Stress Wave propagation in Electro-Magneto-Elastic plate of arbitrary cross-sections

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1 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 1 ISSN Stress Wave propagato Electro-Mageto-Elastc plate of arbtrary cross-sectos P. Pousamy Departmet of Mathematcs Govermet Arts College, Combatore , Tamladu, INDIA. pousamypp@yahoo.com Abstract Wave propagato electro-mageto-elastc plate of arbtrary cross-sectos s studed usg Fourer Expaso Collocato Method. A mathematcal model s developed to study the wave propagato a electro-mageto-elastc plate of arbtrary crosssectos usg the three- dmesoal theory of elastcty. The frequecy equatos are obtaed from the boudary codtos, sce the boudary s rregular shape; t s dffcult to satsfy the boudary codtos alog the surface of the plate drectly. Hece, the Fourer Expaso Collocato Method s appled alog the boudary to satsfy the boudary codtos. The roots of the frequecy equatos are obtaed by usg the secat method, applcable for complex roots. The computed o-dmesoal frequeces are plotted the form of dsperso curves ad ts characterstcs are aalyzed. Keyword: Vbratos of cylders, Geeralzed thermo-elastc cylder/plate, Mechacal vbratos, Stress-stra aalyss, Electromageto-elastc materals, Pezoelectrc plate. I. Itroducto The wave propagato mageto-electro-elastc materals has gaed cosderable mportace sce last decade. The electromageto-elastc materals exhbt a desrable couplg effect betwee electrc ad magetc felds, whch are useful smart structure applcatos. These materals have the capacty to covert oe form of eergy amely, magetc, electrc ad mechacal eergy to aother form of eergy. The composte cosstg of pezoelectrc ad pezomagetc compoets have foud creasg applcato egeerg structures, partcularly smart/tellget structure system. The mageto-electro-elastc materals are used as magetc feld probes, electrc packg, acoustc, hydrophoes, medcal, ultrasoc mage processg, sesors ad actuators wth the resposblty of magetc-electro-mechacal eergy coverso. A method, for solvg wave propagato arbtrary ad polygoal cross-sectoal plates ad to fd out the phase veloctes dfferet modes of vbratos amely logtudal, torsoal ad flexural, by costructg frequecy equatos was devsed by IJSER 1

2 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 ISSN Nagaya [1-3]. He formulated the Fourer expaso collocato method for ths purpose ad the same method s used ths problem.the three-dmesoal behavor of magetoelectroelastc lamates uder smple support has bee studed by Pa [4] ad Pa ad Heylger [5]. A exact soluto for magetoelectroelastc lamates cyldrcal bedg has also bee obtaed by Pa ad Heylger [6]. Pa ad Ha [7] studed the exact soluto for fuctoally graded ad layered mageto-electro-elastc plates. Feg ad Pa [8] dscussed the dyamc fracture behavor of a teral terfacal crack betwee two dssmlar mageto-electroelastc plates. Buchaa [9] developed the free vbrato of a fte mageto-electro-elastc cylder. Da ad Wag [1,11] have studed thermo-electro-elastc traset resposes pezoelectrc hollow structures ad hollow cylder subjected to complex loadgs. Later Kog et al [1] preseted the thermo-mageto-dyamc stresses ad perturbato of magetc feld vector a ohomogeeous hollow cylder. Ager et al [13-15], studed respectvely, the free vbrato of clamped-clamped mageto-electroelastc cyldrcal shells, free vbrato behavor of multphase ad layered mageto-electro-elastc beam, free vbratos of smply supported layered ad multphase mageto-electro-elastc cyldrcal shells. Ho et al [16] aalyzed a pot heat source o the surface of a sem-fte trasversely sotropc electro-mageto-thermo-elastc materals. Sharma ad Mohder Pal [17] developed the Raylegh-Lamb waves mageto-thermo-elastc homogeeous sotropc plate. Later Sharma ad Thakur [18] studed the effect of rotato o Raylegh-Lamb waves mageto-thermo-elastc meda. Gao ad Noda [19] preseted the thermal-duced terfacal crackg of magetoelectroelastc materals. B et al [] studed the wave propagato o-homogeeous mageto-electroelastc plates. Pousamy [1-3] have studed the wave propagato geeralzed thermo-elastc cylder of arbtrary cross secto, thermoelastc ad geeralzed thermo elastc plates of arbtrary ad polygoal cross-sectos respectvely. Pousamy ad Rajagopal [4,5] have studed, the wave propagato a geeralzed thermo elastc sold cylder of arbtrary cross-secto ad a homogeeous trasversely sotropc thermo elastc sold cylder of polygoal cross-sectos respectvely usg the Fourer expaso collocato method.. Formulato of the Problem We cosder a homogeeous trasversely sotropc mageto-electro-elastc plate of arbtrary cross-sectos. The system dsplacemets ad stresses are defed by the cyldrcal co-ordates r, ad z. The goverg equatos of moto, electrc ad magetc coducto equato the absece of body force are IJSER 1

3 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 3 ISSN r r u rr, r r, rz, z rr, tt 1 1 r r v 1 1 r, r, z, z r, tt r r w. (1) 1 1 rz, r z, zz, z rz, tt The electrc coducto equato s D r D r D D. () 1 1 r, r, r, z, z The Magetc coducto equato s B r B r B B. (3) 1 1 r, r r, z, z Where c e c e c e e E q H rr 11 rr 1 13 zz 31 z 31 z c e c e c e e E q H 1 rr zz 31 z 31 z c e c e c e e E q H zz 13 rr 13 zz z z c e r 66 r c e e E q H z 44 z c e e E q H (4) rz 44 rz 15 r 15 r D e e E m H r 15 rz 11 r 11 r D e e E m H 15 z D e e e e e E m H (5) z 31 rr zz z z ad IJSER 1

4 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 4 ISSN B q e m E H r 15 rz 11 r 11 r B q e m E H 15 z B q e e q e m E H. (6) z 31 rr zz z z Where rr,, r, rz, z are the stress compoets, 11, 1, 13,, 44 are the delectrc costats, 11, c c c c c ad c c c are elastc costats, 11, are the magetc permeablty coeffcets, e31, e, e15 are the pezoelectrc materal coeffcets, q31, q, q15 are the pezomagetc materal coeffcets, m11, m are the mageto-electrc materal coeffcets, s the mass desty of the materal, Dr, D ad compoets. D are the electrc dsplacemets, B, B ad z r Bz are the magetc dsplacemets The stra ej are related to the dsplacemets correspodg to the cyldrcal coordates r,, z are gve by e rr u, r, e 1 r v, u, ezz w, z 1 ez v r w , z,, er r u, v, r r v, erz u, z w, r 1 (7) where u, v ad w are the mechacal dsplacemets alog the radal, crcumferetal ad axal drectos respectvely. The Electrc feld vector E, r,, z s related to the electrc potetal E as E r E, r E 1 E ad E,z r E (8) z Smlarly, the magetc feld vector H, r,, z s related to the magetc potetal H as H r H, r H 1 H ad H z r H (9) z Substtutg the Eqs. (4)-(9) the Eqs. (1)-(3), we obta the set of dsplacemet equatos as follows; IJSER 1

5 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 5 ISSN rr r zz r c u r u r u c r u c u c c r v c c r v ,, 66, 44, 66 1, 11 66, c c w e e E q q H u 44 13, rz 31 15, rz 31 15, rz, tt c c r u c c r u c v r v r v c v c r v , r 11 66, 66, rr, r 44, zz 11, c c r w e e r E q q r H v , z 31 15, z 31 15, z, tt c c u r u r v c w r w w c w e E 44 13, rz, z, z 44, rr, r,, zz, zz q H e E r E r E q H r H r H w, zz 15, rr, r, 15, rr, r,, tt zz rr r rr r e w r w r w e e u r u r v e w E 15, rr, r, 31 15, rz, z, z, zz, zz m H E r E r E m H r H r H, 11,,, 11,,, H zz m E rr r E r r E H rr r H r r H q w r w r w q q u r u r v q w m E 15, rr, r, 31 15, rz, z, z, zz, zz, 11,,, 11,,, (1a) (1b) (1c) (1d) (1e) 3. Soluto of the Problem The Eq. (1) s a coupled partal dfferetal equato wth three dsplacemets ad magetc ad electrc coducto compoets. To ucouple the Eq. (1), we follow Sharma ad Sharma [6] ad seek the solutos the followg form 1 1,,,, r u r r r 1 1,,,, r v r r r W W W, z z, E E E, z z,, z z, (11) H H H IJSER 1

6 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 6 ISSN where 1 for, 1 for 1,,, W r,, E r,, H, r, potetals for the symmetrc mode ad r,, r,, W r,, E r, ad H, potetals for the at symmetrc mode of vbratos. r, r are the dsplacemet r are the dsplacemet Substtutg the Eq.(11) (1), we get W E H c11 1 c44 c13 c44 e31 e15 q31 q15 z t z z z W E H c11 1 c44 c13 c44 e31 e15 q31 q15 z t z z z (1a) (1b) c c W c c e e E z t z z q q H z 15 1 (1c) e e W e e E m m H z z z z q q W q q m m E H z z z (1d) (1e) ad c z t 661 c44 (13) where 1 1 r r r r 1 IJSER 1

7 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 7 ISSN The Eq. (13) gves purely trasverse wave, whch s ot affected by the electrc ad magetc feld. Ths wave s polarzed the plaes perpedcular to the z-axs ad t may be referred as the smple harmoc wave. We assume that the dsturbace s tme harmoc through the factor t e, s agular velocty ad hece, the system of Eqs. (1a)-(1e) becomes W E H c c c c e e q q z z z z W E H c c c c e e q q z z z z (14a) (14b) c c W c c e e E z z z q q H z 15 1 (14c) e e W e e E m m H z z z z (14d) q q W q q m m E H z z z z (14e) We cosder the free vbrato of arbtrary cross-sectoal plate, so we assume that r,, z, t r cos m z L cos W r,, z, t W r s m z L cos c 44 Er,, z, t Ers m z Lcos q c 44 Hr,, z, t Hrs m z Lcos q (15) IJSER 1

8 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 8 ISSN ad r,, z, t r cos m z L s (16) Itroducg the dmesoless quattes such as r x a, tl a, m L c j c j, c 44 e j e j, e q j q j, q a c, 44 m j mc j 44, q e c jc44 j, j, L s the legth of the plate ad usg the Eqs. (15) ad (16) the Eqs. (14) ad (13), we get e j 44 q c t 1 c t W e e t E q q t H L L L L c t W 1 c t e t E q t H L L L L e t W e e t t E m t m H L L L L q t W q q t m t m E t H (17) 15 L L L L ad 66 L c t (18) 1 Where x x x x. The Eq. (17) s a homogeeous lear equato whch has a trval soluto to obta the o-trval soluto, the determat of the coeffcet matrx s equal to zero. Thus we get IJSER 1

9 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 9 ISSN c g g t g t g t 11 1 L 3 L 4 L L L g t g e t q t L 5 L L g t e t g g m 3 L 6 7 g t q t g m g 4 L 7 8, W, E, H (19) where g t, g c13 1 L, g3 e31 e15, g4 q31 q15, g5 ctl 1, g t, 6 L g m t, 7 L g t 8 L Evaluatg the determat gve Eq. (19), we obta the partal dfferetal equato of the form A B C D E W E H,,, () where A c q 15 11e15 m11 m11 e15 q 15 IJSER 1

10 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 1 ISSN B c11{ g6 11 g8 11 g7 m11 g m 11 e15 g8e15 tl 11 m q 15 g3m11 g4 11 q 15 g6 q15 tl m g7 e q15 } g1 q15 e m m e q 15 gtl g 1111 m 11 e15 g3 11 g4 m11 g3tl g 11e15 m11 q 15 g3 11 g4 m11 q 15 g3q15 g4e15 g4tl g m11e15 11q 15 g3m11 g411 e15 g3q15 g4e15 11 L C c11{ g6g8 g7 g 5 g6 11 g811 g7m11 t L e15 g8 g7 q15 g7 g 8 4 t L 11 m11 11 } g1{ g6 11 g8 11 g7 m11 g m 11 e15 g8e15 tl m11 q 15 g6 q15 t m11 11 g7 e15 q15} gt L{ g g6 11 g811 g7 m11 15 e g3g8 g4g7 q15 g3g7 g4g6 t g g m g m g } g t { g g e g q t m g g L L L g g g g g m q t g g t g q g e } L L g4tl{ g g7 e15 g6 q15 tl m11 11 g3g7 g4g6 g5 g3m11 g411 4 L L{ L 4 e t g g t g q g e 15 L 3 4 L } D c g g g g t g g g g t g t g g g g g g g t g g } g { g g g g g g g m L t L e15 g8 g7 q15 g7 g 8 t L 11 m11 11 } g t g g g g t g g g g g g gg 4 6 L L g t g t g g g g g g g t g g 4 3 L L L 3 4 E g g g g g t g g g L Solvg the Eq. (), the soluto for the symmetrc mode obtaed as 4 1 A J r cos 4 1 W a A J r cos 4 1 E b A J r cos IJSER 1

11 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 11 ISSN H c A J r cos (1a) The solutos to the at symmetrc modes of vbratos, (1a), we get W, E, H are obtaed by chagg cos by s the Eq. 4 1 A J r s 4 1 W a A J r s 4 E b AJ rs H c AJ r s. (1b) where J s the Bessel fucto of frst kd of order.the costats a, b ad c defed the Eq. (1) s calculated usg the followg equatos g t a g t b g t c c g L 3 L 4 L g a e t b q t c g t 5 L L L e t a g b g m c g t L L q t a g m b g c g t () 15 L L Solvg the Eq. (), we obta IJSER 1

12 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 1 ISSN L g3 g7 m g4 g6 a g g g e t 6 3 b c g g t c g g m t g g g e t 3 4 L 1 7 L 6 3 L c11 g1 e15 tl gg3tl L L t g g g e t Solvg the Eq. (13), we obta the soluto for symmetrc mode as A5J 5r s (3a) ad the soluto for at symmetrc mode s obtaed by chagg s by cos the Eq. (3a), we get A5J 5r cos (3b) Where J s the Bessel fuctos of frst kd of order, ad 5 tl c66. If 1,,3,4,5, the the a Bessel fucto J s replaced by the modfed Bessel fucto I. 4. Boudary codtos ad frequecy Equatos I ths problem, the vbrato of arbtrary cross-sectoal plate s cosdered. Sce the boudary s rregular shape, t s dffcult to satsfy the boudary codtos alog the surface of the plate drectly. Hece, the Fourer expaso collocato method s appled to satsfy the boudary codtos. For the plate, the ormal stress xx ad shearg stresses,, the electrc feld D x ad the xy xz magetc feld Bx s equal to zero for stress free boudary. Thus the followg types of boudary codtos are assumed for the plate of arbtrary cross-secto s IJSER 1

13 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 13 ISSN D B (4) xx xy xz x x where s the value at the th segmet of the boudary, f the agle betwee the ormal to the segmet ad the referece axs s assumed to be costat, the the trasformed expresso for the stresses are gve by 1 cos s s cos 1 c c u r c c u v xx 11 1, r 11 1, c r v u v s c W e E q H 66,, r 13, z 31, z 31, z 1 1 c u r v u s r u v v cos xy 66, r,,, r 1 c u W cos v r W s e E q H xz 44, z, r, z, 15, r 15, r D e u W E m H x 15, z, r 11, r 11, r B q u W m E H (5) x 15, z, r 11, r 11, r Substtutg the Eqs. (1) ad (3) the Eq. (4) ad performg the Fourer seres expaso to Eq.(4) alog the boudary as dscussed Pousamy [1-3], the boudary codto alog the boudary of the surfaces are expaded the form of double Fourer seres. Whe the plate s symmetrc about more tha oe axs, the boudary codtos the case of symmetrc mode ca be wrtte the form of a matrx as gve below: E E E E E1 E N E1 EN E1 EN E1 EN E1 EN A1 A EN EN EN EN EN1 ENN EN1 ENN EN1 ENN EN1 ENN EN1 E A NN F F F F F1 F N F1 F N F1 F N F1 F N F1 F 5 N A4 A FN FN FN FN FN 1 FNN FN 1 FNN FN 1 FNN FN 1 FNN FN 1 FNN A G G G G G1 G N G1 G N G1 G N G1 G N G1 G N A1 N GN GN GN GN GN1 GNN GN1 GNN GN1 GNN GN1 GNN GN1 GNN A H H H H H1 HN H1 HN H1 H N H 1 H N H 1 H N A N H N H N H N H N H N1 H NN H N1 H NN H N1 H NN H N1 H NN H N1 H NN I I I I I1 IN I1 IN I1 I A N I1 IN I1 IN I N I N I N I N I A N1 I NN I N1 I NN I N1 I NN I N1 I NN I N1 I NN 5N (6) where IJSER 1

14 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 14 ISSN I j Em e R m d j j j, cos, m I F f R, sm d 1 I j Gm g R m d I H h R, cos m d, j j j, cos, m I j j m 1 1 I R, cos m d (7) 1 The coeffcets e are gve the Appedx A. Smlarly, the matrx for the at symmetrc mode s obtaed as E1 E11 E1N E11 E1N E11 E1N E11 E1N E11 E 1N A A E N E N1 E NN E N1 E NN E N1 E NN E N1 E NN E N1 E NN F1 F11 F1N F11 F1N F11 F1N F11 F1N F11 F A1 1N A F N F N1 F NN F N1 F NN F N1 F NN F N1 F NN F N1 F NN G1 G11 G1N G11 G1N G11 G1N G11 G1N G11 G A 1N A G N G N1 G NN G N1 G NN G N1 G NN G N1 G NN G N1 G NN A H 1 H 11 H 1N H 11 H 1N H 11 H 1N H 11 H 1N H 11 H 1N A H N H N1 H NN H N1 H NN H N1 H NN H N1 H NN H N1 H NN A I 1 I 11 I 1N I 11 I 1N I 11 I 1N I 11 I 1N I 11 I 1N A I N I N1 I NN I N1 I NN I N1 I NN I N1 I NN I N1 I NN 5 A 5 11 N 1 N 31 N 41 N 51 N (8) where I j E m e R m d j j j, s, m I F f R, cos m d 1 I j Gm g R m d j j j, s, m I H h R, s m d 1 IJSER 1

15 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 15 ISSN I j j m 1 1 I R, s m d (9) where j=1,,3,4 ad 5, I s the umber of segmets, R s the coordate r at the boudary ad N s the umber of trucato of the Fourer seres. The frequecy equatos are obtaed by trucatg the seres to N+1 terms, ad equatg the determat of the coeffcets of the ampltude A ad A (=1,,3,4 ad 5), for symmetrc ad at symmetrc modes of vbratos. 5. Sold crcular Plate The frequecy equato for sold crcular plate ca be wrtte the form A (3) where A s the 5 5 matrx wth elemets, 1,,3,4,5 a j are gve by j a [ c ( 1) J ax ax J ax x [ c a t c a e b q c J ax ], 1,,3, L a15 c66 ( 1) J 5ax 5ax J 1 5ax a 1 J ax ax J ax, 1,,3,4 1 1 a5 c66 J 5ax 5ax J 1 5ax 5ax J ax 15 a3 tl a e b q15 c J ax ax J 1 ax, 1,,3, 4 a t J ax 35 L 5 a e t a b m c J ax ax J ax 4 15 L , 1,,3,4 a e t J a ax L a5 q15 tl a m b c J ax ax J 1 ax, 1,,3, 4. IJSER 1

16 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 16 ISSN a q t J a ax L 5 6. Numercal results ad Dscussos The electro-magetc materal costats based o graphcal results of Aboud, 1[7] used for the umercal calculatos s gve the Table 1. Table. 1 The materal propertes of the electro-magetc materal based o graphcal results of Aboud [7] compostes c 11 c 1 c 13 c c 44 c e 15 e 31 e q 15 q 31 q m 11 m Uts: /, 1 /, / 6 9 /, 1 /, 1 / c N m C Vm e C m j j j q N Am Ns C m Ns VC j j j I the umercal calculato, the agle s take as a depedet varable ad the coordate R at the th segmet of the boudary s expressed terms of. Substtutg R ad the agle, betwee the referece axs ad the ormal to the th boudary le, the tegratos of the Fourer coeffcets e, f, g, h,, e, f, g, h ad ca be expressed terms of the agle. Usg these coeffcets to the Eqs. (7) ad (9), the frequeces are obtaed for electro-mageto-elastc plate of arbtrary cross-sectos. I the preset problem, there are two kds of basc depedet modes of wave propagato have bee cosdered, amely, the logtudal ad flexural at symmetrc modes of vbratos. 6.1 Ellptc cross-secto The ellptc cross secto of a plate s show Fgure 1 ad ts geometrc relatos used for umercal calculatos gve below are due to Nagaya [] as, s 1 cos R b a b a b 1 ba ta ta,for, IJSER 1

17 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 17 ISSN , for 1 ta ba ta, for. (31), where a s the semmajor axs ad b s the semmor axs of the ellptc plate ad boudary, 1, s the agle betwee the ormal to the segmet ad the referece axs at the th boudary. R s the coordate r at the Fgure 1 Ellptc cross-sectoal plate 6. Parabolc cross-secto The geometry of the parabolc cross secto s gve Fgure. The geometrc relatos of the parabolc cross secto gve by Nagaya [1] are as follows: 1 R c e c cos cos c e s s ta s 1 e c 1 c 1 ec R ta for the parabolc curve, ad (3a) R c 1 cos, l where for the straght le boudary (3b) 1, R s the coordate r at the boudary ad th boudary. The parameters e ad c used Eq. (3) are defed Fgure. s the agle betwee the ormal to the segmet ad the referece axs at IJSER 1

18 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 18 ISSN Fgure Parabolc cross-secto 6.3 Logtudal mode The geometrcal relatos for the ellptc cross-sectos gve Eq. (31) are used drectly for the umercal calculatos, ad three kds of basc depedet modes of wave propagato are studed. I case of the logtudal mode of ellptcal crosssecto, the cross-secto vbrates alog the axs of the plate, so that the vbrato ad dsplacemets the cross-secto s symmetrcal about both major ad mor axes. Hece, the frequecy equato s obtaed by choosg both terms of ad m as,,4,6,... Eq. (6) for the umercal calculatos. Durg the logtudal moto of parabolc ad cardodal cross-sectos, the vbrato ad dsplacemets are symmetrcal about the major axs ad hece the frequecy equato s obtaed from Eq. (6) by takg m,,1,,3.... Sce the boudary of the cross-sectos amely, ellptc, cardod ad parabolc are rregular shape, t s dffcult to satsfy the boudary codtos alog the curved surface, ad hece Fourer expaso collocato method s appled. I ths method, the curved surface, the rage ad s dvded to segmets, such that the dstace betwee ay two segmets s eglgble ad the tegratos s performed for each segmet umercally by usg the Gauss fve pot formula.the o-dmesoal frequeces are computed for 1., usg the b-secto method (applcable for the complex roots Ata [8]) 6. 4 Flexural mode I the case of flexural mode of ellptcal cross-secto, the vbrato ad dsplacemets are at symmetrcal about the major axs ad symmetrcal about the mor axs. Hece, the frequecy equatos are obtaed from Eq. (8) by choosg m, 1,3,5,.... Sce the vbrato ad dsplacemets are at symmetrcal about the major axs for the parabolc ad cardodal cross-sectoal plates, the frequecy equato s obtaed by takg m, 1,,3... Eq. (8). IJSER 1

19 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 19 ISSN The geometrc relatos for the ellptc ad parabolc cross-sectos are gve respectvely the Eqs. (31), (3) ad for the cardods cross-sectoal plate, the geometrc relatos are cosdered from the Eqs. (4) ad (6) of the referece Nagaya [3] are used for the umercal calculatos. The otatos amely, S1, S, S 3... ad A1, A, A 3... used the graphs ad Table respectvely represets the symmetrc ad at symmetrc modes vbrato, ad the subscrpts 1,, 3 etc.. represets the frst, secod, thrd, fourth ad ffth modes vbratos. The frequecy equato for the sold crcular plate s obtaed by exact method s gve Eq. (3), from the equato., the frequeces are obtaed by exact method are used to compare the results of the preset method. The dmesoless frequeces are computed for 1. for dfferet aspect ratos a/b=1., 1.5 ad. for logtudal ad flexural at symmetrc modes of vbratos usg b-secto method. The frequeces obtaed for the logtudal ad flexural at symmetrc modes of vbrato by the exact method s matches well wth the frequeces obtaed for the aspect rato a/b=1. for a ellptc cross-sectoal plate s gve the Table.The o-dmesoal frequecy of ellptc cross secto for the aspect rato a/b=1. wll represet a crcular plate. Sce the frequecy for the ellptc cross secto for the aspect rato a/b=1. matches wth the frequecy obtaed by the exact method s show the Table. So the problem s exteded for ellptc, cardods ad parabolc cross sectoal plates. A graph s draw betwee the geometrc rato L/a=.5 wth the aspect rato a/b=1.5 versus o-dmesoal frequecy for a logtudal mode of electro-mageto-elastc plate of ellptc cross-sectoal plate s show Fg.3. From the Fg.3, t s observed that the o-dmesoal frequecy creases for dfferet modes of vbratos. The smlar behavor s observed for a flexural at symmetrc modes of ellptc cross-sectoal electro-mageto-elastc plate s show Fg.4. A dsperso curve s draw betwee the geometrcal rato L/a=.5 wth the aspect rato a/b=.5, 1., 1.5 ad. versus the o-dmesoal frequecy for a logtudal modes of electro-mageto-elastc plate of ellptc cross-sectoal plate s show Fg.5. From the Fg. 5, t s observed that the o-dmesoal frequeces are decreases by creasg the aspect ratos, ths s the proper physcal behavor of plate wth respect to ts aspect ratos. The Fgs.6 ad 7 respectvely represets the relato betwee the geometrc rato L/a=.5 wth geometrc parameter s=.,.5 ad the o-dmesoal frequecy for a logtudal ad flexural atsymmetrc modes of a cardod cross-sectoal plate. From the Fgs.6 ad 7 t s observed that the o-dmesoal frequecy creases learly wth respect to the Geometrc rato L/a both logtudal ad flexural atsymmetrc modes of vbratos. The cross-over pots show the logtudal modes of cardodc cross sectoal plates represets the trasfer of electro-magetc eergy betwee the modes of vbrato. IJSER 1

20 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 ISSN A graph s draw betwee the geometrc rato L/a=.5 wth e/c=1.5 versus o-dmesoal frequecy for a logtudal mode of electro-mageto-elastc plate of parabolc cross-sectoal plate s show Fg.8. From the Fg.8, t s observed that the o-dmesoal frequecy creases frst ad the t starts to decreases for a partcular perod. The cross-over pots betwee the waves represets the trasfer of electro-magetc eergy betwee the modes of vbrato. The smlar behavor s observed for a flexural atsymmetrc modes of parabolc cross-sectoal electro-mageto-elastc plate s show Fg Coclusos I ths paper, the wave propagato a electro-mageto-elastc plate of arbtrary cross secto are aalyzed by satsfyg the boudary codtos o the rregular boudary usg the Fourer expaso collocato method ad the frequecy equatos for the logtudal ad flexural at symmetrc modes of vbratos are obtaed. Numercally the frequecy equatos are aalyzed for the plate of dfferet cross-sectos such as ellptc, cardods ad parabolc cross sectoal plates. The computed dmesoless frequeces are plotted graphs for logtudal ad flexural at symmetrc modes of vbratos. The problem ca be aalyzed for ay other cross-secto by usg the proper geometrc relato. 8. Ackowledgemet Dr. P. Pousamy s thakful to Uversty Grats Commsso, New Delh, for fudg to udertake ths research work, Ref. F. No / 1 (SR), ad the Drectorate of Collegate Educato, Taml Nadu, for the permsso redered towards the same. Hs grattude also exteds to Govt. Arts College (Autoomous), Combatore-18, for provdg wth the facltes to take up ths work. Appedx A e [ c cos ( 1) J ax ax J ax 66 1 [ cos s ]cos 11 1 x a c c t c a e b q c J ax t cos L L L c ( 1) J ax ax J ax s cos t s, 1,, 3, (A1) L e [c cos 1 J a ax a ax J a ax cos t cos L c [ 1 J a ax a ax J a ax a ax J a ax ]cos t s s (A) 1 1 L f 1 J ax ax J ax ax J ax cos t cos s L 1 J ax ax J ax cos t s cos, 1,, 3, 4. (A3) IJSER 1

21 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 1 ISSN J a5ax a5ax J 1 a5ax a5ax J a5ax costl s cos f 1 J a ax a ax J a ax cos t cos s 5 L (A4) t a J axs t s s, 1,, 3, 4. g t a cos e b q c J ax ax J ax s t cos L L L L 5 s cos cos J a5ax a5ax J 1 a5axt L s tl s s g t J a ax t 5 L L (A5) (A6) h s cos, 1,,3, 4 e tl a b m c J ax ax J 1 ax tl (A7) h e t J a ax 5 15 L 5 (A8) q15 tl a m b c J ax ax J ax, 1,,3, 4. (A9) q t J a ax 5 15 L 5 (A1) Refereces [1] K. Nagaya, Method for solvg vbrato problems of plate wth arbtrary shape, J. of Acoust. Soc. of Am. 66(6) (198) 9-. [] K. Nagaya, Stress wave propagato a bar of arbtrary cross-secto, Trasacto of the ASME 49 (198) [3] K. Nagaya, Drect Method o the Determato of Ege frequeces of Arbtrary shaped plates, Trasacto of the ASME 15 (1983) [4] E. Pa, Exact soluto for Smply Supported ad Multlayered Mageto-electro-elastc plates, Trasactos of the ASME 68 ( ) [5] E. Pa ad P.R. Heylger, Free vbrato of Smply Supported ad Multlayered Mageto-electro-elastc Plates, J. of Soud ad Vb. 5 () [6] E. Pa ad P. R. Heylger, Exact solutos for Mageto-electro-elastc Lamates cyldrcal bedg, It. J. of Sold ad Struct. 4 (5) IJSER 1

22 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 ISSN [7] E. Pa ad F. Ha, Exact soluto for fuctoally graded ad layered mageto-electro-elastc plates, It. J. of Egg. Sc. 43 (5) 1-9. [8] W. J. Feg ad E. Pa, Dyamc fracture behavor of a teral terfacal crack betwee two dssmlar mageto-electroelastc plates, J. of Egg. Fracture Mech. 75 (8) [9] G. R. Buchaa, Free vbrato of a fte mageto-electro-elastc cylder, J. of Soud ad Vb. 68 (3) [1] H.L. Da ad X. Wag, Thermo-electro-elastc traset resposes pezoelectrc hollow structures, It. J. of Sold ad Struct. 4 (5) [11] H.L. Da ad X. Wag, Mageto-thermo-electro-elastc traset respose a pezoelectrc hollow cylder subjected to complex loadgs, It. J. of Sold ad Struct. 43 (6) [1] T. Kog, D. X. L ad X. Wag, Thermo-mageto-dyamc stresses ad perturbato of magetc feld vector ohomgeeous hollow cylder, Appl. Mathematcal Modelg, (9) [13] A. R. Ager, N. Gaesa, ad S. Swarama, Free vbrato of clamped-clamped mageto-electro-elastc cyldrcal shells, J. of Soud ad Vb. 9 (1) [14] A.R. Ager, N. Gaesa, ad S. Swarama, Free vbrato behavor of multphase ad layered mageto-electro-elastc beam, J. of Soud ad Vb. 99 (7) [15] A. R. Ager, N. Gaesa, ad S. Swarama, Free vbratos of smply supported layered ad multphase mageto-electroelastc cyldrcal shells, J. Smart Materal ad Struct. 15 (6) [16] P.F. Ho, A.Y. Leug, ad H.F. Dg, A pot heat source o the surface of a sem-fte trasversely sotropc electromageto-thermo-elastc materal, It. J. of Egg. Sc. 46 (8) [17] J.N. Sharma ad Mohder Pal, Raylegh-Lamb waves mageto-thermoelastc homogeeous sotropc plate, It. J. of Egg. Sc. 4 (4) [18] J.N. Sharma ad M.D. Thakur, Effect of rotato o Raylegh-Lamb waves mageto-thermoelastc meda, J. of Soud ad Vb. 96 (6) IJSER 1

23 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 3 ISSN [19] G. F. Gao ad N. Noda, Thermal-duced terfacal crackg of magetoelectroelastc materals, It. J. of Egg. Sc. 4 (4) [] W. B, Y. Jagog ad H. Cufu, Wave propagato o-homogeeous mageto-electro-elastc plates, J. of Soud ad Vb. 317 (8)5-64. [1] P. Pousamy, Wave propagato a geeralzed thermo elastc sold cylder of arbtrary cross- secto, It. J. of Sold Struct. 44 (7) [] P. Pousamy, Wave propagato thermo-elastc plate of arbtrary cross-sectos, Multdscple Modelg Materals ad Struct. 7 (11) [3] P. Pousamy, Dsperso aalyss of geeralzed thermo elastc plate of polygoal cross-sectos cross-sectos, J. of Appl. Mathematcal Modelg 36 (1) [4] P. Pousamy ad M. Rajagopal, Wave propagato a homogeeous trasversely sotropc thermo elastc sold cylder of polygoal cross-secto, J. of Vb. ad cotrol 16 (5) (1) [5] P. Pousamy ad M. Rajagopal, Wave propagato a trasversely sotropc thermo elastc sold cylder of arbtrary cross-secto, J. of Acta Mechaca Solda Sca 4(6) (11) [6] J.N Sharma ad P. K. Sharma, Free vbrato aalyss of homogeeous trasversely sotropc thermo elastc cyldrcal pael, J.of Thermal Stresses 5 () [7] J. Aboud, Mcromechacal aalyss of fully coupled electro-mageto-thermo-elastc multphase compostes, J. Smart Materal ad Struct. 1 (1) [8] H. M. Ata, Numercal Methods for Scetsts ad Egeers, Hdusta Book Agecy, New Delh,. Table. 1 The materal propertes of the electro-magetc materal based o graphcal results of Aboud[7] compostes c 11 c 1 c 13 c c 44 c IJSER 1

24 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 4 ISSN e 15 e 31 e q 15 q 31 q m 11 m Uts: /, 1 /, / 6 9 /, 1 /, 1 / c N m C Vm e C m j j j q N Am Ns C m Ns VC j j j Table.Comparso betwee the frequeces Ω obtaed from the exact ad preset methods of logtudal ad flexural at symmetrc modes of vbratos Aspect rato a/b Mode Exact Method Preset Method Preset Method Preset Method S Logtudal S Mode S S S A Flexural A at symmetrc A Mode A A IJSER 1

25 Dmesoless frequecy Ω Dmesoless frequecy Ω Dmesoless frequecy Ω Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 5 ISSN S1 S S Aspect rato a/b=1.5, L/a=.5 Fg A1 A A a/b=.5 a/b=1. a/b=1.5.. A Fg. 4.5 IJSER Aspect rato a/b=.5, 1., 1.5,., L/a=.5

26 Dmesoless frequecy Ω Dmesoless frequecy Ω Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 6 ISSN Fg S=. S= Geometrc parameter s=.,.5, L/a=.5 Fg S=. S= Geometrc parameter s =.,.5, L/a=.5 Fg.7 IJSER 1

27 Dmesoless frequecy Ω Dmesoless frequecy Ω Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 7 ISSN S1 S3 S S Aspect rato e/c=1.5, L/a=1. Fg A1 A A3 A Aspect rato e/c=1.5, L/a=1. Fg.9 Table Capto Table. 1 The materal propertes of the electro-magetc materal based o graphcal results of Aboud[7] compostes Table.Comparso betwee the frequeces Ω obtaed from the exact ad preset methods of logtudal ad flexural at symmetrc modes of vbratos Fgure Captos Fg. 3 Aspect rato a/b=1.5, L/a=.5 versus dmesoless frequecy Ω for logtudal modes of ellptc cross-sectoal plate IJSER 1

28 Iteratoal Joural of Scetfc & Egeerg Research Volume 3, Issue 8, August-1 8 ISSN Fg. 4 Aspect rato a/b=1.5, L/a=.5 versus dmesoless frequecy Ω of flexural atsymmetrc modes of ellptc crosssectoal plate Fg. 5 Aspect rato a/b=.5, 1., 1.5,., L/a=.5 versus dmesoless frequecy Ω for logtudal modes of ellptc cross-sectoal plate Fg.6 Geometrc parameter s =.,.5, L/a=.5 versus dmesoless frequecy Ω for logtudal modes of cardodal cross-sectoal plate Fg.7 Geometrc parameter s =.,.5, L/a=.5 versus dmesoless frequecy Ω for flexural at symmetrc modes of cardodal cross-sectoal plate Fg. 8 Aspect rato e/c=1.5 wth L/a=1. versus dmesoless frequecy Ω of logtudal modes of parabolc crosssectoal plate Fg. 9 Aspect rato e/c=1.5 wth L/a=1. versus dmesoless frequecy Ω of flexural atsymmetrc modes of parabolc cross-sectoal plate IJSER 1

Modeling and analysis of waves in a heat conducting thermo-elastic plate of elliptical shape

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