Modeling the Radiation Field of Normal-Beam Ultrasonic Transducers by Multi-Gaussian Beam Technique

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1 Modelig the Radiatio Field of Normal-Beam Ultrasoic Trasducers by Multi-Gaussia Beam Techique Ami Yaghootia, Farhag Hoarvar, ad Mehdi Zeighami NDE Lab., Faculty of Mechaical Egieerig, K. N. Toosi Uiversity of Techology, ardis St., Molasadra Ave., Vaak Sq., Tehra, Ira. Abstract The modelig of ultrasoic tests ca help i gaiig a better uderstadig of the test process. The model ca also predict the test results ad provide a basis for choosig the optimum parameter settigs. Oe of the first steps i modelig a ultrasoic test is to uderstad the characteristics of the ultrasoic wave field. I this paper, the multi-gaussia beam () techique is used for modelig the radiatio field of a ormal-beam ultrasoic trasducer. By usig this approach, oe ca avoid the computatioal difficulties ecoutered i other modelig techiques such as Rayleigh-Sommerfeld ad Gree fuctio itegrals. By modelig the radiatio field of a MHz trasducer, the beam profile of the reflected sigal have bee calculated. Experimets cofirm the computatioal results. Keywords: Ultrasoic Modelig, Multi-Gaussia Beam, Wave Field. Itroductio Modelig of wave field geerated by a trasducer ca help i odestructive evaluatio of materials by predictio of echoes, simulatio of defects, ad also i sigal processig. Moreover, a good model ca be very helpful i the desig ad optimizatio of a testig procedure ad i the iterpretatio of experimetal results. It is also much simpler ad cheaper to perform parametric studies with by usig a good model compared to coductig experimets. By usig the model, the probability of detectio of various types of defects ca be assessed. I the earliest works of wave field modelig the trasducer surface was cosidered to act as a source of poit forces (Cherry 96). This method had computatio difficulties. By usig Johso's Gree fuctio, Tag et. al. (99) derived itegral expressios of the elastic displacemets for both compressio ad shear pisto sources actig o the surface of a elastic half-space. The shortcomig of this method is its iability to model the ear-field of the wave field as well as oblique icidece modelig. Harris (98) reviewed three approximatios icludig Rayleigh surface itegral, Kig itegral, ad Schoch solutio to fid the pressure field of a plaar probe o a ifiite half-space. Other umerical methods were also developed to show the displacemet field i a solid media or the pressure field i a fluid due to wave radiated from ormal or agular trasducers (Alia et. al. 4). Based o a elastodyamics solutio, Bostrom ad Wirdelius (995) modeled the wave field for elliptical ad rectagular trasducers i a ormal or agular positio, emittig wave ito a solid half space. Although the exact o-axis field ad a approximatio to the far field ca be expressed i a closed form, describig the ear field, the trasiet zoe, ad off-axis beams are difficult ad time cosumig i umerical methods by itegratio over poits of the probe surface.

2 We ad Breazeale (988) proposed a alterative approach. They computed the total field by superimposig a umber of Gaussia beam solutios. They have show that by superimpositio of oly Gaussia solutios, the field radiated by a circular pisto trasducer ca be accurately modeled. I this paper, the multi-gaussia beam techique is used for modelig the radiatio field of a ormal-beam ultrasoic trasducer. By usig this approach, oe ca avoid the computatio difficulties ecoutered i other techiques icludig the Rayleigh-Sommerfeld ad Gree fuctio itegrals. I some ultrasoic tests, it is required to use ultrasoic beams with large spread agles. Based o the locatio of defects i the material, the wave ca be reflected from ay poit o the beam cross sectio. By modelig the radiatio field of a MHz trasducer, the beam profile ad frequecy spectrum of the reflected sigal have bee calculated. Multi-Gaussia Beam Modelig Modelig the fields radiated by ultrasoic trasducers is a challegig task because of the large umber of possible trasducer types, sizes, ad cofiguratios that are used i practice. The multi-gaussia beam model ca describe the wave field of a circular pisto trasducer by superpositio of a umber of idividual Gaussia beams with a proper set of weightig factors. Although ultrasoic trasducers do ot geerate Gaussia beams but this modelig works well i cosiderig the trasducer to act as a pisto source. Oe of the advatages of multi-gaussia beam modelig is sigificat reductio i computatio time as well as ability i modelig the wave field i aisotropic materials. I Gaussia beam modelig of circular trasducers, it is first assumed that the trasducer is a time harmoic Gaussia source with time depedecy ad waves are emitted perpedicular to the trasducer surface ( x directio) as a quasi-plae wave. The displacemet field i Gaussia beam is obtaied from the followig formulatio (Kim et. al. 4), u ( x, x, x [ M ( x )] T det ( ), ω iv M x ik x x ) = exp( ik ) exp x ρc det M () ω () where v is the velocity o the trasducer surface, k = ω / c is the wave umber, ω is frequecy i radias per secod, c is logitudial wave speed i specime, ad ρ is the T desity of the meduium i which the wave radiates. Moreover, x = [ x, x ] ad M is a complex-valued symmetric matrix. The multi-gaussia beam modelig ca fially be writte as (Schmerr ad Sog 7), u ( x, x, x [ M ( x )] N T ia iωx x, ) = exp( )exp ω ik x + () = ib x / DR where N is the umber of Gaussia beams, DR = ka / is the Rayleigh distace, a is the radius of the trasducer ad

3 ib / c DR + ib = x / DR M () ib / cdr + ib x / DR [ ( x )] A ad B are complex-valued expasio coefficiets that eed to be determied to match the velocity field o the face of the trasducer. We ad Breazeale (988) foud te coefficiets by a optimizatio method for circular plaar pisto trasducers. Experimet The results of multi-gaussia beam model have to be verified by compariso with other models or with experimets. A experimet was carried o a 69 mm thick alumium block. The compressioal ad shear velocity i alumium are 6 m/s ad 5 m/s, respectively. The wave was geerated by a 6.5 mm diameter trasducer havig a ceter frequecy of MHz. The test was performed by ultrasoic pulse-echo techique i which the sigle trasducer acted as both the trasmitter ad receiver. Results ad Discussio I the first step, to show the ability of multi-guassia beam for modelig both the ear field ad far field of the trasducer, the computed pressure field of a circular trasducer with?? mm diameter ad 5 MHz cetral frequecy is illustrated i Fig.. Fig. - Modelig of the trasducer wave field wave field The backwall echo obtaied from the alumium test block is show i Fig a. The frequecy spectrum of this echo is also show i Fig. b. Both the multi-gaussia model ad the Gree fuctio mdoel were used for modelig the backwall echo. The Gree fuctio model was developed followig the approach used by Tag et. al. (994). The modeled echo obtaied from this approach is show i Fig. c alog with its frequecy spectrum show i Fig. d. The echo obtaied from the multi-gaussia model is show i Fig. e ad its reletive spectrum frequecy is show i Fig. f. Both modeled echoes are i good agreemet with the experimetal result.

4 (a) (b) (c) (d) (e) (f) 5 Time (µs) 4 5 Frequecy (MHz) Fig. - Comparsio of experimetal sigal (a), Gree fuctio model (c) ad multi-gaussia beam modelig (e). Frequecy spectra of three echo is show i (b), (d) ad (f) respectively. Next, we compare the capability of the two models i modelig the radiated ultrasoic wave at various agles. We cosider a 5 MHz trasduce with 6.5 mm diameter to be trasmittig the ultrasoic wave ito a semi-circular specime havig a radius of 75 mm. The wave is the received by a idetical trasducer o the cured surface of the specime as show i Fig.. Fig. - Cofiguratio of experimetal set-up 4

5 .8 θ = 5 o Tag(994).8 θ = o Tag(994) frequecy (MHz) 5 5 frequecy (MHz).8 θ = 5 o Tag(994).8 θ = o Tag(994) frequecy (MHz) 5 5 frequecy (MHz) Fig. 4- Frequecy spectrum comparssio of (solid lie) ad Gree's fuctio (dashed lie) As show i Fig. 4, by icreasig i agle with respect to the ceteral axis of the trasmittig trasducer, the frequecy spectrum of the received echoes chage. This chage of frequecy spectra is due to the destructive iterferece of waves origiatig from various poits o the trasducer surface. Agai the two models are i good agreemet. Experimeal verificatio of the two models is curretly uderway. Coclusio I this paper, the multi-gaussia beam () approach was used to model the wave field of a ormal beam trasducer. The advatage of usig this method is modelig the ear-field as well as the far-field of the trasducer i both fluids ad solids. It also avoids the computatioal difficulties ecoutered with other modelig techiques. To examie the capability of this model, a backwall echo was modeled by the multi-gaussia beam approach ad compared with the experimetal as well as Gree fuctio modelig results. The result obtaied from the model was i good agreemet with the experimetal result. Moreover, the frequecy spectra of beam radiated from a ormal trasducer was modeled by the approach ad compared with Gree's fuctio results. Agai, the two models were i good agreemet. The techique ca be used i modelig the refractio ad reflectio from iterfaces without ay computatio difficulties. 5

6 Referece [] J. T. Cherry, Jr., "The Azimuthal ad olar Radiatio atters Obtaied From a Horizotal Stress Applied at the Surface of Elastic Half Space", Bull. Seismol. Soc. Am. 5, 7-6 (96). [] X. M. Tag, M. N. Toksoz, ad C. H. Cheg, "Elastic wave radiatio ad diffractio of a pisto source", J. Acoust. Soc. Am. 87, 894-9, (99). [] G. R. Harris, "Review of trasiet field theory for a bared plaar pisto", J. Acoust. Soc. Am. 7, - (98). [4] A. Alia, H. Djelouah, N. Bouaoua, "Fiite differece modelig of the ultrasoic field radiated by circular trasducers", J. Comput. Acoust. (4), , (4). [5] A. Bostrom ad H. Wirdelius, "Ultrasoic probe modelig ad odestructive crack detectio", J. Acoust. Soc. Am. 97(5), , May (995). [6] J.J. We ad M.A. Breazeale, "A diffractio beam field expressed as the superpositio of Gaussia beams", J. Acoust. Soc. Am. 8 (5), , (988). [7] H. J. Kim, J. S. ark, S. J. Sog, ad L.W. Schmerr Jr, "Modelig Agel Beam Ultrasoic Testig Usig Multi-Gaussia Beam", Joural of Nodestructive Evaluatio, 4,(),8-9. [8] L.W. Schmerr, S.J. Sog, "Ultrasoic Nodestructive Evaluatio Systems: Models ad Measuremets", Spriger, 7. 6