FREE VIBRATION ANALYSIS OF ROTATING PIEZOELECTRIC BAR OF CIRCULAR CROSS SECTION IMMERSED IN FLUID

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1 Mateials Physics ad Mechaics 4 (15) 4-34 Received: Mach 1 15 FREE VIBRATION ANALYSIS OF ROTATING PIEZOELECTRIC BAR OF CIRCULAR CROSS SECTION IMMERSED IN FLUID R. Selvamai Depatmet o Mathematics Kauya Uivesity Coimbatoe Tamil Nadu Idia selvam179@gmail.com Abstact. Fee vibatio aalysis o otatig pieoelectic ba o cicula coss-sectio immesed i luid is discussed usig thee-dimesioal theoy o pieoelecticity. The equatios o motio o the ba ae omulated usig the costitutive equatios o a pieoelectic mateial. The equatios o motio o the luid ae omulated usig the costitutive equatios o a iviscid luid. Thee displacemet potetial uctios ae itoduced to ucouple the equatios o motio electic coductio. The peect-slip bouday coditios ae applied at the solid-luid iteaces to obtai the equecy equatio o the coupled systems. The equecy equatios ae obtaied o logitudial ad lexual (symmetic ad atisymmetic) modes o vibatio ad ae studied umeically o PZT-4 mateial. The computed wave umbe ad electo mechaical couplig is peseted i the om o dispesio cuves. The secat method is used to obtai the oots o the equecy equatio. 1. Itoductio The developmet i pieoelectic sesos ad actuatos is impotat o the desig ad costuctio o light weighted ad high peomace smat stuctues. Pieoelectic polymes allow thei use i a multitude o compositios ad geometical shapes o a lage vaiety o applicatios om tasduces i acoustics ultasoic s ad hydophoe applicatios to esoatos i bad pass iltes powe supplies delay lies medical scas ad some idustial o-destuctive testig istumets. Some o the applicatios o these polymes iclude Audio device-micophoes high equecy speakes toe geeatos ad acoustic modems; Pessue switches positio switches acceleometes impact detectos low metes ad load cells; Actuatos- electoic as ad high shuttes. The otatig pieoelectic ba o cicula coss sectio has gaied impotace i costuctio o gyoscope to measue the agula velocity o a otatig body. Most o the studies i elastic wave popagatio i cylidical waveguides ae coceed with isotopic cylides. The wave popagatio i elastic solid has bee discussed extesively i details by Ga [1]. The popagatio o compessioal elastic waves alog a aisotopic cicula cylide with hexagoal symmety was ist studied by Mose []. Theoetical studies o electoelastic wave popagatio i aisotopic pieoceamic cylides have also bee pusued o may yeas. The appoach usually applied o pieoelectic solids is the simpliicatio o Maxwell s equatios by eglectig magetic eects coductio ee chages ad displacemet cuets. Studies by Tieste [3] should be metioed amog the ealy otable cotibutios to the topic o the mechaics o pieoelectic solids. Electoelastic goveig equatios o pieoelectic mateials ae peseted by Pato ad Kudyavtsev [4]. Shul ga [5] studied the popagatio o axisymmetic ad o-axisymmetic waves i aisotopic pieoceamic cylides with vaious pepolaiatio diectios ad bouday coditios. 15 Istitute o Poblems o Mechaical Egieeig

2 Fee vibatio aalysis o otatig pieoelectic ba o cicula coss sectio immesed i luid 5 Paul ad Vekatesa [6-7] studied the wave popagatio i iiite pieoelectic solid cylides o abitay coss-sectio usig Fouie expasio collocatio method omulated by Nagaya [8]. Rajapakse ad Zhou [9] solved the coupled electoelastic equatios o a log pieoceamic cylide by applyig Fouie itegal tasoms. Pape by Wag [1] should be metioed amog the studies o cylidical shells with a pieoelectic coat. Ebeee ad Ramesh [11] aalyed axially polaied pieoelectic cylides with abitay bouday coditios o the lat suaces usig the Bessel seies. Beg et al. [1] assumed electic ield ot to be costat ove the thickess o pieoceamic cylidical shells. Late Botta ad Cei [13] exteded this appoach ad compaed thei esults with those i which the eect o vaiable electic potetial was ot cosideed. Kim ad Lee [14] studied pieoelectic cylidical tasduces with adial polaiatio ad compaed thei esults with those obtaied expeimetally ad umeically by the iite-elemet method. Belie ad Solecki [15] have studied the wave popagatio i a luid loaded tasvesely isotopic cylide. I that pape Pat I cosists o the aalytical omulatio o the equecy equatio o the coupled system cosistig o the cylide with ie ad oute luid ad Pat II gives the umeical esults. Guo ad Su [16] discussed the popagatio o Bleustei - Gulyaev wave i 6mm pieoelectic mateials loaded with viscous liquid usig the theoy o cotiuum mechaics. Qia et al [17] aalyed the popagatio o Bleustei- Gulyaev waves i 6mm pieoelectic mateials loaded with a viscous liquid laye o iite thickess. Dayal [18] ivestigated the ee vibatios o a luid loaded tasvesely isotopic od based o ucouplig the adial ad axial wave equatios by itoducig scala ad vecto potetials. Nagy [19] studied the popagatio o logitudial guided waves i luid-loaded tasvesely isotopic od based o the supepositio o patial waves. Guided waves i a tasvesely isotopic cylide immesed i a luid wee aalyed by Ahmad []. Pousamy ad Selvamai [1 ] have studied espectively the thee dimesioal wave popagatio o tasvesely isotopic mageto themo elastic cylidical pael ad lexual vibatio i a heat coductig cylidical pael embedded i a Wikle elastic medium i the cotext o the liea theoy o themo elasticity. The dyamic espose o a heat coductig solid ba o polygoal coss sectio subjected to movig heat souce is discussed by Selvamai [3] usig the Fouie expasio collocatio method (FECM). Zhag [4] ivestigated the paametic aalysis o equecy o otatig lamiated composite cylidical shell usig wave popagatio appoach. Body wave popagatio i otatig themo elastic media was ivestigated by Shama ad Gove [5]. The popagatio o waves i coductig pieoelectic solid is studied o the case whe the etie medium otates with a uiom agula velocity by Waue [6]. Roychoudhui ad Mukhopadhyay [7] studied the eect o otatio ad elaxatio times o plae waves i geealied themo visco elasticity. Hua ad Lam [8] has studied the equecy chaacteistics o a thi otatig cylidical shell usig geeal dieetial quadatue method. Segiu et al. [9] studied the eegy dissipatio ad citical speed o gaula low i a otatig cylide ad they oud that the coeiciet o ictio have the geatest sigiicace o the cetiugig speed. The aim o the peset aticle is to study the ee wave popagatio i a otatig pieoelectic ba o cicula coss-sectio immesed i luid. The equecy equatios ae obtaied om the solid-luid iteacial bouday coditios. The computed wave umbe ad electomechaical couplig with espect to equecy ae plotted i the om o dispesio cuves o logitudial ad lexual modes o vibatios o the mateial PZT-4.. Goveig ield equatios The liea costitutive equatios o coupled elastic ad electic ield i a pieoelectic medium ae give by T Ce E T D e E (1)

3 6 R. Selvamai whee the stess vecto the stai vecto e the electic ield vecto E ad the electic displacemet vecto D ae give i the cylidical coodiate system (Fig. 1) by T E T e e e e e e e D T E E E T D D D () whee C ad deotes the matices o elastic costats pieoelectic costats ad dielectic costats espectively. Fig. 1. Rotatig ba immesed i luid. The matices C ad o the tasvesely isotopic mateial is give by c11 c1 c13 c1 c11 c13 c13 c13 c33 C c44 c44 c66 e15 e15 e31 e31 e (3) 33 By cosideig a homogeeous tasvesely isotopic pieoelectic otatig cicula ba o iiite legth immesed i luid the equatios o motio i the absece o body oce ae 1 u ( ( u) ( u t )) t 1 u t 1 u ( ( u) ( u t )) t The electic displacemets D D ad D satisy the Gaussia equatio is. (4) 1 1 D D D. (5) The elastic the pieoelectic ad dielectic matices o the 6mm cystal class the pieoelectic elatios ae c e c e c e e E c1e c11e c13e e31e c e c e c e e E c66e c44e e15 E c44e e15 E (6)

4 Fee vibatio aalysis o otatig pieoelectic ba o cicula coss sectio immesed i luid ad D e e E D e15e 11E D e e e e e E (7) whee ae the stess compoets e e e e e e ae the stai compoets c11 c1 c13 c33 c 44 ad c66 c11 c1 ae the ive elastic costats e31 e15 e 33 ae the pieoelectic costats ae the dielectic costats is the mass desity. The comma i the subscipts deotes the patial dieetiatio with espect to the vaiables. The displacemet equatio o motio has the additioal tems with a time depedet cetipetal acceleatio ( u) ad ( u t ) whee u ( u w) is the displacemet vecto ad ( ) is the agula velocity. The stai e ij ae elated to the displacemets ae give by e 1 u e u u e u (8a) 1 1 e u u e u u u e u u. (8b) The comma i the subscipts deotes the patial dieetiatio with espect to the vaiables. Substitutig the Eqs. (6) (7) ad (8) i the Eqs. (4) ad (5) esults i the ollowig thee-dimesioal equatios o motio electic coductios as ollows: 1 1 e31 e15 V u w t u tt c u u u c c u c u c u c c u c c u (9a) c c u c c u c u u u c u c u c c u e e V u (9b) tt t tt c u u u c c u u c c u c u e V e V V V w u u (9c) V V V e u u u e e u u u e u V (9d) 3. Solutios o the ield equatio To obtai the popagatio o hamoic waves i pieoelectic cicula solid ba we assume the solutios o the displacemet compoets to be expessed i tems o deivatives o potetials which ae take om Paul [6]. Thus we seek the solutio o the Eqs. (9) i the om o Paul [6] ae 1 ikt 1 ikt u t e u t e V t ive ikt ikt E t E e t 1 i k E t E e i u t We a t i k E t E e (1) i k is the wave umbe is the agula equecy W whee 1 i k t 7

5 8 R. Selvamai ad E ae the displacemet potetials ad V is the electic potetials ad a is the geometical paamete o the ba. By itoducig the dimesioless quatities such as x a ka a c 44 c11 c11 c44 c13 c13 c44 c33 c33 c44 c66 c66 c44 R ad substitutig Eq.(9) i Eqs.(1) we obtai c11 1 c13 W e15 e31 V c c W e V e e e W V (11) ad c (1) 1 whee x x x x The Eq. (11) ca be witte as. c g g g e g g e W V g (13) whee g1 c33 g3 g4 11 e15 g5 e31 e15 ad g6 1 c13. Evaluatig the detemiat give i Eq. (13) we obtai a patial dieetial equatio o the om A 6 B 4 C D W V (14) whee A c 11 e15 11 B 1 c e15 e31 e15 c13e c33 c11 c13 11 c13 11 e15 c11 e13 C c c11 e31 e15 e15 { c11 1 c33 33 e31 e e31 1 c13 c1333 c33 c13 e15 } D c c Solvig the Eq. (14) we get solutios o a cicula ba as 3 A J axcos W a A J axcos i1 i i Hee i 1 3 i a 3 i1 i i i ae the oots o the algebaic equatio V b A J ax cos. (15) i1 i i i A a B a C a D. (16) The solutios coespodig to the oot a i is ot cosideed hee sice J is used whe the oots i 13 eo except o. The Bessel uctio i a J is ae

6 Fee vibatio aalysis o otatig pieoelectic ba o cicula coss sectio immesed i luid I is used whe the oots i 13 eal o complex ad the modiied Bessel uctio ae imagiay. The costats a b deied i the Eq. (15) ca be calculated om the equatios i i c a e e b c a 1 13 i i 11 i ia c33 ai e15 ia bi c13 1 ia. (17) Solvig the Eq. (1) we obtai A4J 4ax si (18) whee 4 a. I a the Bessel uctio J is eplaced by the modiied Bessel uctio I Equatios o motio o the luid I cylidical pola coodiates ad the acoustic pessue ad adial displacemet equatio o motio o a i viscid luid ae o the om Achebach [3] 1 p B u u u u (19) ad cu () tt espectively whee B is the adiabatic bulk modulus is the desity acoustic phase velocity i the luid ad 1 i a is the c B u u u u. (1) Substitutig u u 1 ad u ad seekig the solutio o Eq.(19) i the om () t i k t cos e. (3) The luid that epesets the oscillatoy wave popagatig away is give as A H ax (4) whee 5 a B i which uctio o ist kid. I 5a B B c44 1 H is the Hakel the the Hakel uctio o ist kid is to be eplaced by K whee K is the modiied Bessel uctio o the secod kid. By substitutig Eq. (3) i Eq. (19) alog with Eq. (4) the acoustic pessue o the luid ca be expessed as p A H ax e 1 i Ta 5 5 cos (5) 9

7 3 R. Selvamai 4.1. Electo mechaical couplig. The electomechaical couplig ( ) o a cylidical ba is impotat o alteatio o stuctual esposes though applied electic ields i the desig o sesos ad suace acoustic dampig wave iltes. By Testo et al. [31] the electo mechaical couplig is deied as V e V (6) V e whee V e ad V ae the Phase velocities o the wave ude electically shoted ad chage ee bouday coditios at the suace o the ba. 5. Bouday coditios ad equecy equatios I this poblem the ee vibatio o tasvesely isotopic otatig pieoelectic solid ba o cicula coss-sectio immesed i luid is cosideed. I the solid-luid iteace poblems the omal stess o the ba is equal to the egative o the pessue exeted by the luid ad the displacemet compoet i the omal diectio o the lateal suace o the cylide is equal to the displacemet o the luid i the same diectio. These coditios ae due to the cotiuity o the stesses ad displacemets o the solid ad luid boudaies. Sice the iviscid luid caot sustai shea stess the shea stess o the oute luid is equal to eo. Fo the solid-luid poblems the cotiuity coditios equie that the displacemet compoets the suace stess compoets ad electic potetial must be equal. The bouday coditios ca be witte as V u p u a. (7) i Substitutig the solutios give i the Eqs. (15) (18) ad (6) i the bouday coditio i the Eq. (7) we obtai a system o ive liea algebaic equatios as ollows: AX (8) whee A is a 5 5 matix o ukow wave amplitudes ad X is a 5 1 colum vecto o the ukow amplitude coeiciets A1 A A3 A4 A 5. The compoets o A ae deied i the Appedix A. The solutio o Eq. (8) is otivial whe the detemiat o the coeiciet o the wave amplitudes X vaishes that is A. (9) Eq. (9) is the equecy equatio o the coupled system cosistig o a tasvesely isotopic otatig pieoelectic solid cicula ba immesed i iviscid luid. 6. Numeical esults ad discussio The equecy equatio give i Eq. (8) is tascedetal i atue with ukow equecy ad wave umbe. The mateial chose o the umeical calculatio is PZT-4. The mateial popeties o PZT-4 is take om Belicout et al. [3] ae used o the umeical calculatio 1 is give below: c Nm 1 c Nm 1 c Nm 1 c Nm 1 c Nm 1 c Nm e31 5.Cm e Cm e15 1.7Cm C N m C N m 75 Kgm ad o luid the desity ad used o the umeical calculatios. 1Kgm 3 phase velocity c 15msec 1

8 Fee vibatio aalysis o otatig pieoelectic ba o cicula coss sectio immesed i luid 31 I this poblem thee ae two kids o basic idepedet modes o wave popagatio have bee cosideed amely the logitudial ad lexual modes. By choosig espectively = ad =1 we ca obtai the o-dimesioal equecies o logitudial ad lexual modes o vibatios. The dispesio cuves ae daw i Figs. -4 o wave umbe vesus the equecy o logitudial ad lexual (symmetic ad ati symmetic) modes o pieoelectic cicula ba immesed i luid. Fom the Figs. -4 it is obseved that the wave umbes ae iceased with espect to its equecies. The icease i agula velocities i all the modes is sigiicat ad the lexual modes ae gettig dispesed compaed with logitudial mode. Fig.. Dispesio o wave umbe with equecy o logitudial modes o pieoelectic otatig ba immesed i luid. Fig. 3. Dispesio o wave umbe with equecy o lexual symmetic modes o pieoelectic otatig ba immesed i luid. Fig. 4. Dispesio o wave umbe with equecy o lexual atisymmetic modes o pieoelectic otatig ba immesed i luid.

9 3 R. Selvamai A compaiso is made amog the electo mechaical couplig with espect to equecy o logitudial ad lexual (symmetic ad atisymmetic) modes o vibatio i the Figs. 5-7 espectively. Fom the Figs. 5-7 it is clea that the modes o electo mechaical couplig ae meges o a paticula peiod o equecy ate that it stats icease ad deceases. Fig. 5. Dispesio o electo mechaical couplig with equecy o logitudial modes o pieoelectic otatig ba immesed i luid. Fig. 6. Dispesio o electo mechaical couplig with equecy o lexual symmetic modes o pieoelectic otatig ba immesed i luid. Fig. 7. Dispesio o electo mechaical couplig with equecy o lexual atisymmetic modes o pieoelectic otatig ba immesed i luid. The coss-ove poits betwee the modes o electo mechaical couplig with espect to the iceasig agula velocities shows that thee is eegy tase betwee the modes o vibatios due to the exta added oce o otatig ad hostig luid.

10 Fee vibatio aalysis o otatig pieoelectic ba o cicula coss sectio immesed i luid Coclusio The popagatio o waves i a pieoelectic solid ba o cicula coss-sectio immesed i luid is discussed usig thee-dimesioal theoy o pieoelecticity. Thee displacemet potetial uctios ae itoduced to ucouple the equatios o motio electic coductio. The equecy equatio o the coupled system cosistig o ba ad luid is developed ude the assumptio o peect-slip bouday coditios at the luid-solid iteaces. The equecy equatios ae obtaied o logitudial ad lexual modes o vibatio ad ae studied umeically o PZT-4 mateial ba immesed i luid. The eect o otatio ad the hostig luid is poouced i the dispesio o wave umbe ad electomechaical couplig. Appedix A 1 a 15 H 5a 1 a c 1 c a c a e b J a c a J a i 1 3 1i i 13 i 31 i i 66 i 1 i a14 c66 1 J 4a 4a J 1 4a a 1 J a a J a i 13 i i i i a4 4a 1J 4a 4a J1 4a a5 a34 J 4a a 13 3 i ai e b 15 i J ia ia J 1 ia i a4 i bi J ia i 13 a44 a45 1 a J a a35 1 a5i H ia ia H 1 ia i a H a a H a Reeeces [1] K.F. Ga Wave Motio o Elastic Solids (Dove Publicatios New Yok 1991). [] R.W. Mose // Joual o the Acoustical Society o Ameica 6 (1954) 11. [3] H.F. Tieste Liea Pieoelectic Plate Vibatios (Pleum Pess New Yok 1969). [4] V.Z. Pato B.A. Kudyavtsev Electomagetoelasticity (Godo ad Beach New Yok 1988). [5] N.A. Shul ga // Iteatioal Applied Mechaics 38(8) (8) 953. [6] H.S. Paul M. Vekatesa // Joual o the Acoustical Society o Ameica 8(6) (1987) 1. [7] H.S. Paul M. Vekatesa // Joual o the Acoustical Society o Ameica 85(1) (1989) 17. [8] K. Nagaya // Joual o the Acoustical Society o Ameica 7(3) (1981) 763. [9] R.K.N.D. Rajapakse Y. Zhou // Smat Mateials ad Stuctues 6 (1997) 169. [1] Q. Wag // Iteatioal Joual o Solids ad Stuctues 39 () 33. [11] D.D. Ebeee R. Ramesh // Joual o the Acoustical Society o Ameica 113(4) (3) 19. [1] M. Beg P. Hagedo S. Gutschmidt // Joual o Soud ad Vibatio 74(4) 91. [13] F. Botta G. Cei // Iteatioal Joual o Solids ad Stuctues 44 (7) 61. [14] J.O. Kim J.G. Lee // Joual o Soud ad Vibatio 3 (7) 41. [15] M.J. Belie R. Solecki // Joual o the Acoustical Society o Ameica 99(4) (1996) [16] F.L. Guo R. Su // Iteatioal Joual o Solids ad Stuctues 45 (8) [17] Z. Qia F. Ji P. Li // Iteatioal Joual o Solids ad Stuctues 47 (1) [18] V. Dayal // Joual o the Acoustical Society o Ameica 93(3) (1993) 149. [19] B. Nagy // Joual o the Acoustical Society o Ameica 98(1) (1995) 454. [] F. Ahmad // Joual o the Acoustical Society o Ameica 19(3) (1) 886.

11 34 R. Selvamai [1] P. Pousamy R. Selvamai // Mateials Physics ad Mechaics 17() (13) 11. [] R. Selvamai P. Pousamy // Mateials Physics ad Mechaics 14(1) (1) 64. [3] R. Selvamai // Mateials Physics ad Mechaics 1() (14) 177. [4] X.M. Zhag // Compute Methods i Applied Mechaics ad Egieeig 191 () 7. [5] J.N. Shama D. Gove // Mechaics Reseach Commuicatios 36 (9) 715. [6] J. Waue // Joual o the Acoustical Society o Ameica 16() (1999) 66. [7] R.S. Roychoudhui S. Mukhopadhyay // Iteatioal Joual o Mathematics ad Mathematical Scieces 3(7) () 497. [8] L.I. Hua K.Y. Lam // Iteatioal Joual o Mechaical Scieces 4(5) (1998) 443. [9] Segiu C. Dagomi Mathew D. Siott S. Ee Semecigil Öde F. Tua // Joual o Soud ad Vibatio 333(5) (14) [3] J.D. Achebach Wave Popagatio i Elastic Solids (Noth-Hollad Amstedam 1973). [31] F. Testo C. Cheu N. Felix M. Lethiecq // Mateials Sciece ad Egieeig C 1 () 177. [3] D.A. Belicout D.R. Cua H. Jae Pieoelectic ad Pieomagetic Mateials ad Thei Fuctio i Tasduces. I: Physical Acoustics (Academic Pess New Yok ad Lodo 1964) Vol. 1 Pt. A.