Tite: rack Diagnostics via ourier Transform: ea and maginary omponents vs. Power pectra Density Authors: Leonid M. Geman Victor Giurgiutiu van V. Petrunin 3 rd nternationa Workshop on tructura Heath Monitoring (WHM), eptember -4,, tanford University, tanford, A
ABTAT A new generic approach to damage detection is advocated. The new generic approach is optima for the anaysis of frequency domain data for heath monitoring, damage detection, and diagnostics. The new approach consists of using new features based on the rea and imaginary parts of the ourier transform. This approach is more generic that conventiona approaches based on power spectra density, and it can be shown that the power spectra density approach is a particuar case of the proposed new generic approach. Numerica exampes based on the anaysis of synthetic signas generated using the noninear mode of a cracked part of machinery are given. The synthetic signas simuated the forced vibroacoustica osciation under narrow band Gaussian excitation. Our new generic approach is used for detecting damage consisting of a fatigue crack of various reative sizes. The numerica exampes show that the detection, based on isher s criterion, was definitey more effective when using our new generic features that when using the conventiona power spectra density feature. The proposed new generic approach to damage detection offers a cear improvement in effectiveness over the conventiona approach based on the power spectra density. NTODUTON A new genera optima approach was proposed [] for those cases where one and mutidimensiona ourier transforms are used for heath monitoring and diagnostics. This approach consists of using simutaneousy the rea and imaginary components of the ourier transforms as diagnostics features. This is in contrast to Leonid M. Geman, and Victor Giurgiutiu, University of outh aroina, Mechanica Engineering Department, 3 Main treet, oumbia,, 98, UA. van V. Petrunin, Nationa Technica University of Ukraine, Kiev, 56, Ukraine.
most pubished appications concerning heath monitoring, diagnostics, and pattern recognition where the power spectra density () is used. n the present paper we consider the forced osciation diagnostics method with the narrowband Gaussian excitation for fatigue crack detection. The part of the machinery is the singe degree of freedom noninear osciators with the noninear stiffness due to the fatigue cracks. The method consists of exciting the tested obects into the resonance osciations. The diagnostics decision is then based on the osciation parameter estimates. Usuay, the diagnostics information of the forced and free osciation method of the crack diagnostics is contained in the ourier transform of the high harmonic resonant osciations [,3]. The purpose of the paper is the comparison of the diagnostics features based on the approach [] with the traditiona power spectra density for fatigue crack diagnostics. THE DYNAM MODEL O MAHNEY PAT WTH AK The basis of the considered forced osciation method of crack diagnostics is the fact that noninearity eve in a cracked part of the machinery changes with the change of the crack size. Assuming that the part of the machinery under anaysis can be approximated with a singe degree of freedom noninear osciator, the equations of the motion can be expressed as foows [, 3]: () X + hx + ω X = A t cos( ω f t + ϕ), X + hx + ω X = A() t cos( ω f t + ϕ), x, x <, x where X =, x is dispacement, h is damping coefficient, m c k k h =, ω =, ω =, m and c are the obect mass and damping m m m respectivey, k and k are stiffness at stretching (crack opening) and compression (crack cosing) respectivey. At compression the crack is cosed and the materia behaves ike a continuum and the stiffness is the same as that of the materia without crack k = k. At stretching the crack is opened and the materia is discontinuous, hence the stiffness decreases with the quantity k = k k. A () t is the ayeigh enveope of narrowband Gaussian process, ω f and ϕ are constant excitation frequency and random initia phase [4]. The random initia phase is uniformy distributed in the interva [ ; π ] natura frequency of the system with the crack is given by [,3]: (). The
k ω ω k ω = = ω n, () ω + ω k + k where ω n is the natura frequency of the un-cracked part, ω = k n m. The reationship between the reative stiffness change and reative crack size is given by [3] = k k, where is the reative crack size; is the crack size, is part size in the direction of the crack. DAGNOT EATUE oowing [], we utiize simutaneousy the rea X and imaginary X components of the most genera short time ourier transform of the high harmonic of the resonant osciations as diagnostics features: t X = X( t)cos( iω t) dt, (3) t X = X( t)sin( iω t) dt, where t is upper time imit, i is the harmonic number. for hypotheses We consider the two-cass diagnostics of the reative crack size: H, =,. = The optima noninear transformations of the components (3) of proposed feature vector and of the of high harmonic resonance osciations are the ikeihood ratios respectivey: W( X, X L( X, X ) = W( X, X H ) H ),
where ( X X H ) ( X ) L ( H ) ( H ) W X =, W X W, is the conditiona two-dimensiona probabiity distribution function (pdf) of feature vector for hypothesis H, W ( X H ) is conditiona onedimensiona pdf of the for hypothesis H. NUMEAL MULATON ince Equation () does not accept cosed form soution, we used numerica simuation to study its response and crack detection opportunities. Using Equation (), we generated numericay synthetic signas that represent the resonant response of the cracked part of the machinery under random narrowband Gaussian excitation centered on ω. These synthetic vibration signas were used to iustrate a nove crack diagnostics method based on new features. Using the histogram method [5], we estimate the conditiona twodimensiona pdf of the features described by Equation (3) and the conditiona onedimensiona pdf of the of the synthetic vibration signa. To compare the diagnostics effectiveness of the proposed new features vs. the conventiona feature, we are going to use isher's criterion [5]: [ m ( u/ H ) m ( u/ H )] σ ( u / H ) + σ ( u/ H ) =, (4) where u / H is conditiona quantity for hypothesis H, =,, m and σ are the mean and the variance operators respectivey. Empoying Monte-aro simuation procedure for training and testing of the diagnostics method, we cacuate the isher criterion for our new features and the conventiona feature, : NEW [ m ( L{ X, X } H ) m ( L{ X, X } H )] σ L{ X, X } H + σ L X, X H = (5) ( ) ( { } ) [ m ( L{ X} H ) m ( L{ X} H )] σ L{ X } H + σ L X H ( ) ( { } ) The foowing parameters were used in the simuation. = (6)
TABLE. ELATVE AK ZE VALUE O TETED HYPOTHE Test case H H ase (a)..5 ase (b).5. TABLE. ELATVE EQUENE O THE AK MAHNEY PAT O VAOU AK ZE...5. ω.97.96.94 ω n The racked Part of Machinery We consider a part of machinery with natura frequency f n = Hz ( ω n = 4π rad/s). n this part of machinery, we considered the presence of a crack of various reative crack sizes (Tabe ). As shown in Tabe, two cases were considered: one with a mid crack (ase (a)), the other with a more severe crack (ase (b)). The Excitation The excitation consisted of the narrowband Gaussian excitation of Equation (), tuned to the natura frequency of the crack part of machinery ( ω f = ω ). As shown in Tabe, we obtained, for each reative crack size, a different reative ω natura frequency,. n generating the Gaussian excitation signa, the vaue ω n 3 b =.3 was used for the parameter of the pdf of the excitation signa enveope. The igna Processing The simuated steady state resonant response signa was processed with an T agorithm. Then, attention was focused on the second harmonic ( i = ). The choice of the second harmonic was ustified due to its higher sensitivity [3] to the presence of the crack than the other harmonics. This aspect is specific to the noninear nature of the descriptive Equation (). As diagnostic features, we use: (i) the new features, rea and imaginary ourier components; (ii) the conventiona features,. 3 amping interva is t =.5 s ; number of time history periods varies from n = 8 to n = 9 in case (a) and from n = 9 to n = 3 in case (b) with the variation of the signa duration from t =.49s to t =.54s. The aiasing parameter, ξ, was varied in the range [,]. or ξ = and ξ =, there is no aiasing. or
ξ (,) aiasing presents. n our study, we investigated the effect of the aiasing on the sensitivity of our diagnostics features. The number of simuation experiments was equa to for every hypothesis ( H or H ) and for every vaue of the aiasing parameter ξ..8.6.4.8.6.4....4.6.8 ξ..4.6.8 ξ (a) igure : Variation of the isher criterion,, with aiasing parameer ξ. The symbo ( ) represents resuts obtained with the new diagnostics features, whie ( ) represents resuts for the conventiona feature. ase (a) and (b) of Tabe appy. (b) imuation esuts igure presents the isher s criterions (5, 6) vs. aiasing parameter ξ for the new optima features (stars) and the conventiona feature, (circes). The resuts for the simuation cases (a) and (b) are presented separatey. n both cases, and for a vaues of the aiasing parameter ξ (,) the new diagnostics features proposed by us provide a higher vaue than the conventiona feature,. This indicates that the new diagnostics features have a better diagnostics effectiveness than the conventiona feature,. We consider the diagnostics efficiency ratio of the isher s criterions for the optima new features NEW and vs. aiasing parameter ξ (igure ): NEW =, (7) One can see from igure that the ratio given by Equation (7) is aways greater than and reach maximums near the midde of the aiasing parameter range, i.e., at ξ =.5. Over the whoe aiasing parameter range, ξ [, ], the proposed new approach based on the features given by Equation (3) are shown to be more efficient for crack diagnostics than the conventiona approach based on feature. The effect of aiasing is to strengthen even more this diagnostic capabiity.
.6..8.4.6..8.4..4.6.8 ξ..4.6.8 (a) (b) igure : Variation of diagnostics efficiency ratio,, with aiasing parameter, ξ, for cases (a) and (b) as defined in Tabe. ξ ONLUON A new generic approach to damage detection has been presented. The new generic approach is optima for the anaysis of frequency domain data for heath monitoring, damage detection, and diagnostics. The new approach consists of using new features based on the rea and imaginary parts of the ourier transform. This approach is more generic that conventiona approaches based on power spectra density, and it can be shown that the power spectra density approach is a particuar case of the proposed new generic approach. Numerica exampes based on the anaysis of synthetic signas generated using the noninear mode of a cracked part of machinery were given. The synthetic signas simuated the forced vibroacoustica osciation under narrow band Gaussian excitation. ince the vibration of cracked parts of machinery is significanty noninear, our approach utiized the second harmonic of the ourier transform, which is known to be more sensitive to interna damage than the other harmonic. The new generic approach was used for detecting damage that consisted of a fatigue crack of various reative sizes. The numerica exampes showed that the detection was definitey more effective when using our new generic features that when using the conventiona power spectra density feature. t is shown using isher s criterion, that the power spectra density is not optima feature for crack diagnostics for a cases. The optima features proposed here provide better diagnostic effectiveness gains than the power spectra density. This resut confirms previousy pubished findings [,6]. Thus, we recommend that both the rea and imaginary parts of the ourier transforms shoud be considered as fundamenta diagnostics features, incuding crack diagnostics. The fact that the diagnostic effectiveness of these new features increases with the degree of signa aiasing is aso worth mentioning. AKNOWLEDGMENT The financia support of the Twinning Program from Nationa esearch ounci (enior Program Officer Key obbins) is thankfuy acknowedged.
EEENE. Geman, L. M., and. G. Braun.. The Optima Usage of the ourier Transform for Pattern ecognition, Mechanica ystems and igna Processing, 5(3): 64-646.. Geman, L. M., and. V. Gorpinich.. Non-Linear vibroacoustica free osciation method for crack detection and evauation, Mechanica ystems and igna Processing, 4(3): 343-35. 3. Bouraou, N.. and L. M. Geman. 997. Theoretica bases of the free osciation method for acoustica nondestructive testing, presented in Proceedings of Noise-on'97, the Pennsyvania tate University, 997. 4. Geman, L. M., Yu.. Zinkovsky and. V. Petrunin.. pectra Anaysis of Nonstationary Narrowband andom Excitation, adioeectronics and ommunication ystems, No.8: 35-4. 5. Young, T.Y. and K.-. u. 986. Handbook of Pattern ecognition and mage Processing, Academic Press, nc. 6. Geman, L. M.,. G. Braun., A. N. Miroposky and. V. Petrunin.. Diagnostics via ourier Transforms: ea and maginary omponents vs. Power pectra Density, presented at the 4th nternationa onference Acoustica and Vibratory urveiance Methods and Diagnostic Techniques, ETM - enis, rance, eptember -4,.