Vibration analysis on rotary machines using unconventional statistical methods

Transcription

1 Master s Thesis Vibration analysis on rotary machines using unconventional statistical methods Authors: Eduard Abella Alberto Tullio Instituto Tecnológico de Buenos Aires Supervisor: Dr. Ingeniero Francisco Redelico Departamento de Ingeniería Industrial Buenos Aires, Julio 2014

2 1. Introduction Normally, the status of rotary machines cannot be evaluated directly. Instead, the vibration signals produced by these machines must first be measured and then analyzed in detail [1-3]. The information about a fault in the rotary machines is reflected by singular points of abrupt changing signals and, because of that, it is important to detect dynamic changes of the signal in time to get an anticipatory information about the fault [4]. Friction, strikes, clearance between mobile parts, fractures in the bench or broken fasteners may occur during the lifetime of the machine. These are all sources of non-stationary and non-linear vibrations and, therefore, the traditional study of the vibrations through linear methods may not be effective at all to detect dynamic changes on the signal. Many methods have been presented in order to detect these changes since the development of nonlinear dynamic theory. This paper studies the utility of using the Shannon entropy [5] and the Permutation Entropy [6], the Fisher Information [7] and the Statistical Complexity to detect dynamic changes on the vibration signals. The Shannon entropy, H p, characterizes the uncertainty of the expected outcome in a time series. When the transmission is totally random all outcomes are equally probable and the entropy of information is maximum. Conversely, if there is only a single possibility for the outcome, H=0 [9]. The Shannon entropy gives global information, because it is not sensible to the reordering of the values in the series. To avoid that, Permutation Entropy is used instead. Permutation Entropy is promoted by Bandt and Pompe and is considered one of the simplest symbolization techniques. It quantifies the regularity of finite length time series. Moreover, it considers the temporal causality in the studied process [8]. The Fisher Information may be interpreted as the measure of disorder in a system or phenomena. It is considered a local information measure and it is sensible to little perturbations in the probability distribution. The amount of F[f] decreases when equiprobability is getting closer. It reaches an absolute minimum for a totally flat distribution. As can be observed, the Fisher Information has the opposite behavior to the Shannon entropy. In physics, an universal agreement for the complexity definition is not given. Due to that, several methods have been proposed to calculate it. This statistical measure allows to differentiation between different grades of periodicity and chaos. This paper uses the complexity definition promoted by [8]. In this paper the relations between the previously mentioned parameters are studied under different working conditions of the rotary machine. Specifically, relationships have been studied for the case in which the motor drives a balanced shaft and the case in which it is unbalanced. 2. Theoretical frameworks 2.1. Shannon Entropy Considering a discrete source, for each possible state i there will be a set of probabilities P = {p i ; i = 1,..., N} of N producing the different possible symbols N, the only restriction being : i=1 p i = 1. For that source, the Shannon entropy is given by [5] N S[P] = p i ln(p i ) i=1 (1) 1

3 Shannon entropy is considered a gauge of uncertainty in a physical process. As the formula describes, when S[P] = S min = 0, it is possible to predict the outcome, since it means there is only one of p i = 1, (p j = 0, j i). On the other hand, the uncertainty is maximum when S[P] = S max, and it is impossible to predict the outcome. This happens when the set of probabilities is a uniform distribution P = {1/N,...,1/N}. It is useful to normalize the Shannon Entropy in order to get the unity as the highest value allowed. Therefore, the normalized Shannon Entropy is described as H[P] = S[P]/S max = H (2) where S max = ln(n). Thereby, 0 H 1. Reordering the p i would not change the value of H, therefore, the Shannon entropy is considered a global gauge instead of local Fisher information Fisher information was prompted by R. A. Fisher in 1922, in the statistical inference field [10], and it is used in the analysis of complex signals [11]. Considering a discrete probability distribution, P = {p i ; i = 1,..., N}, N being the number of states in the system, N and satisfying the condition i=1 p i = 1, the Fisher Information is given by [12] N 1 F[P] = (p i+1 p i ) 2 i=1 p i (3) Since reordering the p i in the P distribution would change the F[P] value, the Fisher Information is considered a local gauge Statistical Complexity Several methods have been proposed to calculate the Complexity Information [12-21]. The complexity notion is related with the possibility of finding a hidden structure or pattern in a series characterizing the dynamic of the signal. Therefore, statistical complexity is turned into a gauge that indicates whether a pattern exists out of sight. The entropies measurements may not be effective to quantify intern structures, correlations or patterns in the dynamics of a process [18]. The Statistical Complexity used in this paper is widely explained in [8]. The way it is calculated in [8] allows to distinguish between different grades of periodicity or chaos. According to [8], the statistical complexity in a series is given by C C JS [P] = H S [P] Q J [P, P e ] (4) where H[P] is the normalized Shannon Entropy, and Q is defined in terms of the Jensen-Shannon divergence [21], so that Q J [P, P e ] = Q 0 J[P, P e ] (5) 2

4 J[P, P e ] = S[P + P e /2] S[P]/2 S[P e ]/2 Q 0 = 2{( N+1 )ln(n + 1) ln(2n) + ln N} 1 N (6) (7) 2.4. Permutation Entropy Using of the Permutation Entropy to analyze dynamical systems was promoted by C. Bandt and B. Pompe in [6]. The theorem is completely described in [6,22]. Concurring with [24], let us embed a scalar time series {x(i), i = 1,2,..., N} to a m-dimensional space. According to the Takens-Maine theorem, it may be reconstructed as X(1) = {x(1), x(1 + τ),..., x(1 + (m 1)τ)}... X(i) = {x(i), x(i + τ),..., x(i + (m 1)τ}... X(N (m 1)τ) = {x(n (m 1)τ), x(n (m 2)τ),..., x(n)} (8) where m is called the embedded dimension and τ is the time delay. As it is described in [24], the m number of real values for each X(i) can be arranged in an increasing order {x(i + (j 1 1)τ) x(i + (j 2 1)τ... x(i + (j m 1)τ)} (9) If there is an equality, and two or more X(i) have the same value, e.g., x[i + (j i1 1)τ = x[i + (j i2 1)τ, the quantities may be ordered according to their j s [22]. Therefore, if j i1 < j i2, it is written x[i + (j i1 1)τ] x[i + (j i2 1)τ]. According to [25], any vector X(i) is mapped onto a group of symbols S(l) = (j 1, j 2, j m ) (10) where l = 1,2,.. k. Due to m! being the largest number of distinct symbols, k m!. S(l) is one of the m! permutations of m distinct symbols (j 2,...,j m ). For each symbol sequence, respectively, the probability distribution is k denoted as P 1, P 2,..., P k. Whether l=1 P l = 1, the permutation entropy of order m for each time series {x(i), i = 1,2,..., N} is written as the Shannon entropy for each S(l) symbol sequence as H p (m) = P l ln P l k l (11) The maximum value of H p is ln(m!), and it is reached when all the symbol sequences, S(l), have the same probability distribution, P l = 1/m!. Therefore, in order to normalize the value of H p, its value is divided by the maximum reachable value ln(m!) as 0 H p = H p /ln(m!) 1 (12) H p gives information about how random a time series is. The highest the value of H p is 1 and the smallest is 0. When the value of H p becomes 0, it means that the signal is very regular [6]. On the other hand, when the value of 3

5 H p becomes 1, it means that all the permutations have the same probability, which means the signal is white noise [25]. A subtle transformation of the time series may be easily noticed through a H p change [25]. In [25], it is seen that if m is too small, such as 1 or 2, the strategy of using the permutation entropy to detect dynamical changes may not be effective, since there are too few states. On their original paper, Bandt and Pompe [6] choose τ=1, and recommend m=3,, 7. On [25] the effect of dimension m and time delay τ is studied, as well as the effect of the data length w. It is concluded that a data length, w, greater than 256 points gives a stable value of the permutation entropy. The variation of PE value becomes insignificant when the data length w exceeds 256 points. Also, it is concluded that when the dimension m is below 4, PE may not be effective to detect changes of the mechanical signals, and, on the other hand, when m is greater than 8 the calculation of PE becomes computationally expensive. When the effect of the time delay τ is studied, it is seen that when it is greater that 5 the results cannot detect small changes in signals. However, in this study a time delay τ=1, a embedded dimension m=3 and a data length w=512 have been chosen Relations between statistical parameters C.Vignat and J.F.Bercher [34] showed that the simultaneous study of the Fisher Information and a function of the differential entropy, called entropy power and given by N = (1/2πe) e 2h may be used for the detection of the nonstationarity status in a signal. O. A. Rosso, and his team members, prompted the entropy-complexity chart, H x C in order to be able to characterize and determine the difference between deterministic and stochastic dynamical systems [8, 26, 27,28]. For the purpose of staying on the line of inquiry, F. O. Zamora suggests in his PhD thesis the study of the Shannon entropy - Fisher information plane. This H x C plane gives global information, while the H x F plane gives, global and local information at the same time. As a result, [8] shows that important improvements are integrated when the locality is incorporated into the graphical information. The objective for this paper is to research the usefulness of the Complexity - Fisher plane (Q x F), and the Shannon entropy - Permutation entropy (H x H) plane for the detection of faults in rotary machines. 3. Application to unbalanced status of rotary machines Rotary machine are formed by a number of components which interact with each other when the machine is in operating condition. The vibration signal measure obtained at this state is complex due to strong interference noises. The resulting vibration becomes more complex when there are faults on a component [25]. A series of experiments have been done in order to investigate the utility of unconventional statistical methods (Permutation Entropy, Shannon Entropy, Fisher and Complexity) for detecting faults in the rotary machine. Machine samples have been taken at different working conditions: - No-load condition: the machine is working in a normal condition. The shaft is balanced. It is labeled as Type 0. - Unbalanced shaft: the motor shaft drives two aluminum disks where it is possible to add masses in order to separate the center of gravity from the system's rotation axis. Consequently, it produces vibrations on the machine. Masses (screws and nuts) have been added to disc 1 and/or 2 to create a misalignment. 4

6 Fig. 1. Main components used in the study. (1) Digital sampler. (2) Frequency drive. (3) Bench. (4) Electric motor. (5) Bearing 1. (6) Accelerometer. (7) Disc 1. (8) Disc 2. (9) Bearing 2 Frequency: the assay has been performed at 20 Hz. Accelerometer: it has always been placed in the first bearing, the nearest to the engine. For each condition, a sample of 1024 points has been taken. The proposed misalignments effectuated on the rotary machine are: Type 1: two additional masses, located at the external holed-rings coaxially. Type 2: two additional masses, located at the external holed-rings, with a phase displacement of 90 o. Type 3: one additional mass located at the external holed-ring. Only in one of the discs. Type 4: one additional mass located at the internal holed-ring. Only in one of the discs. Type 5: like Type 3 but the mass located at a hole with a phase displacement of 90 o. Type 6: like Type 4 but the mass located at a hole with a phase displacement of 90 o. For each Type the Shannon entropy and the Permutation Entropy, the Fisher Information and the Statistical Complexity have been calculated. Values of m = 3, τ = 1 and w = 1024 have been used. 5

7 Fig. 2. Shannon entropy X Permutation entropy Fig. 3. Shannon entropy X Fisher Fig. 4. Complexity X Fisher 6

8 In all three graphs, the no-load condition point (Type 0) is completely separated from the others. The next step is to study the behavior of a no-load condition signal plus a misalignment: the vibration signal is measured with the machine at a no-load condition status and, afterwards, with the misalignment Type 1. With the 2048 points obtained, the statistical parameters are calculated. These parameters have been calculated using windows of 512 points. Shannon entropy and the Permutation Entropy, the Fisher Information and the Statistical Complexity. Moreover, Shannon X Fisher, Shannon X Permutation entropy and Complexity X Fisher graphs are done in order to find some relations. Finally, in order to check if the process is reversible, the next experiment is done. The permutation entropy for a 512-point window is calculated and, after that, the order of the values is inverted. In this way, the first value turns into the last one, and the last one passes to the first position. Afterwards, its permutation entropy is calculated. Once both permutation entropies are calculated, they are subtracted from each other. If the process is reversible, the subtraction of entropies is 0. If it is not, the subtraction is not null. The misalignment studied is Type 1, because it is the one that presents more difference between its Shannon entropy and Type 0 Shannon entropy Study for no-load condition adding type 1fault The statistical parameters are calculated considering the next signal. It is composed by a "healthy" measure and a sick measure. The first zone corresponds to the measure taken when the machine is working on a no-load condition. The second part is given when the machine is working with the fault. Fig. 5. Test signal formulated for quantitative measure evaluation. When the parameters are calculated, three different zones are distinguished. The first one corresponds to the calculation only done with the "healthy" values. The 512-points window only takes "healthy" values to calculate the parameter. This occurs from the first window to the 512th. The second zone, called transition zone is given when the 512-point window takes "healthy" and "sick" values at the same time to calculate the corresponding parameter. This occurs when from the 513th window to the 1536th. The third zone occurs when the calculation window takes only sick values. 7

9 In any case, the target of the study is the detection of changes in the statistical parameters depending on the zone they are calculated. Below, the results are shown. Fig. 6. Permutation entropy Fig. 7. Shannon entropy Fig. 8. Fisher Fig. 9. Complexity As can be seen in the previous figures, there is agreement between the feature of vibration signals and the change in value of the different parameters. Hence, the permutation entropy, the Shannon entropy, the Fisher Information and the Complexity are valid to detect dynamical changes in vibration signals. Moreover, the following charts have been proposed to observe whether relationships exist between the studied parameters. 8

10 Fig. 10. Shannon entropy X Fisher Fig. 11. Shannon entropy X Permutation entropy Fig. 12. Complexity X Fisher In the previous figures, a relation between the parameters can be seen. Depending on the feature of the signal, the parameter has a stable value. When the calculation is done in the transition zone, a path is drawn in the plane, connecting both, healthy and sick, zones. Finally, the chart with the subtracted permutation entropies is drawn. 9

11 Fig. 13. Permutation entropies subtract As the chart shows, the difference of permutation entropies is null for the first zone. That means that the process is reversible, so that the bearing is working properly. Afterwards, in both the transition zone and the "sick" zone, the difference of permutation entropies is not null and, therefore, the process is not reversible. In physics, when a process is not reversible it cannot work backwards. Mechanically, the explanation is that when the bearing starts its breakage it cannot work as it did in the initial state. 4. Conclusion In this paper, statistical parameters, such as permutation entropy, Shannon entropy, Fisher Information and Complexity have been applied in order to study their validity to detect dynamical changes in the vibrations of a rotary machine. The rotary machine studied has been tested under different working conditions. Specifically, it has been tested under the no-load condition and adding an eccentricity on its shaft. The experimental study concludes that these applied parameters have different values for each condition and both conditions are clearly differentiated, as well as the transition zone between the healthy values and the sick ones. The extraction of the entropies has also been studied. It concludes that the process is reversible when it is working in no-load conditions, but it is not when critical misalignments on the shaft are introduced. Nevertheless, a minimum number of sick values is needed to detect its irreversibility. Acknowledgments This work has been possible due to the guidance and encouragement of Dr. Ing. Francisco Redelico, deputy manager on the Industrial Engineering Department at Instituto Tecnológico de Buenos Aires. We whole heartedly thank him for his support. We would like to sincerely thank CEMAT and, specially, Nicolás Andrés Oryazabal for his patience and for making possible the research using the machinery on his office. Last but not least, we are beholden to Andrés Alonso for letting us use his rotary machine and his essential help. 10

12 References [1] S. Wang, W. Huang, Z. K. Zhu, Transient modeling and parameter identification based on wavelet and correlation filtering for rotating machine fault diagnosis, Mech. Syst. Signal Process. 25 (4) (2011) [2] X. Wang, Y. Zi, Z. He, Multiwavelet denoising with improved neighboring coefficient for application on rolling bearing fault diagnosis, Mech. Syst. Signal Process. 25 (1)(2011) [3] I. Trendafilova, An automated procedure for detection and identification of ball bearing damage using multivariate statistics and pattern recognition, Mech. Syst. Signal Process. 24 (6)(2010) [4] S. Sheng, L. Zhang, R. Gao, A systematic sensor placement strategy for enhanced defect detection in rolling bearings, IEEE Sensors J. 6 (2006) [5] C. Shannon, A Mathematical Theory of Communication. The Bell System Technical Journal. 27 (1948), , [6] C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (2002) [7] R. A. Fisher. Theory of Statistical Estimation. Mathematical Proceedings of the Cambridge Philosophical Society 22, (1925). [8] F. O. Zamora," Medida de información de Fisher y distribución causal de probabilidad de Bandt y Pompe: aplicaciones al análisis de sistemas dinámicos caóticos," Ph.D. dissertation, Dept. Physics, Universidad de La Plata, La Plata, [9] D. Bonchev. Shannon's information and Complexity in, Mathematical Chemistry Series: Complexity in Chemistry, Vol. 7, D. Bonchev and D. H. Rouvray (eds.), Taylor & Francis London (2003) pp [10] R. A. Fisher, On the Mathematical Foundations of Theoretical Statistics. Philosophical Transactions of the Royal Society of London, Series A 222, (1922). [11] B. Roy Frieden, Science from Fisher Information: A unification (Cambridge University Press, Cambridge, UK, 2004). [12] A. N. Kolmogorov, Three approaches to the quantitative definition of information. Problems in Information Transmission 1, 1 7 (1965). [13] P. Grassberger, How to measure self-generated complexity. Physica A 140, (1986). [14] P. Grassberger, Toward a quantitative theory of self generated complexity. International Journal of Theoretical Physics 25, (1986). [15] J. P. Crutchfield, K. Young, Inferring statistical complexity. Physical Review Letter 63, (1989). [16] R. Wackerbauer, R. A. Witt, H. Atmanspacher, J. Kurths, H. Scheingraber, A comparative classification of complexity measures. Chaos, Solitons and Fractals 4, (1994). [17] L. López-Ruiz, L. Mancini, X. Calbet, A statistical measure of complexity. Physics Letter A 209, (1995). [18] D. P. Feldman, J. P. Crutchfield, Measures of statistical complexity: Why?. Physics Letter A 238, (1998). [19] J. S. Shiner, M. Davison, P. T. Landsberg, Simple measure for complexity. Physical Review E 59, (1999). [20] M. T. Martín, A. Plastino, O. A. Rosso. Statistical complexity and disequilibrium. Physics Letter A 2003, (2003). [21] P. W. Lamberti, M. T. Martín, A. Plastino, O. A. Rosso, Intensive entropic nontriviality measure. Physica A 334, (2004). [22] Y. Cao, W. Tung, J.B. Gao, V.A. Protopopescu, L.M. Hively, Detecting dynamical changes in time series using the permutation entropy, Phys. Rev. E70 (2004) [23] Y.G. Xu, L.L. Li, Z.J. He, Approximate entropy and its applications in mechanical fault diagnosis, Inf. Control 31 (2002) [24] R. Yan, Y. Liu, R. X.Gao, Permutation entropy: A nonlinear statistical measure for status characterization of rotary machines, Mech. Syst. Signal Process. 29 (2012) [25] O. A. Rosso, L. De Micco, H. A. Larrondo, M. T. Martín, A. Plastino, Generalized statistical complexity measure. International Journal of Bifurcation and Chaos (2010). [26] O. A. Rosso, H. A. Larrondo, M. T. Martín, A. Plastino, M. A. Fuentes, Distinguishing noise from chaos. Physical Review Letters 99, (2007). 11

13 [27] O. A. Rosso, M. T. Martín, H. Larrondo, A. M. Kowalski, A. Plastino, Generalized Statistical Complexity: a new tool for dynamical systems. En Concepts and Recent Advances in Generalized Information Measures and Statistics. A. M. Kowalski, R. Rossingoli and E. E. F Curado (Editores) (Bentham e Books, Bentham Publishers, 2012, accepted for publication). 12

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