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

Transcription

1 589 Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape Abstract Wave propagato heat coductg thermo elastc plate of ellptcal cross-secto s studed usg the Fourer expaso collocato method based o Suhub s geeralzed theory. The equatos of moto based o two-dmesoal theory of elastcty s appled uder the plae stra assumpto of geeralzed thermo elastc plate of ellptcal cross-sectos composed of homogeeous sotropc materal. The frequecy equatos are obtaed by usg the boudary codtos alog outer ad er surface of ellptcal cross-sectoal plate usg Fourer expaso collocato method. The computed o-dmesoal frequecy, velocty ad qualty factor are plotted dsperso curves for logtudal ad flexural (symmetrc ad atsymmetrc) modes of vbratos. Keywords Wave propagato plate; vbrato of thermal plate; Fourer collocato method; geeralzed thermo elastc plate; thermal relaxato tmes. R. Selvama a P. Pousamy b a Departmet of Mathematcs, Karuya Uversty, Combatore-64 4, Taml Nadu, Ida. b Departmets of Mathematcs, Govt Arts College, Combatore, 64 08, Taml Nadu, Ida. a Correspodg author: selvam79@gmal.com Receved.0.04 I revsed form Accepted Avalable ole INTRODUCTION The plates of ellptcal cross-sectos are frequetly used as structural compoets ad ther vbrato characterstcs are mportat for practcal desg. The propagato of waves thermoelastc materal has may applcatos varous felds of scece ad techology, amely, atomc physcs, dustral egeerg, thermal power plats, submare structures, pressure vessel, aerospace, chemcal ppes, ad metallurgy. The mportace of thermal stresses causg structural damages ad chages fuctog of the structure s well recogzed wheever thermal stress evromets are volved. A method, for solvg wave propagato doubly coected arbtrary ad polygoal crosssectoal plates ad to fd out the phase veloctes dfferet modes of vbratos amely logtudal, torsoal ad flexural, by costructg frequecy equatos was devsed by Nagaya (98a; 98b; 983a; 983b; 983c). He formulated the Fourer expaso collocato method for

2 590 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape ths purpose, the same method s adopted ths problem. The geeralzed theory of thermo elastcty was developed by Lord ad Shulma (967) volvg oe relaxato tme for sotropc homogeeous meda, whch s called the frst geeralzato to the coupled theory of elastcty. These equatos determe the fte speeds of propagato of heat ad dsplacemet dstrbutos, the correspodg equatos for a sotropc case were obtaed by Dhalwal ad Sheref (980). The secod geeralzato to the coupled theory of elastcty s what s kow as the theory of thermo elastcty, wth two relaxato tmes or the theory of temperature-depedet thermoelectrcty. A geeralzato of ths equalty was proposed by Gree ad Laws (97). Gree ad Ldsay (97) obtaed a explct verso of the costtutve equatos. Ths theory cotas two costats that act as relaxato tmes ad modfy ot oly the heat equatos, but also all the equatos of the coupled theory. The classcal Fourer s law of heat coducto s ot volated f the medum uder cosderato has a ceter of symmetry. Erbay ad Suhub (986) studed the logtudal wave propagato a geeralzed thermoplastc fte cylder ad obtaed the dsperso relato for a costat surface temperature of the cylder. Sharma ad Pathaa (005) vestgated the geeralzed wave propagato crcumferetal curved plates. Asymptotc of wave moto trasversely sotropc plates was aalyzed by Sharma ad Kumar (0). Tso ad Hase (995) have studed the wave propagato through cylder/plate uctos. Heylger ad Ramrez (000) aalyzed the free vbrato characterstcs of lamated crcular pezoelectrc plates ad dsc by usg a dscrete-layer model of the weak form of the equatos of perodc moto. Thermal deflecto of a verse thermo elastc problem a th sotropc crcular plate was preseted by Gakward ad Deshmukh (005). Verma ad Hasebe (00) vestgated the wave propagato plates of geeral asotropc meda geeralzed thermo elastcty. Later, Verma (00) has preseted the propagato of waves layered asotropc meda geeralzed thermo elastcty a arbtrary layered plate. The free vbrato of o-homogeeous trasversely sotropc mageto-electro-elastc plates was studed by Che et al. [9]. Kumar ad Partap (007) preseted the free vbrato of mcrostretch thermoelastc plate wth oe relaxato tme. Pousamy ad Selvama (0) have studed the dsperso aalyss of geeralzed mageto-thermo elastc waves a trasversely sotropc cyldrcal pael usg the wave propagato approach. Later, Selvama (0) performed mathematcal modelg ad aalyss for dampg of geeralzed thermoelastc waves a homogeeous sotropc plate. I ths paper, the -plae vbrato of heat coductg thermo elastc ellptcal cross-sectoal plate of homogeeous sotropc materal s studed. The solutos to the equatos of moto for a sotropc medum s obtaed by usg the two dmesoal theory of elastcty. To satsfy the boudary codtos, the Fourer expaso collocato method s performed to the equatos of the boudary codtos ad the frequecy equatos are obtaed for logtudal ad flexural (symmetrc ad atsymmetrc) modes of vbratos. The computed o-dmesoal frequecy, velocty ad qualty factor are plotted dsperso curves for logtudal ad flexural (symmetrc ad atsymmetrc) modes of vbratos. FORMULATION OF THE PROBLEM We cosder a homogeeous, sotropc, thermally coductg elastc plate of ellptcal cross - sectos wth uform temperature T 0 the udsturbed state tally. The system dsplace- Lat Amerca Joural of Solds ad Structures (04)

3 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 59 mets ad stresses are defed polar coordates r ad q a arbtrary pot sde the plate ad deote the dsplacemets u r the drecto of r ad u q the tagetal drecto q. a a y 0 x b b Fgure : Geometry of rg-shaped ellptcal plate. The two dmesoal stress equatos of moto, ad stra dsplacemet relatos ad heat coducto equato the absece of body force for a learly elastc medum are ( ) rr, r r rqq, r rr - qq = ur, tt rq, r r qq, q r rq = uq, tt - r, rr, r, qq = r, t rt, tt b é 0 r, rt r, t q, qt s s s s r s s s r K c ( T r T r T ) c T T T u r ( u u ) ù () ad ( e eqq ) e ( T T, ) ( e e ) e ( T T ) s = l m - b h rr rr rr t s = l m - b h s qq rr qq qq rq = me rq, t () where s rr, s qq, s rq are the stress compoets, e rr, e qq, e rq are the stra compoets, T s the temperature chage about the equlbrum temperature T 0, r s the mass desty, c v s the specfc heat capacty, b s a couplg factor that couples the heat coducto ad elastc feld equatos, K s the thermal coductvty, h, t s the thermal relaxato tmes, t s the tme, l ad m are Lame costats. The stra e related to the dsplacemets are gve by e rr = u, e r ( u u ) r, r qq =, e u r ( u u ) r q, q = - - (3) rq q, r q r, q whch u r ad u q are the dsplacemet compoets alog radal ad crcumferetal drectos, respectvely. The comma the subscrpts deotes the partal dfferetato wth respect to the varables. Substtutg Eqs. (3) ad () Eq. (), the followg dsplacemet equatos of motos are obtaed Lat Amerca Joural of Solds ad Structures (04)

4 59 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape ( )( ) qq ( ) q q ( 3 ) q q ( ), r ( ) ( ) ( 3 ) ( ) ( ), - Krc ( T r T r T ) = rc T rtt bt éu r ( u,, ) ù l m u r u - r u mr u r l m u r l m u - b T ht = ru rrr, rr, r r,, r,, t rtt, m u r u - r u r l m u r l m u r l m u -- b T ht = ru q, rr q, r q q, qq r, q r, rq, t q q, tt, rr, r, qq, t, tt 0 r, tr rt u q t q (4) 3 SOLUTIONS OF THE PROBLEM Eq. (4) s a coupled partal dfferetal equato wth two dsplacemets ad heat coducto compoets. To ucouple Eq.(4), we follow Mrsky (964) by assumg the vbrato ad dsplacemets alog the axal drecto z equal to zero. Hece assumg the solutos of Eq. (4) the form r r = 0 = 0 (,, q ) ( r,, q ) u (, r q, z,) t = e é f r y f r y ù å úe (, q, ) (, q r, ) uq(, r q, z,) t = e é r f y r r f y ù å - - úe ( ) ( ) Tr (, q, zt,) = l m ba e T T e å = 0 wt wt wt (5) where e = for = 0, e = for ³, f r, q, y ( r, q ), T (, ) r q, f ( r, q ), y ( ) r, q ad T ( r, q ) are the dsplacemet potetals. Itroducg the dmesoless quattes such as T a = t mra, x = r a, c = ( l m) r, a = ca K, W = w a c, ( ) e = Ta 0 b r cck v, e = ct ( c K), ad e 3 = c h a ad usg Eq. (5) Eq. (4), we obta where 4 ( ) = -, w s the frequecy, ( ) A B C f = 0 (6) A =, B ( ee ) ( 3 e a e) =W - W, ( c e ) a =W - W (7) ad ( ( ) ) l W - V y = 0 (8) - where º x x x x q The parameters defed Eq. (7) amely, e couples the equatos correspodg to the elastc wave propagato ad the heat coducto whch s called the couplg factor; the coeffcet e, whch s troduced by the theory of geeralzed thermo elastcty, may reder the goverg system of equatos hyperbolc. The parameter e 3 s the coeffcet of the term dcatg the dfferece betwee emprcal ad thermodyamc temperatures. Solvg the partal dfferetal equato (6), the solutos for symmetrc mode s obtaed as Lat Amerca Joural of Solds ad Structures (04)

5 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 593 å éa J ( ax) B Y ( ax) ùcos (9.a) f = a a q = å é ( a ) ( a ) ùcos q (9.b) T = d A J ax B Y ax = ad the soluto for the atsymmetrc mode f s obtaed by replacg cosq by sq (9a) ad (9b), we get Eqs. f = é a a ù ú q å AJ ( ax ) BY ( ax ) s (0.a) = T = d éa J ax BJ ax ù ú å ( a ) ( a ) s q (0.b) = where J s the Bessel fucto of frst kd of order ad Y s the Bessel fucto of secod kd of order. Solvg Eq. (8), we obta ( ) ( ) for symmetrc mode, ad for the atsymmetrc mode sq by cosq. y = é A 3 J a3 ax B 3 Y a3 ax ù s q (.a) y s obtaed from Eq. (a) by replacg y = é A3J ( a3ax) B3J ( a3ax) ù úcosq (.b) where ( 3a ) ( ) a = l W - V. If ( a ) a < 0 ( =,,3) replaced by the modfed Bessel fucto I ad K respectvely., the the Bessel fuctos J ad Y are 4 BOUNDARY CONDITIONS AND FREQUENCY EQUATIONS I ths problem, the vbrato of thck arbtrary cross-sectoal plate s cosdered. Sce the boudary s rregular shape, t s dffcult to satsfy the boudary codtos alog both outer ad er surface of the plate drectly. Hece, the Fourer expaso collocato method s appled to satsfy the boudary codtos. For the plate, the ormal stress s xx, shearg stress s xy ad thermal feld T alog the er surface of the plate s equal to zero. Smlarly, ormal stress s xx, shearg stress s xy ad thermal feld T alog the outer surface of the plate s equal to zero. Thus, the boudary codtos alog the outer boudary of the plate are ( s ) ( s ) ( T ) 0 xx = = = () xy ad for the er boudary, the boudary codtos are Lat Amerca Joural of Solds ad Structures (04)

6 594 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape ( sxx ) ( sxy ) ( T ) = = = 0 (3) where x s the coordate ormal to the boudary ad y s the coordate tagetal to the boudary, s xx, s xx are the ormal stresses, s xy, s xy are the shearg stresses, T, T are the thermal felds ad ( ) s the value at the - th segmet of the outer ad er boudary respectvely. Sce the cross-secto of the plate s rregular, t s dffcult to fd the trasformed expresso of the stresses the thck arbtrary cross-sectoal plates because the coordate x ad y are vary wth the agle q. Therefore, the er ad outer boudary of the thck arbtrary cross-sectos s dvded to small segmets such that the varatos of the stresses are assumed to be costat. Assumg the agle g, betwee the ormal to the segmet ad the referece axs to be costat, the trasformed expressos for the stresses are followed by Nagaya (983b) as - - ( u, r ( u uqq, )) [ u, cos ( ) r ( u uqq, ) s ( ) 0.5( r ( uq -ur, q )-uq, r ) s ( q - g )] - b( T ht, t ) xx = r r r r r - r - s l m q g q g ( qq ) ( ) ( ( q q) q ) ( ) xy = ur, r - r u, ur - r u r, - u u, r - s m[ s q g s q g ] (4) Substtutg Eqs. (9)-() Eqs. () ad (3), the boudary codtos are trasformed as follows: é æ ö ù WT ( S a xx ) Sxx e = 0 ç è ø ú é æ ö ù WTa ( Sxy ) Sxy e = 0 ç è ø ú é æ ö ù WTa ( St ) St e = 0 ç è ø ú (5) for the er surface ad é T S S ù W a e = 0 ú é T S S ù W e a = 0 ú é WTa t = 0 ( xx ) ( xx ) ( xy ) ( xy ) ( S ) ( S ù t ) úe (6) for the outer surface, where 3 4 æ ö æ ö xx = ç å è ø = è ç ø 3 4 æ ö 3 4 xy = ç A å 6 3 fb3 è ø = è 3 æ ö æ ö t = çè å ø ç = è ø S e A e B e A e B e A e B e A e B e A e B æ ö S f A f B f A f B f A f B f A f B f ç ø S g A g A g A g A g A g B g A g B g A g B Lat Amerca Joural of Solds ad Structures (04) (7.a)

7 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape ( ) å( 3 3 ) S = 0.5 e A e B e A e B e A e B e A e B e A e B xx = ( ) å( 3 3 ) S = 0.5 f A f B f A f B f A f B f A f B f A f B xy = t = 0.5( ) å( gb ga gb ga 3 gb 3 ) = S g A g B g A g B g A æ 3 4 ö æ ö 0.5 xx = ç ç 3 3 å S e A e B e A e B e A e B e A e B ç è ø çè ø æ ö æ ö S f A f B f A f B f A f B f A f B è ø è ø = xy = å 3 3 ç ç = æ 3 4 ö æ ö 0.5 t = å çè ø ç = è ø S g A g B g A g B g A g B g A g B (7.b) (8.a) xx = 0.5ç ç 3 3 æ ö æ ö S e A e B e A e B e A e B e A e B ç è ø å çè ø xy = = ç å ç = 3 t 3 4 æ ö = 0.5ç 0 0 æ ö 0 0 ga gb ga gb ga3 gb 3 çè ø å ç è ø = æ S f A f B ö æ ö f A f B f A f B f A f B è ø è ø S g A g B (8.b) The coeffcets for e - g are gve the Appedx A. Performg the Fourer seres expaso to Eqs. () ad (3) alog the boudary, the boudary codtos alog the er ad outer surfaces are expaded the form of double Fourer seres. Whe a 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: é ù E00 E00 E00 E00 E0 E0N E0 E0N E0 E0N E0 E0N E0 E0N E0 E0N é A ù 0 B EN0 EN0 EN0 EN0 EN ENN EN ENN EN ENN EN ENN EN ENN EN E A NN B F 0 F0 F0 F0 F FN F FN F FN F 0 FN F FN F F N A F N0 FN0 FN0 FN0 FN FNN FN FNN FN FNN FN FNN FN FNN FN FNN A N G B 00 G00 G00 G00 G0 G0N G0 G0N G0 G0N G0 G0N G0 G0N G0 G0N B G N N 0 GN0 GN0 GN0 GN GNN GN GNN GN GNN GN GNN GN GNN GN GNN = E00 E00 E00 E00 E0 E0N E0 E0N E0 E0 N E0 E0 N E0 E0 N E0 E0 N EN0 EN0 EN0 EN0 EN ENN EN ENN EN ENN EN ENN ENN ENN EN E NN F0 F0 F0 F0 F F N F F N F F N F F N F F N F F A N FN 0 FN 0 FN 0 FN 0 FN FNN FN FNN FN FNN FN FNN FN FNN FN F A 3N NN B G00 G00 G00 G00 G00 G0N G00 G0N G00 G0N G00 G0 N G00 G0 N G00 G 3 0N B ú 3N GN0 GN0 GN0 GN0 GN GNN GN GNN GN GNN GN GN GN GN GN G NN (9) Lat Amerca Joural of Solds ad Structures (04)

8 596 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape where m q ( ) å ò ( ) E = e p e R, q cosmqdq m ( ) å ò ( ) F = e p f R, q smqdq m ( ) å ò I = q I q = q I q ( ) G = e p g R, q cosmqdq = q (0.a) I q m = = q I q m = = q I q m = = q ( ) å ò ( ) E e p e R, q cosmqdq ( ) å ò ( ) F e p f R, q smqdq ( ) å ò ( ) G e p g R, q cosmqdq (0.b) Smlarly, the matrx for the atsymmetrc mode s obtaed as é ù E0 E0 E EN E EN E EN E EN E EN E EN é A ù 30 B EN 0 EN 0 EN ENN EN ENN EN ENN EN ENN EN ENN EN ENN F A 00 F00 F0 F0N F0 F0N F0 F0N F0 F0N F0 F0N F0 F0N A N FN 0 FN 0 FN FNN FN FNN FN FNN FN FNN FN FNN FN FNN B G 0 G0 G GN G GN G GN G GN G GN G GN B N G N 0 GN 0 GN GNN GN GNN GN GNN GN GNN G A N GNN GN GNN = E0 E0 E EN E EN E EN E EN E EN E EN A N B EN 0 EN 0 EN ENN EN ENN EN ENN EN ENN EN ENN EN ENN B N F00 F00 F0 F0N F0 F0N F0 F0N F0 F0N F0 F0N F0 F0N A FN 0 F0 FN FNN FN FNN FN FNN FN FNN FN FNN FN F NN A3 N G0 G0 G GN G GN G GN G GN G GN G G B B 3N GN 0 GN 0 GN GNN GN GNN GN GNN GN GNN GN GNN GN G ú NN () where Lat Amerca Joural of Solds ad Structures (04)

9 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 597 q ( ) å ò ( ) E = e p e R, q smqdq m = q I q ( ) å ò ( ) F = e p f R, q cosmqdq m = q I q ( ) å ò I ( ) G = e p g R, q smqdq m m = q I q ( ) å ò ( ) E = e p e R, q smqdq m = q I q ( ) å ò ( ) F = e p f R, q cosmqdq m = q I q ( ) å ò ( ) G = e p g R, q smqdq = q (.a) (.b) ad where =,, 3, 4, 5 ad 6, I s the umber of segmets, R s the coordate r at the er boudary, R s the coordate r at the outer boudary ad N s the umber of trucato of the Fourer seres. For the otrval soluto of the system of equatos gve Eqs. (9) ad (), the determat of the coeffcet matrx must vash ad these determats gve the frequeces of symmetrc ad atsymmetrc modes of vbratos respectvely. 4. Ellptc cross-sectoal plate The geometry of ellptc cross-sectoal rg shaped plate s show Fgure.The geometrcal relatos of a ellptc rg shaped plate gve by Nagaya (98b) are used for umercal calculato ad are gve below: é ù = ( ) cos q ( ) s q é * ù * ( b a) ú R b a b a b g = p - ta ta q, for q < p g p q p * =, = é * ù * ( b a) ú g = p ta ta q, for q > p (3.a) for the outer surface ad é ù s q é * ù * ( b a) ú R b = ( a b ) cos q ( a b ) g = p - ta ta q, for q < p g p q p * =, = é * ù * ( b a) ú g = p ta ta q, for q > p (3.b) Lat Amerca Joural of Solds ad Structures (04)

10 598 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape for the er surface, where a ad a are the legth of er ad outer sem maor axs, ad b * ad b are the legth of sem mor axs of a ellptc cross-secto. Also ql = ( ql ql - ) ad R s the coordate r at the - th boudary, g s the agle betwee the referece axs ad the ormal to the segmet. For -plae vbrato problem, there exst two types of vbratos; oe of whch s geerated maly by flexural moto ad the other by logtudal moto. I the preset problem, there are three kds of basc depedet modes of wave propagato have bee cosdered, amely, the logtudal ad two flexural (symmetrc ad atsymmetrc) modes. 5 NUMERICAL ANALYSIS The umercal aalyss of the frequecy equato s carred out for heat coductg thermo elastc ellptc cross-sectoal plates. The materal propertes of copper at 4 K are take approxmately as Posso rato = 0.3, desty r = 8.96x0 3 kg/m 3, the Youg s modulus E =.39x0 N/m, l = 8.0x0 kg/m*s, m = 4.0x0 0 kg/m*s, c = 9.x0 - m /K*s, ad K = 3x0 - kg*m/k*s. The thermal parameters such a, e, e, ad e 3 are chose by the followg argumets are gve by Erbay ad Suhub (986). I the umercal calculato, the agle q s take as a depedet varable ad the coordate R ad R are at the - thsegmet of the boudary s expressed terms of q. Substtutg R, R ad the agle g, betwee the referece axs ad the ormal to the -th boudary le, the tegratos of the Fourer coeffcets e, f, g, e, f, ad g ca be expressed terms of the agle q. Usg these coeffcets to equatos (0) ad (), the frequeces are obtaed for heat coductg thermo elastc ellptcal cross-sectoal plate. 5. Logtudal mode The geometrcal relatos for the ellptc cross-sectos gve Eq. (3) 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 cross-secto, the cross-secto vbrates alog the axs of the cylder, so that the vbrato ad dsplacemets the cross-secto s symmetrcal about both maor ad mor axes. Hece, the frequecy equato s obtaed by choosg both terms of ad m as 0,, 4, 6,... Eq. (9) for the umercal calculatos. Sce the boudary of the ellptcal cross-sectos s 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 q = 0 ad q = p s dvded to 0 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 0 <W., usg the secat method. 5. Flexural mode I the case of flexural mode of ellptcal cross-secto, the vbrato ad dsplacemets are atsymmetrcal about the maor axs ad symmetrcal about the mor axs. Hece, the frequecy equatos are obtaed from Eq. () by choosg m=,,3,5,.... Two kds of flexural (symmet- Lat Amerca Joural of Solds ad Structures (04)

11 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 599 rc ad atsymmetrc) modes are cosdered. The computed o-dmesoal frequeces are preseted the form of dsperso curves. 5.3 Qualty factor The rato of the total elastc eergy ad the eergy loss oe cycle of a materal s defed as the qualty factor. I egeerg ad physcs, the qualty factor of a materal s a dmesoless parameter that compares the tme costat for decay of a oscllatg physcal structure s ampltude to ts oscllato perod. Equvaletly, t compares the frequecy at whch a structure oscllates to the rate at whch t dsspates ts eergy. The qualty factor s defed as Q F w = VQ where V ad Q represets the phase velocty ad atteuato coeffcet, respectvely. 5.4 Dsperso curve The varato of o-dmesoal frequecy versus dmesoless wave umber of logtudal modes of thermo-elastc crcular cross sectoal plate wth ad wthout thermal feld for the two values of aspect rato a b = a b = 0.5,.5 s show Fgures ad 3 respectvely. From Fgure, t s observed that the dsperso learly crease wth respect to the wave umber. For wthout thermal feld, the relato betwee the wave umber ad frequecy both the cases of aspect ratos are alke, but for the thermal cluso the behavor s oscllatg throughout the full rage of wave umber. Fgures 4 ad 5 shows the dsperso characterstc of flexural (symmetrc) modes of ellptc cross-sectoal plate wth ad wthout thermal feld. It shows that, the dsperso s commo for both the plate wth ad wthout thermal cluso, where the hgher aspect rato attas hgher frequecy compared wth lower aspect rato. Fgure : No-dmesoal wave umber versus dmesoless frequecy of logtudal modes of crcular cross sectoal plate. Lat Amerca Joural of Solds ad Structures (04)

12 600 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape Fgure 3: No-dmesoal wave umber versus dmesoless frequecy of logtudal modes of thermo-elastc crcular cross sectoal plate. Fgure 4: No-dmesoal wave umber versus dmesoless frequecy of flexural symmetrc modes of ellptcal cross sectoal plate. Fgure 5: No-dmesoal wave umber versus dmesoless frequecy of flexural symmetrc modes of thermo-elastc ellptcal cross sectoal plate. Lat Amerca Joural of Solds ad Structures (04)

13 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 60 Fgure 6: No-dmesoal wave umber versus dmesoless frequecy of flexural at symmetrc modes of ellptcal cross sectoal plate. Fgure 7: No-dmesoal wave umber versus dmesoless frequecy of flexural at symmetrc modes of thermo-elastc ellptcal cross sectoal plate. A graph s draw betwee o-dmesoal wave umber versus dmesoless frequecy of flexural (atsymmetrc) modes of ellptc cross-sectoal plate wth ad wthout thermal evromet are show Fgures 6 ad 7 respectvely. I both fgures, t s observed that as the wave umber creases, the o-dmesoal frequeces also creases, also t could be ferred that, the dsperso characterstc are promet oly lower orders wth eglgble varatos hgher rag. From the fgures, t s observed that the o-dmesoless frequecy creases wth respect to ts wave umber. It s also observed that, the cross over pots betwee the modes of dfferet aspect ratos of logtudal ad flexural symmetrc ad atsymmetrc modes represets the trasportato of eergy betwee the two dfferet vbrato medum. The varato of velocty versus dmesoless frequecy of a flexural symmetrc modes of ellptc cross-sectoal plate wth ad wthout thermal evromet are show Fgures 8 ad 9, Lat Amerca Joural of Solds ad Structures (04)

14 60 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape respectvely. From the Fgures 8 ad 9, t s observed that, for the creasg values of the aspect rato modes are merges ad osclatg through etre rage of frequecy. The mergg of thermal modes ad oscllato of pot betwee the vbratoal modes shows that, there s a eergy trasportato betwee the modes of vbratos by the effect of aspect rato ad asotropc of the materal. Fgures 0 ad reveals that the varato of qualty factor wth the dmesoless frequecy for the flexural symmetrc modes wth ad wthout thermal sgal. The qualty factor s qute hgh at lower rage of frequecy ad starts to decay wth creasg frequecy. The qualty factor profle are dspersve tred for flexural symmetrcal modes wth thermal sgal tha flexural symmetrcal modes wthout thermal sgal ad experece oscllato the wave umber rage 0. V 0.6 for flexural symmetrc thermal modes of vbrato. The cross over pots betwee the vbrato modes represets the trasfer of eergy betwee the thermal modes. Fgure 8: Varato of velocty versus dmesoless frequecy of flexural symmetrc modes of ellptcal cross sectoal plate. Fgure 9: Varato of velocty versus dmesoless frequecy of flexural symmetrc modes of thermo-elastc ellptcal cross sectoal plate. Lat Amerca Joural of Solds ad Structures (04)

15 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 603 Fgure 0: Varato of qualty factor versus dmesoless frequecy of flexural symmetrc modes of ellptcal cross sectoal plate. Fgure : Varato of qualty factor versus dmesoless frequecy of flexural symmetrc modes of thermoelastc ellptcal cross sectoal plate. 6 CONCLUSIONS I ths paper, the wave propagato a heat coductg thermo elastc plate of ellptcal shaped s aalyzed by satsfyg the boudary codtos o the rregular boudary usg the Fourer expaso collocato method ad the frequecy equato for the logtudal ad flexural (symmetrc ad atsymmetrc) modes of vbratos are obtaed. The computed dmesoless frequecy, velocty ad qualty factor are plotted graphs for logtudal ad flexural (symmetrc ad atsymmetrc) modes wth ad wthout thermal sgals. From the graphcal represetato, the effect of thermal eergy wth thermal relaxato tmes ad the asotropy of the materal o the varous cosdered wave characterstcs are more sgfcat ad domat the flexural modes of ellptcal cross sectoal plate. Refereces Dhalwal, R.S.. Sheref, H.H., (980). Geeralzed thermo elastcty for asotropc meda. Q. Appl. Math.,8(): - 8. Lat Amerca Joural of Solds ad Structures (04)

16 604 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape Erbay, E. S., Suhub, E., S., (986). Logtudal wave propagato thermo elastc cylder. J. Thermal Stresses 9: Gree, A. E., Laws, N., (97). O the Etropy Producto Iequalty. Arch. Ratoal Mech. Aal., 45: Gree, A. E., Ldsay, K.A., (97). Thermoelastcty. J. Elastcty : -7. Gakwad, M.K., Desmukh, K.C., (005). Thermal deflecto of a verse thermo elastc problem a th sotropc crcular plate. J. Appled Mathematcal modelg 9: Heylger, P.R., Ramrez, G., (000). Free vbrato of Lamated crcular pezoelectrc plates ad dsc. J. Soud ad Vb., 9(4): Kumar, R., Partap, G., (007). Free vbrato of mcrostretch thermo elastc plate wth oe relaxato tme. J. theoretcal ad appled fracture mechacs. 48: Lord, H. W., Shulma, Y., (967). A geeralzed dyamcal theory of thermo elastcty. J. Mech Phys Solds., 5: Mrsky, I., (964). Wave propagato trasversely sotropc crcular cylders. J. of Acoust. Soc. of Am. 36: 4-5. Nagaya, K., (98a). Smplfed method for solvg problems of plates of doubly coected arbtrary shape, Part I: Dervato of the frequecy equato. Joural of Soud ad Vbrato74(4): Nagaya, K., (98b). Smplfed method for solvg problems of plates of doubly coected arbtrary shape, Part II: Applcatos ad expermets. Joural of Soud ad Vbrato 74(4): Nagaya, K., (983a). Vbrato of a thck walled ppe or rg of arbtrary shape ts plae. Joural of Appled Mechacs, Vol.50: Nagaya, K., (983b). Vbrato of a thck polygoal rg ts plae. Joural of Acoustcal Socety of Amerca 74(5): Nagaya, K., (983c). Drect method o the determato of ege frequeces of arbtrary shaped plates. Joural of App. Mech. 05: 336. Selvama, R., Pousamy, P., (0). Dampg of geeralzed thermoelastc waves a homogeeous sotropc plate. Mater.Phys.Mechs. 4: Sharma, J.N., Pathaa, V, P., (005). Geeralzed thermo elastc wave propagato crcumferetal drecto of trasversely sotropc cyldrcal curved plates. J. Soud ad Vb. (74): Pousamy, P., Selvama, R., (0). Dsperso aalyss of geeralzed mageto-thermo elastc waves a trasversely sotropc cyldrcal pael. J.Therm Stresses 35(): 94. Sharma, J.N., Kumar, R., (004). Asymptotc of wave moto trasversely sotropc plates. J. Soud ad Vb. 74: Tso, Y.K., Hase, C.H., (995). Wave propagato through cylder/plate uctos. J. Soud ad Vb. 86(3): Verma, K.L., Hasebe, N., (00). Wave propagato plates of geeral asotropc meda geeralzed thermoelastcty. It. J. Eg. Sc. 39: Verma, K. L., (00). O the propagato of waves layered asotropc meda geeralzed thermo elastcty. J. Appl. Acoustcs 68: 440. Lat Amerca Joural of Solds ad Structures (04)

17 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape 605 Appedx A The expressos e k used Eqs. (0) ad () are gve as follows: { } é ù ( 3 ) { } e = ( - ) J ( aax) ( aax) J ( aax) cos( q - g )cosq { } ( ) - x ( aa) l cos ( q - g ) ld We J aax cosq ( ) J ( aax) -( aax) J ( aax) sqs( q - g ), =, { } { } 3 = - a - a a q q - g e ( ) J ( ax) ( ax) J ( ax) cos cos( ) [ ( )-( a ax) ] J ( a ax) ( a ax) J ( a ax) sqs( q - g ) { } { } 4 = - a - a a q q - g e ( ) Y ( ax) ( ax) Y ( ax) cos cos( ) [ ( )-( a ax) ] Y ( a ax) ( a ax) Y ( a ax) sqs( q - g ) { } é ù ( 3 ) { } e = ( - ) Y ( aax) ( aax) Y ( aax) cos( q - g )cosq { } ( ) - x ( aa) l cos ( q - g ) ld We Y aax cosq ( ) Y ( aax) -( aax) Y ( aax) sqs( q - g ), = 5,6 { } { } f = [ ( )-( aax) ] J ( aax) ( aax) J ( aax) cosqs( q - g ) ( aax) J ( aax) -( ) J ( aax) sqcos( q - g ), =, { } { } 3 = - a - a a q q - g f ( ) J ( ax) ( ax) J ( ax) cos s( ) - ( a ax) J ( a ax) -[( a ax) - ( )] J ( a ax) s qcos ( q - g ) { } { } 4 = - a - a a q q - g f ( ) Y ( ax) ( ax) Y ( ax) cos s( ) - ( a ax) Y ( a ax) -[( a ax) - ( )] Y ( a ax) s qcos ( q - g ) { } { } f = [ ( )-( aax) ] Y ( aax) ( aax) Y ( aax) cosqs( q - g ) ( aax) Y ( aax) -( ) Y ( aax) s qcos ( q - g ), = 5, 6 (A.) (A.) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) { ( q g ) ( a ) ( a ) ( a ) q g q} k = d cos - J ax - ax J ax cos( - ) cos, =, (A.9) k = 0.0, k 4 = 0.0 (A.0) 3 { ( q g ) ( a ) ( a ) ( a ) q g q} k = d cos - Y ax - ax Y ax cos( - ) cos, = 5, 6 (A.) { } é ù ( 3 ) { } e = ( - ) J ( aax) ( aax) J ( aax) cos( q - g )sq { } ( ) - x ( aa) l cos ( q - g ) d We J aax cosq - ( ) J ( aax) -( aax) J ( aax) cosqs( q - g ), =, (A.) Lat Amerca Joural of Solds ad Structures (04)

18 606 R. Selvama ad P. Pousamy / Modelg ad aalyss of waves a heat coductg thermo-elastc plate of ellptcal shape { } { } 3 = - a - a a q q - g e ( ) J ( ax) ( ax) J ( ax) s cos( ) - [ ( )-( aax) ] J ( a ax) ( a ax) J ( a ax) cosqs( q - g ) { } { } 4 = - a - a a q q - g e ( ) Y ( ax) ( ax) Y ( ax) s cos( ) - [ ( )-( aax) ] Y ( a ax) ( a ax) Y ( a ax) cosqs( q - g ) { } é ù ( 3 ) { } e = ( - ) Y ( aax) ( aax) Y ( aax) cos( q - g )sq { } ( ) - x ( aa) l cos ( q - g ) d We Y aax cosq - ( ) Y ( aax) -( aax) Y ( aax) cosqs( q - g ), = 5,6 { } { } f = [ ( )-( aax) ] J ( aax) ( aax) J ( aax) sqs( q - g ) 3 - ( aax) J ( aax) -( ) J ( aax) cosqcos( q - g ), =, { } { } f = ( ) J ( a ax) -( a ax) J ( a ax) sqs( q - g ) 4 ( a ax) J ( a ax) -[( a ax) - ( )] J ( a ax) cos qcos ( q - g ) { } { } f = ( ) Y ( a ax) -( a ax) Y ( a ax) sqs( q - g ) ( a ax) Y ( a ax) -[( a ax) - ( )] Y ( a ax) cos qcos ( q - g ) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) { ( q g ) ( a ) ( a ) ( a ) q g q} k = d cos - J ax ax J ax cos( - ) s, =, (A.9) 3 k = 0.0, k 4 = 0.0 (A.0) { ( q g ) ( a ) ( a ) ( a ) q g q} k = d cos - Y ax ax Y ax cos( - ) s, = 5, 6 (A.) Lat Amerca Joural of Solds ad Structures (04)