TRANSVERSE SHEAR AND NORMAL DEFORMATION THEORY FOR VIBRATION ANALYSIS OF CURVED BANDS

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1 Jounal of Computational and Applied Mechanics, Vol. 5., No. 1., (2004), pp TANSVESE SHEA AND NOMAL DEFOMATION THEOY FO VIBATION ANALYSIS OF CUVED BANDS Béla Kovács Institute of Mathematics, Univesity of Miskolc 3515 Miskolc-Egyetemváos, Hungay [eceived: May 5, 2003] Dedicated to Pofesso József FAKAS on the occasion of his seventy-fifth bithday Abstact. A new laminate model is pesented fo the dynamic analysis of laminated cuved bands. The collocation cuved band is used to denote a cylinde panel in the plane stain state. The diffeential equations which goven the fee vibations of a cuved band and the associated bounday conditions ae deived by Hamilton s pinciple consideing bending, shea and nomal defomation of all layes. The autho used a new iteative pocess to successively efine the stess/stain field in the sandwich cuved band. The model includes the effects of tansvese shea and otay inetia. The iteative model is used to pedict the modal fequencies and damping of simply suppoted sandwich cuved band. The solutions fo a thee-laye cuved band ae compaed to a thee-laye appoximate model. Mathematical Subject Classification: 74H45, 74K10 Keywods: dynamic analysis, vibations, layeed band, damping 1. Intoduction Laminated composite cuved beams have been used in engineeing applications fo many yeas. Design applications of isotopic and cuved bas, ings and aches of abitay shape ae assisted by a well-developed theoy and poven design guidelines [1 4]. The development of the theoy and design guidelines fo composite cuved beams is much less satisfactoy. Ealie woks ae elated to sandwich beams o closed composite ings [5 9]. Thefinite element method was used to study the dynamic esponse of sandwich cuved beams by Ahmed [5 6]. Fee and foced vibations of a thee-laye damped ing wee investigated by Di Taanto [7]. LuandDouglas [8] investigate the damped thee-layeed sandwich ing subjected to a time hamonic adially concentated load. The pape gives an analytical solution fo the mechanical impedance at an abitay point on the suface of the damped stuctue as a function of the focing fequency. Futhemoe, an expeimental pocedue is employed to measue the diving point mechanical impedance as a veification of the calculated c 2004 Miskolc Univesity Pess

2 50 B. Kovács esults. Tansient esponse was studied fo thee-laye closed ings by Sagatz [9]. Damping popeties of cuved sandwich beams with viscoelastic laye wee studied by Tatemichi et al. [10]. Viscoelastic damping in the middle coe laye was emphasized. Nelson and Sullivan [11] analyzed the complete cicula ing consisting of a laye soft viscoelastic mateial sandwiched between two had elastic layes. The equations which goven the foced vibation of a damped cicula ing wee solved by the method of damped foced modes. The essence of the damped foced mode method is the use of hamonic focing functions which ae in-phase with local velocity and popotional to local inetia loads. The constant of popotionality is the loss facto of the composite stuctue, η n. A clea altenative to a damped foced mode solution is to set all the focing functions to zeo and solve the esulting complex eigenvalue poblem. Isvan and Nelson [12] investigated the natual fequencies and composite loss factos of fee vibation of a soft coed cicula ach simply suppoted at each end. Although hamonic motion is assumed, what is not stated is that some hamonic excitation is equied to maintain such motion in the pesence of damping. The dynamic eigenvalue poblem is then posed fo an unfoced system. Kovacs [13] solved the poblem of fee vibations of a stiff coed sandwich cicula ach. All the tangential displacement components ae assumed to be piecewise linea acoss the thickness, thus implying the inclusion of shea defomations and otay inetia. The incemental equations of motion based on the pinciple of vitual displacements of a continuous medium ae fomulated using the total Lagangian desciption by Liao and eddy [14]. They developed a degeneate shell element with a degeneate cuved beam element as a stiffene fo the geometic non-linea analysis of laminated, anisotopic, stiffened shells. Bhimaaddi et al. [15] pesented a 24-d.o.f. of isopaametic finite element fo the analysis of geneally laminated cuved beams. The otay inetia and shea defomation effects wee consideed in this study. Qatu developed a consistent set of equations fo laminated shallow [16] anddeepaches[17].exact solutions ae pesented fo laminated aches having geneal bounday conditions by Qatu and Elshakawy [18]. The in-plane fee vibational analysis of symmetic cossply laminated cicula aches is studied by Yildiim [19]. The fee vibation equations ae deived based on the distibuted paamete model. The tansfe matix method is used in the analysis. The otay inetia, axial and shea defomation effects ae consideed in the Timoshenko analysis by the fist-ode shea defomation theoy. Vaswani, Asnani and Naka [20] deived a closed fom solution fo the system loss factos and esonance fequencies fo a cuved sandwich beam with a viscoelastic coe by the itz method. ao and He [21] used the enegy method and Hamilton s pinciple to deive the govening equation of motion fo the coupled flexual and longitudinal vibation of a cuved sandwich beam system. Both shea and thickness defomations of the adhesive coe ae included. Equations fo obtaining the system modal loss factos and esonance fequencies ae deived fo a system having simply suppoted ends by the itz method. It is well-known that the accuate detemination of the stess fieldinthelami- nate configuations is paticulaly impotant fo stess citical calculations such as damping and delamination. Zapfe and Lesieute [22] developed an iteative pocess to

3 A defomation theoy fo the vibation analysis of cuved bands 51 efine successively the shape of the stess/stain distibution fo the dynamic analysis of laminated beams. The iteative model is used to pedict the modal fequencies and damping of simply suppoted beams with integal viscoelastic layes. The eigenpoblem of the plane bending of cicula ach shaped layeed beams was investigated by using the finite element method [23]. Thefinite element model of the stuctue has two-two elements along the face thickness and thee elements along the thickness of the coe. The two edges of the cicula ach ae simply suppoted. The coesponding model is fomed by eight node hexahedon elements (280 pcs.). Flexue of the thee-laye sandwich ach esults in enegy dissipation due to stains induced in the viscoelastic laye. In a symmetical aangement with identical elastic layes, most of the damping is due to shea of the viscoelastic laye. In an unsymmetical aangement, with dissimila elastic layes, one might expect damping due to diect stain as well as shea in the viscoelastic laye, the fome being known as extensional damping and the latte as shea damping. Both these effects have been included by Kovacs [24]. Howeve, the stess-stain law assumed fo the viscoelastic laye was not stictly coect and was only an appoximation if extensional effects wee consideed. An analysis of the vibation of tansvesely isotopic beams, which have small constant initial cuvatue was pesented in ossettos [25], ossettos and Squies [26]. A closed-fom geneal solution to the govening equations was deived. Natual modes and fequencies wee detemined fo both clamped and simply suppoted end conditions. In Khdei and eddy [27], an analysis of the vibation of slightly cuved coss-ply laminated composite beams is pesented. Hamilton s pinciple is used to deive the equations of motions of fou theoies. Exact natual fequencies ae detemined fo vaious end conditions using the state space concept. The combined effects of initial cuvatue, tansvese shea defomation, othotopicity atio, stacking sequence and bounday conditions ae evaluated and discussed. Yildiim [28] offes a compehensive analysis of fee vibation chaacteistics of symmetic coss-ply laminated cicula aches vibating pependicula to thei planes. Govening equations of symmetic laminated cicula aches made of a linea, homogeneous, and othotopic mateial ae obtained in a staightfowad manne based on the classical beam theoy. The tansfe matix method is used fo the fee vibation analysis of the continuous paamete system. The pesent eseach offes a new laminated model fo the dynamic analysis of laminated cuved bands, which includes both tansvese shea and tansvese nomal effects. The diffeential equations which goven the fee vibations of a cuved band and the associated bounday conditions ae deived by Hamilton s pinciple consideing bending, shea and nomal defomation of all layes. The autho used a new iteative pocess to successively efine the stess/stain fields in the sandwich band. The model includes the effects of tansvese shea and otay inetia. The cuent model is developed fo the specific case of a simply suppoted cuved band with unifom popeties along the length.

4 52 B. Kovács 2. Govening equations of motion The geomety of inteest and the notations used ae shown in Figue 1. As indicated in the Figue, the cuved band ends ae simply suppoted. The collocation cuved band is used to denote a cylinde panel in the plane stain state. Conside the cuved band with a cylinde middle suface and the adius of cuvatue of the middle suface. The cuved band consists of thee diffeent layes of homogeneous mateials bonded togethe to fom a composite stuctue. Subscipt i, whee i =1, 2, 3 is used to denote quantities in the vaious layes, stating fom the outemost laye, so that layes 1,3 epesent the elastic layes while laye 2 epesents the viscoelastic laye. A state of plane stain is assumed, as well as the fact that the mateials in each laye of the band ae homogeneous and isotopic. Pefect bonding of the layes and linea elasticity ae also assumed in the analysis. The composite band is lightly damped and it is assumed that all the enegy dissipated is dissipated in the viscoelastic laye. thid laye cente line e e ϕ w ( ϕ,t) 2h 3 2h2 2h1 e e z b second laye 3 ϕ 4 ϑ 2 1 fist laye ( ϕ t) v, O Figue 1. Geomety of the laminated cuved band The fom of the displacement field ove the domain of the cuved band is t(, ϕ, t) =u(, ϕ, t)e ϕ + w(, ϕ, t)e = = v 0 (ϕ, t) µ w0 ϕ v 0(ϕ, t) + f()v 1 (ϕ, t) e ϕ +[w 0 (ϕ, t)+g () w 1 (ϕ, t)] e (2.1) whee f () =0and g () =0, so (v 0,w 0 ) denote the displacement of a point (, ϕ) of the cente-suface along the cicumfeential and adial diections, espectively. The tems f()v 1 (ϕ, t) and g () w 1 (ϕ, t) can be thought to be coection to account fo tansvese shea and nomal defomation effects, espectively. The functions f() and g () epesent the shape of the coections though the thickness of the cuved

5 A defomation theoy fo the vibation analysis of cuved bands 53 band, while v 1 (ϕ, t) and w 1 (ϕ, t) detemine its distibution along the cicumfeential diection. The solution of a given poblem equies the detemination of the unknown functions v 0 (ϕ, t),v 1 (ϕ, t),w 0 (ϕ, t),w 1 (ϕ, t),f() and g (). By using the standad expessions t = we + ue ϕ, ε ϕ = 1 u ϕ + w, γ ϕ = 1 w ϕ + u u, ε = w, the stain tenso of each laye can be computed fom equations (2.1): ε ϕ = 1 v0 ϕ µ 2 w 0 ϕ 2 v 0 + f() v 1 ϕ ϕ + w 0 + g () w 1 (ϕ, t), (2.2) df γ ϕ = d f () v 1 (ϕ, t)+ g () w 1 ϕ, (2.3) ε = dg d w 1(ϕ, t), (2.4) whee f 1 () if 1 2 f() = f 2 () if 2 3 f 3 () if 3 4 and g 1 () if 1 2 g() = g 2 () if 2 3 g 3 () if 3 4 ae single-valued functions definedateachpointthoughthethickness. Fom equation (2.3), it can be seen that the functions df d f and g epesent the shape of the tansvese shea stain field though the thickness of the cuved band at agivenϕ -location. While the assumed fom of the functions, f() and g () changes fom one iteation to the next, at any given iteation they can be teated as known functions. Thecuvedbandvibatesintheϕ-plane. It is assumed that the plane stain state occus within the stuctue and thus within each i-th laye. By teating the poblem as a plane stain one and assuming that the mateials in each laye ae homogeneous and isotopic, we have the following stess-stain elations: σ i = E i (1 2ν i )(1+ν i ) ((1 ν i) ε i + ν i ε ϕi ), (2.5) E i σ ϕi = (1 2ν i )(1+ν i ) ((1 ν i) ε ϕi + ν i ε i ), (2.6) τ ϕi = G i γ ϕi, (2.7) whee E i is the Young s modulus and G i is the shea modulus within the i-th laye of the stuctue. Also ν i is the Poisson s coefficient, which chaacteises the compession in the adial diection due to tension in the cicumfeential diection, and vice vesa. In addition the stesses τ ϕzi,τ zi ae equal to zeo.

6 54 B. Kovács The stain enegy stoed in the cuved band is given by: U = b 2 3X ϑz i=1 ϕ=0 Z i+1 i [σ i ε i + σ ϕi ε ϕi + τ ϕi γ ϕi ] ddϕ. (2.8) The kinetic enegy, which includes components associated with tansvese, in plane and otay inetia, is given by T = b 2 3X ϑz Z i+1 i=1 ϕ=0 i ρ i µ ti 2 ddϕ, (2.9) whee the dots ove t 1, t 2 and t 3 denote the patial deivative with espect to time. The diffeential equations of motion and bounday conditions ae deived using Hamilton s pinciple. The equations of motion fo the fou unknown functions, w 0 (ϕ, t), w 1 (ϕ, t),v 0 (ϕ, t) and v 1 (ϕ, t) ae 4 w 0 A 11 ϕ 4 + A 2 w 0 12 ϕ 2 + A 3 v 0 13 ϕ 3 + A 3 v 1 14 ϕ 3 + A v 0 15 ϕ + A v 1 16 ϕ + A 2 w 1 17 ϕ w 0 +A 18 w 0 + A 19 w 1 = D 11 ϕ 2 t 2 + D 3 v 0 12 ϕ t 2 + D 3 v 1 13 ϕ t 2 + D 2 w 0 2 w 1 14 t 2 + D 15 t 2, (2.10) 3 w 0 A 21 ϕ 3 + A w 0 22 ϕ + A 2 v 0 23 ϕ 2 + A 2 v 1 24 ϕ 2 + A w 1 25 ϕ + A 26v 1 = 3 w 0 = D 21 ϕ t 2 + D 2 v 0 22 t 2 + D 2 v 1 23 t 2, (2.11) 2 w 0 A 31 ϕ 2 +A 2 w 1 32 ϕ 2 +A v 0 33 ϕ +A v 1 34 ϕ +A 2 w 0 35w 0 +A 36 w 1 = D 31 t 2 +D 2 w 1 32 t 2, (2.12) 3 w 0 A 41 ϕ 3 + A w 0 42 ϕ + A 2 v 0 43 ϕ 2 + A 2 v 1 44 ϕ 2 + A w 1 45 ϕ = 3 w 0 = D 41 ϕ t 2 + D 2 v 0 42 t 2 + D 2 v 1 43 t 2, (2.13) whee A ij and D ij ae given in the Appendix. K 1 18 and M 1 8 ae section stiffness and mass coefficients, given by K [1,...,7] = b K [8,17,18] = b 3X Z i+1 i=1 i 3X Z i+1 i=1 i E i (1 ν i ) (1 2ν i )(1+ν i ) G i 1,, 1,f i,g i, 1 f i, 1 g i d, (2.14) " µ 1 dfi g2 i, d f 2 µ i dfi, d f # i g i d, (2.15)

7 A defomation theoy fo the vibation analysis of cuved bands 55 i+1 3X Z " E i (1 ν i ) 1 (1 2ν i )(1+ν i ) f i 2, 1 g2 i, 1 µ # 2 f 1 dgi ig i, d (2.16) d K [9,...,12] = b K [13,...,16] = b i=1 i 3X Z i+1 i=1 i M [1,2,3,4,5,6,7,8] = b E i ν i (1 2ν i )(1+ν i ) 3X Z i+1 i=1 i ρ i dg i d, dg i d,f dg i i d,g i dg i d, (2.17) d, 2, 3,f i, 2 f i,fi 2,g i,gi 2 d, (2.18) whee f i = f i () and g i = g i (). The kinematic and natual bounday conditions specified at ϕ =0and ϕ = ϑ,aegivenby KINEMATIC NATUAL v 0 =0 o F 2 w 0 v 11 ϕ + F ϕ + F 13 v1 ϕ + F 14w 0 + F 15 w 1 =0, w 0 =0 o F 21 3 w 0 ϕ 3 + F 22 2 v 0 ϕ 2 + F 23 2 v 1 ϕ 2 + F 24 3 w 0 ϕ 2 t + v 1 =0 o F 31 2 w 0 +F 2 v 0 25 t + F 2 v t =0, 2 v ϕ + F ϕ + F 33 v 1 ϕ + F 34w 0 + F 35 w 1 =0, w 0 ϕ =0 o F 41 2 w 0 v ϕ + F ϕ + F 43 v1 w w 1 =0 o F 1 51 ϕ + F 52v 1 =0, ϕ + F 44w 0 + F 45 w 1 =0, (2.19) whee F ij ae constants. Fo the special case of a simply suppoted cuved band, the fist, thid and fouth natual bounday conditions ae combined with the kinematic condition, w 0 = w 1 =0. 3. Solution fo a simply suppoted cuved band Sinusoidal mode shapes that satisfy the bounday conditions ae assumed. Consequently, the assumed displacements ae: w 0 (ϕ, t) =W 0 sin(k n ϕ)e iωnt, (3.1) w 1 (ϕ, t) =W 1 sin(k n ϕ)e iωnt, (3.2) v 0 (ϕ, t) =V 0 cos(k n ϕ)e iωnt, (3.3) v 1 (ϕ, t) =V 1 cos(k n ϕ)e iωnt, (3.4) whee k n =(nπ) /ϑ. Since the motion is now hamonic, it is justified to admit hysteetic damping into the viscoelastic laye by putting complex moduli. The Young s and shea modulus of the constituent mateials ae epesented by the complex quantities E2 = E 2(1 + iα 2 ), G 2 = G 2(1 + iβ 2 ), (3.5) whee α 2 and β 2 denote the mateial loss factos in extension and shea, espectively. Since G 2 and E2 ae used as complex moduli of the middle laye, the diffeential equations of motion will have complex coefficients. The substitution of equations (3.1),

8 56 B. Kovács (3.2), (3.3) and (3.4) into equations ( ), will esult in a set of fou simultaneous, homogeneous algebaic equations with symmetic and complex coefficients. In matix fom, these equations ae ω 2 n [M]+[Y] {U} =0, {U} = {V 0,V 1,W 0,W 1 } (3.6) whee M ij and Y ij ae in the Appendix. The complex eigenvalues give the desied natual fequencies and mode shapes with thei phase elations. The natual fequency is appoximately equal to the squae oot of the eal pat of the eigenvalue. The modal loss facto fo the n-th mode is appoximately equal to the atio of the imaginay pat of the eigenvalue to the eal pat of the eigenvalue η n =Im(ωn)/ 2 e(ωn) 2. (3.7) 4. Impoved estimate fo shea coection functions f() and g() An impoved estimate fo the coection functions f() and g () is deived fom the equation of elementay stess equilibium. The equations of motion in plane stain 2 σ ϕ τ ϕ + ϕ = 2 ρ 2 u t 2, (4.1) σ + σ + τ ϕ ϕ σ ϕ = ρ 2 w t 2, (4.2) applied to the layeed cuved band with σ ϕi = E i (ε ϕi + ν i ε i )/ 1 νi 2 expessions, aenowinthefom (2 τ ϕi )+ E i (1 ν i ) (1 2ν i )(1+ν i ) 2 v 0 +f i () 2 v 1 ϕ 2 + w 0 ϕ + g i() w 1 ϕ 3 w 0 ϕ 3 + ϕ 2 E i ν i + (1 2ν i )(1+ν i ) µ 3 w 0 ϕ t 2 2 v 0 t 2 dg i w 1 d ϕ = 2 = 2 v 0 ρ i t 2 + f i () 2 v 1 t 2, (4.3) σ + σ 1 E i (1 ν i ) v 0 (1 2ν i )(1+ν i ) ϕ 2 w 0 ϕ 2 + f i() v 1 ϕ + w 0 + g i ()w 1 E i ν i dg i (1 2ν i )(1+ν i ) d w 1 + τ µ ϕi 2 ϕ = ρ w 0 i t 2 + g i() 2 w 1 t 2, (4.4) whee i =1, 2, 3 and i i+1. Using equations (2.3), (3.1), (3.2), (3.3) and (3.4), it is obvious that equations (4.3) and (4.4) can be witten the following fom d d (2 τϕi)+ E i (1 ν i ) (1 2ν i )(1+ν i ) k2 nv 0 + k3 nw 0 f i ()knv knw g i ()knw 1 2 E i ν i + (1 2ν i )(1+ν i ) dg i d k nw 1 =

9 dσ i d A defomation theoy fo the vibation analysis of cuved bands 57 = 2 ρ i ωn 2 V 0 (k nw 0 V 0 )+f i ()V 1, (4.5) + σ i () 1 E i (1 ν i ) (1 2ν i )(1+ν i ) k nv 0 + k2 nw 0 f i ()k n V 1 + E i ν i dg i (1 2ν i )(1+ν i ) d W 1 k n τ ϕi () = ρ i ωn 2 (W 0 + g i ()W 1 ), (4.6) +W 0 +g i ()W 1 whee τ ϕi (, ϕ, t) = τ ϕi ()cos(k nϕ)e iωnt and σ i (, ϕ, t) = σ i ()sin(k nϕ)e iωnt. The shape of the shea stess distibution can be found by integating equation (4.5) though the thickness τ ϕi () = 1 2 Z i E i ν i + (1 2ν i )(1+ν i ) i ½ 2 ρ i ω 2 n V 0 dg i d k nw 1 E i (1 ν i ) (1 2ν i )(1+ν i ) (k nw 0 V 0 )+f i ()V 1 + k2 nv 0 k3 nw 0 + f i ()k 2 nv 1 k 2 nw 0 g i ()k 2 nw 1 ¾ d c i, (4.7) whee c 1 =0, c 2 = 2τ 2 ϕ1 ( 2 ), c 2 = 3τ 2 ϕ2 ( 3 ). (4.8) Then if equation (4.7) is used in equation (4.6), the shape of the nomal stess distibution can be found by integating equation (4.6) though the thickness σ i () = 1 Z ½ ρ i ω 2 E i ν i dg i n (W 0 + g i ()W 1 ) (1 2ν i )(1+ν i ) d W 1 k n τ ϕi () 1 E i (1 ν i ) (1 2ν i )(1+ν i ) k nv 0 + ¾ k2 nw 0 f i ()k n V 1 + W 0 + g i ()W 1 d+ 1 d i, (4.9) whee d 1 =0, d 2 = 2 σ 1 ( 2 ), d 2 = 3 σ 2 ( 3 ). (4.10) The shape of the tensile stain distibution ε i is calculated using equation (4.9) and the constitutive equation (2.5) ε i () = 1 νi 2 σ i () ν i ε ϕi (), (4.11) E i whee ε i (, ϕ, t) =ε i ()sin(k nϕ)e iωnt and ε ϕi (, ϕ, t) =ε ϕi ()sin(k nϕ)e iωnt. Upon substitution of equation (4.11) into equation (2.4), the new estimate fo the nomal coection function g () obtained by integating equation (2.4) though the thickness, is given by

10 58 B. Kovács g 1 () = g 2 () = g 3 () = Z 2 Z Z 3 Z 1 ε 1 2 () d + ε 1 W 1 2 W () d, 1 1 2, (4.12) 1 W 1 ε 2 () d, 2 3, (4.13) 1 W 1 ε 2 () d + Z 3 1 W 1 ε 3 () d, 3 4. (4.14) Evidently g () =g 2 () =0 at the efeence axis. The shape of the shea stain distibution is calculated using equation (4.7) and the constitutive elation γ ϕi () = τ ϕi (), i =1, 2, 3 (4.15) G i whee γ ϕi (, ϕ, t) =γ ϕi ()cos(k nϕ)e iωnt. Upon substitution of equation (4.15) into equation (2.3) and using equations ( ), the new estimate fo the shea coection function f () obtained by integating equation (2.3) though the thickness, is given by f 1 () = " Z 2 whee 1 2 f 2 () = Z 1 " 1 γ ϕ2 () # Z " W 1 g 2 () 1 γ ϕ1 k n d + () # # W 1 g 1 () k n d, G G 2 G G (4.16) " γ ϕ2 () W 1 g 2 () k n G G # d, 2 3 (4.17) and " Z " # 3 Z " # # 1 γ ϕ2 () W 1 g 2 () 1 γ ϕ3 () W 1 g 3 () f 3 () = k n d + k n d G G 3 G G (4.18) whee 3 4 and evidently f () =f 2 () =0 at the efeence axis. The integals in equations ( ) ae evaluated numeically using a tapezoidal method and f () and g () can be complex quantities. This new estimates of f () and g () ae used as the coection functions fo the next iteation. As with any smeaed laminate model, thee ae two distinct ways to calculate the shea stess distibution: fom the mateial constitutive elations; o by the elementay stess equilibium. The ultimate goal of the iteative analysis is the detemination of the functions, f () and g (), that cause the two stess distibutions to be equal. This defines the convegence point fo the iteative functions f () and g (), the point at which the stesses and stains ae self-consistent.

11 A defomation theoy fo the vibation analysis of cuved bands 59 Table 1 Vaiation of the lowest fequency and the loss facto with adhesive shea modulus G 2 N/m 2 [20] Pesent theoy z } { z } f [Hz] η f [Hz] η { 6, ,898 0,0644 7,508 0,0624 6, ,36 0, , , ,94 0, , , ,8 0, , , ,47 0, ,23 0, esults and discussion Numeical esults wee geneated to obseve the effects of cuvatue, coe thickness and adhesive shea modulus on the system natual fequencies ω n and modal loss factos η n. Vaswani et al. [20] assembled a seies of design cuves fo the dynamic chaacteization of a thee-laye damped cicula ing segment which is simply suppoted at each end, see Fig.1. The model assumes that all tansvese shea defomation and enegy dissipation occus in the coe mateial. The dissipation is modeled using a complex modulus fomulation. The esonant fequencies and the associated system loss facto have been expeimentally detemined fo fou sandwich beam specimens and the values compaed with those obtained theoetically. easonably good ageement is seen between the theoetical and expeimental esults. Howeve, the model of Vaswani et al. ovepedicts natual fequencies by 5%, appoximately. The pesent smeaed laminate model was compaed to the design cuves of Vaswani et al. fo the fist tansvese modes pesented in the pape by Vaswani et al., with simply suppoted bounday conditions. Table 2 Vaiation of the lowest fequency with adhesive thickness z [20] } { Pesent theoy z } { 2h 2 [mm] f [Hz] f [Hz] 1,0 17,981 17,028 2,0 19,1 18,125 3,0 20,09 19,04 4,0 20,94 19,81 5,0 21,66 20,47 The adhesive shea modulus plays a vey impotant ole in the damping of a sandwich cuved band. The vaiations of the lowest natual fequency and the associated loss facto with espect to the shea modulus G 2 (= eal pat of G 2) ae given in Table 1 fo the thee laye ach using the design cuves of Vaswani et al. and the pesent laminate model. The input data used hee wee h 1 = h 2 = h 3 =2.0 mm, υ =1.0, α 2 = β 2 =0.5, =1.0 m, E 1 = E 3 = N/m 2,G 1 = G 3 = N/m 2,

12 60 B. Kovács ρ 1 = ρ 3 = kg/m 3,ρ 2 = ρ 1 /2. G 2 vaied fom N/m 2 to N/m 2 and E 2 =2.5 G 2. The pesent smeaed laminate model fequency pedictions ae geneally consistent with the esults of Vaswani et al. The slight discepancy is due to facesheet shea and otay inetia, effects which the model of Vaswani et al. does not conside. The model of Vaswani et al. ovepedicts natual fequencies by 5%, appoximately. The modal loss factos pedicted by the pesent laminate model ae also in good ageement with the esults of Vaswani et al. The vaiation of the system loss facto η with the shea modulus G 2 is simila to that obtained fo staight sandwich beams. Fo each coe thickness, a maximum is obseved which inceases as the coe thickness inceases and is also seen to occu at highe values of the shea modulus. At low values of the shea modulus, although the defomations ae lage, shea stiffness is small, hence low damping is obseved. At vey high values of the shea modulus, the shea stiffness is high and the defomations ae small, again esulting in low damping. Table 3 Vaiation of the loss facto with adhesive thickness z [20] } { Pesent theoy z } { 2h 2 [mm] η η 1,0 0, ,0 0, ,0 0, ,0 0, ,0 0,196 0,199 The effects of the adhesive thickness 2h 2 on the system natual fequencies and loss factos ae also studied. The input data in this case wee h 1 = h 3 =2.0 mm, υ =1.0, α 2 = β 2 =0.5, =1.0 m, E 1 = E 3 = N/m 2,G 1 = G 3 = N/m 2,ρ 1 = ρ 3 = kg/m 3,ρ 2 = ρ 1 /2, G 2 = N/m 2,E 2 =2.5 G 2. The thickness 2h 2 was inceased fom 1.0 mm to 5.0 mm in steps of 1.0 mm. The vaiations of f and η with 2h 2 ae given in Tables 2-3. It can be seen fom these Tables that both f and η incease with 2h 2. Table 4 Vaiation of the lowest fequency with adius z [20] } { Pesent theoy z } { [mm] f [Hz] f [Hz] ,75 28, ,78 23, ,94 19, ,9 16, ,469 14,658

13 A defomation theoy fo the vibation analysis of cuved bands 61 The thid paamete which effects the system natual fequencies and modal loss factos is the adius of cuvatue of the middle suface of the adhesive laye. In this case, the angle υ is kept constant, while changing. This means the total length of the sandwich ach system will change with. The vaiations of f and η with ae shown in Tables 4-5. The input data was h 1 = h 2 = h 3 =2.0 mm, υ =1.0, α 2 = β 2 =0.5, E 1 = E 3 = N/m 2,G 1 = G 3 = N/m 2,ρ 1 = ρ 3 = kg/m 3, ρ 2 = ρ 1 /2, G 2 = N/m 2,E 2 =2.5 G 2.vaied fom 800 mm to 1200 mm in steps of 100 mm. It can be seen that f deceases with. The vaiations of f with ae obvious, as the total length of the cuved sandwich beam system inceases with any incease in. Table 5 Vaiation of the loss facto with adius. z [20] } { Pesent theoy z } { [mm] η f [Hz] 800 0,211 0, ,1895 0, ,1696 0, ,1516 0, ,1357 0, Conclusions A new iteative laminate model has been pesented fo a thin sandwich ach that can accuately detemine the dynamic stess distibution in soft as well as had coed sandwich aches. This epesents an advance ove pevious smeaed laminate models, in which accuate estimates of the stess fieldweeonlypossibleiftheassumed displacement field was a easonable appoximation of the actual diplacement field. EFEENCES 1. Chidampaam, P. and Leissa, A.W.: Vibations of plana cuved beams, ings, and aches. Applied Mechanics eview, 46, (1993), Laua, P.A.A. and Mauizi, M.J.: ecent eseach on vibations of ach-type stuctues. The Shock and Vibation Digest, 19, (1987), Makus, S. andnanasi, T.: Vibations of cuved beams. The Shock and Vibations Digest, 7, (1981), Davis,., Henshell,.D. and Wabuton, G.B.: Constant cuvatue beam finite elements fo in-plane vibation. Jounal Sound and Vibations, 25, (1972), Ahmed, K.M.: Fee vibations of cuved sandwich beams by the method of finite elements. Jounal Sound and Vibations, 18, (1971), Ahmed, K.M.: Dynamic analysis of sandwich beams. Jounal Sound and Vibations, 21, (1972),

14 62 B. Kovács 7. Ditaanto,.A.: Fee and foced esponse of a laminated ing. Jounal of Acoustical Society of Ameica, 53, (1973), Lu, Y.P. and Douglas, B.E.: On the foced vibations of thee-laye damped sandwich ing. Jounal Sound and Vibations, 32, (1974), Sagatz, M.J.: Tansient esponse of thee-layeed ings. Jounal Applied Mechanics, 44, (1977), Tatemichi, A., Okazaki, A. andhikayama, M.: Damping popeties of cuved sandwich beams with viscoelastic laye. Bulletin of Nagaya Institute, 29, (1980), Nelson, F. C. andsullivan, D.F.: The Foced vibations of a thee-laye damped cicula ing. Jounal of Ameican Society of Mechanical Enginees, 154, (1977), Isvan, O. and Nelson, F. C.: Fee vibations of a thee-laye damped cicula ing segment. Mecanique Mateiaux Electicite, , (1982), Kovacs, B.: Fee vibations of a layeed damped ach. Publications of the Univesity of Miskolc, 36, (1996), Liao, C. and eddy, J.N.: Analysis of anisotopic, stiffened composite laminates using a continuum-based shell element. Computes and Stuctues, 34, (1990), Bhimaaddi, A., Ca, A.J. and Moss, P.J.: Genealized finite element analysis of laminated cuved beams with constant cuvatue. Computes and Stuctues, 31, (1989), Qatu, M.S.: In-plane vibation of slightly cuved laminated composite beams. Jounal of Sound and Vibations, 159, (1992), Qatu, M.S. and Elshakawy, A.A.: Vibation of laminated composite aches with deep cuvatue and abitay boundaies. Computes and Stuctues, 47, (1993), Qatu, M.S.: Equations fo the analysis of thin and modeately thick laminated composite cuved beams. Intenational Jounal of Solid and Stuctues, 30, (1992), Yildiim, V.: otay inetia, axial and shea defomation effects on the in-plane natual fequencies of symmetic coss-ply laminated cicula aches. Jounal of Sound and Vibation, 224, (1999), Vaswani, J., Asnani, N.T.and Naka, B.C.: Vibation and damping analysis of cuved sandwich beams with a viscoelastic coe. Composite stuctues, 10, (1988), He, S. andao, M.D.: Pediction of loss factos of cuved sandwich beams. Jounal of Sound and Vibation, 159, (1992 ), Zapfe, J. A. andlesieute, G. A.: Vibation analysis of laminated beams using an iteative smeaed laminate model. Jounal of Sound and Vibation, 199, (1997), Kovacs, B., Szabo, F.J. and Doboczoni, A.: Fee vibations of a layeed cicula ing segment. micocad-system, 93 Intenational Compute Science Meeting Poceding, Miskolc, Hungay, (1993), Kovacs, B.: Vibation and stability of layeed cicula ach, Ph.D. thesis, Hungaian Academy of Sciences, Budapest, (1992). (in Hungaian) 25. ossettos, J.N.: Vibation of slightly cuved beams of tansvesely isotopic composite mateials. Ameican Institute of Aeonautics and Astonautic Jounal, 9, (1971),

15 A defomation theoy fo the vibation analysis of cuved bands ossettos, J.N. and Squies, D.C.: Modes and fequencies of tansvesely isotopic slightly cuved Timoshenko beams. Jounal of Applied Mechanics, 40, (1973), Khdei, A.A. and eddy, J.N.: Fee and foced vibation of coss-ply laminated composite shallow aches. Intenational Jounal of Solids and Stuctues, 34, (1997 ), Yildiim, V. : Out-of-plane fee vibation chaacteistics of symmetic coss-ply laminated composite aches with deep cuvatue. Jounal of the Mechanical Behavio of Mateials, 10, (1999), Appendix A. NOMENCLATUE b E i E2 e e ϕ e z ε ϕi ε i f() g () γ ϕ i G 2 G i h i ϕ n T σ ϕi σ i τ ϕ i t i α 2 β 2 η n ω n f n ρ i width of the cuved band elastic modulus of laye i complex modulus in tension unit vecto in the adial diection unit vecto in the tansvese diection unit vecto in the z-diection tensile stain of laye i in the tansvese diection tensile stain of laye i in the adial diection shea coection function nomal coection function shea stain of laye i complex modulus in shea shea modulus of laye i half-thickness of laye i cicumfeential coodinate mode numbe cylindical coodinate adius of middle suface of the cuved band kinetic enegy tensile stess of laye i in the tansvese diection tensile stess of laye i in the adial diection shea stess of laye i displacement vecto of laye i adius at the bottom of the fist laye adius at the top of the fist laye adius at the bottom of the thid laye adius at the top of the thid laye mateial loss facto in tension of the second laye mateial loss facto in shea of the second laye composite loss facto fo the n-th mode fequency of oscillation in adians fo the n-th mode fequency of oscillation in Hetz fo the n-th mode density of laye i

16 64 B. Kovács ϑ v 0 w 0 opening angle of the cuved band tangential displacement of the middle suface adial displacement of the middle suface Appendix B. Definitions fo the vaious coefficients Equations (2.10) to (2.13) in the main text contain cetain A ij and D ij tems which ae defined as follows: A 11 = K 3 + K 2 / 2 2K 1 /, A 12 =2K 3 2K 1 /, A 13 = K 2 / 2 + K 1 / A 14 = K 6 K 4 /, A 15 = K 1 /, A 16 = K 6, A 17 = K 14 + K 7 K 13 / K 5 / A 18 = K 3, A 19 = K 14 + K 7, A 21 = K 4 / K 6, A 22 = K 6, A 23 = K 4 /, A 24 = K 9, A 25 = K 18 K 15 K 11, A 26 = K 17 A 31 = K 13 / + K 14 K 5 / + K 7, A 32 = K 8, A 33 = K 13 / + K 5 / A 34 = K 15 + K 11 K 18, A 35 = K 7 + K 14, A 36 = K 12 +2K 16 + K 10 A 41 = K 2 / 2 K 1 /, A 42 = K 1 /, A 43 = K 2 / 2, A 44 = K 4 / A 45 = K 13 / K 5 / D 11 = M 1 2M 2 / + M 3 / 2, D 12 = M 2 / M 3 / 2, D 13 = M 4 M 5 / D 14 = M 1, D 15 = M 7, D 21 = M 4 + M 5 /, D 22 = M 5 / D 23 = M 6, D 31 = M 7, D 32 = M 8, D 41 = M 2 / + M 3 / 2 D 42 = M 3 / 2, D 43 = M 5 /. Equation (3.6) in the main text contain Y ij and M ij tems which ae defined as follows: Y 11 = 2kn 2 2kn 4 K1 / + knk 4 2 / kn 2 + kn 4 K3 Y 12 = Y 21 = k n K 6 + kn 3 (K 6 K 4 /), Y 14 = Y 41 = k n K 1 / + kn 3 K1 / K 2 / 2, Y 13 = Y 31 = K 14 + K 7 kn 2 (K 14 + K 7 K 13 / K 5 /), Y 22 = K 17 + knk 2 9 Y 23 = Y 32 = k n (K 18 K 15 K 11 ), Y 24 = Y 42 = k 2 nk 4 /, Y 44 = k 2 nk 2 / 2 Y 33 = K 12 +2K 16 + K 10 + knk 2 8, Y 34 = Y 43 = k n (K 13 / + K 5 /) M 11 = M 1 + kn 2 M1 + M 3 / 2 2M 2 /, M 12 = M 21 = k n (M 4 M 5 /) M 13 = M 31 = M 7, M 14 = M 41 = k n M2 / M 3 / 2, M 22 = M 6 M 24 = M 42 = M 5 /, M 33 = M 8, M 44 = M 3 / 2 M 23 = M 32 = M 24 = M 42 =0.