FREE VIBRATION ANALYSIS OF CLAMPED-FREE COMPOSITE CYLINDRICAL SHELLS WITH AN INTERIOR RECTANGULAR PLATE

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1 FREE VIBRATION ANALYSIS OF CLAPED-FREE COPOSITE CYLINDRICAL SHELLS WITH AN INTERIOR RECTANGULAR PLATE Young-Shn Lee and young-hwan Cho Department of echancal Degn Engneerng, Chungnam Natonal Unverty, 0 Kung-Dong, Yuong-Gu, Taejon, , Korea SUARY: Th paper decrbe the free vbraton charactertc of clamped-free compote cylndrcal hell wth a longtudnal nteror rectangular plate ung the receptance method. To formulate the frequency equaton of the combned hell, the natural frequence and mode hape functon for ndvdual component of the plate and hell wth clamped-free boundary condton at both end of the hell are obtaned ung Love hell theory and clacal plate theory. The natural frequence of combned hell can be calculated from the charactertc of the plate and hell two component ytem. Thee calculaton are needed for component ytem under dynamc force and moment exerted at the jont. The analytcal reult are compared wth thoe of the experment and fnte element analy. The effect of the thckne of the nteror plate and the fber orentaton angle on the natural frequence of the combned compote hell are alo dcued. KEYWORDS: Receptance ethod, Cylndrcal Shell, Free Vbraton, Natural Frequency, Dynamc Loadng INTRODUCTION Cylndrcal hell are ued to approxmate more complex tructure uch a n aeropace, ubmarne and nuclear preure veel ued n many ndutral applcaton. The combned hell wth a longtudnal nteror rectangular plate eem to be the mot realtc model, and an arcraft fuelage wth t floor tructure may be the dealzed model of a combned hell and plate []. The vbraton problem of combned hell wth nteror plate are beng tuded by everal reearcher ung varou analytcal method uch a extended Raylegh-Rtz, tranfer matrx, dynamc tffne and Raylegh-Rtz method [-5]. Recently, a one of the analytcal method for vbraton analy of combned ytem, the receptance method ha been appled to the analy of free vbraton for contnuou rectangular plate [6] and the cylndrcal hell wth crcular plate attached at arbtrary axal poton [7,8]. In reference [9] a mply upported compote cylndrcal hell wth an nteror plate analyzed ung the receptance method and experment. In th tudy, n order to analyze the free vbraton of a clamped-free compote cylndrcal hell wth a longtudnal nteror plate, the receptance method ued. For two plate and hell tructure wth clamped-free edge condton, the natural frequence and mode hape

2 functon are obtaned through the Raylegh-Rtz procedure baed on the energy prncple. The dynamc force and moment exerted at the jont due to the contrant of the dplacement of two ytem can be aumed ung the Drac delta functon. The natural frequence of combned hell can be calculated from charactertc of the component ytem of the hell and plate. Thoe have to be calculated for component ytem under dynamc force and moment exerted at the jont. The analytcal reult are compared wth thoe of the experment and fnte element (FE) analy. The effect of the thckne of the plate and the fber orentaton angle on the natural frequence of the combned hell are alo dcued. FORULATION The geometry of the compote crcular cylndrcal hell wth a longtudnal, nteror plate and the coordnate ytem are hown by Fg.. The dplacement component of the plate and hell n each drecton are preented a u P, u P, u P 3 and u S, u S, u S 3, repectvely. Where, ubcrpt (or upercrpt) and p ndcate the hell and the plate, repectvely. The plate attached at θ and θ poton of the hell baed on the vertcal centerlne. Fg. : Geometry of the compote crcular cylndrcal hell wth a longtudnal, nteror plate Receptance of Combned Structure A receptance defned a the rato of a dplacement or lope repone at a certan pont to a harmonc force or moment nput at the ame or dfferent pont. When the two ytem are joned and no force (or moment) external to the two ytem are appled, t mut be equal becaue of dplacement (or lope) contnuty. Thu, the natural frequence of combned tructure can be found from [0] β = 0. () j + j Where, the j and β j are the receptance of the hell and plate, repectvely. The dplacement of the hell ubjected to dynamc loadng can be expreed by modal dplacement and mode partcpaton factor. Fk jωt u (,, t) = Uk (, ) e () k= ( ωk ω ) Where, the mode component of the plate and hell n three prncpal drecton, and U k

3 ω k the natural frequency of two ndependent ytem whch are calculated by Love' hell theory and clacal plate theory, repectvely. The dynamc forcng term are gven a below. F = k f U k A A dd ρhn (3) Where, N k k k = U A A d d (4) f (=,,3) are the nput forcng functon appled along two jont lne. The tranvere mode hape atfyng clamped-free edge condton of the hell ued a followng from neglectng the dplacement component n axal and crcumferental drecton. U S 3 = φ( x) co( nθ ) (5) Where φ (x) the beam modal functon whch atfy clamped-free edge condton of the hell and plate n the axal drecton, and can be expreed a followng; φ ( x) = coh( λ x / L) co( λ x / L) σ {nh( λ x / L) + n( λ x / L)} (6) and λ and σ can be obtaned from equaton (7) and (8) below. nh λ n λ σ = (7) coh λ + co λ coh λ co λ + = 0 (8) The mode hape of the plate ued n wdth and normal drecton are expreed a below. U p = φ( x) co( nπy / b) (9) U p = φ( x) n( nπy / ) (0) Dplacement Due to Dynamc Loadng 3 b When a rectangular plate attached at θ and θ poton of the hell, the tranvere dynamc exctaton exerted at the jont due to the contrant of the dplacement of the hell by the plate are hown n Fg., and can be aumed a Eq.() ung the Drac delta functon. f S jωt θ, t) = F φ( x) δ ( θ θ ) e =, () Fg. : Force and moment appled on the hell at the jont Subttutng Eq.() to Eq.(3) allow u to calculate the dynamc forcng functon Eq.(3) (5), the dplacement of the hell can be expreed a followng form. F k. Ung

4 u3 θ ) = u3 θ ) F + u (, ) 3 x θ F () The crcumferental lope of the hell can be obtaned from Eq.() by dfferentaton wth repect to the crcumferental coordnate βθ θ ) = βθ θ ) F + β θ ) θ F (3) The dynamc moment loadng exerted at the jont due to the contrant by the plate can be aumed a followng form. S jωt m θ, t) = φ( x) δ ( θ θ ) e =, (4) The dynamc force due to the moment loadng can be obtaned wth [0] ( A ) ( A ) Fk = U A A dd ρhn k + (5) k A A Therefore, the dplacement of the hell by moment loadng can be expreed a follow; u3 θ ) = u3 θ ) + u (, ) 3 x θ (6) and the lope of hell by moment loadng at the jont can alo be obtaned from Eq.(6) by dfferentaton wth repect to the crcumferental coordnate βθ θ ) = βθ θ ) + β θ ) θ (7) A a mlar method, when the two jonted edge of the rectangular plate are mply upported and the other edge have a clamped-free n the axal drecton, one can conder the receptance for a rectangular plate by force and moment exerted at the jont, y ) and ( x, y ). To formulate the receptance that one need, the dplacement and lope a the functon of axal coordnate due to force and moment are calculated. Detaled procedure of the formulaton wll not be decrbed here. Frequency Equaton for Combned Shell For the combned hell wth an nteror rectangular plate, conderng the contnuty condton at the hell/plate jonng pont can derve the frequency equaton. Here, only the lope n the wdth drecton of the plate and the normal dplacement of the hell due to dynamc force are condered. Alo the normal dplacement of the plate and the lope of the hell n the crcumferental drecton due to dynamc moment are taken nto conderaton becaue the other component of dplacement can be gnored. By applyng the contnuty condton at the jont, the frequency equaton can be expreed a Eqn. (8). + β + β 3 + β3 4 + β4 F + β + β 3 + β β 4 = 0 (8) 3 + β β β β 34 F 4 + β β β β 44 Neglectng the n-plane dplacement due to moment ( u p / ) and the lope n the normal drecton due to force ( β 3p / F ) for the rectangular plate, the frequency equaton of the combned hell wll be equaton (9). + β 3 + β3 4 + β β 4 = 0 (9) 3 + β β β β 44

5 RESULTS AND DISCUSSIONS To compare the analytcal reult, a modal tet ung the mpact exctng method and FE analy ung a commercal code ANSYS [] are performed. Specmen teted are o o o fabrcated n plan weave gla/epoxy compote wth [ 03 / ± 453 / 903 ] S tackng equence. Table preent the dmenon of the combned hell, and the hell have the ame o dmenon a the plate. The plate attached at the center ( θ = 90 ) of the hell and fabrcated wth the ame materal properte a the hell. The materal properte of the GFRP plan weave compote are obtaned by unaxal tenle tet and are a follow; E = E = 6. GPa, G = 4.9 GPa, ρ = kg / m, υ =0. For the vbraton tet, the exctaton method by mpact hammer ued wth the Fat Fourer Tranform (FFT) analyzer, an mpact hammer and an accelerometer. The chematc dagram for the expermental modal tet hown n Fg. 3. ateral Length (L S ) Table : Dmenon of combned compote hell Shell Radu (a) Thckne (h S ) Length (L P ) Plate Wdth (b) (Unt : mm) Thckne (h P ) GFRP Impact Hammer Charge Amp. FFT Analyzer Specmen Acc. Charge Amp. Natural Frequence ode Shape Bed Structure Fg. 3: Schematc dagram of the expermental modal tet Table preent reult of the frt eght natural frequence from analytcal, expermental and FE analy. The expermental reult are repreented ung the mode equence number of the combned hell, where the P and S ndcate the mode of the plate and hell, repectvely. The fundamental frequency of the combned hell wth clamped-free edge condton 08.7 Hz, and t how the frt bendng mode, P(,) of the nteror plate. The dcrepancy between the analytcal and FE reult.4% for the lowet fundamental frequency, but the expermental reult about % lower than that of the analytcal method

6 due to the uncertanty of the expermental procedure and the pecmen teted. The frt frequency for the hell 36 Hz at S(,) mode, but t doe not appear n the experment. Accordng to ncreang frequence, the mode of the plate and hell couple wth each other. Fg. 4 how the expermental mode hape of the GFRP compote combned hell wth clamped-free edge condton. Table : Comparon of the natural frequence of analytcal, expermental and FE reult o of the GFRP plan weave compote cylndrcal hell wth nteror plate at θ 90 locaton ethod Natural frequency (Hz) ode Analy Exp. ode FE t nd 3 rd 4 th 5 th 6 th 7 th 8 th : Frequency acendng order 8.0 P(,) P(,) P(3,) P(,3) S(,3) S(,3) 60.0 P(,) S(,4) P(4,) - = Hz, P(,) mode 30 Hz, P(,) mode 460 Hz, S(,3) mode 60 Hz, S(,4) mode Fg. 4: Typcal expermental mode hape of the GFRP compote combned hell wth clamped-free edge condton In the cae of the GFRP plan weave compote combned hell wth nteror plate at o θ = 90 locaton, the effect of the thckne (h p ) of the plate on the frequence are tuded and hown n Fg. 5 for two mode of the plate and hell. For th calculaton, the thckne of the plate vare from.0mm to 8.0mm. The fundamental frequency of the combned hell prmarly exhbt plate moton wth one half wave n each drecton. In th cae, the frequence of the frt two mode of the plate are lnearly ncreaed a the thckne of the plate ncreaed, even f the frequency ncrement n the cae of the thck plate lghtly

7 maller than the thn plate. If the thckne of the plate larger than 5.0mm, the frequency of the plate mode, P(,) hgher than the one of the hell mode, S(,). For the hell mode, the frt frequency appear at the crcumferental wave number n=, and the frequence become le entve to change n plate thckne. 600 h =3.5mm ode S(,3) Natural Frequency (Hz) P(,) S(,) P(,) Plate Thckne (mm) Fg. 5: Effect of the plate thckne( h P ) on the frequence of the GFRP plan weave o compote cylndrcal hell wth nteror plate at θ 90 locaton = To tudy the nfluence of fber orentaton angle ( Θ ) of the compote on the frequence of the combned hell, the CFRP compote made of T300/N508 choen and the materal properte are a below []: E = 8.0 GPa, E = 0.3 GPa, G = 7.7 GPa, ρ = kg / m, υ =0.8 The compote hell condered lamnated wth the [(Θ /- Θ ) ] tackng equence, and the thckne of each ply aumed to be 0.5 mm. The plate attached at the center of the hell, and the geometrcal parameter of the hell are: L /a= and a/h =00. Fgure 6 how the fundamental frequency of the CFRP compote combned hell wth varou fber orentaton angle. The frequency of the plate alone wth mple upport at the two jont and clamped-free at the other two edge ha the hghet value at the fber orentaton angle, Θ= 90 o. For the hell wth the clamped-free boundary condton, the hghet frequency 650 Hz at Θ= 40 o. Parenthezed number n Fg. 6 ndcate the crcumferental wave number (n) for the axal mode, m=. The crcumferental wave number on the fundamental frequency decreae a the fber orentaton angle ncreae. Th ndcate that the fber angle ha a greater effect on the tffne n the crcumferental drecton. When the plate and hell are joned, the fundamental frequency of the combned hell of a lghtly hgher value than the frequency of the nteror plate alone. Alo, ncreang the fber orentaton angle ncreae the dfference n the fundamental frequency between the plate alone and the combned hell. In the cae of the fber angle, Θ= 90 o, the combned hell ha a hghet frequency, 03.4 Hz, whch 80 Hz hgher than that of the plate alone.

8 700 Fundamental Frequency (Hz) (5) (6) (5) (4) (5) (4) Combned Shell Plate Only Shell Only (3) (3) (3) (3) L /a= a/h =00 h =h =.0mm p Fber Orentaton Angle (Θ ) Fg. 6: Fundamental frequence of the CFRP lamnated compote cylndrcal hell wth the [( Θ /- Θ ) ] tackng equence for varou fber orentaton angle CONCLUSIONS The major concluon from th tudy are a follow:. The fundamental frequency of the combned hell wth clamped-free edge condton the frt bendng mode of the nteror plate, and the plate vbrate a a mply upported plate at two jont.. For the effect of the thckne of the nteror plate, the frequence are lnearly ncreaed a the thckne of the plate ncreae. Thoe of hell mode become le entve to change a plate thckne ncreae. 3. The combned hell ha the hghet fundamental frequency n the cae of the fber orentaton angle, Θ=90 o, and the fundamental frequency of the combned hell of a lghtly hgher value than the frequency of the nteror plate alone. REFERENCES. Petyt,. and We, J., "Free Vbraton of an Idealzed Fuelage Structure", Proceedng of the 5th Internatonal odal Analy Conference, Japan, 997, pp. 647<653.. Peteron,.R. and Boyd, D.E., Free Vbraton of Crcular Cylnder wth Longtudnal, Interor Partton, Journal of Sound and Vbraton, Vol. 60, No., 978, pp.45<6. 3. Ire, T., Yamada, G. and Kobayah, Y., "Free Vbraton of Non-Crcular Cylndrcal Shell wth Longtudnal Interor Partton", Journal of Sound and Vbraton, Vol. 96, No., 984, pp.33<4. 4. R.S. Langley, "A Dynamc Stffne Technque for the Vbraton Analy of Stffened Shell Structure", Journal of Sound and Vbraton, Vol. 56, No. 3, 99, pp. 5<540.

9 5. aou, J., Cheng, L. and Rchard,.J., "Free and Forced Vbraton of a Cylndrcal Shell wth a Floor Partton", Journal of Sound and Vbraton, Vol. 90, No., 995, pp.< Azm, S., Hamlton, J.F. and Soedel, W., "The Receptance ethod Appled to the Free Vbraton of Contnuou Rectangular Plate", Journal of Sound and Vbraton, Vol. 93, No., 984, pp. 9<9. 7. Huang D.T. and Soedel, W., "Study of the Forced Vbraton of Shell-Plate Combnaton Ung the Receptance ethod", Journal of Sound and Vbraton, Vol. 66, No., 993, pp. 34< Ym, J.S., Shon, D.S. and Lee, Y.S., "Free Vbraton of Clamped-Free Crcular Cylndrcal Shell wth a Plate Attached at an Arbtrary Axal Poton", Journal of Sound and Vbraton, Vol. 3, No., 998, pp. 75< Lee, Y.S. and Cho,.H., Free Vbraton of a Compote Cylndrcal Shell wth a Longtudnal, Interor Rectangular Plate, The Frt Aan-Autralan Conference on Compote ateral, Oaka, Japan, 998, pp. 30-< Soedel, W., Vbraton of Shell and Plate, nd ed., arcel Dekker, Inc., Swanon Analy Sytem Inc. : ANSYS Uer' anual: Volume Element, 99.. Ta, S.W., Compote Degn, Thnk Compote, Dayton, Oho, U.S.A., Lam, K.Y. and Loy, C.T., Influence of Boundary Condton and Fber Orentaton on the Frequence of Thn Orthotropc Lamnated Cylndrcal Shell, Compote Structure, Vol. 3, 995, pp. < Jone, R.., echanc of Compote ateral, Hemphere Publhng Co., New York, Whtney, J.., Structural Analy of Lamnated Anotropc Plate, Technomc Publhng Co., Lancater, Pennylvana, 987.