Mde-Frequency Analyss f Lamnated Sphercal Shell Umut Tpal Department f Cvl Engneerng Karadenz Techncal Unversty 080, Trabzn, Turkey umut@ktu.edu.tr Sessn ENG P50-00 Abstract Ths paper deals wth mde-frequency analyss f a smply-supprted equal-sded sectr f a lamnated sphercal shell. The prblem s mdelled usng fnte element package prgram ANSYS. The frmulatn based n frst-rder shear defrmatn thery. Fur elements are chsen alng each edge f the sectr. The reduced methd f egenvalue slutn s chsen fr the undamped mde-frequency analyss. The frst fve mdes are extracted t btan the fundamental frequency (frst mde natural frequency). The numercal studes are cnducted t determne the effects f wdth-t-thckness rat (b/h), degree f rthtrpy ( E / E ), fber rentatns ( θ ) n the the nn-dmensnal fundamental frequency. The results are gven n graphcal frm and the btaned results are cmpared. Intrductn Fbre renfrced lamnated cmpste materals are beng ncreasngly used n aerspace and ther applcatns due t ther hgh specfc strength, hgh specfc stffness and lw specfc densty. Cmpstes, n the frm f shells, fnd applcatn n aerspace and ther ndustres. Sphercal shells are used fr many structures such as aerspace vehcles, rf dmes, pressure vessels and submarnes. Thus, the free vbratn f cmpste sphercal shell s an mprtant prblem t be nvestgated. The free vbratn f lamnated shell f revlutn has been cnsdered n lteratures [ -7 ]. Ths paper deals wth mde-frequency analyss f smply supprted lamnated sphercal shell usng frst-rder shear defrmatn thery (FSDT). The reduced methd f egenvalue slutn s chsen fr the analyss. The frst fve mdes are extracted t btan the fundamental frequency (frst mde natural frequency).the numercal studes are cnducted t determne the effects f wdth-t-thckness rat (b/h), degree f rthtrpy ( E / E ), fber rentatns ( θ ) n the nndmensnal fundamental frequency. The results are gven n graphcal frm and the btaned results are cmpared. Statement f the prblem Fgure shws a lamnated dubly curved panel (pen shell) f rectangular planfrm, f ttal thckness h. x and x represent the drectns f the lnes f curvature f the mddle surface, whle the x -axs s a straght lne perpendcular t the mddle surface. (=, ) dentes the Prceedngs f The 00 IJME.-. INTETECH Cnference
prncpal rad f curvature f the mddle surface. The thckness f the kth layer s dented by ) h = x x, n whch x and x, k=,..,n are the dstances frm the reference (k ) (k surface t the uter (tp) and nner (bttm) faces, respectvely, f the kth lamna, wth N beng the ttal number f layers. The dsplacement feld, based n frst-rder shear defrmatn thery, s gven by u = ( + x / )u + xφ, =,, u u = () n whch u (=,, ) represents the cmpnents f dsplacement at a pnt x (=,, ), whle u dentes the same fr the crrespndng pnt at the md-surface. Assumptns f shallwness ( ξ ), vanshng gedesc curvatures, transverse nextensblty and the frst-rder shear defrmatn thery, crrespndng knematc relatns f a dubly curved shell are gven: ε = ε + xκ, = ε + xκ ε 5 = ε 5, = ε + xκ ε, ε 4 = ε 4, ε () where u u u ε = u, +, ε = u, +, ε 4 = u, + φ, u ε 5 = u, + φ, ε = u, + u,, κ = φ,, = φ, κ, = φ + φ ( u ) κ,,, u, () Fgure. A lamnated dubly curved panel f rectangular planfrm Prceedngs f The 00 IJME.-. INTETECH Cnference
The equatns f mtn can be derved by substtutng the abve knematc relatns n an expressn fr vrtual wrk, the detals f whch are mtted n the nterest f brevty f presentatn. These can be wrtten as fllws: where Q N, + N, + M, C + = (4) Q N, + N, M, C + = (5) N N Q, + Q, = C (), M, Q C 4 M + = (7), M, Q C5 M + = (8) C C P P = P + u,tt P φ,tt + + (=,), P u, tt P C = (9) + = P + u,tt + Pφ,tt (=,), (0) n whch surface-parallel and rtatry nertas are ncluded. P (=,, ) are as presented belw N = (k ) x (, x, x ) (P, P, P ) = ρ dx k x k () where ρ represents the densty f the layer materal. N, N, N are the surface-parallel stress resultants, whle M,M, M are mment resultants (stress cuples), and Qand Q are the transverse shear stress resultants, all per unt length. The stress resultants and cuples N, M, Q ) are gven as fllws: ( u u N = A u, + A u, + Bφ, + Bφ, + +, (!) N = A ( u, + u, ) + B φ, + φ, ( u, u, ), () Prceedngs f The 00 IJME.-. INTETECH Cnference
u u M = B u, + B u, + Dφ, + Dφ, + +, (4) M = B ( u, + u, ) + D φ, + φ, ( u, u, ) (5) u Q A55 u, K = + φ () where A j, Bj and D j (, j=,, ) are extensnal, cuplng, and bendng rgdtes, respectvely, whle A j (, j=4, 5) dentes transverse shear rgdtes. N, M and Q can be btaned frm the expressns fr N, M and Q, respectvely, by replacng subscrpt by, 5 by 4, by -, and vce versa. K and K are shear crrectn factrs. Fr the fnte element analyss, f the dampng s neglected, the equatn f mtn f the structure fr free vbratn can be wrtten as [ M ]{ D& } + [ K ]{ D} = { 0} & (7) where { D } s a vectr cntanng the unrestraned ndal degrees f freedms, [ ] mass matrx, [ K ] s a structural stffness matrx. Snce { } vectrs { } D & becme where { } D and { } { D} = { D} snωt, { D& } = ω { D} snωt D vectr cntans the ampltudes f { } (7) can be wrtten n as where [ ] [ M ] ( K ){ D} = 0 M s a structural D underges harmnc mtn, the & (8) D vectr and ω s the frequency. Therefre, Eq. λ (9) λ = ω s the egenvalue and { D } becmes the egenvectr Numercal esults and Dscussns Fr ths study, a square plate wth the fllwng dmensns and mechancal prpertes s selected: a=b=00mm, = = 00mm, h = mm, E = 5x0 Pa, = x0 Pa, G = G = 0,5x0 Pa, G = x0 Pa, ν = 0,5, ρ = gm / mm E Prceedngs f The 00 IJME.-. INTETECH Cnference
Effect f Materal Anstrpy Three-layer crss-ply (0/90/0) and angle-ply (45/-45/45) lamnates wth b/h=00 are analysed t study the effect f materal anstrpy. An ncrease f E / E rat, keepng E same, leads t an ncrease n the dmensnless fundamental frequency n the case f bth crss-ply and angleply lamnate (Fgure ). Als as seen, the fundamental frequences n the case f angle-ply lamnate are hgher than thse n the case f bth crss-ply lamnate. 4 0 ω 8 (0/90/0) (45/-45/45) 4 0 5 0 0 5 0 40 50 Fgure. Varatn f fundamental frequency wth materal anstrpy Effect f Fbre Orentatn Fur-layer symmetrc ( α / α / α / α) and ant-symmetrc ( α α / α / α) / lamnates wth the angle f fbre rentatn varyng frm 0 t 45 wth b/h=00 are analysed. As can be seen frm Fgure, a change n fbre rentatn angle frm 0 t 45 leads t an ncrease n the fundamental frequency f vbratn. Als as seen, the fundamental frequences n the case f symmetrc layup are hgher thse n the case ant-symmetrc layup fr 5 and 0, but t s reverse fr 45. 4 0 E / E ω 8 4 (α/-α/-α/α) (α/-α/α/-α) 0 0 5 0 45 α Fgure. Varatn f fundamental frequency wth fbre rentatn angle Prceedngs f The 00 IJME.-. INTETECH Cnference
Effect f Wdth-t-thckness at Fur-layer crss-ply and angle-ply lamnates wth symmetrc and ant-symmetrc arrangement f layers and havng dfferent wdth-t-thckness rats are analysed. As can be seen frm Fgure 4, as a/h ncreases, the dmensnless frequency ncreases. Symmetrc layup has hgher frequency as cmpared t ant-symmetrc layup n the case f crss-ply lamnates whereas the reverse s the case wth angle-ply lamnates. 4 0 ω ATICLE IN PESS 8 4 (0/90/90/0) (0/90/0/90) (45/-45/-45/45) (45/-45/45/-45) 0 0 0 50 00 b/h Fgure 4.Varatn f fundamental frequency wth a/h rat Cnclusns In ths paper, mde-frequency analyss f lamnated sphercal shell usng a fnt element mdel, based n frst-rder shear defrmatn thery s presented. The nn-dmensnal fundamental frequency f vbratn s fund t ncrease wth ncrease n wdth-t-thckness rat, materal anstrpy and angle f fbre rentatn. The fundamental frequences n the case f angle-ply lamnate are hgher than thse n the case f bth crss-ply lamnate fr the effect f the materal anstrpy. The fundamental frequences n the case f symmetrc layup are hgher thse n the case ant-symmetrc layup fr 5 and 0, but t s reverse fr 45. Symmetrc layup has hgher frequency as cmpared t ant-symmetrc layup n the case f crss-ply lamnates whereas the reverse s the case wth angle-ply lamnates fr the effect f the wdth-t0-thckness rat. Ths paper can be nvestgated fr dfferent fnte element frmulatns, bundary cndtns, aspect rats and number f layers. eferences [] Cugnn, J., Gmür Th. and Schrderet, A. Identfcatn by Mdal Analyss f Cmpste Structures Mdelled wth FSDT and HSDT Lamnated Shell Fnte Elements, Cmpstes: Part A 5, 004, Vl. 5, pp. 977-987. [] Chaudhur,. A. and Kabr, H.. H. Effect f Bundary Cnstrant n the Frequency espnse f Mderately Thck Dubly Curved Crss-Ply Panels Usng Mxed Furer Slutn Functns, Jurnal f Sund and Vbratn, 005, Vl. 8, pp. -9. Prceedngs f The 00 IJME.-. INTETECH Cnference
[] Kabr, H.. H. and Chaudhur,. A. Free Vbratns f Ant-Symmetrc Angle-Ply Fnte Dubly Curved Shells, Internatnal Jurnal f Slds and Structures, 99, Vl. 8, pp. 7. [4] Qatu, M.S. and Lessa, A. W. Free Vbratns f Cmpletely Free Dubly Curved Lamnated Cmpste Shallw Shells, Jurnal f Sund and Vbratn, 99, Vl. 5, pp. 9-9. [5] Dng, K. and Tang, L. Exact Slutn fr Axsymmetrc Thck Lamnated Shells, Cmpste Structures, 999, Vl. 4, pp. 5-9. [] Lew, K. M., Peng, L. X. and Ng, T. Y. Three-Dmensnal Vbratn Analyss f Sphercal Shell Panels Subjected t Dfferent Bundary Cndtns, Internatnal Jurnal f Mechancal Scences, 00, Vl. 44, pp. 0 7. [7] am, K. S. S. and Babu, T. S. Free Vbratn f Cmpste Sphercal Shell Cap wth and wthut A Cutut, Cmputers and Structures, 00, Vl. 80, pp. 749-75. [8] Hajang, D and Wequ, C. Exact Shell Thery Analyss f Free Vbratns f Submerged Thn Sphercal Shells, Int. J. Slds Structures, 998, Vl. 5, N., pp. 48-489. [9] Tan, D. Y. Free Vbratn Analyss f Shells f evlutn, Jurnal f Sund and Vbratn, 998, Vl., N., pp. 5-. [0] Yung, P. G. A Parametrc Study n The Axsymmetrc Mdes f Vbratn f Mult- Layered Sphercal Shells wth Lqud Cres f elevance t Head Impact Mdellng, Jurnal f Sund and Vbratn, 00, Vl. 5, N. 4, pp. 5-80. [] Lee, J. J., Oh, I. K., Lee, I. and hu J. J. Nn-lnear Statc and Dynamc Instablty f Cmplete Sphercal Shells Usng Mxed Fnte Element Frmulatn, Internatnal Jurnal f Nn-Lnear Mechancs, 00, Vl. 8, pp. 9 94. [] Engn, A. E. and Lu, Y. K. Axsymmetrc espnse f A Flud-Flled Sphercal Shell n Free Vbratns, Jurnal f Bmechancs, 970, Vl., N., pp. -. [] Nagheh M. snd Hayek, S. J. Vbratn f Fber-enfrced Sphercal Shells, Fbre Scence and Technlgy, 97,Vl. 4, N., pp. 5-7. [4] De Suza, V. C. M. and Crll, C. G. A. Free Vbratns f Orthtrpc Sphercal Shells Engneerng Structures, 98, Vl., N., pp. 7-84. [5] Nrdsn, F. I., Free Vbratns f Thn Elastc Sphercal Shells, Internatnal Jurnal f Slds and Structures, 984, Vl. 0, N. 7, pp. 7-87. [] Evrgen, H and Ertepnar, A. Stablty and Vbratns f Layered Sphercal Shells Made f Hyperelastc Materals, Internatnal Jurnal f Engneerng Scence, 989, Vl. 7, N., pp. -. [7] Narasmhan, M. C. and Alwar,. S. Free Vbratn Analyss f Lamnated Orthtrpc Sphercal Shells, Jurnal f Sund and Vbratn, 99, Vl. 54, N., pp. 55-59. Bgraphes TOPAL UMUT s a research assstant at Mechancal Deparment at Karadenz Techncal Unversty n TUKEY. He s a PhD. student. He wrks n lamnated cmpste structures. Fr example, vbratn, bucklng, strength and ptmzatn f lamnated cmpste structures. Prceedngs f The 00 IJME.-. INTETECH Cnference