1 INTRODUCTION. Dao Van Dung a Hoang Thi Thiem a, *

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1 679 Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil Tuncted Conicl Shells with Eccentic Functionlly Gded Mteil Stinge nd Ring Stiffenes Abstct In this esech wok, n ect nlyticl solution fo feuency chcteistics of the fee vibtion of otting functionlly gded mteil (FGM) tuncted conicl shells einfoced by eccentic FGM stinges nd ings hs been investigted by the displcement function method. Mteil popeties of shell nd stiffenes e ssumed to be gded in the thickness diection ccoding to simple powe lw distibution. The chnge of spcing between stinges is consideed. Using the Donnell shell theoy, eckhnisky smeed stiffenes techniue nd tking into ccount the influences of centifugl foce nd Coiolis cceletion the govening eutions e deived. Fo stiffened FGM conicl shells, it is difficult tht fee vibtion eutions e couple set of thee vible coefficient ptil diffeentil eutions. By suitble tnsfomtions nd pplying Glekin method, this difficulty is ovecome in the ppe. The sith ode polynomil eution fo w is obtined nd it is used to nlyze the feuency chcteistics of otting ESFGM conicl shells. Effects of stiffene, geometics pmetes, cone ngle, vibtion modes nd otting speed on feuency chcteistics of the shell fowd nd bckwd wve e discussed in detil. The pesent ppoch poves to be elible nd ccute by comping with published esults vilble in the litetue. Do Vn Dung Hong Thi Thiem, * Vietnm Ntionl Univesity, Hnoi, Viet Nm Coesponding utho: * hongthithiem@gmil.com Received 5..6 In evised fom Accepted.8.6 Avilble online.8.6 Keywods Rotting tuncted conicl shell Fee vibtion Feuency chcteistics FGM stiffene Functionlly gded mteil (FGM). INTRODUCTION Revolution shell stuctues involving functionlly gded mteil (FGM) conicl shells potentilly hve wide ppliction in mny moden industy fields such s eospce, iplne, missile, booste

2 68 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil nd othe eospce vehicles (Koizumi M. 99, Shen HS. 9). Theefoe, the fee vibtion of otting tuncted conicl shells is one of inteesting nd impotnt poblems nd hs eceived consideble ttention of eseches. Fo unstiffened conicl shells, much significnt esults e obtined. Chndsekhn nd Rmmuti (98 nd 98) studied isymmetic nd symmetic fee vibtions of lminted conicl shells using the RyleighRitz pocedue. Shu (996) pesented n efficient ppoch fo nlyzing the fee vibtion of conicl shells. The sme utho (996) investigted the fee vibtion of composite lminted conicl shells by genelized diffeentil udtue. By using the Glekin nd hmonic blnce methods, Xu et l. (996) studied the nonline fee vibtion of symmeticlly lminted, cossply, geometiclly pefect thick conicl shell with its two ends both clmped nd both simply suppoted. m nd Hu (997 nd 999) nlyzed the vibtion nd influences of boundy conditions on the feuency chcteistics of otting tuncted cicul conicl shells. Following this diection, the feuency nlysis of otting tuncted cicul othotopic conicl shells with diffeent boundy conditions ws studied by Hu (). Tht sme utho (), bsed on the ove fist ppoimtion theoy nd Glekin pocedue, nd tking into ccount the influences of centifugl nd Coiolis cceletions, investigted feuency chcteistics of otting tuncted cicul lyeed conicl shell. With the impoved genelized diffeentil udtue method, m et l. () nlyzed fee vibtion chcteistics of tuncted conicl pnels. Using the clssicl thin shell theoy nd the elementfee kpritz method, iew et l. (5) investigted the fee vibtion of thin conicl shells. Civlek (6) poposed the discete singul convolution method fo nlyzing the fee vibtion of otting conicl shells in which egulized Shnnon s delt kenel is selected s the singul convolution to illustte his lgoithm. Sofiyev et l. (), by using the Donnell shell theoy nd Glekin method, pesented n nlyticl pocedue to study the fee vibtion nd stbility of homogeneous nd nonhomogeneous othotopic tuncted nd complete conicl shells with clmped edges unde etenl pessue. Fo unstiffened FGM conicl shells, thee e mny vilble esults. Tonbene (9) nd Tonbene et l. (9), bsed on the fist ode she defomtion theoy nd D diffeentil udtue solution, studied the fee vibtion nlysis of functionlly gded conicl, cylindicl nd nnul pltes stuctues using D diffeentil udtue solution. Sofiyev (9) nlyzed the vibtion nd stbility behvio of feely suppoted FGM conicl shells subjected to etenl pessue by Glekin method. The sme utho () nlyzed the nonline vibtion of FGM tuncted conicl shells by nlyticl ppoch. Fo unstiffened FGM conicl pnels, Bich et l. () investigted by nlyticl method the line mechnicl buckling of tht stuctue using the clssicl shell theoy nd Glekin method. The investigtion on the line buckling of tuncted hybid FGM conicl shells with piezoelectic lyes subjected to combined ction of theml nd electicl lods ws epoted by Tobi et l. (). Bsed on the Fist ode she defomtion theoy (FSDT), Mlekzdeh nd Heydpou () studied effects of centifugl nd Coiolis, of geometicl nd mteil pmetes on the fee vibtion behvio of otting FGM unstiffened tuncted conicl shells subjected to diffeent boundy conditions. As cn be seen the bove intoduced esults only elte to unstiffened stuctues. Howeve, in pctice, pltes nd shells including conicl shells usully e einfoced by stiffenes system to povide the benefit of dded lod cying cpbility with eltively smll dditionl tin Ameicn Jounl of Solids nd Stuctues (6) 67975

3 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 68 weight. Thus, the study on dynmic behvio of those stuctues is significnt pcticl poblem. Fo stiffened isotopic o othotopic conicl shells, Weingten (965) studied the fee vibtion of ingstiffened simply suppoted conicl shell by consideing n euivlent othotopic shell nd using Glekin method. Applying the enegy ppoch, Cenwelge nd Muste (969) investigted the esonnt feuencies of simply suppoted ingstiffened, nd ing nd stingestiffened conicl shells. Ro nd Reddy (98) studied the minimum weight design of illy loded simply suppoted stiffened conicl shells with ntul feuency constints. The influence of plcing the stiffenes inside s well s outside the conicl shell on the optimum design is consideed. The epessions fo the citicl il buckling lod nd ntul feuency of vibtion of conicl shell lso e obtined. By pplying stuctul symmety techniues, Mustff nd Ali (987) investigted fee vibtion chcteistics of stiffened cylindicl nd conicl shells. Sinivsn nd Kishnn (989), using the integl eution fo the spce domin nd mode supeposition fo the time domin, obtined the esults on the dynmic esponse nlysis of stiffened conicl shell pnels in which the effect of eccenticity is tken into ccount. Mecitoglu (996) studied the vibtion chcteistics of stiffened tuncted conicl shells bsed on the DonnellMushti thin shell theoy, the stiffenes smeed techniue nd the colloction method. The poblem on the fee vibtion of otting composite conicl shells with stinge nd ing stiffenes ws solved by Tlebitooti et l. (). Fo stiffened FGM shells, Njfizdeh et l. (9) with the lineized stbility eutions in tems of displcements studied buckling of FGM cylindicl shell einfoced by ings nd stinges unde il compession. The stiffenes nd skin, in thei wok, e ssumed to be mde of functionlly gded mteils nd its popeties vy continuously though the thickness diection. Following this diection, Dung nd Ho (5) obtined the esults on the sttic nonline buckling nd postbuckling nlysis of eccenticlly stiffened FGM cicul cylindicl shells unde tosionl lods in theml envionment. The mteil popeties of shell nd stiffenes e ssumed to be continuously gded in the thickness diection. Glekin method ws used to obtin closedfom epessions to detemine citicl buckling lods. Dung et l. () investigted the sttic buckling of FGM conicl shells einfoced by FGM stiffenes unde il compessive lod nd etenl pessue by nlyticl method. The chnge of distnce between stinges is consideed in tht wok. By consideing homogenous stiffenes, Bich et l. () obtined the esults on the nonline sttic nd dynmic nlysis of eccenticlly stiffened FGM cylindicl shells nd doubly cuved thin shllow shells bsed on the clssicl shell theoy. The govening eutions of motion wee deived by using the smeed stiffenes techniue nd the clssicl shell theoy with von Kmn geometicl nonlineity. The nonline citicl dynmic buckling lod is found ccoding to the BudinskyRoth citeion. Dung et l. () studied mechnicl buckling of eccenticlly stiffened functionlly gded (ESFGM) thin tuncted conicl shells subjected to il compessive lod nd unifom etenl pessue lod bsed on the smeed stiffenes techniue nd the clssicl shell theoy nd consideing homogenous stiffenes. Dung nd Nm () solved the poblem on the nonline dynmic nlysis of eccenticlly stiffened functionlly gded cicul cylindicl thin shells unde etenl pessue nd suounded by n elstic medium. The nonline citicl dynmic buckling lod is found ccoding to the BudinskyRoth citeion. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

4 68 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil Fom the eview of the litetue, s cn be seen the fee vibtion of otting FGM tuncted conicl shell einfoced by eccentic stiffene system nd tking into ccount the influences of centifugl foce nd Coiolis cceletion still is not investigted. The im of this ppe is to investigte just mentioned poblem. We focus on thee new contibutions s: FGM tuncted conicl shells e einfoced by FGM stinges nd ings A chnge of spcing between stinge stiffenes is consideed A centifugl foce nd Coiolis cceletion e tken into ccount. The govening eutions e deived using Donnell shells theoy nd smeed stiffenes techniue. Fo stiffened FGM conicl shells, it is difficult tht these eutions e couple set of thee vible coefficient ptil diffeentil eutions, while fo stiffened cylindicl shells those govening eutions only e couple set of constnt coefficient ptil diffeentil eutions. This difficulty is ovecome in the ppe. The sith ode polynomil eution fo feuency w is obtined by pplying Glekin method. Numeicl simultions e been done to show effects of geometics pmetes, vibtion modes nd otting speed, einfocement stiffene on feuency chcteistics of the shell. The pesent esults e vlidted by comping with those in the litetue. MATERIA PROPERTIES OF SHE AND STIFFENERS Conside thin cicul tuncted conicl shell with the semivete ngle, thickness h, length nd smll bse dius s shown in Fig.. The cuviline coodinte system is defined s ( z),,, whee the oigin is locted in the middle sufce of the shell, is in the geneti diection mesued fom the vete of shell, is in the cicumfeentil diection nd the es z is pependicul to the plne (, ) nd its diection is the outwds noml diection of the conicl shell. Denotes the distnce fom the vete to smll bse, nd u, v nd w the displcement components of point in the middle sufce in the diection, nd z, espectively. Figue : Geomety nd coodinte system of stiffened FGM tuncted conicl. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

5 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 68 Assume tht the FGM conicl shell ottes bout its veticl symmeticl is with constnt ngul velocity denoted by W. Also ssume tht the shell is stiffened by closely spced cicul ings nd longitudinl stinges nd the stiffenes nd skin e mde of functionlly gded mteils vying continuously though the thickness diection of the shell with the powe lw. Two cses e investigted in this wok. Cse : Conicl shell with cemic outside sufce nd metl inside sufce nd inside stiffene. Cse : Conicl shell with cemic outside sufce nd metl inside sufce nd outside stiffene. Fo cse, Young Modulus nd densities of FGM shell nd FGM stiffenes e given by (Njfizdeh et l., 9 Dung et l., ) k æz hö Esh = Em E cm, h h / z h /, k ³ () è æz hö sh = m cm, è h k h / z h /, k ³ () æ z hö Es = Em E cm è h, k h/ h z h/, k ³ () æ z hö s = m cm è h k, h/ h z h/, k ³ () æ z hö E = Em E cm è h k, h/ h z h/, k ³ (5) æ z hö = m cm è h whee nsh = ns = n = n = const,, k, h/ h z h/, k ³. (6) k k nd k e volume fctions indees of shell, stinge nd ing, espectively nd subscipts c, m, sh, s nd denote cemic, metl, shell, longitudinl stinges nd cicul ing, espectively. It is evident tht, fom Es. ()(6), continuity between the shell nd stiffenes is stisfied. Note tht the thickness of the stinge nd the ing e espectively denoted by h, nd h nd E c, E m e Young Modulus of the cemic nd metl nd E sh, E s, E, sh, s, e Young Modulus nd densities of shell, of stiffene in the diection nd diection, espectively. The coefficient n is Poison s tio. Note k = k = k, when k, The Poisson s tio n is ssumed to be constnt. Young Modulus nd densities fo the cse e given in Appendi I. k leds to homogeneous stiffene. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

6 68 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil FUNDAMENTA EQUATIONS Accoding to the Donnell shell theoy, the stin components t distnce z fom the efeence sufce of shell e of the fom (Bush nd Almoth, 975 Reddy, Volmi, 97) e e m k e e m z k = g g m k û û û (7) whee e m nd e m e the middle sufce stins in meidionl nd cicumfeentil diection, nd g m is the she stin of the middle sufce of the shell, nd k, k nd k e the chnge of cuvtues nd twist, espectively. They e elted to the displcement components uvw,, s (Bush nd Almoth, 975 Volmi, 97 Hu, ) u e, m u w e m v, cot = sin g m û v u, v, sin û w, k cos w, k w, v =,. sin sin k û cos cos w, w, v, v sin sin sin sin û (8) Using Hooke s w, the stessstin eltions e epessed by: Fo the conicl shell Esh sh s n sh Esh s = e ne sh n s û Esh g ( n ) ( e ne ) ( ). û (9) Fo the stinge nd ing stiffenes st s Ese t = s E e û û () whee the subscipts sh nd st denote shell nd stiffenes, espectively, nd EE s, e Young Modulus of stinge nd ing stiffene espectively. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

7 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 685 The contibution of stiffenes is tken into ccount by using the smeed stiffene techniue. In ddition, the chnge of spcing between stinges in the meidionl diection lso is consideed. Integting the bove stessstin eutions nd thei moments though the thickness of the shell, the epessions fo foce nd moment esultnts of ESFGM tuncted conicl shells e defined s (Bush nd Almoth, 975 Njfizdeh et l., 9 Dung et l., ) æ ö æ ö E sb C A em Ae m B k è ø Bk d è ø N æ ö = E b N Aem A e m Bk ( B C) k è d ø N û A66gm B66k û æ ö æ ö C E sb B e m Be m D k è ø Dk è ø d M æ ö = E b M Bem ( B C) e m Dk D k è d ø M û B66gm D66k û () () whee C, C, d, d, E s, E, E s, E, Aij, Bij, D ij cn be found in Appendi I. The fundmentl eutions fo the vibtion of otting tuncted conicl shells, bsing on the Donnell shell theoy, e s (Hu, Chen et l., 99) N o o N N N æ u wö cossin sin ( N ) sin è N ø sin v sin u W = t t N N cot M cos M sin sin N o æ u u sin sin sin v ö N sin è ø æ u wö W sin cos v è t t = ø t M M M M M sin sin æ w uö sin cos N o ( wcos usin cos ) cot N è ø sin () tin Ameicn Jounl of Solids nd Stuctues (6) 67975

8 686 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil v cos w W =. t t whee cn be found in Appendi I, nd the ulity N is defined s the initil hoop tension due to the centifugl foce effect nd given by (Hu, ) o N = W sin. () Note tht the system of eutions () not only consists of thee tems of eltive cceletions u/ t, v/ t nd w/ t but lso fou tems of Coiolis cceletions W sin v / t, W sin u / t, W cos w / t nd W cos v / t. Intoducing Es. (7) into Es. (), then substituting the esulting eutions nd E. () into E. (), the vibtion eutions in tems of displcements fo otting ESFGM tuncted conicl shell e obtined s R () u R () v R () w = (5) R () u R () v R () w = (6) R () u R () v R () w = (7) whee R ij e ptil diffeentil opetos nd defined s following R æ Ebö æ ö A 66 W sin è û A æ E b ö A æ ö è d ø è ø t s = A è l (8) R = ( A A66 ) cot ( B sin B66) æ E b ö û A sin A 66 sin è d ø cot ( B B66 B C) æ ö sin Wsin û è ø t R æ C ö = B è ( ) B B 66 B C sin B ( B B66) A cot ( B C ) æ Ebö A cot è d ø sin æ ö cossin W è ø û (9) () tin Ameicn Jounl of Solids nd Stuctues (6) 67975

9 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 687 æ R = ( A A66) cot ö ( B B66) W sin sin sin è û æ E b A A ö æ cot ö 66 sin è ( B d C B66) W sin ø sin è û æ ö Wsin è ø t R A66 B66 cot D 66 cot æ E = b ö A cot û sin ( B è d C) ø sin cot æ E b ö D cot cot A sin è d øû 66 B 66 D 66 æ ö cot W sin cot A è 66 B66 D æ û 66 ö û è t R = ( 66) sin B B cot ( 66) sin D D û æ ( B C) cot E b ö D sin sin è d øû ( ) sin B C æ cot E b ö D66 D sin è d øû cot æ E b ö A cot cot sin D ( B è d 66 ø C) æ ö sin sin cos û W è t R æ ö è C = B B ( B B ) sin 66 ( B C B66) sin cot A æ ö ( B C ) sin cos W è û cot æ E b ö A æ ö è d W sinc os ø è cot = ( 66) ( 66) ( B ) C R B B D D sin sin û cot æ E b ö ( B C) D sin sin d è ø û () () () () (5) tin Ameicn Jounl of Solids nd Stuctues (6) 67975

10 688 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil cot æ E b ö ( D D66) D ( B C B66) sin d sin è û cot æ E b ö D D66 D ( cot )( B C) B66 sin cot è d ø sin sin æ Ebö A æ ö è d cos øû W è t R æ E b ö æ E b ö D è ø sin è d s = D l ( D D ) sin 66 cot æ E D ( D D66) b ö B D sin d è ø û cot æ E b ö æ ö ( B C) D D66 D W sin sin d è è ø û æ E b ö D cot cot æ E è ( d ) b ö æ ö B C A cos ø W è d ø è ø û æ ö è ø t (6) The system of Es. (57) is used to nlyze the feuency chcteistics of otting ESFGM tuncted conicl shells. It is difficult tht these eutions e couple set of thee vible coefficient ptil diffeentil eutions. This is min diffeence between the fee vibtion nlyses of otting conicl shell nd cylindicl shell. This is lso eson why the investigtions on the vibtion of otting ESFGM conicl shell e vey limited. This difficulty will be got ove below. BOUNDARY CONDITIONS AND SOUTION OF THE PROBEM Assuming tht the stiffened FGM tuncted conicl shell is simply suppoted t both ends. Thus the boundy conditions e epessed in the following fom v= w =, N =, M = = t, =. (7) The displcement components stisfying ccutely the bove mentioned geometic boundy conditions nd the foce boundy conditions in the vege sense, my be chosen s u v w ( ) cos m p = Y cos( n wt) ( ) sin m p = Y sin( n wt) ( ) sin m p =Y cos( n wt) (8) tin Ameicn Jounl of Solids nd Stuctues (6) 67975

11 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 689 whee m is the numbe of hlfwves long geneti nd n is the numbe of cicumfeentil fullwves, w (d/s) is the ntul cicul feuency of the otting ESFGM conicl shell, nd YY, nd Y e unknown constnts. As bove noted, it is difficult to use the til function (8) nd Es. (57) to obtin diectly the polynomil eution of feuency. Theefoe, diffeent pocedue is poposed s follows. Becuse p, so we cn cy out following euivlent tnsfomtions. Fistly multiplying E. (5) by esulting eutions, leds to nd Es. (6, 7) by, then pplying Glekin method fo the p w p ò ò ò p w p ò ò ò p w p ò ò ò R() u R() v () R w û R() u R() v () R w û R () u R () v () û R w p( ) cos( n wt) cos m p( ) sin( n wt) sin m p( ) cos( n wt) sin m sin dddt = sin dddt = sin dddt =. (9) Substituting epessions (8) into Es. (57) then into E. (9), fte integting longe nd some engements, we obtin whee the coefficients ( H Hw ) Y ( H Hw ) Y ( H Hw ) Y ( H Hw ) Y ( H H w) HY ( H Hw ) Y ( ) H Hw H ij, ij H e given in Appendi II. H Y = Y = Y= Fo E. () to hve nontivil solution, the deteminnt of the chcteistic mti should be set eul to zeo. Developing tht deteminnt nd solving esulting eution fo feuency, yields () 6 HHHw H H H HHH HHH HHH HHH û HHH H HH HHH HHH û H H H HHH HHH H HH HHH H HH HHH HHH HHH H HH û H H H H HH HH H HHH HHH () tin Ameicn Jounl of Solids nd Stuctues (6) 67975

12 69 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil H HH HHH û H H H H H H H H H H H H HHH = HHH HHH. E. () is the sith ode polynomil eution fo w nd it is used to nlyze the feuency chcteistics of otting ESFGM conicl. Numeicl esults below will show tht this eution will hve two oots whose bsolute vlues e smllest nd they e el numbes, one positive nd the othe negtive. The positive vlue coesponds to the fowd wve nd the negtive vlue coesponds to the bckwd wve. 5 NUMERICA RESUTS AND DISCUSSION 5. Veifiction of the Pesent Method Befoe stting the nlysis of shell feuency chcteistics, the vlidity of the pesent study should be ensued. Tble compes the feuency pmete esults of this ppe fo unstiffened isotopic tuncted conicl shell with the feuency pmete given by (Hu, ) f = wr A Computtions hve been cied out fo sttony isotopic conicl shell with the following dt bse s: k =, W =, m =, h / R =., =.5 R/ sin, E =.865 ( P), n =., = (kg/m ). 9 Feuency w is found fom E. (). It cn be obseved good geement is obtined in this compison. f = = 5 = 6 N Pesent Hu [] Pesent Hu [] Pesent Hu [] Tble : Compison of feuency pmete f fo sttiony isotopic conicl shell with simplesuppoted boundy conditions. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

13 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil ESFGM Tuncted Conicl Shells E = GP, = 8 kg/m In the following subsections, the mteils used e Alumin with c 8 c nd Aluminum with E m = 7 GP, m = 7kg/m nd ν=.. The stiffenes e FGM. 5.. Effect of Cicumfeentil Wve Numbe n Tble nd Fig. nd b show effects of cicumfeentil wve numbe n on feuency ω in cse fowd wve (ine()) nd bckwd wve (ine(b))with volume fction inde k = 5 nd semivete ngle = 5 6. Conside stiffenes ttched to inside of the shell. The input pmetes e tken s h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, Ω=5 d/s, k=k=k=. As cn be seen tht t the sme otting velocity W when n the feuency w deceses with the incese of cicumfeentil wve numbe n. But the feuency w inceses with the incese of cicumfeentil wve numbe n when n ³ 5. We lso cn be seen tht feuency ω hs the minimum vlue t mode (m, n)=(,). α= α=5 α=6 n= ().58e5 (b).56e5 ().98e5 (b).97e5 ().7e5 (b).e5.67e5.656e5.e5.89e5.987e5.98e5.56e5.e5.e5.99e5.7e5.78e5.e5.e5.868e5.86e5.67e5.67e5 5.76e5.68e5.989e5.98e5.79e5.789e5 6.7e5.7e5.87e5.8e5.9e5.7e5 7.99e5.9e5.696e5.69e5.7e5.7e e5.98e5.9e5.88e5.77e5.769e5 9.77e5.769e5.76e5.757e5.9e5.8e5.656e5.65e5.e5.98e5.75e5.7e5 Fowd wveine(), Bckwd wveine(b). Tble : Effect of cicumfeentil wve numbe n on feuency ω. 5 k=k =k = k=k =k =5 h=. m, /h=, /=.5 b =b =. m, n s =n = h =h =.m, m= =5 d/s 5 k=k =k = k=k =k =5 h=. m, /h=, /=.5 b =b =. m, n s =n = h =h =.m, m= =5 d/s : = o : = o : =5 o : =5 o : =6 o : =6 o n 6 8 n 6 8 ine () Fowd wve ine (b) Bckwd wve Figue : Effects of cicumfeentil wve numbe n on feuency (m=). tin Ameicn Jounl of Solids nd Stuctues (6) 67975

14 69 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 5.. Effect of Rotting Speed W Tble descibes effects of otting speed W on the citicl feuency. It is cle tht the citicl feuency ω inceses with the incese of W. Fo emple, in Tble, fo the fowd wve (with = 6 ) when the otting speed W vies the vlues fom to (d/s), the vlue ωc inceses fom.67e5 to.97e5. This incese is consideble bout.9 %. Gphiclly, Figs. nd lso show tht the citicl feuency ω inceses with the incese of W. In ddition when the conicl shell is sttiony stte (fo W =) stnding wve occus. Howeve, fowd nd bckwd wves will ppe when the conicl shell stts to otte. ωc = 5 = = 5 = 6 Ω= d/s.9e5 ().7 e5 ().86e5 ().67e5 () Ω=5 d/s.9e5 ()*.88e5 ().9e5 ().6e5 ().86e5 ().86e5 ().67e5 ().67e5 () Ω= d/s.97e5 ().86e5 ().e5 ().5e5 ().866e5 ().86e5 ().67 e5 ().67e5 () Ω= d/s.e5 ().8e5 ().7e5 ().e5 ().87e5 ().86e5 ().677e5 ().67e5 () Ω=5 d/s.6e5 ().7e5 ().7e5 ().8e5 ().896e5 ().87e5 ().7e5 ().69e5 () Ω= d/s.77e5 ().67e5 ().e5 ().7e5 ().969e5 ().9e5 ().789e5 ().766e5 () Ω=5 d/s.e5 ().77e5 ().8e5 ().e5 ().75e5 ().6e5 ().88e5 ().89e5 () Ω= d/s.e5 ().e5 ().58e5 ().e5 ().6e5 ().5e5 ().97e5 ().96e5 () * Cicumfeentil wve numbe n h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, k k k. Tble : Effect of otting speed W on citicl feuency ωc (inside FGM stiffene)..6. Fowd wve Bckwd wve.6. Fowd wve Bckwd wve c..8 =5 o = o =5 o =6 o k=k =k = Figue : Effects of Ω on citicl feuency ωc (α chnges). c. k=.5.8 k= k=5 Metl = o Figue : Effects of Ω on citicl feuency ωc (k chnges). 5.. Effect of SemiVete Angle Tble pesents effects of semivete ngle on citicl feuency ωc. The geometicl pmetes of shell nd stiffene e given by h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, Ω=5 d/s, k=k=k. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

15 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 69 As cn be obseved tht the citicl feuency of shell deceses when the semivete ngle inceses. This decese is significnt. Fo emple, fo the fowd wve with Ω=5 d/s, k= when the semivete ngle vies the vlues fom to 8, the citicl feuency ωc deceses fom.5e 5 to.e5. Gphiclly, Fig. 5 lso shows tht citicl feuency deceses when inceses. w c k=.5 k= k=5 =.95e5 ().78e5 ().5e5 ().98e5 ().97e5 ().957e5 () =5.8e5 ().67e5 ().e5 ().8e5 ().956e5 ().9e5 () =.58e5 ().5e5 ().9e5 ().76e5 ().96e5 ().9e5 () =.5e5 ().95e5 ().e5 ().e5 ().88e5 ().87e5 () =5.9e5 ().95e5 ().868e5 ().86e5 ().7e5 ().69e5 () =6.75e5 ().7e5 ().67e5 ().67e5 ().57e5 ().5e5 () =75.6e5 ().6e5 ().e5 ().e5 ().55e5 ().9e5 () =8.6e5 ().6e5 ().e5 ().e5 ().8e5 ().87e5 () Tble : Effect of semivete ngle on citicl feuency ωc (inside FGM stiffene). c 8 6 Fowd wve h=. m, /h=, m= /=.5, n s =n = b =b =. m, =5 d/s h =h =.m :k=k =k =.5 :k=k =k = :k=k =k =5 6 8 Stiffened Unstif fened (): /h= (b): /h= 8 Fowd wve b =.8 m, /=, m= b =b =. m, n s =n = 6 h =h =.m, =5 d/s k=k =k =, =5 o.98e (,).6e (,9).89e (,7).86e (,) b 5 n 7 9 Figue 5: Effects of on the citicl feuency ωc. Figue 6: Effects of stiffenes on feuency ω. 5.. Effect of Stiffenes Tble 5 descibes effects of stiffene on feuency with /h=, =.8 m, /=, b=. m, h=. m, b=. m, h=. m, m=, Ω=5 d/s, k=k=k=, α=5. Fom obtined esults s cn be seen with the sme stiffene numbes, the citicl feuency w c of othogonlly stiffened shell is the biggest nd the citicl feuency w c of the shell with the stinge is the smllest. Figs 6 descibes effects of stiffene on feuency with two cses /h= nd /h=. As cn be obseved the w n cuves of stiffened FGM otting conicl shell e lowe thn those of un tin Ameicn Jounl of Solids nd Stuctues (6) 67975

16 69 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil stiffened FGM otting conicl shell when the vlue of n is smlle ny vlue n * (In this emple n * = ) nd invese tend fo n n *. ω 5 Stinge ns=8, n= Ring ns=, n=8 Othogonl ns=, n= n= n= n= n= n= n= n= n= n= n= Tble 5: Effect of stiffenes on the feuency ω. 8 6 Stif f ened Unstif fened Bckwd wve =.8 m, /=, m= b =b =. m, n s =n = h =h =.m, =5 d/s k=k =k =, =5 o.9e (,) (): /h= (b): /h=.85e (,7).85e (,) b n Figue 6b: Effects of stiffenes on the feuency ω. b.69e (,9) c : /= : /=.5 : /= : /= Fowd wve h=. m, /h=, m= b =b =. m, n s =n = h =h =.m, =5 d/s = o, k=k =k k Figue 7: Effects of k on the citicl feuency ωc (/ chnges) Effect of Volume Fction Inde k Tble 6 consides effects of inde volume k on the citicl feuency fo stiffened FGM tuncted conicl shell when the otting speed Ω= d/s, 5 d/s nd d/s. Fig. 7 plots lines c Fig.8 plots lines c w k coesponding to /=.5. w k coesponding to = tin Ameicn Jounl of Solids nd Stuctues (6) 67975

17 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 695 It is found tht the citicl feuency ωc deceses when k incese. This is esonble becuse the vlue incese of k implies the incese of metl constituent in shell. So the stiffness of shell deceses nd tht leds to the citicl feuency ωc deceses. w c Ω= d/s Ω=5 d/s Ω= d/s Cemic.85e5 ().9e5().8e5 ().9e5 ().8e5 () k=.5.98e5 ().5e5 ().95e5 ().e5 ().95e5 () k=.7e5 ().e5 ().e5 ().6e5 ().5e5 () k=5.8e5 ().88e5 ().87e5 ().868e5 ().8e5 () k=.76e5 ().768e5 ().757e5 ().79e5 ().76e5 () Metl.6e5 ().6e5 ().6e5 ().66e5 ().69e5 () h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, α=, k=k=k. Tble 6: Effect of volume fction inde k on the citicl feuency (inside FGM stiffene). 6 Fowd wve h=. m, /h=, m= b =b =. m, n s =n =, k=k =k h =h =.m, =5 d/s c 8 6 : =5 o : = o : =5 o : =6 o k Figue 8: Effects of k on citicl feuency ωc ( chnges)..5.5 c.5 : /= : /=.5 : /= : /= Fowd wve 5 /h Figue 9: Effects of /h on the citicl feuency ωc Effect of /h nd / Tbles 7, 8 nd Fig. 9 illustte effects of the diustothickness tio /h nd lengthtodius tio / on citicl feuency ωc of stiffened FGM tuncted conicl shells with the pmetes given by =.8 m, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, Ω=5 d/s, k=k=k=, α=. Figs. nd show the effects of /h tio nd / tio on feuency ω of the shell. It is obseved tht the citicl feuency ωc s well w decese mkedly with the incese of those tios. This decese is consideble. Fo emple in Tble 7, the vlue w c =.775e5 (with /h=, /=) deceses bout.8 times in compison with w c =.5e5 (with /h=, /=). tin Ameicn Jounl of Solids nd Stuctues (6) 67975

18 696 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil ωc /=.5 5 /h=.775e5 ().66e5 ().78 ().5e5 ().5e5 ().9e5 () 5.6e5 ().69e5 ().9 ().7e5 ().98e5 ().765e5 ().5e5 ().7e5 (). ().997e5 ().85e5 ().67e5 ().6e5 ().9e5 ().899 ().8e5 ().66e5 ().56e5 () 5.e5 ().765e5 ().668 ().597e5 ().97e5 ().8e5 () Tble 7: Effect of /h nd / on the citicl feuency w c (fowd wve). ωc /=.5 5 /h=.765e5 ().5e5 ().768 ().9e5 ().e5 ().889e5 () 5.6e5 ().67e5 ().8 ().95e5 ().95e5 ().75e5 ().e5 ().95e5 ().8 ().98e5 ().79e5 ().66e5 ().6e5 ().7e5 ().887 ().789e5 ().69e5 ().55e5 () 5.e5 ().75e5 ().656 ().585e5 ().8e5 ().5e5 () Tble 8: Effect of /h nd / on the citicl feuency ω (bckwd wve) Fowd wve =.8 m, /=.5, m= b =b =. m, n s =n = h =h =.m, =5 d/s k=k =k =, = o 7 Fowd wve 6 =.8 m, /h=, m= b =b =. m, n s =n = h 5 =h =.m, =5 d/s k=k =k =, = o.e.9e.899e : /h=5 : /h= : /h= 5 n e : /= : /=.5.e : /=.85e n Figue : Effects of /h on the feuency ω Figue : Effects of / on the feuency ω Compison Between Outside FGM Stiffene nd Inside FGM Stiffene Tble 9 compes effects of outside FGM stiffene nd inside FGM stiffene on the citicl feuency with k=.5 5 nd α= nd h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, Ω=5 d/s, k=k=/k. We cn see with the sme input pmete the citicl feuency ωc fo inside FGM stiffene is bigge thn outside FGM stiffene when 5 while 6 75 the citicl feuency ωc fo inside FGM stiffene is smlle thn outside FGM stiffene. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

19 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 697 w c k=.5 k= k=5 =.6e5 ().8e5 ().e5 ().5e5 ().9e5 ().885e5 () b.95e5 ().78e5 ().5e5 ().98e5 ().97e5 ().957e5 () =.5e5 ().e5 ().956e5 ().9e5 ().76e5 ().79e5 () b.5e5 ().95e5 ().e5 ().e5 ().88e5 ().87e5 () =5.99e5 ().9e5 ().8e5 (5).87e5 (5).67e5 ().66e5 () b.9e5 ().95e5 ().868e5 ().86e5 ().7e5 ().69e5 () =6.78e5 ().7e5 ().669e5 ().665e5 ().5e5 ().5e5 () b.75e5 ().7e5 ().67e5 ().67e5 ().57e5 ().5e5 () =75.55e5 ().5e5 ().6e5 ().58e5 ().7e5 ().7e5 () b.6e5 ().6e5 ().e5 ().e5 ().55e5 ().9e5 () Outside FGM stiffene, b Inside FGM stiffene Tble 9: Compison of citicl feuency ωc between outside FGM stiffene nd inside FGM stiffene when the semivete ngle α vies. w c Ω= d/s Ω=5 d/s Ω= d/s k=.5 k= k=5 k=.8e5 ().5e5 ().e5 ().6e5 ().e5 () b.98e5 ().5e5 ().95e5 ().e5 ().95e5 ().98e5 ().956e5 ().9e5 ().976e5 ().9e5 () b.7e5 ().e5 ().e5 ().6e5 ().5e5 ().75e5 ().76e5 ().79e5 ().78e5 ().75e5 () b.8e5 ().88e5 ().87e5 ().868e5 ().8e5 ().7e5 ().79e5 ().697e5 ().7e5 ().7e5 () b.76e5 ().768e5 ().757e5 ().79e5 ().76e5 () Outside FGM stiffene, b Inside FGM stiffene h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, α=, k=k=/k. Tble : Compison of citicl feuency ωc between outside FGM stiffene nd inside FGM stiffene when the volume fction inde k vies. Tble compes effects of inside FGM stiffene nd outside FGM stiffene on the citicl feuency with k=.5 5 nd Ω= 5 d/s. As cn be seen tht the citicl feuency ωc of n inside FGM stiffene ttched shell is bigge thn one of outside FGM stiffene ttched shell Compison Between FGM Stiffene nd Homogeneous Stiffene (Inside Stiffene) Tble compes the citicl feuencies of homogeneous stiffene ttched shell with those of FGM stiffene ttched shell when the volume fction inde k=.5 5 nd α vies the vlues fom o to 6 o. Tble lso compes the citicl feuencies of homogeneous stiffene ttched shell with those of FGM stiffene ttched shell when the volume fction inde k=.5 5 nd the otting speed Ω= 5 (d/s). It is found tht the citicl feuency coesponding to FGM stiffene ttched shell is bigge thn one of homogeneous stiffene ttched shell in these two cses. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

20 698 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil w c k=.5 k= k=5 α=.7e5 ().59e5 ().5e5 ().e5 ().9e5 ().896e5 () b.95e5 ().78e5 ().5e5 ().98e5 ().97e5 ().957e5 () α=5.e5 ().87e5 ().e5 ().7e5 ().95e5 ().9e5 () b.8e5 ().67e5 ().e5 ().8e5 ().956e5 ().9e5 () α=.55e5 ().e5 ().99e5 ().98e5 ().85e5 ().8e5 () b.58e5 ().5e5 ().9e5 ().76e5 ().96e5 ().9e5 () α=.99e5 ().98e5 ().98e5 ().97e5 ().77e5 ().76e5 () b.5e5 ().95e5 ().e5 ().e5 ().88e5 ().87e5 () α=5.89e5 (5).8e5 (5).796e5 (5).79e5 (5).66e5 ().65e5 () b.9e5 ().95e5 ().868e5 ().86e5 ().7e5 ().69e5 () α=6.67e5 ().67e5 ().68e5 ().65e5 ().5e5 ().5e5 () b.75e5 ().7e5 ().67e5 ().67e5 ().57e5 ().5e5 () Homogeneous stiffene, b FGM stiffene h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, Ω=5 d/s. Tble : Effect of semivete ngle α on citicl feuency ωc. w c Ω= d/s Ω=5 d/s Ω= d/s k=.5 k= k=5.985e5 ().99e5 ().98e5 ().e5 ().98e5 () b.98e5 ().5e5 ().95e5 ().e5 ().95e5 ().9e5 ().98e5 ().97e5 ().98e5 ().99e5 () b.7e5 ().e5 ().e5 ().6e5 ().5e5 ().765 e5 ().77e5 ().76e5 ().79e5 ().765e5 () b.8e5 ().88e5 ().87e5 ().868e5 ().8e5 () Homogeneous stiffene, b FGM stiffene h=. m, /h=, /=.5, b=. m, h=. m, ns=, b=. m, h=. m, n=, m=, α=. Tble : Effect of the volume fction inde k on the citicl feuency. 6 CONCUSIONS An nlyticl solution is pesented, in this ppe, to investigte the fee vibtion of otting eccenticlly stiffened functionlly gded tuncted conicl. Some new contibutions e obtined s follows: i. FGM tuncted conicl shells e einfoced by FGM stinges nd ings in which chnge of spcing between stinge stiffenes is consideed ii. A centifugl foce nd Coiolis cceletion e tken into ccount. iii. The sith ode polynomil eution fo ω is obtined nlyticlly nd it is used to nlyze the feuency chcteistics of otting ESFGM conicl shells. iv. Effects of stiffene, geometics pmetes, cone ngle, vibtion modes nd otting speed on feuency chcteistics of the shell fowd nd bckwd wve e discussed in detil. Acknowledgements This esech is funded by Vietnm Ntionl Foundtion fo Science nd Technology Development (NAFOSTED) unde Gnt No tin Ameicn Jounl of Solids nd Stuctues (6) 67975

21 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 699 Refeences Bich, D.H., Dung, D.V., Nm, V.H. (). Nonline dynmic nlysis of eccenticlly stiffened impefect functionlly gded doubly cuved thin shllow shells. Compos Stuct 96: 895. Bich, D.H., Dung, D.V., Nm, V.H., Phuong, N.T. (). Nonline sttic nd dynmic buckling nlysis of impefect eccenticlly stiffened functionlly gded cicul cylindicl thin shells unde il compession. Int J Mech Sci Compos 7: 9. Bich, D.H., Phuong, N.T., Tung, H.V. (). Buckling of functionlly gded conicl pnels unde mechnicl lods. Compos Stuct 9: 798. Bush,D.O., Almoth, B.O. (975). Buckling of bs, pltes nd shells. Mc GwHill, New Yok. Chndsekhn, K., Rmmuti. (98). Aisymmetic fee vibtion of lminted conicl shell. In Poceedings of Int Symposium on the Mech Behviou of Stuctul Medi, Ottw 8. Chndsekhn, K., Rmmuti. (98). Asymmetic fee vibtion of lyeed conicl shells. J Mech Design : 56. Chen, Y., Zho, H.B., Shen, Z.P. (99). Vibtion of high speed otting shells with clcultions fo cylindicl shells. J Sound Vib 6: 76. Civlek, O. (6). An efficient method fo fee vibtion nlysis of otting tuncted conicl shells. Int J Pessue Vessels Piping 8:. Cenwelge, O.E., Muste, D. (969). Fee vibtion of ing nd stinge stiffened conicl shells. J Acoust Soc Am 6: Dung, D.V., Ho,.K. (5). Nonline tosionl buckling nd postbuckling of eccenticlly stiffened FGM cylindicl shells in theml envionment. Compos Pt B. 69: Dung, D.V., Ho,.K., Ng, N.T. (). On the stbility of functionlly gded tuncted conicl shells einfoced by functionlly gded stiffenes nd suounded by n elstic medium. Compos Stuct 8:779. Dung, D.V., Ho,.K., Ng, N.T., Anh,.T.N. (). Instbility of eccenticlly stiffened functionlly gded tuncted conicl shells unde mechnicl lods. Compos. Stuct 6:. Dung, D.V., Nm, V.H. (). Nonline dynmic nlysis of eccenticlly stiffened functionlly gded cicul cylindicl thin shells unde etenl pessue nd suounded by n elstic medium. Euopen J Mech A/Solids 6: 5. Hu,. (). Feuency nlysis of otting tuncted cicul othotopic conicl shells with diffeent boundy conditions. Compos Sci Tech 6: Hu,. (). Feuency chcteistics of otting tuncted cicul lyeed conicl shell. Compos Stuct 5: Koizumi, M. (99). The concept of FGM cemic tsctions. Funct Gdient Mte :. m, K.Y., Hu,. (997). Vibtion nlysis of otting tuncted cicul conicl shell. Int J Solids Stuct ():8 97. m, K.Y., Hu,. (999). Influence of boundy conditions on the feuency chcteistics of otting tuncted cicul conicl shell. J Sound Vib :7 95. m, K.Y., i, H., Ng, T.Y., Chu, C.F. (). Genelized diffeentil udtue method fo the fee vibtion of tuncted conicl pnels. J Sound Vib 5(): 98. iew, K.M., Ng, T.Y., Zho, X. (5). Fee vibtion nlysis of conicl shells vi the elementfee kpritz method. J Sound Vib 8: Mlekzdeh, P., Heydpou, Y. (). Fee vibtion nlysis of otting functionlly gded tuncted conicl shells. Compos Stuct 97: Mecitoglu, Z. (996). Vibtion chcteistics of stiffened conicl shell. J Sound Vib 97():96. tin Ameicn Jounl of Solids nd Stuctues (6) 67975

22 7 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil Mustff, B.A.J., Ali, R. (987). Fee vibtion nlysis of multisymmetic stiffened shells. Comput Stuct 7: 8. Njfizdeh, M.M., Hsni, A., Khzeinejd, P. (9). Mechnicl stbility of functionlly gded stiffened cylindicl shells. Appl Mth Model :557. Ro, S.S., Reddy, E.S. (98). Optimum design of stiffened conicl shells with ntul feuency constints. Comput Stuct ():. Reddy, J.N. (.). Mechnics of lminted composite pltes nd shells: Theoy nd Anlysis, Boc Rton CRC Pess. Shen, H.S. (9). Functionlly gded mteils Nonline nlysis of pltes nd shells. C CRC Pess. Shu, C. (996). An efficient ppoch fo fee vibtion nlysis of conicl shells. Int J Mech Sci 8: Shu, C. (996). Fee vibtion nlysis of composite lminted conicl shells by genelized diffeentil udtue. J Sound Vib 9: Sofiyev, A.H. (9). The vibtion nd stbility behvio of feely suppoted FGM conicl shells subjected to etenl pessue. Compos Stuct 89: Sofiyev, A.H. (). The nonline vibtion of FGM tuncted conicl shells. Compos Stuct 9: 75. Sofiyev, A.H., Kuuoglu, N., Hlilov, H.M. (). The vibtion nd stbility of nonhomogeneous othotopic conicl shells with clmped edges subjected to unifom etenl pessues. Appl Mth Modelling (7): 87. Sinivsn, RS., Kisnn, P.A. (989). Dynmic nlysis of stiffened conicl shell pnels. Comput Stuct (): 8 7. Tlebitooti, M., Ghyou, M., ZieiRd, S., Tlebitooti, R. (). Fee vibtions of otting composite conicl shells with stinge nd ing stiffenes. Ach Appl Mech 8: 5. Tobi, J., Kini, Y., Eslmi, M.R. (). ine theml buckling nlysis of tuncted hybid FGM conicl shells. Compos Pt B 5: 657. Tonbene, F. (9). Fee vibtion nlysis of functionlly gded conicl, cylindicl nd nnul shell stuctues with foupmete powelw distibution. Comput Methods Appl Mech Engg 98: 95. Tonbene, F., Viol, E., Inmn, D.J. (9). D diffeentil udtue solution fo vibtion nlysis of functionlly gded conicl, cylindicl nd nnul shell stuctues. J Sound Vib 8: 599. Volmi, A.S. (97). Nonline dynmic of pltes nd shells. Science Edition (in Russin). Weingten, V.I. (965). Fee vibtion of ing stiffened conicl shells. AIAA J : 758. Xu, C.S., Xi, Z.Q, Chi, C.Y. (996). Nonline theoy nd vibtion nlysis of lminted tuncted thick conicl shells. Int Nonline Mech ():95. APPENDIX I h h h h E b E sb A = bh, A = bh, z =, z =, C =, C =, d = l, d =, d l n l psin E ne E E =, A = A =, A =, A =, B = B = n ( ) n n n n 66 s ne E E ne E B =, B66 =, D = D =, D =, D66 = ( ) n n n n ( n) Fo shell tin Ameicn Jounl of Solids nd Stuctues (6) 67975

23 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil 7 E = = Emh Ecmh /( k ), E Ecmh /( k ) /(k ) û E = E m h / E cm h /( k ) /( k ) /(k ) û = Fo inside stiffene æ h ö hh h = = h hh E s Emh Ecm, E s Em Ecm k èk k æ h ö h 6h h h h h h hh E = s Em Ecm èk k k æ h ö hh h = = h hh E Emh Ecm, E Em Ecm k èk k æ hh ö 6h h h = h h h hh E Em Ecm èk k k æ ö æ ö æ ö c m = c m A = c m A m h m m è k ø è k ød è k l Fo outside stiffene æ ö z h E = s Ec Emc, h / z h / h è ø h k æ ö = z h E Ec Emc, k = = è k / k, h / z h / h h ø k æ h ö h hh h hh E = = s Ech Emc, E s Ec Emc k èk k æ hh ö 6hh h = h hh hh E s Ec Emc èk k k æ h ö h hh h hh E = = Ech Emc, E Ec Emc k èk k æ hh ö 6hh h h hh hh E = Ec Emc èk k k æ ö æ ö æ ö c m = m c A = m c A m h c c è k ø è k ød è k l in which n s, n is the numbe of stinge nd ing stiffene h nd b e the thickness nd width of stinge (diection) h nd b e the thickness nd width of ing (diection). Also, tin Ameicn Jounl of Solids nd Stuctues (6) 67975

24 7 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil d d( ) = nd d e the distnce between two stinges nd two ings, espectively. The untities z, z epesent the eccenticities of stiffenes with espect to the middle sufce of shell (Fig. ). APPENDIX II m p ( ) ( ) m p E sb ( ) H = A sin sin m l n p A66 W ( ) ( ) ( ) n p sin m sin m æ ö W ( ) p E b p n p sin A sin ( è ) A m d ø sin ( ) = ( ) ( ) ( ) H psin p sin m m mnp = ( ) mnp np H ( A A66) cot ( B B66)( ) m m æ E b ö A è A66 d W = W p H p sin [ ( ) ] sin ( ) mp m m 5 5 m p = ( ) ( ) m p ( ) H B sin C sin m m mn p B B66 mp ( ) mp ( ) A cos ( B C ) sin m mp W ( ) [ ( ) ] mp sin ( ) sin cos 5 m p m W ( ) ( ) mp p sin cos B sin ( ) cos m m æ Ebö A è d mnp = ( ) ( ) mnp H ( A A66) ( B B 66) cot m tin Ameicn Jounl of Solids nd Stuctues (6) 67975

25 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil ( ) mnp ( ) 5 [ ( ) ] W sin m 6 m p ( 5 ) mnp W ( ) ( ( ) ) np sin m 5 m p m m æ ö E b pn A A66 ( W è ) cot ( B C B66) np sin d ø m [ ( ) ] ( ) W [ ( ) ] np sin mp m mp m [ ( ) ] ( ) [ ( ) ] H = pw sin pwsin mp m mp m p û m p = ( ) [ ( ) ] m p H A66 sin B 66 cos 5 m p m ( ) ( ) m p ( ) n p D66 cot sin m m sin æ ö E b ( ) n p cot n p cot A ( B è C) ( ) d ø m sin sin æ ö E b ( ) p D p A è 66 sin B66 cos ( ) 6p D66 d ø m mp [ ( ) ] p cot sin A66 sin B66cos ( ) mp m mp W [ ( ) ] 5 [ ( ) ] 5 mp sin W sin 5 5 mp m p m [ ( ) ] ( ) mp m ( ) [ ( ) ] ( ) H = psin p sin 5 m p m ( ) m mnp = ( ) ( ) mnp H ( B B66) cot ( D D 66) m æ ö ( ) n p B C n p cot E b ( ) D è m sin sin d ø tin Ameicn Jounl of Solids nd Stuctues (6) 67975

26 7 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil æ ö E b ( ) np np cot A np D è 66 cot cot ( B C) d ø m æ ö np E b np ( ) cot D66 D ( B è C) ( ) d ø = W ( ) [ ( ) ] H p sin( ) pwsin( ) 5 m p m ( ) ( ) m m p = ( ) [ ( ) ] m p H B sin C sin 5 m p m ( ) ( ) mn p B B66 ( ) m sin m mp ( ) ( ) mp ( ) Acos ( B C )sin m mp W ( ) 5 [ ( ) ] 5 ( ) sin cos m 6 m p m mp W ( ) [ ( ) ] m p sin cos B sin 5 m p m [ ( ) ] pn B C B66 p p ( ) ( B C )sin mp m m sin m m æ ö E b W [ ( ) ] ( ) cos A p W è sin cos p sin cos d ø mp m [ ( ) ] mp m mnp = ( ) ( ) mnp H ( B B66) ( D D 66) cot m æ ö ( ) n p B C cot E b ( ) n p D p n è m sin sin d ø æ ö E b np cot D D66 D ( cot ) ( B è C) ( ) np B66 ( ) d ø æ ö ( ) cot E b np np A è cot ( D D66) D E b d m d û np ( B C B66) ( ) tin Ameicn Jounl of Solids nd Stuctues (6) 67975

27 D.V. Dung nd H.T. Thiem / Resech on Fee Vibtion Feuency Chcteistics of Rotting Functionlly Gded Mteil = W ( ) [ ( ) ] H p sin( ) pwsin( ) 5 m p m ( ) ( ) m m p = ( ) [ ( ) ] m p E sb H D sin sin 5 m p m l æ ö ( ) ( ) E b m n p D D66 n p D è û ø m p sin d sin ( ) m p ( ) ( ) m p Bcos sin m m æ ö E b ( ) cot n p D n p ( B è C) ( ) d ø m sin sin æ ö E b W ( ) [ ( ) ] D D66 D è n p sin d ø 5 m p m W ( ) ( ) p n p sin cos ( B C) ( ) pw m 5 ( ) [ ( ) ] ( ) cos sin p W cos sin 5 m p m æ ö ( ) E b ( ) D D66 p A cot sin n p è û ø m p d m sin æ ö 5 p E b m p sin D D è sin [ ø ( ) ] d mp m ( ) [ ( ) ] ( ) H = psin p sin 5 m p m ( ). m 5 5 tin Ameicn Jounl of Solids nd Stuctues (6) 67975