Numerical simulations for long range guided waves Nondestructive Testing by a wavelet based two port equivalent.

Transcription

1 Numercal smulatons for long range guded waves Nondestructve Testng by a wavelet based two port equvalent. F. BERTONCINI, A. MUSOLINO, M. RAUGI, F. TURCU Department of Electrc Systems and Automaton Unversty of Psa Va Dotsalv Psa, ITALY Abstract: - In ths paper the effectveness of a procedure that allows the detecton of a defect by the analyss of parameters of a two port equvalent of a NDT ultrasound guded waves system s nvestgated. A wavelet expanson of the nput (transmtted) and output (reflected) sgnals has been carred out and a constant coeffcent matrx behavng as a transfer functon between the nput and output port has been defned. The two port equvalent representaton correctly smulates the complete system under a wde range of nput waveforms n the actual operatng condton. The effectveness of ths method to detect and locate defects n the ppes has been nvestgated. Key-Words: - Ultrasound guded waves, Two port equvalent, Wavelets 1 Introducton In many ndustres ppe corroson s one of the major problems for plant mantenance. Thus, non-destructve detecton and classfcaton of ppes ntegrty s of actual nterest. Recently, the possblty to nspect large part of ppes from a sngle locaton by means of the so called Lamb guded waves has recalled nterest from the potentalty of the method [1]-[3]. The Lamb guded waves are generated n the ppes va a pezoelectrc or magnetostrctve transducer, called transmtter that s drven by proper electrcal sgnals. Another transducer, called recever s located at a proper dstance from the transmtter and gves an electrc sgnal when t detects the elastc wave that s travelng n the ppe. The defect wthn an ncdent guded wave path mpacts the sgnature of the scattered felds by exctng all possble modes for the gven frequency value. The study of the scattered feld (reflected and transmtted) provdes essental features for defect detecton. By way of phase velocty and frequency tunng, defect detecton senstvty and locaton analyss can be good despte dstance and envronmental constrants. Technques that are based on the Fourer Method are often used to perform the analyss of the sgnals nvolved n ths procedure. The ablty to go beyond detecton and locaton to classfcaton and szng, however, s very dffcult. The queston of whch mode and frequency should be selected to mpnge onto a defect and have the best chance of classfyng or szng that defect s stll a challengng problem. Indeed one of the dffcult problems about ths knd of nondestructve test (NDT) nspecton s the mult-mode coexstence. The dfferent modes cannot be dstngushed by the waveforms n the tme doman. However the Fourer methods need a large number of spatal samples of Lamb waveform. Ths requrement makes the Fourer methods dffcult to be used n practce []-[]. In ths paper we have consdered the transmtter and the recever transducers respectvely as the nput and the output ports of the testng devce. A two port representaton, smlar to those used to schematze electronc crcuts, has been obtaned for the above devce. Ths method, not yet nvestgated for these knds of applcatons, has been studed for detecton and classfcaton of defects. Problem Formulaton Recently, Wavelet Theory [7], [] has receved a great nterest n many areas of engneerng. In partcular Wavelet Expanson (WE) technques have been used for the numercal soluton of Multconductor Transmsson Lnes [9] and have demonstrated to be a powerful tool especally n terms of smulaton effcency when fast transents are analyzed. As a matter of fact the man feature of wavelet bases s that the wavelet functons are compactly supported, hence they effcently represent sgnals that are compactly supported; furthermore they self concentrate where the frequency content of the analyzed sgnal requres to be represented by a hgher number of functons (self adaptve zoomng). In partcular WE s here used as a tool through whch a transfer functon represented by a matrx of constant coeffcents s constructed. The dentfcaton of ths matrx can be obtaned from data n the tme of frequency doman. Let us consder a two-port system, represented by ts nput and output sgnal, respectvely defned as x(t) and y(t).

2 In the frequency doman we have the transfer functon relaton between the x and y phasors as: Y = W( jω) X (1) where the transfer functon W s a complex number representng the relaton between nput and output quanttes. Ths transfer functon n the consdered problems s not a pror known but t can be obtaned from the tme doman mpulse response of the studed system obtaned ether expermentally or by numercal smulaton. Gven a wavelet bass d(t)=[d1(t), d(t),, dn(t)] of dmenson n wth n power of two, the WE of x and y are a n-dmensonal vector y() t = d() t y, and x() t = d() t x, where y and x are vectors of coeffcents n terms of wavelet bass. The relatonshp between nput and output quanttes can be wrtten n the wavelet doman as a matrx vector product, as follows: y = W x () where W s a matrx of constant coeffcents relatng WE of nput and output quanttes. In applcatons where a mathematcal model of the studed system s known the matrx W s known, whle n ths case the matrx W has to be determned. The determnaton of W s then performed as follows: the th column of W s equal to y n the wavelet doman when x= x [,..., 1,1, 1,,...,] T = +. Therefore x s nverse transformed n the tme doman obtanng x() t ; then ts Fourer transform X (jω), can be easly obtaned by a FastFourerTransform and used as nput for equaton (1). Then, from the obtaned Y =W(jω)X, y (t) s determned by InverseFastFourerTransform and successvely by WE of y (t) we get the y = W x correspondng to the -th column of the matrx W. It has to be noted that the nverse transform of x s actually the -th element of the chosen wavelet bass. Ths scheme s performed for all the columns of W determnng the complete representaton of the two-port equvalent matrx n the wavelet doman for a gven bass dmenson n. To fully determne the matrx W the prevous explaned operatons must be performed for a number of tmes correspondng to the value n correspondent to the n-dmenson of the WE of the x and y quanttes. In the author experence ths usually vares among and n most applcatons. However ths number can be sgnfcantly reduced. Indeed, from the Multresoluton Analyss theory [7] the n functons d(t) (scalng functons or wavelets) composng the bass d(t) belong to dfferent subspaces. A hgher number of the above mentoned subspaces (n other words a hgher dmenson of the bass d(t)), s related to a better representaton of a sgnal; ths s referred to as resoluton of the bass. The bass functons d(t) of each subspace are translated versons of a sngle functon; hence n case of a tme nvarant system a sngle functon of a subspace can be used to determne the output for all the functons of the subspace. Ths smple consderaton allows a strong reducton of the number of smulatons requred for the characterzaton. Due to ther good characterstcs n terms of effcent representaton of fast sgnals, Daubeches wavelets on the nterval wth N = vanshng moments (see [7] for detals about ther constructon) are chosen for ths problem. Daubeches wavelets on the nterval are characterzed by havng N (where N s the number of vanshng moments) functons defned on the left border and N on the rght border of the nterval. And ths holds for each subspace of the Multresoluton Analyss. Furthermore the coarser resoluton s also related to the number of vanshng moments N. In the case under analyss the lower resoluton leads to 3 functons. Hgher resolutons mean hgher CPU tmes (and n ths case also an hgher number of 3D smulatons n order to characterze the system) but, obvously, an hgher accuracy. Dmensons of the bass of n = and n = are a reasonable number of functons leadng to low CPU tmes and desred accuracy. Furthermore the number of wavelet coeffcents (the number of bass functons) s equal to the samples of the analyzed tme doman functon, hence the ncrease of n leads also to a better tme representaton of the nput and output sgnals. Lookng at a bass of functons n detal, t has been found out that t s composed by 3 subspaces, respectvely characterzed by 3 scalng functons at resoluton 3, 3 wavelets at resoluton 3 and wavelets at resoluton. Among those functons the only ndependent ones for each subspace are the functons (equal to N) on the border and the central functon (the one that s translated n order to form the bass). Hence referrng to the bass of dmenson the ndependent functons to smulate are N fun = 3 N + 3 = 39. In the same way for the bass composed by functons the ndependent smulatons to run are only N fun = N + =. 3 Problem Soluton It has been evaluated the effects that dfferent types of asymmetrcal ppe defect have on the matrx W. For ths purpose, t has been computed the norm of W correspondng to each sngular defect. Table 1 shows the parameters (characterstcs of each defect) nvolved n smulatons. The ncdent wave used n the smulatons s a torsonal wave havng the wavelength λ=.9m. For the

3 evaluaton t has been used the crcumferental component only, as beng the most representatve. Table1 Thckness (%) Length (mm) Sold angle (deg.) 1 thckness=% thckness=3% thckness=% thckness=7% thckness=9% 1-norm of W for dfferent values of thckness (length=3mm) The norms used to evaluate the wavelet matrx are: - Norm1(W)=max(sum(abs(W))), that s the largest column sum of W; - Norm(W)=sqrt(sum(dag(W *W))),.e. the Forbenus norm; - Norm3(W)=max(svd(W)),.e. the largest sngular value of W. Fgures 1 to 7 llustrate the behavor of norm1 of matrx W for dfferent sets of defects. A result to be noted s that the larger the defect volume, the hgher becomes the senstvty of the W matrx. In addton, axal dmenson (.e. length of defect), show mnor nfluence on the norm of W matrx at small sold angle or thckness (fg. to ). Hgher values of sold angle and thckness (fg.1,, 7) also brng to a dfferent norm trend. Fgure summarzes the mpact that each of the types of defect has on the wavelet matrx. It confrms once agan that the W matrx s more senstve at varaton of sold angle and thckness of defects. norm1 of W for dfferent values of thckness (length=mm) thckness=% thckness=3% thckness=% thckness=7% thckness=9% 3 3 sold angle (deg) Fg. Behavor of W matrx at varatons of sold angle and thckness of defect norm1(w) 1 norm1 of W for dfferent sold angles (length=3mm) theta= deg theta= deg theta=3 deg theta= deg thckness(%) Fg.3 Behavor of W matrx at varatons of thckness and sold angle of defect. norm1 of W at dfferent values of sold angle (thckness=3% ) theta= deg theta= deg theta=3 deg theta= deg sold angle (deg) Fg.1 Behavor of W matrx at varatons of length and thckness of defect length (mm) Fg. Behavor of W matrx at varatons of length and sold angle of defect

4 norm1 of W for dfferent values of length (thckness=3%) length=3 mm length= mm length= mm length=1 mm length= mm 3 3 Norm of W for dfferent varatons of defects angle-thckness-length length-angle-thckness thckness-angle-length norm1(w) =max(svd(w)) 3 3 sold angle(deg) Fg. Behavor of W matrx at varatons of sold angle and length of defect defects Fg. Behavor of W matrx at dfferent varatons of all types of defect norm1 of W for dfferent values of length (sold angle= deg) length=3mm length=mm length=mm length=1mm length=mm Concluson In order to classfy a partcular ppe defect by the means of an NDT ultrasound guded waves system, a WE of the two waves (ncdent and reflected) has been performed. The resulted W matrx (treated as a transfer functon between nput and output) behaves dfferent for each type of defect. The results show that W matrx can be a useful nstrument n detectng the ppe defects. It can also nform about the volume of defect thckness(%) Fg. Behavor of W matrx at varatons of thckness and length of defect norm1 of W for dfferent values of thckness (theta=deg) thckness=% thckness=3% thckness=% thckness=7% thckness=9% length(mm) Fg.7 Behavor of W matrx at varatons of length and thckness of defect Acknowledgements Ths work was supported n part by the Mnstry of Unversty under a Program for the Development of Research of Natonal Interest PRIN 39. References: [1] Alleyne, D.N. and Cawley, P. The exctaton of Lamb waves n ppes usng dry coupled pezoelectrc transducers, J NDE, Vol, pp11-, 199. [] H. Kwun and A. E. Holt, Feasblty of Underlaggng Corroson Detecton n Steel Ppe Usng the Magnetostrctve Sensor Technque, NDT&E Internatonal, pp (199) [3] Böttger, W., Schneder, H., and Wengarten, W. Prototype EMAT system for tube nspecton wth guded ultrasonc waves, Nuclear Eng. and Desgn, Vol, pp3-37, 197 [] Alleyne, D.N. and Cawley, P. A two-dmensonal Fourer transform method for measurements of propagatng multmode sgnals, J. Acoust. Soc. Am., Vol 9, pp 19-11, [] Ln S., Ito T., Kawashma K., Okade M., Nagamzo N. Szng of axal defects n ppes wth FEM smulaton of wave propagaton and wavelet transformaton 199 IEEE Ultrasoncs Symposum, pp 77-.

5 [] Sun Z., Mao Y., Jang W., Zhang D. Investgaton on Interacton of Lamb waves and crcumferental notch n ppe by means of wavelet transform, IEEE Ultrasoncs Symposum, pp 7-3. [7] I. Daubeches, Ten lectures on wavelets, SIAM, 9. [] A. Cohen, I. Daubeches, P. Val, Wavelets on the nterval and fast wavelet transforms App. Comput. Harmon. A,. vol. 1, no.1; Dec. 1993; p. -1 [9] S. Barmada, M. Raug, Transent numercal soluton of nonunform MTL equatons wth nonlnear loads by wavelet expanson n tme or space doman, IEEE Trans. on Crc. and Syst., Vol. 7, pp