- Open Access
An approach to performance assessment and fault diagnosis for rotating machinery equipment
© Tao et al.; licensee Springer. 2013
- Received: 7 May 2012
- Accepted: 2 December 2012
- Published: 10 January 2013
Predict and prevent maintenance is routinely carried out. However, how to address the problem of performance assessment maximizing the use of available monitoring data, and how to build a framework that integrates performance assessment, fault detection, and diagnosis are still a significant challenge. For this purpose, this article introduces an approach to performance assessment and fault diagnosis for rotating machinery, including wavelet packet decomposition for extracting energy feature samples from vibration signals acquired during normal and faulty conditions; clustering analysis for demonstrating the separability of the samples; and Fisher discriminant analysis for providing an optimal lower-dimensional representation, in terms of maximizing the separability among different populations, by projecting the samples into a new space. In the new low-dimensional space, the Mahalanobis distance (MD) between the new measurement data and normal population can be calculated for performance assessment. Moreover, this model for performance assessment only requires data to be available in normal conditions and any one of all possible fault conditions, without the necessity for the full life cycle of condition monitoring data. In addition, if monitoring data under different fault conditions are available, the fault mode can be identified accurately by comparing the MDs between the new measurement data and each fault population. Finally, the proposed method was verified to be successful on performance assessment and fault diagnosis via a hydraulic pump test and a ball bearing test.
- Performance assessment
- Fault diagnosis
- Fisher discriminant analysis
- Mahalanobis distance
Currently, driven by the demand to reduce maintenance costs, shorten repair time, and maintain high availability of equipment, maintenance strategies have progressed from breakdown maintenance (fail and fix) to preventive maintenance, then to condition-based maintenance (CBM), and lately toward a prospect of intelligent predictive maintenance (predict and prevent),[1–3]. While the reactively breakdown maintenance and blindly preventive maintenance do sometimes reduce equipment failures, they are more labor intensive, do not eliminate catastrophic failures and cause unnecessary maintenance. This is where CBM steps in. It was reported that 99% of mechanical failures especially rotating machinery are preceded by noticeable indicators. That is to say, with the exception of abrupt, catastrophic failures, most faults of rotating machinery equipment have progression processes to failure. We can view the deterioration as a two-stage process: the first stage as normal operation, and the second stage as a potential failure[5, 6]. Of interest here is when the second stage starts and how it develops. CBM attempts to monitor machinery health based on condition measurements that do not interrupt normal machine operation. Rotating machinery is one of the most common classes of machines. Over the past few years, technologies in condition monitoring, fault diagnostics, and prognosis for rotating machinery, which are important aspects in a CBM program, have been receiving more attention. Because fault diagnosis problems can be considered as classification problems, Fisher discriminant analysis (FDA), which is studied in detail in the pattern classification literature, has been applied to conduct fault diagnosis. However, at present, its application has mainly been concentrated to industrial process (especially chemical processes), but rarely to rotating machinery equipment[8–11]. Moreover, although performance assessment, fault detection, diagnosis, and prognosis have received increased attention with significant progress, currently, few methods can realize those purposes alone. In addition, the incomplete data (monitoring data under normal or fault conditions) have rarely been considered as input data to performance assessment and this has yet to be fully utilized. To fill the gaps, this article uses the combination of FDA and Mahalanobis distance (MD) applied to rotating machinery fault diagnosis, which is further extended to performance assessment and fault detection. Thus, a framework integrating performance assessment, fault detection, and diagnosis is built, which not only solves the problem of when the second stage of potential failure starts and how it develops, but because of its data-driven property also shows appropriate potential and provides a functional interface to performance trend prognosis.
Successful clustering can demonstrate the separability of samples and provide preconditions for FDA. To avoid the influence of the dispersibility of sample data acquired during normal and various fault conditions, the analysis of the separability of sample data is indispensable. Cluster analysis can classify samples into corresponding groups based on the measured parameters, and hierarchical cluster analysis (HCA) is the most commonly used clustering tool[13, 14]. FDA provides an optimal lower dimensional representation, in terms of maximizing the separability among different populations representative of different operational states, by projecting normal and fault populations, and separating them to the limit in the reconstructed space[15–17]. In the reconstructed low-dimensional space, the MD between the new measurement data and the normal population, constructed using normal data, can be calculated for performance assessment. The MD can also be transformed into a normalized confidence value (CV), according to the presupposed threshold. If an abnormal state is detected by performance assessment, the MDs between the new measurement data and the normal and different fault populations are calculated, to identify which population the new data belong to, and thus, the fault mode can be recognized.
The proposed method for performance assessment only needs monitoring data under normal conditions and any one of all possible fault conditions. As for fault diagnosis, if the monitoring data under different fault conditions are available, accurate diagnosis results can be achieved by this model. Moreover, in this model, the algorithm is simple and intuitive and offers good interpretation for the results. The proposed method was also verified to be effective and pragmatic for performance assessment and fault diagnosis via a hydraulic pump test and a ball bearing test.
2.1. Wavelet packet decomposition-based feature extraction
In practice, the characteristic frequencies of rotating machinery equipment or components are usually distributed in both high- and low-frequency bands. In view of this fact, wavelet packet analysis (WPA) is proposed to construct a more sophisticated method of orthogonal decomposition based on multi-resolution analysis, which can divide the full frequency band into multi-levels, so that each band contains information that is more specific[18, 19]. Therefore, wavelet packet decomposition is suitable for extracting both low- and high-frequency features. Statistically analyzing all bands of a signal decomposed by wavelet packet, an energy index of each frequency band can be extracted.
where x jk (j = 0, 1, …, 7; k = 1, 2, …, n) denotes the amplitude of discrete points in the reconstructed signal S 3j .
2.2. Separability analysis of sample dataSeparability analysis of sample data
In order to avoid the influence of the dispersibility of sample data acquired during normal and various fault conditions, analysis of the separability of the sample data is indispensable.
Clustering is an unsupervised pattern classification method, in which the goal is to determine a finite set of categories to describe a dataset according to similarities among its objects[22, 23]. As one of the most popular clustering methods, HCA consists of mathematically treating each sample as a point in multidimensional space described by the chosen variables; it builds a nested partition set called a cluster hierarchy. When a given sample is taken as a point in the space defined by the variables, the distance between this point and all the other points can be calculated, thereby establishing a matrix that describes the proximity between all the samples studied. Based on this matrix of proximity between the samples, one can construct a similarity diagram called a dendrogram. There are many ways of mathematically grouping these points in multidimensional space in order to form hierarchical clusters[24–26].
As an optimal linear dimensionality reduction technique, in terms of maximizing the separation between different populations, FDA has been studied in detail in the pattern classification literature[27–29]. For either performance assessment or fault diagnosis, data collected from the unit during normal and various fault states are categorized into different populations, where each population contains data representing a particular state.
2.3.1. Definition (MD)
where μ is the mean vector of G, then d(x,G) is called the MD between x and the population G.
when x = a k , the maximum value can be achieved as λ k .
For a more detailed discussion of the extremal problem of the quadratic form, refer to.
The mathematical derivation process of FDA follows.
where,, E = ∑ i = 1 k Σ i , n i is the number of samples for G i , n is the total number of samples. Then B is the between-class-scatter matrix, and E is the within-class-scatter matrix.
Thinking along the lines of variance analysis, in order to better separate each population, the choice of a should make B 0 expand as far as possible, while E 0 narrows as far as possible.
According to the lemma given in Section 2.3.2, when a is the standard eigenvector a 1 (standardized into a i ′ Ba i = 1) corresponding to the maximum eigenvalue λ1 of E –1 B, formula (10) achieves the maximum λ1. Thus, the first canonical variable is y 1 = a 1 ′ x.
The second FDA vector is computed to maximize the scatter between classes, while minimizing the scatter within classes, among all axes perpendicular to the first FDA vector a 1. According to the lemma given in Section 2.3.2, the second canonical variable is y 2 = a 2′x and so on for the remaining FDA vectors and canonical variables. Usually, given the first m eigenvalues as λ 1 ≥ λ 2 ≥···≥ λ m , the corresponding standard eigenvectors as a 1,a 2,…,a m , when the accumulation contribution reaches a threshold (such as 85%), we can get m unrelated canonical variables y 1,y 2,…,y m ,y m = a m ′x for discriminant analysis. This is equivalent to mapping a variable from p-dimensional space to m-dimensional space for analysis, where m < p.
2.4. MD for performance assessment
In practice, the whole life cycle of condition-monitoring data acquired from a machine is already scarce due to irregular measurement recording, and/or the huge amount of time they take to accumulate. For example, a bearing may last several years even under harsh operating conditions. Therefore, regardless of whether it is an experimental or practical application, it is hard to acquire condition-monitoring data that are representative of the whole life cycle. More commonly, what we can get are incomplete data, such as normal data and fault data[3, 31]. Usually, the incomplete data are rarely to be considered as input data of performance assessment, and this has yet to be fully utilized. Consequently, maximizing the use of available data to address the problem of performance assessment is a significant challenge.
where c is a scale parameter, which is determined by the averaged MDs under normal state and a predetermined CV benchmark.
2.5. MD for fault diagnosis
If, then x∈G l .
The state represented by the population that has the minimum MD with the real measurement data can be identified as the current operating state and thus fault diagnosis is completed.
Two experimental cases (i) hydraulic pump performance assessment and fault diagnosisand (ii) ball bearing performance assessment and fault diagnosis are presented to validate the effectiveness and practicality of applying the proposed method to performance assessment and fault diagnosis of rotating machinery equipment. The description of these case studies will follow the sequence: (1) experimental setup and data acquisition; (2) signal analysis and feature extraction; (3) analysis of performance assessment and fault diagnosis results.
3.1. Performance assessment and fault diagnosis for hydraulic pump
The hydraulic pump is the heart of a hydraulic system, which determines whether the whole system can run normally or not. Therefore, performance assessment and fault diagnosis of the hydraulic pump is of great importance. Usually, when a hydraulic pump is under an abnormal state, it will be revealed by changes of vibration. Because most mechanical faults are reflected by vibration, the vibration signals of the hydraulic pump are collected and analyzed in this experiment for performance assessment and fault diagnosis.
3.1.1. Experimental setup and data acquisition
3.1.2. Feature extraction by WPA
For those three states, eight feature vector samples were acquired, respectively. The first four samples of each state were used as the FDA learning set (for clearer identification they were numbered as Normal_1 to Normal_4, Fault1_1 to Fault1_4, and Fault2_1 to Fault2_4, corresponding to the normal state, slipper loose state, and valve plate wear conditions, respectively), while the others were used as the testing set (they were also numbered in the same manner as: N_1 to N_4, F1_1 to F1_4, and F2_1 to F2_4, respectively).
Feature vector samples for learning (hydraulic pump)
Feature vector samples for testing (hydraulic pump)
3.1.3. Analysis of states separability
Clustering membership of learning set (hydraulic pump)
3.1.4. Analysis of performance assessment results
In this study, for the purpose of performance assessment, the normal samples numbered as Normal_1 to Normal_4 were used to construct and characterize the normal population noted as G _N. Through FDA, a space conversion method was achieved, by which, the different populations and the new measurement data can be projected from the original eight-dimensional space into a new two-dimensional space.
3.1.5. Analysis of fault diagnosis results
MDs and diagnosis results of samples in testing set (hydraulic pump)
3.2. Performance assessment and fault diagnosis for ball bearing
Bearings are critical components in rotating machines because their failure could lead to serious damage in machines. In recent years, bearing fault diagnosis has received increasing attention[34–36]. In this case, vibration signals are acquired from the ball bearing housing for performance assessment and fault diagnosis.
3.2.1. Experimental setup and data acquisition
3.2.2. Feature extraction by WPA
Through implementation of FFT on the acquired vibration data, it was found that the characteristic frequencies under normal, inner-race fault, outer-race fault, and ball fault states are distributed separately in different frequency bands, and that there is little overlap between any two adjacent frequencies in the spectrum. Therefore, three-layer wavelet packet decomposition can also be applied to the vibration data, and thus, eight-dimensional feature vectors can be acquired by calculating and normalizing the energy of each band.
For those four states, eight feature vector samples were acquired. The first four samples of each state were used as the FDA learning set (for clearer identification they were numbered as N_1 to N_4, I_1 to I_4, O_1 to O_4, and B_1 to B_4, corresponding to the normal state, inner-race fault state, outer-race fault state, and ball fault state, respectively), while the others were used as the testing set (they were also numbered in the same manner as N_T1 to N_T4, I_T1 to I_T4, O_T1 to O_T4, and B_T1 to B_T4, respectively).
Feature vector samples for learning (rolling bearing)
Feature vector samples for testing (rolling bearing)
3.2.3. Analysis of states separability
Clustering memberships of learning set (rolling bearing)
3.2.4. Analysis of performance assessment results
In this study, the normal samples numbered N_1 to N_4 were used to construct and characterize the normal population noted as G_N. Through FDA, the original eight-dimensional space was projected into a new three-dimensional space.
3.2.5. Analysis of fault diagnosis results
MDs and diagnosis results of samples in testing set (rolling bearing)
In this article, addressing the challenging issues on performance assessment, fault detection, and fault diagnosis, a novel method based on FDA and MD is introduced, and an integrated framework for performance assessment, fault detection, and fault diagnosis is built.
In this method, FDA is applied as an optimal linear dimensionality reduction technique, in terms of maximizing the separation between different populations. In the new low-dimensional space, MD, which can be transformed into normalized CV, is calculated for performance assessment, and abnormal states can be detected with the presupposed threshold. Furthermore, once various fault data are available, the unknown fault mode can be identified accurately by comparing the MDs between the new data and each normal/fault population.
However, how to transform MD into CV for performance assessment and to determine an adaptive threshold for fault detection is still a challenge for future work. In the future, we are going to acquire sequential online degradation measurements for real-time performance degradation assessment and detection, and enlarge the number of fault and learning samples for more accurate fault diagnosis. Moreover, we plan to apply this approach to different components, such as gearboxes, shafts, etc., to further verify the effectiveness and evaluate the possibility of generalizing the proposed approach.
This research was supported by the National Natural Science Foundation of China (Grant nos.61074083, 50705005, and 51105019) as well as the Technology Foundation Program of National Defense (Grant no. Z132010B004). The authors are very grateful for the valuable suggestions from the reviewers and editor.
- Lee J, Ni J, Djurdjanovic D, Qiu H, Liao HT: Intelligent prognostics tools and e-maintenance. Comput. Ind. 2006, 57: 476-489. 10.1016/j.compind.2006.02.014View ArticleGoogle Scholar
- Jardine AKS, Lin D, Banjevic D: A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mech. Syst. Signal Process. 2006, 20: 1483-1510. 10.1016/j.ymssp. 2005.09.012View ArticleGoogle Scholar
- Heng A, Zhang S, Tan ACC, Mathew J: Rotating machinery prognostics: state of the art, challenges and opportunities. Mech. Syst. Signal Process. 2009, 23: 724-739. 10.1016/j.ymssp. 2008.06.009View ArticleGoogle Scholar
- Bloch HP, Geitner FK Machinery Failure Analysis and Trouble Shooting. In Practical Machinery Management for Process Plants. 3rd edition. Gulf Professional Publishing, Houston; 1997.Google Scholar
- Wang WB: Modelling the probability assessment of system state using available condition information. IMA J. Manage. Math. 2006, 17(3):225-234. 10.1093/imaman/dpi035MATHView ArticleGoogle Scholar
- Wang WB: A two-stage prognosis model in condition based maintenance. Eur. J. Oper. Res. 2007, 182: 1177-1187. 10.1016/j.ejor.2006.08.047MATHView ArticleGoogle Scholar
- Ma LL, Zhang Z, Wang JZ: Combination method of support vector machine and fisher discriminant analysis for chemical process fault diagnosis. In Paper presented at the 29th Chinese Control Conference. Beijing; 2010:4000-4003.Google Scholar
- Chiang LH, Kotanchek ME, Kordon AK: Fault diagnosis based on Fisher discriminant analysis and support vector machines. Comput. Chem. Eng. 2004, 28: 1389-1401. 10.1016/j.compchemeng.2003.10.002View ArticleGoogle Scholar
- He QP, Qin SJ, Wang J: A new fault diagnosis method using fault directions in Fisher discriminant analysis. AICHE J. 2005, 51(2):555-571. 10.1002/aic.10325View ArticleGoogle Scholar
- He XB, Wang W, Yang YP, Yang YH: Variable-weighted Fisher discriminant analysis for process fault diagnosis. J. Process. Control. 2009, 19: 923-931. 10.1016/j.jprocont.2008.12.001View ArticleGoogle Scholar
- Fuente MJ, Garcia-Alvarez D, Sainz-Palmero GI: Fault detection and identification method based on multivariate statistical techniques. In Paper presented at the Proceedings of Emerging Technologies & Factory Automation. Mallorca; 2009:1-6.Google Scholar
- Liao LX, Lee J: Design of a reconfigurable prognostics platform for machine tools. Expert Syst. Appl. 2010, 37: 240-252. 10.1016/j.eswa.2009.05.004View ArticleGoogle Scholar
- Gurrutxaga I, Albisua I, Arbelaitz O, Martin JI, Muguerza J, Perez JM, Perona I: An efficient method to find the best partition in hierarchical clustering based on a new cluster validity index. Pattern Recognit 2010, 43: 3364-3373. 10.1016/j.patcog.2010.04.021MATHView ArticleGoogle Scholar
- Goldberger J, Tassa T: A hierarchical clustering algorithm based on the Hungarian method. Pattern Recognit. Lett. 2008, 29(11):1632-1638. 10.1016/j.patrec.2008.04.003View ArticleGoogle Scholar
- Safayani M, Shalman MTM: Matrix-variate probabilistic model for canonical correlation analysis. EURASIP J. Adv. Signal Process. 2011., 7: Article ID 748430Google Scholar
- Eftekhari A, Moghaddam HA, Forouzanfar M, Alirezaie J: Incremental local linear fuzzy classifier in fisher space. EURASIP J. Adv. Signal Process. 2009. Article ID 360834Google Scholar
- Liao LX: An Adaptive Modeling for Robust Prognostics on a Reconfigurable Platform. PhD, University of Cincinnati, Engineering: Industrial Engineering. University of Cincinnati, Cincinnati; 2010.Google Scholar
- Zhao XZ, Ye BY: Convolution wavelet packet transform and its applications to signal processing. Dig. Signal Process. 2010, 20: 1352-1364. 10.1016/j.dsp. 2010.01.007View ArticleGoogle Scholar
- Wu JD, Liu CH: An expert system for fault diagnosis in internal combustion engines using wavelet packet transform and neural network. Expert Syst. Appl. 2009, 36: 4278-4286. 10.1016/j.eswa.2008.03.008View ArticleGoogle Scholar
- Pan YN, Chen J, Li XL: Bearing performance degradation assessment based on lifting wavelet packet decomposition and fuzzy c-means. Mech. Syst. Signal Process. 2010, 24: 559-566. 10.1016/j.ymssp. 2009.07.012View ArticleGoogle Scholar
- Kotnik B, Kacic Z: A comprehensive noise robust speech parameterization algorithm using wavelet packet decomposition-based denoising and speech feature representation techniques. EURASIP J. Adv. Signal Process. 2007. Article ID 64102Google Scholar
- Forestier G, Wemmert C, Gancarski P: Multisource images analysis using collaborative clustering. EURASIP J. Adv. Signal Process. 2008. Art ID 374095Google Scholar
- Wang ZS, Maier A, Logothetis NK, Liang HL: Single-trial classification of bistable perception by integrating empirical mode decomposition, clustering, and support vector machine. EURASIP J. Adv. Signal Process. 2008. Article ID 592742Google Scholar
- Kojadinovic I: Hierarchical clustering of continuous variables based on the empirical copula process and permutation linkages. Comput. Stat. Data Anal. 2010, 54: 90-108. 10.1016/j.csda.2009.07.014MATHMathSciNetView ArticleGoogle Scholar
- Almeida JAS, Barbosa LMS, Pais AACC, Formosinho SJ: Improving hierarchical cluster analysis: a new method with outlier detection and automatic clustering. Chemometrics Int. Lab. Syst. 2007, 87: 208-217. 10.1016/j.chemolab.2007.01.005View ArticleGoogle Scholar
- Song ML, Song YQ, Yu HY, Wang ZY: Calculation of China’s environmental efficiency and relevant hierarchical cluster analysis from the perspective of regional differences. Math. Comput. Model. 2012, . 10.1016/j.mcm.2012.04.003Google Scholar
- Bouveyron C, Brunet C: Probabilistic Fisher discriminant analysis: a robust and flexible alternative to Fisher discriminant analysis. Neurocomputing 2012, 90: 12-22. 10.1016/j.neucom.2011.11.027View ArticleGoogle Scholar
- Khemchandani R, Jayadeva , Chandra S: Learning the optimal kernel for Fisher discriminant analysis via second order cone programming. Eur. J. Oper. Res. 2010, 203: 692-697. 10.1016/j.ejor.2009.09.020MATHMathSciNetView ArticleGoogle Scholar
- Shin H: An extension of Fisher’s discriminant analysis for stochastic processes. J. Multivariate Anal. 2008, 99: 1191-1216. 10.1016/j.jmva.2007.08.001MATHMathSciNetView ArticleGoogle Scholar
- Rao CR: Linear Statistical Inference and Its Application. 2nd edition. Wiley, New York; 1973.View ArticleGoogle Scholar
- Heng A, Tan A, Mathew J, Yang BS: Machine prognosis with full utilization of truncated lifetime data. In Proceedings of the Second World Congress on Engineering Asset Management. Harrogate; 2007:775-784.Google Scholar
- Niu G, Singh S, Holland SW, Pecht M: Health monitoring of electronic products based on Mahalanobis distance and Weibull decision metrics. Microelectron. Reliab. 2011, 51: 279-284. 10.1016/j.microrel.2010.09.009View ArticleGoogle Scholar
- Wang JP, Hu HT: Vibration-based fault diagnosis of pump using fuzzy technique. Measurement 2006, 39: 176-185. 10.1016/j.measurement.2005.07.015View ArticleGoogle Scholar
- Prabhakar S, Mohanty AR, Sekhar AS: Application of discrete wavelet transform for detection of ball bearing race faults. Tribol. Int. 2002, 35(12):793-800. 10.1016/S0301-679X(02)00063-4View ArticleGoogle Scholar
- Al-Raheem KF, Roy A, Ramachandran KP, Harrison DK, Grainger S: Rolling element bearing fault diagnosis using Laplace-Wavelet envelope power spectrum. EURASIP J. Adv. Signal Process. 2007. Article ID 073629Google Scholar
- Ibrahim A, Bonnardot F, El Badaoui M, Guillet F: Detection of bearing damage using stator current, and voltage to cancel electrical noise. EURASIP J. Adv. Signal Process. 2011. Article ID 235236Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.