THE MATHEMATICAL ANALYSIS OF ULTRASONIC WAVES PROPAGATION APLICATED TO REALIZATION OF HIGH POWER PIEZOELECTRIC TRANSDUCERS

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1 Journal of cience and Arts ear, No. 4(7), pp , 0 OIGINAL AE THE MATHEMATIAL ANALI OF ULTAONI WAVE OAGATION ALIATED TO EALIATION OF HIGH OWE IEOELETI TANDUE GIGOE LIVIU ODOBEU Manuscript receied:.07.0; Accepted paper: ublished online: Abstract. The high energy ultrasound is generated by piezoelectric eleents which are excited fro electric signal obtained fro an electronic generator. The transission of this high energy is obtained by transission or concentrator eleents, in function of what we want to do with this energy utilized especial in no conentional technologies. The efficiency work is ery iportant aspect because the energy utilizes has high alues. In this paper is presented ethods to calculate the eleents coponents necessary to generate and transits this high ultrasound energy: piezoelectric transducers and transission/concentrator eleents. Is presented the experiental results obtained with the theory presented. Original contribution consists by ethod used and suggestie graphics for appreciation of paraeters ariations. eywords: piezoelectric transducer, ultrasound power, diension design. Matheatics ubject lassification 00: 74J05. INTODUTION In paper [] is described a few ethods to increase the efficiencies of acoustic systes especial by point of iew of electronic generator and its adaptation with acoustic syste. In papers [] and [3] is taken in account a arious types of adaptation schees between electronic generator and acoustic coponents. In paper [4] is presented a ethod to design a few resonances transforation/adaptation bodies in function of work conditions and acoustic charge. In paper [5] is presented a few applications of high ultra acoustic energy utilized in fabrication of soap. In this paper is utilizes the general principles described in [], [3] and [4] for to design and characterization the piezoelectric transducers. Is find out a work procedure to calculate the physical diensions of eleent which constitutes the piezoelectric transducer in function of coposition aterial, work paraeters and analyzing there influences on characteristic paraeters of piezoelectric transducer. In practice, [4], we eet two iportant cases in propagation of ultrasonic waes. One of these is the case of the transission/concentrator eleents, when the ultrasound waes pass through one propagation ediu (usual aluinu alloy or titan) and the second is the case of propagation of ultrasound waes through three edius (usual steel - piezoelectric eleentaluinu alloy). This is the case of ultrasound piezoelectric transducer. Institute of olid Mechanics, oanian Acadey Bucharest, oania. E-ail: g_odobescu@yahoo.co. IN:

2 396 The atheatical analysis of Liiu G. Odobescu. THE INILE OF OEATION In case of a piezoelectric transducer, [], fored by two pastilles, which are catch between two etallic blocks (naed director and reflector) by a central screw. The used aterials in reflector coposition are: steel, aluinu alloy, bronze, naal brass. The used aterials in director coposition are: titaniu, aluinu alloy or agnesiu coposite. The long of transducer, by resonance conditions, is / Fig. ( represents the wae long of ultrasound wae through aterial). Is noted: f i i i i () f where: - i, are constants which referrers to the end piece respectiely to the ceraic aterial. - i and represents the waes long of ultrasound wae through end aterial and ceraic aterial, []; - i and represents propagation speed through end aterial and ceraic aterial, [/s]; - f represents frequency of excited wae [Hz] and =f represents its pulsation. / eflector Director / 4 Neutral / 4 plane Fig. The piezoelectric transducer. The propagation speed through aterial, noted, depends by: - the aterial density, [g/ 3 ]; - the elasticity odule, [N/ ]; - the oisson coefficient. Is considered a piezoelectric transducer presented in Fig., where, below of transducer, is represented, [3], the eolution of ibration aplitude y(x) in function of distance x. l l l Neutral plane End piece-p Midle piece-n x /4 M M M y (x) y (x) y (x) L Fig.. ei transducer with three propagation edius.

3 The atheatical analysis of Liiu G. Odobescu 397 (null aplitude ibration at neutral plane of iddle piecen) and We hae: y l 0 y l l M, where: l - represents the total thickness of whole (in general two pieces) ceraics package, []. Is noted by wae factor the expression: i i i where i = p, c, referrers to end piece, ceraic piece respectiely iddle piece. We note the aplitude of ibration, y,, expressed in [], at end, ceraic, respectiely iddle pieces: yx Mcos xl with continuity condition y l M yx Msin x L with continuity condition y L 0 with continuity condition yl l 0 y x= sin x l l M where L represents the distance on x axe where the aterial points will hae null displaceent if we hae only ceraic aterial. o, we hae y L 0. The conditions, at separation liit between of these three propagation edius, because the propagation is linear and continuous between these edius, will be: y 0 y 0 a) for displaceents... ) ) y l y l b) for forces...3) F 0 F 0 4) F l F l dy( x) We hae the relation: FxA. where: - F(x) represents the force along x axe, [N]; - - represents the aplitude ibration, []; - A- represent the contact area between these three edius, [ ]; dy( x) - represents the elocity of aterial particles in sei-transducer coponents, [/s] o, we hae: () dy ( x) FxA AM sin[ xl] - at end piece dy ( x) Fx A A M cos[ xl] - at ceraic piece dy( x) FxA AM cos[ xl l] - at iddle piece (3) It puts the liit conditions for displaceents - a) conditions - and is obtained: ) y 0 y 0 or cos x l sin x L M sin l cos l M M M (4) IN:

4 398 The atheatical analysis of Liiu G. Odobescu ) y l y l or M sin x L M sin x l l M sin l M sin l L It puts the liit conditions for forces - b)/3 conditions - and is obtained: 3) F 0 F 0. (5) It is obtained: - A M sin( l )= A M cos(- L ) M sin l A (6) M cos L A M eplacing the ratio M with the alue finds out fro a)/ condition, relation (4): sin cos l l sin cos l L A A (7) Taken l L and is obtained: l L A A A A A A (8) Because we hae, fro definition: = (for bars) and i = i i ( i =f i ). In relation (8) = and = represents the acoustic ipedances for ceraics and for the end piece. It puts the second condition, b)/4 for forces and obtained relations (9) and (0): 4). F (l ) = F (l ). (9) A M cos (l - L ) = AM cos (-l ) M cos l L A (0) cos l A M M eplacing the ratio M sin sin l l L with the alue finds out fro a)/ condition, relation (4): cos l L cos l A A ()

5 The atheatical analysis of Liiu G. Odobescu l c (l - L ) = A A In conclusion we hae obtained: L l = - l c (l - L ) = () where: = ; = (3) It notes: L = L ; l = ; l = ; l = The equation () becoes: L = and c( - L )=- (4) Is obtained: L = L = ; L L (5) eplacing L in the second equation [5], is obtained: (6) Or relation: + + = (7) Fro si expressions is obtained: (8) This relation is noted with and shows the transition fro etal (-director) to end piece (p-reflector) fig.. This relation perits to calculate the diensions of sei- IN:

6 400 The atheatical analysis of Liiu G. Odobescu transducer haing 3 propagation edius [4]. It starts fro neutral plane which is deterined (l is deterined). The deterination of l, l or l diensions when other sizes are know, ay be effected fro one of relations gien down, which are obtained fro the sae equation (7): arc p l = arc l = arc l = (9) where, l represents the total thickness of whole package ceraics of sei-transducer considered. Because ceraics isn t placed into nodal plane (where we hae the null displaceents y=0 and axiu echanical tensions T=ax) for to obtain the sae piezoelectric effect it shall to increase the thickness of piezoelectric eleents package. This is disadantageous by econoic point of iew. Is preferred to placed nodal plane in proxiity of piezoelectric eleents but situates in director piece where it realized the cached of assebly transducer, because the displaceents is null in this zone. To calculate the transersal diensions, necessary for a gien leel of power, we take in account two paraeters: the axiu stress and dynaic deforations adissible for aterials fro which it is constitute the piezoelectric transducer. The stress (T), respectiely dynaic deforations (=l/l) into a resonant echanical eleent are in function of perissible displaceent at end eleent. This displaceent is deterined by acoustic charge. Let s find out a relation between displaceent at end eleent and axiu tension which take place at neutral plane fro the transducer. Let s note: the axiu aplitude displaceent, situated at radiation face of transducer; - y - the aplitude displaceent at x distance fro nodal plane. The nodal plane represents the section of transducer where the displaceents are nulls and echanical tensions are axius. It can writes fig. 3: y (x)= 0 sinx= 0 sin x. (0) The alue of relatie deforation for any x alue: esults: x x x y x y dy ( x) li x 0 () x cos x (x)= 0 () The axiu relatie deforation (x) in nodal plane (x=0):

7 The atheatical analysis of Liiu G. Odobescu 40 ax (x=0)= 0 f 0 0 (3) where: represents the speed sound through aterial, [/s]; =f ibrations pulsation, [Hz]; =/f the length wae of ibrations through aterial, []. The axiu aplitude of dynaic ibrations T (or the axiu aplitude of dynaic pressure ) will gie by Hooke law: F T = = A, where: - represents the elasticity odule of aterial, [N/ ]; F - represents the force, [N]; A - represents the section area, [ ]. Also, we hae: in the case of bars (haing the sae diaeter fro it along). where; - represents the sound speed through aterial, [/s]; - represents, the aterial density of transducer pieces, [kg/ 3 ]. N E U T A L L A N E / 4 A y /4 0 x Fig. 3. Transersal diensioning. The axiu aplitude of ibration speed (noted ) at end of eleent is gien by relation: dy ( x) y (x)= 0 sin x (x)= = 0 cos x esults, with condition cos x = and utilizing relation (3) for 0 : T T T = 0 = ax = or: T = = 0 o, the echanical tension (T ) or acoustic pressure ( ) at nodal plane is gien by product between axiu speed of particles ( ) at this end and specific acoustic ipedance (=): T = = =. The acoustic intensity I, represents the acoustic energy flux which pass through unit surface perpendicular on propagation direction of waes. It is gien by relation: IN:

8 40 The atheatical analysis of Liiu G. Odobescu I where: represents the axiu speed aplitude in iddle piecedirector, at end face, [/s]; - represents the pressure aplitude in iddle piece-director, [N/ ]. By oertaking alue for I ax it ay destroy the transducer ia: - by oertaking alue for echanical tensions T [N/ ], usual dan/c or - by oertaking alue for teperature T 0. We hae the relations: = T ; = (4) eplacing, results: I= I= (expressed by ); (expressed by ) (5) Is noted: = - the transducer ipedance; = - the ediu ipedance where it is transit the energy flux. At separation liit transducer/ediu these two acoustic intensities are equals: I =I =I. eplacing in relation (5) relation (4), will obtain: I= T (6) The relation is alid in case of a hoogeneous transducer. In case of a coposite transducer, we hae an aplification coefficient G gien by relation: T G G I= (7) The debited power by transducer through end surface A will be: =IA= T G A (8) This is the power in function of aterial paraeters. For it calculates we proceed: - It iposes T - which not pass the adissible axiu tensions (by point of iew of fatigue); - debited power. - It knows the end of transducer (director); the work ediu;

9 The atheatical analysis of Liiu G. Odobescu 403 G - the aplification. esults A the transersal area of transducer necessary to generated the ipose power. The power which ay be debits fro transducer it is depending fro acoustic ipedance charge =. For a sall acoustic 0, the director will ibrate with a large aplitude and so not can appliqué high power by point of iew of safety. If we hae: water like acoustic charge = =,50 6 g / s; and air like acoustic charge = = 40 g / s. Therefore, the adissible power, in case of air, is uch sall. The transersal diensions aren t being too big. It is ipose to be ½ fro longitudinal diension of transducer (/) for not perit to appear the transersal resonance belong of piezoelectric transducer. 3. EXEIMENTAL EULT A piezoelectric transducer, haing 500W power at 0 khz work frequency is gien in Fig. 4. In these situations is preferable to use the elocity transforations for to agnify the ibration aplitude. These elocity transforations are coposed fro aluinu alloy resonators haing a long gien by / (=56 in case of a 0 khz frequency) - Fig. 5. Is built two acoustic systes and with 000V excitation signal is obtained: a. The acoustic syste I, haing D=5 ; D3=5 diaeters. The ibration aplitude was 50. b. The acoustic syste II, haing D=59 ; D3= diaeters. The ibration aplitude was 80. l l l N E U T A L L A N E EFLETO D I E T O Fig W / 0 khz transducer. In both systes it is utilizes an acoustic transforation haing D 4 =60 and D 5 =6 diaeters. The aplification obtained with these two acoustic systes is gien by: D D4 D ax G G D3 D 5 in conforities with general forulas D in. eplacing is obtained: - The acoustic syste I G = 3; - The acoustic syste II G =39,3 These coefficients will represent the aplifications realized fro these acoustic ipedances in these two cases. In both cases we hae an aplification of acoustic ipedance and an aplification of transducer charge. IN:

10 404 The atheatical analysis of Liiu G. Odobescu eflector Director D D oncentrator D3 D4 Velocity transforation D5 ceraics 3 Fig. 5. Acoustic chain-transducer (), concentrator (), elocity-transforation (3). 4. ONLUION With help of the relations presented in paper we can calculate the diensions of a piezoelectric transducer used to produce the power ultrasonic field necessary at unconentional technology [5] like washes, solders etc. The results were erified in practice and help us to quickly find the diensions and the influence of diensions on global perforances. With aterial and financial support gien by..tehnofina.a. and used theory presented is realized and hoologated a few planar transducers in power range (50-50)W whose ay be used at washing and iscellaneous ultrasonic technology. It was realized and hoologated, also, a piezoelectric transducer haing 000W power, which ay be utilized at washing and solders based by ultrasonic field. EFEENE [] Odobescu, G., ugina, I., Lala,., Metode de creştere a eficienţei sisteelor acustice de are putere. esiunea de counicǎri ştiinţifice oisia de Acustica Acadeia oanǎ Bucureşti, 3, 000. [] Odobescu G., Lala,., The coparatie study about energetic efficiency for iscellaneous types of adaptation schees generator-piezoelectric transducers, The Annual yposiu of the Institute of olid Mechanics, IOM 00, 6-7 May, Bucharest 83, 00. [3] Odobescu, G., The propagation study of ultrasound high intensity waes through resonance bodies transforation with a new adaptation schee with acoustic charge, First onference of the oanian Acoustical ociety, 5-7 October 00, Bucharest, 75, 00. [4] Odobescu, G., The propagation study of ultrasound high intensity waes through resonance bodies transforation/adaptation with acoustic charge, The 7 th International ongress on Acoustic, epteber -7, 00, oe, Italy, ID 5, 7A.0.07 Nonlinear acoustic, 00. [5] Eddy, N.O., Ukpong, I.J., ossible relationship between the soap and the lye, Applied ciences, A, 8(), 60,