼ y1 ,y2 ,. . .,ym an observation of the test group with unknown condition. Then the Mahalanobis distance D2 calculated for the test group observation is given by 2 3 m X m   X 2 aij yi yj 5=m D ¼4

ð11Þ

i¼1j¼1

When an observation belongs to the reference group, D2 E1, otherwise D2 should be considerably greater than one and increases as the difference between the observation and the reference group increases. According to the degree of difference between each degradation state (incipient fault, severe degradation and complete failure) and normal state of mechanical components, different Mahalanobis distance intervals can be mapped to different stages in the life cycle.

3.1.2. Feature optimization based on two-level orthogonal array According to the multifractal characteristics of mechanical degradation, there could be various vibration features existing in the Mahalanobis space, i.e., DðqÞ, Da, f max , Df , Hm , whereas parts of them are not contributed for degradation identification. Hence, the selection of effective features can improve the identification ability of Mahalanobis space, and save the calculation cost due to feature extraction. According to the number of features participating in the optimization, an appropriate two-level orthogonal array is selected. Orthogonal array is mainly used to identify the most useful features through the fewest experiments. Each feature variable has two levels, i.e., level 1 and level 2, where level 1 represents

390

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

using such feature variables to calculate MD. Level 2 represents not using such feature variables to calculate MD. Meanwhile the appointed features on each row of the orthogonal array are used to construct the Mahalanobis space. One of the feature samples is selected corresponding to one certain degradation state. Then, the Mahalanobis distances D21 ,. . .D22 ,. . .,D2d between each feature vector and the Mahalanobis space are calculated based on eq. (11). The identification performance of the developed Mahalanobis space is evaluated by signal-to-noise ratio (SNR) based on Eq. (12).

Z ¼ 10ln½V ðSV Þ=r

ð12Þ

where

r¼

d X

hP M 2i ,S ¼

i¼1

d 2 i ¼ 1 ðM i Di Þ

i2

r

Pd ,

V¼

D2i S : d1

i¼1

The performance of degradation identification can be improved as the SNR of the Mahalanobis space increases. Considering the SNR of each Mahalanobis space, the average signal-to-noise ratios of features relative to each degradation state on level 1 and level 2 are calculated, respectively. If the average SNR of a certain feature on level 1 is higher than level 2, this feature contributes to better identification, and should be considered to retain rather than eliminate. Finally, a new Mahalanobis space constructed by the reserved features can be used to provide the degradation identification with the best performance. 3.2. Degradation assessment based on SOM Neural networks are known for their predictive capability and the ability to learn patterns from field data which are noisy, imprecise or incomplete. Compared to many neural network models presented in literature, SOM is the one that is quite suitable for unsupervised applications. It has special property of effectively creating spatially organized ‘‘internal representations’’ of various input data features and providing a topology preserving mapping from the high-dimensional space into two-dimensional grid maps [10]. The algorithm of SOM proposed by Kohonen ([6,9]) can be described as a non-linear, smooth mapping of a highdimensional input space onto a low-dimensional output space. In this paper, SOM is applied in the visualization of complex degradation stages of bearings. The SOM consists of M units. Each unit i is associated to an n-dimensional prototype vector mi in the input space and also a position vector on a low dimensional regular grid gi in the output space. The algorithm of SOM can be divided in two steps: (1) A competitive step, where the closest prototype (best matching unit) mc to the input vector is presented as follows: c ¼ arg min 99xðtÞmi ðtÞ99 i

ð13Þ

(2) A cooperative step, where the best matching unit and its neighbors are adapted as follows: mi ðt þ 1Þ ¼ mi ðtÞ þ aðtÞhci ðtÞ½xðtÞmi ðtÞ

ð14Þ

where a(t) is a learning rate and hci(t) is a neighborhood function between units c and i. This adaptation rule has a cooperative nature since not only the best matching unit but also its neighbors are adapted with a factor defined by the neighborhood function hci. A typical choice of hci is the Gaussian neighborhood function shown in Eq. (15), where giand gc are the coordinates of the nodes in the output space. ! 2 99g i g c 99 ð15Þ hci ðt Þ ¼ exp  2s2 As a result of this training algorithm the SOM-based degradation assessment method has several key properties: (1) Dimension reduction. This property results from mapping a high-dimensional input space onto a low-dimensional output space, i.e., the assessment variables with high-dimension can be projected onto a 2-D map by SOM algorithm. When the input is the information about condition states of bearings, such 2-D map from SOM is called degradation mapping on which different BMU areas represent clusters of the same degradation states of bearings. (2) Approximation of the density function. Since prototypes are ‘‘attracted’’ towards high density regions in the input space, after training they will be distributed approximately in the same way as the input data. The best matching unit in the degradation mapping of the closest prototype estimates the degradation distribution for the same failure mode. (3) Topology preservation. The adjacent nodes in the lattice of the map represent similar prototypes in the input space. This property is the consequence of the cooperative nature of the algorithm imposed by the neighborhood function hci. It should be specially noted that prototypes corresponding to adjacent nodes in the output lattice will be subject to a similar adaptation mechanism (since they will always have similar hci) and, hence, they will converge to neighboring regions.

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

391

All these properties make SOM become a powerful tool for the visualization of multidimensional data by transforming complex statistical relationships into simple geometric relationships in a low-dimensional output space (usually 2-D or 3-D) which can be visualized. Particularly, the topology preservation is a key property for grouping similar degradation states of bearings into neighboring regions. Therefore, a dynamic ordered representation of the whole degradation process can be developed by using SOM for degradation trajectory. 3.3. MTS–SOM based dynamic degradation observer A dynamic degradation observer is developed based on the proposed MTS–SOM system in this study. Nine multifractal  features X ¼ ðx1 ,x2 ,. . .,x9 Þ ¼ Dð2Þ, Dð1Þ, Dð0Þ, Dð1Þ, Dð2Þ, Da, f max , Df , Hm Þ of bearing vibration signals are initially selected. Second, MTS is used to optimize the above features, i.e., selecting effective features with greater contribution to the SNR of Mahalanobis spaces. Based on MTS, Mahalanobis distance is used to discriminate bearing degradation states especially to identify the incipient fault stage. Third, by using the optimal features, the assessment mechanism of degradation mapping is trained by SOM, so different BMU areas are able to represent clusters of the same degradation states of bearings. Through plotting the trajectory of current data on a labeled map, the whole bearing degradation process from normal stage until complete failure can be followed over time, which helps to provide important information for the proactive maintenance in advance. The overall work flow of the dynamic degradation observer is shown in Fig. 2. Particularly after the input feature data is normalized, the degradation mapping from SOM should be trained iteratively. In each training step, one sample vector X from the input data set is chosen randomly and the distance between it and all the weight vectors of the SOM, which was originally initialized randomly, is calculated using certain distance measure, such as Euclidian distance. The best matching unit (BMU) is the map unit whose weight vector is the closest to X. After the BMU is identified, the weight vectors of the BMU as well as its topological neighbors are updated so that there are moved closer to the input vector in the input space. At the end of the learning process, depending on their distance in the input space the weight vectors are grouped in several clusters. The trained SOM can be treated as a state space, where different clusters represent different degradation states of bearings. So the condition of the bearings can be described by its matching region in the SOM. And the changes of bearing states can be described by the trajectory of its BMUs in the SOM. In normal stage, the BMUs should follow well-defined paths or trajectories in normal regions. When an incipient fault appears, i.e., the degradation process begins,

Fig. 2. Schematic of the integrated degradation observing process based on MTS–SOM system.

392

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

its corresponding BMU would deviate from the normal region. The level of deviation depends on the severity of the abnormality of the running condition.

4. Case study 4.1. Experiment conditions In this research, accelerated bearing life test is accomplished by the specially designed test rig which is shown in Figs. 3 and 4. Bearing samples are mounted on one shaft of a test stand driven by an AC motor through a rubber belt drive, which compose the test head. It simultaneously hosts four bearings on one shaft. A new one will be installed if one bearing is declared failed, which can improve test efficiency. Data acquisition system includes acceleration sensors, one signal switch instrument and one DAQCard-6023E data acquisition card. During the experiment, four 6208-2RS single row deep groove ball bearings were mounted on one shaft. The rotation speed was kept constant at 3000 rpm. Every bearing had eight balls, with pitch diameter of 34.5 mm, ball diameter of 7.938 mm, and contact angle of 0 degrees. All the bearings were forcibly lubricated by grease lubrication. Four thermocouples were attached to the outer race of each bearing to record the temperature of bearings. During the experiment, the outer races were fixed and the inner races rotated with the shaft. According to the bearing type, the deep groove ball bearing was subjected to a radial load, and the basic dynamic load rate C was set as 31 kN. While the basic load was usually set as P¼0.25C–0.35C¼ 7.75–9.3 kN, so P¼ 8 kN was used in this test. The distribution of the radial load is illustrated in Fig. 3, while other related parameters of the experiment are shown

Load

Thermocouple

Load

Load

Fig. 3. Load conditions of the test rig.

Thermocouple

Data acquisition unit

Accelerometer On-site PC

ABLT-1Atester Weight

Fig. 4. Accelerated bearing life test rig layout.

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

393

in Table 1. In this way, the radial load Fr should be provided as Fr ¼2  P¼ 16 kN by the test rig. The load given by the weights was amplified 100 times by an oil-pressure amplifying unit, i.e., the ratio of the total radial load of the cylinder piston to the weights coupled with hook was set as 100. Therefore the quality provided by the weights on the test rig was then calculated as m¼ Fr/9.8/100  1000  2¼14.3 kg. Accelerometers for vibration detection were installed on the test stand housing. The original data were collected and transformed to the on-site PC and further preprocessed through basic noise reduction. A few features in the time domain were calculated first in the dynamic degradation observer programmed by China University of Petroleum (Beijing), such as RMS, kurtosis, peak to peak value, skewness, etc. Vibration data were generally stored in the disk at every 3 min. The data sampling rate was 20 kHz by the data acquisition unit and the stored data length was 2048 points. The total time of this experiment was more than 680 h, from July 4th to August 1st, 2010. The failure of the bearing is shown in Fig. 5. Four vibration features (i.e., kurtosis, skewness, peak to peak value and RMS) are shown in Fig. 6, which are usually used in traditional methods and will be compared by the proposed indicators. From Fig. 6, the RMS and peak-to-peak value (PP) can roughly manifest the full cycle life of bearing considering several degradation processes, which can be divided into four stages which are normal state, incipient fault, severe degradation and failure stages (the time-waveforms of vibration signals in four degradation stages are shown in Fig. 7), where RMS is the best indicator comparatively. However, the RMS and PP value really appear to have very little relationship to the degradation states except at the final failure stage. They are unable to discover the incipient fault and hard to indicate the severe degradation state. Other features, such as kurtosis and skewness are very sensitive to the early deviation even before the incipient fault stage, while fail to identify the degradation stage which tends to produce false alarms. Therefore, all the parameters in Fig. 6 are impossible to use to assess the different degradation states. And if we only use these parameters as indicators to detect the running condition of bearings, it is usually too late to carry out predictive maintenance, since when they give warning about failure, accidents or unplanned outage will occur soon, and corrective maintenance has to be taken. In the best way, predictive maintenance should be carried out when bearing begins to have incipient fault, not at failure stage. So new degradation indicators are proposed in this research, which are able to differentiate the incipient fault from normal state effectively. The effective results will be presented in Section 4.2 in detail.

Table 1 Parameters of the tested bearings in the experiment.

Parameter

Bearing type 6208-2RS Lubrication method: grease lubrication

Ball number ball diameter (mm) pitch diameter (mm) contact angle Ball-pass frequency of inner ring (Hz) Ball-pass frequency of outer ring (Hz) Ball-spin frequency data sampling rata (kHz) rotation speed (r/min) Load

8 7.938 34.5 0 246.02 153.98 102.9 20 3000 Radial load P¼ 8 kN, 2P¼ 16 kN

Fig. 5. The defective surface of failure bearing (inner race pitting).

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

22

0

20

-0.2

18

-0.4

16

-0.6

SKEWNESS

KURTOSIS

394

14 12

-0.8 -1

10

-1.2

8

-1.4

6

-1.6

4

-1.8 50

100

150

200

250

300

350

400

50

450

100

150

200

250

300

350

400

450

300

350

400

450

time/hour

time/hour 22 220 20

180

18

160

16

RMS

Peak to peak value

200

140

14

120 12 100 10

80

8

60 50

100

150

200

250

300

350

400

450

50

100

150

time/hour

200

250

time/hour

Fig. 6. Vibration features of the tested bearings on its full cycle life. Incipient fault

Severe degradation

Failure 150

100

100

100

100

50 0 -50

50 0 -50

50 0 -50

-100

-100

-100

-150

-150

-150

0

0.05 time/sec

0.1

0

0.05 time/sec

0.1

acceleration/m/s2

150 acceleration/m/s2

150 acceleration/m/s2

acceleration/m/s2

Normal 150

50 0 -50 -100

0

0.05 time/sec

0.1

-150

0

0.05 time/sec

0.1

Fig. 7. Vibration data in time-waveform after noise reduction during bearing’s degradation process.

4.2. Multifractal feature extraction for bearing degradation 4.2.1. Main parameter selection (1) Selection of delay time t. The normal running data of bearing is first analyzed for phase space reconstruction with different delay times t. The evolution process of the phase space trajectory can explain the relationship between the delay time and the degree of phase space extension, which is shown in Fig. 8. When t ¼ 3, 10, the phase trajectory extends to its maximal degree; when t ¼6, 7, the phase trajectory shrinks to its minimal degree along the auxiliary diagonal; when t ¼1, 12, the phase trajectory shrinks to its minimal degree along the main diagonal. So according to the principle of pseudo-phase portrait method, the delay time t should be set as t ¼3. (2) Selection of embedding dimension m. According to the Takens theorem and the trial and error method, when the weight factor is first fixed as q¼2, while m is set from 4 to 20, the corresponding double logarithmic plot lnðrÞlnðcðq,rÞÞcan be drawn which is shown in Fig. 9. The corresponding fractal dimensions D(2) with different values of parameter m are calculated, respectively, which are shown in Table 2. The double logarithmic plot lnðrÞlnðcðq,rÞÞ tends to converge gradually

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

tao=1

tao=2

tao=3

tao=4

395

tao=5

tao=6

500

500

500

500

500

500

0

0

0

0

0

0

-500 -500 -500 -500 -500 -500 -500 0 500 -500 0 500 -500 0 500 -500 0 500 -500 0 500 -500 0 500 tao=7

tao=9

tao=8

tao=10

tao=12

tao=11

500

500

500

500

500

500

0

0

0

0

0

0

-500 -500 -500 -500 -500 -500 -500 0 500 -500 0 500 -500 0 500 -500 0 500 -500 0 500 -500 0 500 Fig. 8. Pseudo-phase portrait of vibration signal with t ¼ 1 to t ¼12.

m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11 m=12 m=13 m=14 m=15 m=16 m=17 m=18 m=19 m=20

2 0 -2

ln(r)

-4 -6 -8 -10 -12 -14 2

3

4

5

6

7

8

ln(c(q,r)) Fig. 9. lnðrÞlnðcðq,rÞÞ double logarithmic plot. Table 2 Fractal dimension with different embedding dimensions. m D(2)

4 2.36

5 2.54

6 2.76

7 3.02

8 3.19

9 3.25

10 3.54

11 3.54

12 3.54

as the embedding dimension m increases from 4. When m¼10, the system becomes steady, and the fractal dimension reaches the stable value as 3.54. Therefore, the embedding dimension should be set as m¼10. 4.2.2. Multifractal features for bearing degradation process Considering the data samples of the above four degradation stages, based on multifractal theory, weight factor is first set as qA[ 20,20], and the generalized dimensions can be calculated by Eqs. (1)–(5). When the weight factor q is set as 2, 1, 0, 1, 2, respectively, the calculated generalized dimensions of four degradation levels are prominently different, which can represent the characteristics of the incipient fault and severe degradation stage of bearings. Hence, the corresponding generalized dimensions as D(  2), D(  1), D(0), D(1), and D(2) serve as five multifractal features. Second, multifractal spectrum is also calculated based on Eqs. (6)–(8). Then the width of multifractal spectrum Da, the difference of the fractal dimensions Df between the maximum probability and minimum probability subsets, the maximum of multifractal spectrum fm, and the multifractal spectrum entropy Hm are considered as four multifractal features as well. Finally, the calculated multifractal feature vectors (D(  2), D( 1), D(0), D(1), and D(2), Da, fm, Df, Hm) are developed for the identification of the degradation process of bearings, which are shown in Table 3, corresponding to the above four different degradation stages. 4.2.3. Feature optimization and degradation identification by MTS The statistical rule of multifractal characteristic distribution is analyzed by Mahalanobis distance. Nine kinds of multifractal features from 50 normal samples are utilized as original data to construct the Mahalanobis space. In the experiment, observational signals during the full cycle life were used to calculate the Mahalanobis distances as

396

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

Table 3 Typical multifractal feature vectors at four different degradation stages. Multifractal features

D(  2)

D(  1)

D(0)

D(1)

D(2)

Da

fm

Df

Hm

Normal state Incipient fault Severe degradation Failure

3.18 4.39 5.11 7.36

2.85 4.32 4.84 7.25

2.72 4.06 4.96 7.18

2.51 3.90 4.74 6.94

2.33 3.57 4.54 5.73

6.89 9.62 11.67 13.37

36.62 50.83 57.95 61.74

 4.19  6.24  7.39  8.10

4.17 4.51 4.72 5.23

(4) Failure

80

Whole Life

Mahalanobis distance

70 (3)Severe degradation

60 (2)Incipient fault

50 40

(1) Normal

30 20 10 50

100

150

200

250

300

350

400

450

time/hour Fig. 10. Mahalanobis distances of observational signals with original features.

degradation indicator, which are shown in Fig. 10. Comparing to Fig. 6, the incipient fault stage is differentiated more obviously. However, the observations of degradation indicator in each stage are unable to concentrate in a certain small range, and there is difficult to set the threshold for each degradation stage. In order to improve the performance of the degradation identification, the above features should be optimized to select the most effective ones. The SNRs of each feature on level 1 and level 2 are computed respectively based on the variable arrangement scheme by the orthogonal array L12(211). The SNR of level 1 related to each feature is subtracted from the SNR of level 2 to describe the difference in whether to select the certain feature. According to the results, the SNRs of features D(0) and D(1) all decrease when they participate in degradation identification, which means that these two features play a negative role in the fault identification process, and should be eliminated from the original reference group. Figs. 11–13 show the Mahalanobis distances as degradation indicators for bearings by using different multifractal feature sets in three cases. In the first case in Fig. 11, only D(0) is eliminated from the reference group because its SNR ( 0.82) is the negative farthest from zero. Compared to Fig. 10, the observations of degradation indicator in each stage are more concentrated in a certain range, which can be easily differentiated from each other. In Fig. 12, the feature set is constructed by only seven features without D(0) and D(1), and the full cycle life can be successfully presented by the constructed Mahalanobis distance as degradation indicator. The four degradation stages are differentiated more clearly. So the thresholds for each stage can be set as (0, 5), [5, 40), [40, 80) and [80, þN) corresponding to the normal, incipient fault, severe degradation and failure stage, respectively. Furthermore D(2) whose SNR is positive is further eliminated from the reference group after the deletion of D(0) and D(1), then the corresponding trend of degradation indicator which composes only six features is shown in Fig. 13, from which the identification results are not as good as Fig. 12. So the feature D(2) should not be eliminated from the reference group, and only the features with negative SNR must be removed for optimization. Therefore, the optimal feature vectors are determined as (D(  2), D(  1), D(2), Da, fm, Df, Hm) which can be utilized to construct the Mahalanobis space as effective degradation indicator. 4.3. Dynamic degradation assessment of bearings In order to get the degradation mapping and trajectory of the degradation process, the training sets should involve at least four different states, which contain the normal state and three degradation states from incipient fault to complete failure stage. With the exception of abrupt catastrophic failure, most of the bearing lives can be considered as continuous processes from normal to failure. Consequently their coordinates in feature space should have traceable trajectories drifting from the normal operation region to various degradation regions. Optimal multifractal features (D(  2), D(  1), D(2), Da, fm, Df, Hm) obtained from Section 4.2 are further used for SOM network training as inputs. During the training,

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

397

(4) Failure

250

Whole Life

Mahalanobis distance

200

(2)Incipient fault

(3)Severe degradation

150

100

(1) Normal

50

50

100

150

200

250 time/hour

300

350

400

450

Fig. 11. Mahalanobis distances of monitoring signals with features without D(0).

140

Mahalanobis distance

(4) Failure

120

Whole Life

100

(3)Severe degradation (2)Incipient fault

80 60

(1) Normal

40 20

50

100

150

200

250 time/hour

300

350

400

450

Fig. 12. Mahalanobis distances of monitoring signals with features without D(0) and D(1) (Dash line: degradation stage threshold).

90 (4) Failure

80 Whole Life

Mahalanobis distance

70

(3)Severe degradation

60 (2)Incipient fault

50 40

(1) Normal

30 20 10 50

100

150

200

250 time/hour

300

350

400

Fig. 13. Mahalanobis distances of monitoring signals with features without D(0), D(1) and D(2).

450

398

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

50 samples of each state are used, and the remaining samples in the experiment are further used to test the accuracy of the trained SOM network. Therefore, the trained SOM based degradation observer can be utilized for detecting the specific degradation stage of bearings with new data and also predicting its future degradation trend, which plays an important role in proactive maintenance. The U-matrix describing the four running stages from trained SOM network are shown in Fig. 14. The U-matrix is considered as the degradation mapping which integrates all the nonlinear multifractal features together to reflect the degradation trend. In Fig. 14, the clusters in blue units represent different degradation states of bearing inner race from the normal states in the top right corner area to the failure stage (inner race pitting) located in the bottom left area. The degradation trajectory can be drawn according to the development of the degradation severity. By continuously tracking the trajectories, degradation observing in optimal multifractal feature space can be accomplished for proactive maintenance. In order to further observer the development of degradation in more detailed trajectory, four different degradation stages (DEG1—DEG 4) are selected from incipient fault (DEG1) to failure stages (DEG4) as SOM output during the training process. The result is shown in Fig. 15. There are some common characteristics both in Figs. 14 and 15, which are summarized as follows: (1) The normal states usually sit in the top right area in these U-matrices, and have a clear bounder between other fault states. So the abnormity detection which is responsible for judging whether the bearing is in good condition or not, can be easily accomplished without any complex calculation. (2) The degradation trend comes from the normal state in the top right area to the bottom left area along the diagonal of the map.

U-matrix

Labels 4.1

Severe Severe

Normal Normal Normal

Severe Severe

Normal Normal Normal

Severe Severe Severe

Normal Normal Normal

Normal

Normal Normal 2.14

Normal

Incipient Incipient Incipient Incipient Incipient Incipient Incipient Incipient Incipient

Failure

Failure

Failure

Incipient Incipient

Failure

Incipient

Failure Failure

Failure

Failure

Failure Failure

Failure

Failure

Failure

Failure

0.178 Fig. 14. U-Matrix for degradation assessment of bearings with degradation trajectory with normal and three degradation stages (‘‘severe’’: severe degradation, ‘‘incipient’’: incipient fault).

U-matrix

Labels 6.23

DEG3

DEG3

DEG3

DEG1

DEG3

Normal

DEG3

Normal Normal Normal

DEG1 DEG1 DEG1 DEG1 DEG1 DEG1 DEG1 DEG1

3.17

Normal

DEG1

DEG2 DEG2 DEG2 DEG2 DEG2 DEG2 DEG2 DEG2

0.109

DEG4

DEG4

DEG4

DEG4 DEG4

DEG4 DEG4 DEG4

DEG4

DEG4

DEG4

DEG4

Fig. 15. U-Matrix of degradation assessment of bearings with degradation trajectory with normal and four degradation stages (DEG1-4: four different degradation stages).

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

399

Table 4 The accuracy of the degradation assessment in the experiment. Practical degradation states of bearing

Results of test Normal Incipient fault Severe degradation Failure Number of misjudgment Number of test samples Accuracy for each states judgment (%) Overall accuracy

Normal

Incipient fault

Severe degradation

Failure

18 2 0 0 2 20 90 85.71%

3 16 1 0 4 20 80

0 2 17 1 3 20 85

0 0 1 9 1 10 90

(3) The severe levels of degradation usually locate in the top left area, so the trajectory will go upwards after it reaches the end of the diagonal of the map. (4) The failure states are located in the bottom areas which indicate the malfunction or the shutdown of machine would happen and the repair or replacement should be prepared in time. So when the new data is obtained in other case in the future, it can be projected on these fault degradation mappings by the trained SOM network and to find out in which area it is located. Therefore the degradation level can be figured out, which provides the information for maintenance decision making. The degradation trajectory can be further used to evaluate the RUL (remaining useful life) of bearings, when the quantitative threshold of the degradation is set up and the corresponding time tag is available. The remaining samples are used to test the effectiveness and accuracy of the proposed method. In the test, 20 samples with unknown degradation level are used, while 10 test samples corresponding to failure, because of the short length of the failure stage. The results are shown in Table 4. Fortunately, all degradation states are identified and alarms are produced for maintenance in the test. However, there are 10 occurrences of misjudgment in total after 70 tests, i.e., two normal states are considered as incipient faults in the test, four incipient faults are misjudged as normal or severe degradation, three severe degradation states are misjudged as incipient fault or failure stage, while only one failure state is considered as severe degradation state. So the overall accuracy of the test is estimated as 85.71%. By further analysis of the test results in Table 4, some explanation should be made as follows, which helps to utilize the proposed MTS–SOM system for degradation observing in the application. (1) The misjudgments in the test happened between adjacent states, for example, the test sample corresponding to incipient fault stage may be misjudged as normal or severe degradation state, but has not been judged as failure state. So actually there is no harm for maintenance decision-making, since the running process of bearings beginning from normal to failure is some kind of progression, and there usually exists no clear boundary between each adjacent state. So the above results can be accepted in the industrial application. (2) The practical degradation states as incipient fault and severe degradation set in the test are mainly determined by MTS-based degradation identification and researcher’s experience with historical data analysis before. Because most experiments have to be in operation until failure happens with neither opportunity nor feasibility to check the degradation states of the running bearings, so the actual accuracy for the judgment of incipient fault and severe degradation states in the test should be higher than the results shown in Table 4.

5. Conclusion In this work we have presented a dynamic degradation observer for degradation identification and assessment of bearing by means of MTS–SOM system. The power of this methodology relies in the efficient synergy between the well known advantages of SOM visualization and the classical feature optimization tool based on MTS, allowing to discovering relationships between degradation-related features and its degradation degree. Multifractal features also play an important role in the degradation observing process for the identification of incipient fault. A case study is illustrated in detail to demonstrate the possibility and effectiveness of the proposed approach. It also showed that MTS–SOM based degradation observer procedure provides a means of enhancing the condition monitoring of bearing, indicating the current degradation state and tracking the degradation trend dynamically. It also suggests further work in several areas, such as visualization of multi-fault condition by SOM mapping, and remaining useful life prediction combined with time series prediction methods.

400

J. Hu et al. / Mechanical Systems and Signal Processing 36 (2013) 385–400

Acknowledgment The project is supported by the National Science Foundation of China (Grant No. 51104168 and Grant No. 51005247); and National Science and Technology Major Project of China (Grant No. 2011ZX05055); and PetroChina Innovation Foundation (Grant No. 2011D-5006-0408); and Excellent Doctoral Dissertation Supervisor Project of Beijing (Grant YB20111141401); and also sponsored by Research Funds Provided to New Recruitments of China University of PetroleumBeijing (Grant No. QD-2010-01). References [1] P. Das, S. Datta, Exploring the effects of chemical composition in hot rolled steel product using Mahalanobis distance scale under Mahalanobis– Taguchi system, Comput. Mater. Sci. 38 (4) (2007) 671–677. [2] P. Grassberger, I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Physica D 13 (1–2) (1984) 34–54. [3] H. Hong, M. Liang, Fault severity assessment for rolling element bearings using the Lempel–Ziv complexity and continuous wavelet transform, J. Sound Vib. 320 (1–2) (2009) 452–468. [4] Z. He, Y.J. Han, A study on the multivariate syatem diagnosis and analysis using Mahalanobis-Taghchi system, Appl. Stat. Manag. 26 (5) (2007) 830–839. in Chinese. [5] J. Hu, L. Zhang, W. Liang, Z. Wang, Mechanical Incipient Fault Detection Based on Multifractal and MTS method, Petrol. Sci. 6 (2) (2009) 208–216. [6] J. Kangas, T. Kohonen, Developments and applications of the self-organizing map and related algorithms, Math. Comput. Simulat. 41 (1–2) (1996) 3–12. [7] P.K. Kankar, S.C. Sharma, S.P. Harsha, Fault diagnosis of ball bearings using machine learning methods, Expert Syst. Appl. 38 (3) (2011) 1876–1886. [8] P.K. Kankar, S.C. Sharma, S.P. Harsha, Fault diagnosis of ball bearings using continuous wavelet transform, Appl. Soft Comput. 11 (2) (2011) 2300–2312. [9] T. Kohonen, Self-organizing neural projections, Neural Networks 19 (6–7) (2006) 723–733. [10] T.S. Li, C.L. Huang, Defect spatial pattern recognition using a hybrid SOM–SVM approach in semiconductor manufacturing, Expert Syst. Appl. 36 (1) (2009) 374–385. [11] D. Logan, J. Mathew, Using the correlation dimension for vibration fault diagnosis of rolling element bearings-I: basic concepts, Mech. Syst. Signal Process. 10 (3) (1996) 241–250. [12] D. Logan, J. Mathew, Using the correlation dimension for vibration fault diagnosis of rolling element bearings-II: select of experimental parameters, Mech. Syst. Signal Process. 10 (3) (1996) 251–264. [13] Y. Pan, J. Chen, X. Li, Bearing performance degradation assessment based on lifting wavelet packet decomposition and fuzzy c-means, Mech. Syst. Signal Process. 24 (2) (2010) 559–566. [14] S. Pedro, F. Nuno, C, Raquel, et al., Fault identification in chemical processes through a modified Mahalanobis–Taguchi strategy, Comput. Aided Chem. Eng. 18 (2004) 799–804. [15] V. Purushotham, S. Narayanan, S.A.N. Prasad, Multi-fault diagnosis of rolling bearing elements using wavelet analysis and hidden Markov model based fault recognition, NDT&E Int. 38 (8) (2005) 654–664. [16] H. Qiu, J. Lee, G. Yu, Robust performance degradation assessment methods for enhanced rolling element bearing prognostics, Adv. Eng. Inform. 17 (2003) 127–140. [17] A. Rojas, A.K. Nandi, Practical scheme for fast detection and classification of rolling-element bearing faults using support vector machines, Mech. Syst. Signal Process. 20 (7) (2006) 1523–1536. [18] B. Samanta, K.R. Al-Balushi, S.A. Al-Araimi, Artificial neural networks and support vector machines with genetic algorithm for bearing fault detection, Eng. Appl. Artif. Intel. 16 (7–8) (2003) 657–665. [19] B.S. Yang, W.W. Hwang, D.J. Kim, et al., Condition classification of small reciprocating compressor for refrigerators using artificial neural networks and support vector machines, Mech. Syst. Signal Process. 19 (2) (2005) 371–390. [20] Y. Yang, D. Yu, J. Cheng, A fault diagnosis approach for roller bearing based on IMF envelope spectrum and SVM, Measurement 40 (9–10) (2007) 943–950. [21] J. Yu, Bearing performance degradation assessment using locality preserving projections and Gaussian mixture models, Mech. Syst. Signal Process. 25 (7) (2011) 2573–2588. [22] J. Yu, Bearing performance degradation assessment using locality preserving projections, Expert Syst. Appl. 38 (6) (2011) 7440–7450. [23] Y.L. Yu, S.L. Xie, H.T. Cai, Introduction of New Signal Processing Methods, Tsinghua University Press, Beijing, 2004 184–190 (in Chinese.