Software Package Evaluation For Lyapunov Exponent And .

DJournal of Energy and Power Engineering 9 (2015) 443-451doi: are Package Evaluation for Lyapunov Exponentand Others Features of Signals Evaluating ConditionMonitoring Performance of Nonlinear Dynamic SystemsJulio César Gomez-Mancilla1, Luis Manuel Palacios-Pineda2 and Valeriy Nosov11. Vibrations & Rotordynamics Laboratory, ESIME (Superior School of Mechanical and Electrical Engineering), IPN (NationalPolytechnic Institute), Zacatenco Professional Center, México, D. F. 07738, México2. Graduated and Research Department, Technological Institute of Pachuca, Pachuca 42083, MéxicoReceived: January 08, 2015 / Accepted: March 02, 2015 / Published: May 31, 2015.Abstract: Efficient use of industrial equipment, increase its availability, safety and economic issues spur strong research on maintenanceprograms based on their operating conditions. Machines normally operate in a linear range, but when malfunctions occur, nonlinearbehavior might set in. By studying and comparing five nonlinear features, which listed in decreasing order by their damage detectioncapability are: LLE (largest Lyapunov exponent), embedded dimension, Kappa determinism, time delay and cross error values; i.e.,LLE performs best. Using somewhat similar ideas from Chaos control, i.e., vary the “mass imbalance” forcing parameters, we aim tostabilize the Lorenz equation. Quite interestingly, for certain imbalance excitation values, the system is stabilized. The previous evenwhen paradigmatically chaotic parameters for Lorenz system are used (plus our forcing terms). This quasi-control approach is validatedstudying signals obtained from the previously mentioned lab test. Finally, it is concluded that analyzing and comparing nonlinearfeatures extracted from baseline vs. malfunction condition (test acquired), one might increase the efficiency and the performance ofmachine condition monitoring.Key words: Modified Lorenz equation, largest Lyapunov exponent, nonlinear features, chaos control, test validation.1. Introduction Technology based on vibration condition monitoringis largely used due to the necessity to increaseequipment availability, safety and economic issues.Machines by their nature are designed to display linearbehavior during normal operation. When machineshave reasonable bounded amplitude and linearbehavior, most of the times are considered as “healthy”systems. Particularly signals obtained from such“healthy system” normally can be analyzed withtraditional techniques as FFT (fast Fourier transform),STFT (short time Fourier transform), wavelet, etc.Nevertheless, machines are always exposed to variableCorresponding author: Julio César Gomez-Mancilla, Ph.D.,professor, research fields: structural and mechanical vibrations,design, rotor non-linear dynamics, chaos. E-mail:gomezmancilla@gmail.com.workloads, insufficient or lack of maintenance andothers aspects leading to the possibility of developingsome kind of malfunction (unbalance, misalignment,steam-whirl, cracks, etc.), as a result depending on themagnitude of such malfunction, the dynamic conditionof the machine changes and might become a systemwith certain degree of nonlinearity.If a vibration signal is analyzed by linear processingtechniques, information revealing malfunctioncharacteristics might get lost, thus the need to usenonlinear processing tools. As an example, a modifiedLorenz equation, where we add an external force, isanalyzed. Nonlinear tools implemented in the Percpackage [1] such as time delay, embedding dimension,error, determinism, stationarity and LLE (largestLyapunov exponent), also time series are analyzed asexplained by Ref. [2], and calculi applied to lab test.

444Software Package Evaluation for Lyapunov Exponent and Others Features of Signals EvaluatingCondition Monitoring Performance of Nonlinear Dynamic SystemsStudies in this area are good but few and seminalbook by Randall [3] helps. Works by Wang, et al. [4, 5]use the pseudo-phase portrait to find the importance ofusing an adequate time delay value to create suchportrait (in Ref. [6], Gomez-Mancilla also emphasizes proper estimation), sensitivity to detect faults inmachinery is also studied. They demonstrated that,these nonlinear features can be effective parameters forfault condition prognosis. In Ref. [7], Antoinecompares the Lyapunov exponent and Jacobian featurevector calculated from an experimental setup.Similar to previous works by Gomez-Mancilla [8, 9]our aim is to discriminate and select nonlinear featuresfrom time signals acquired at normal baselineoperation; while extracting features to characterizechanges in the system dynamic behavior to compare vs.signal features when the machine has malfunctions.2. Lorenz Equation, Original and ModifiedIn 1963, meteorologist Ed Lorenz derived a threedimensional relatively simple set of first-ordernonlinear differential equations. He truncated thepartial differential equations describing thermal drivingof convection in the lower atmosphere (Eq. (1)):x z x y xy x yz xy z(1)For certain parameters values, σ, ρ, β, i.e., σ 10, ρ 25, β 8/3, the system displays an irregular 25, β 8/3, the system displays an irregular deterministicchaotic behavior which may be characterized by theLLE. To study faulty systems, here sinusoidal termsare added aiming at simulate an external excitingrotating force. Such modified Lorenz equation isshown in Eq. (2):x z x F0 sin t y xy x y F0 cos t z xy z F0 sin t (2)Modifying Lorenz equation is some kind of chaoscontrol as described by Refs. [10, 11], yet we are notexactly perturbing nor varying any equation parametervalues. By adding forcing terms, similarly as massimbalance influences a rotating machine system, weexplore and study the resulting system stability.Using Matlab ODE45 (ordinary differentialequation solver45) function, Eqs. (1) and (2) aresolved with sample time equal to 0.01 s (4,000 datasamples). Fig. 1 shows time series corresponding tox(t), y(t) and z(t) modified Lorenz system responseswhich are used to calculate nonlinear features; andfor some parameter values Eqs. (1) and (2) mightbecome chaotic. To calculate their features, nonlinearanalysis package in Ref. [1] is used; datapre-processing is advised as Ref. [6] exposes. Inaccordance to Takens theorem [12], dynamic featuresfrom numerically simulated or acquired stationaryand deterministic processes are extracted.x 0260270280290300220230240250260270280290300y (t)500-50210z (t)500210Fig. 1t (s)Time responses x(t), y(t), z(t) from Eq. (2) for σ 10, ρ 28, β 8/3, F 50, ω 30, two competing attractors co-exist.

Software Package Evaluation for Lyapunov Exponent and Others Features of Signals EvaluatingCondition Monitoring Performance of Nonlinear Dynamic Systems3. Nonlinear Time Series AnalysisNowadays, condition monitoring by different signalprocessing methods (frequency and time domainanalysis, wavelet, etc.), can be realized. Yet, potentialirregular nonlinear behavior arising from presence ofmalfunctions in actual machines motivates the use ofpowerful tools from nonlinear analysis. Followingsteps aim at calculating the LLE [1]. First, obtain aproper τ and embedding dimension m values bymutual information [13] and false nearest neighbormethods [14], respectively. Next, ensure input datarequirements are satisfied by testing if the system isdeterministic [15] and stationary [16]. Then, using asingle time series should verify if the reconstructedphase or embedded space seems congruent. Finally,using Wolf algorithm [17, 18], calculate the LLE anddetermine if the system is chaotic, or not.3.1 Embedded Space Reconstructionobtained about other coordinates is large enough toallow us to introduce values at times (t τ, t 2τ, ,(t (m – 1)τ) to substitute for the original coordinates.An embedding space having same/similar features asthe original system, is therefore recreated (Eq. (3)). p ( t ) xt , xt , xt 2 , ., xt m 1 Time delay (τ): to calculate the time delay, the MI(mutual information) method allows satisfactoryresults, better as compared to other methods (i.e.,linear autocorrelation which are not studied in thiswork). From state xt τ, equation for mutualinformation is:jj Ph , k xt , xt I ( ) - Ph , k xt , xt ln Ph xt Pk xt h 1k 1(4)where, P(xt) and P(xt τ) are the probabilities that avariable assumes a certain value inside the h-th andthe k-th bins, respectively. P(xt, xt τ) is the jointprobability that xt is in bin h and xt τ is in bin k. Ifvariables xt and xt τ are completely independent, itmeans that, these variables are not correlated, and I(τ) 0. Program input to mutual.exe are the number ofbins j and the expected maximal embedding delay.Embedding dimension (m): mainly to provide aEuclidean space with a dimension large enough suchthat, the system dynamics can be unfolded withoutambiguity. FNN (false nearest neighbor) method is an1010101 xX1 τ200(3)3.2 Some Nonlinear Features200 While, inspecting Fig. 2, the necessity for properinput parameters τ and m becomes clear. Notice thequite different generated spaces.201 τxX1 x1 τThe reconstruction generates information about theunobserved (not measured) data, which allowspredicting the rest of the state variables. Mathematicaldescription is based on Takens’ theorem [12], whereusing a single scalar time series x(t) can be enough toreconstruct the state space. The previous since thecoordinates are related to each other through a timedelay (τ) [6, 13]. If τ is very small, then coordinates xtand x(t (m – 1)τ) are numerically so close to each other,that can not be distinguished from each other. On thecontrary, if τ is too large, then xt and x(t (m – 1)τ)become independent of each other, in a statical sense.Having properly determined τ, the information4450-10-10-10-20-20-20(a)(b)(c)Fig. 2 State space reconstruction for the modified Lorenz system (Eq. (2)): σ 10, β 8/3, ω 30, F 50, m 3; τ applied togenerating signal x(t) are: (a) τ 18; (b) τ 58; (c) τ 80.