Dynamics of shear deformable laminated composites using Raleigh Ritz method

Transcription

1 Cahun: The NPS Institutina Archive Facuty and Researcher Pubicatins Facuty and Researcher Pubicatins 2003 Dynamics f shear defrmabe aminated cmpsites using Raeigh Ritz methd Kar, Ramech Mnterey, Caifrnia. Nava Pstgraduate Sch

2 Dynamics f Shear Defrmabe Laminated Cmpsites Using Raeigh Ritz Methd Ramesh Kar a Department f Aernautics & Astrnautics, Nava Pstgraduate Sch, 699 Dyer Rad, Bdg 234, Rm 245, Mnterey, CA 93943, U.S.A. ABSTRACT Layered cmpsites have attracted attentin fr their high specific stiffness, high specific strength, and appicatin specific tairing f their prperties. It is as recgnized that ayered cmpsites are prne t deaminatin faiure in additin t ther faiure mdes. Cnsideratin f transverse shear n the defrmatin behavir f the cmpsites is an imprtant aspect in the study f deaminatin mde faiure f such pates. In this paper, we cnsider the effects f incuding the transverse shear defrmatin n the vibratin characteristics f ayered cmpsites. The frmuatin is based n the Raeigh-Ritz methd using the beam characteristic functins. MATLAB based symbic math t bx is used in evauating th eintegras resuting frm the Raeigh Ritz apprach. Varius cmmny ccuring bundary cnditins are discussed. Resuts are prvided shwing the effects f the shear defrmatin n the dynamics f ayered aminated cmpsites. The effects f aminate thickness, fiber rientatin, and the pate aspect ratis n the free vibratin characteristics f the cmpsite aminates are given t demnstrate the methdgy described. Keywrds: Dynamics f Cmpsites, Laminated Cmpsites, Transverse Shear, Vibratins f Shear Defrmabe Pates Further authr infrmatin: (Send crrespndence t R.K. a R.K.: E-mai: rkar@nps.navy.mi, Teephne: INTRODUCTION: Laminated cmpsites have received increasing attentin due t their high specific stiffness and high specific strength prperties. In additin, the structura tairing prperties assciated with these materia systems ffer attractive design ptins. Hwever, ne f the drawbacks f amiated cmpsites is the ptentia fr deaminatin faiure. In this respect, transverse shear defrmatin becmes imprtant. There are severa studies attributed t the dynamic anaysis f cmpsites incuding transver shear defrmatin. 1 5 In this paper, we extend the shear defrmatin thery f aminated cmpsites t frmuate the dynamics f ayered cpmpsite aminates using Raeigh Ritz apprximatin. Symbic mathematics is used in cacuating the integras invved in the methd rather than using the tabuated vaues, thereby autmating the cmputatin prcess and imprving the accuracy f the resuts. 2. MINIMUM TOTAL POTENTIAL ENERGY FOR LAMINATED PLATES: The principe f minimum tta ptentia energy fr an anistrpic pate is given by where V = R δv =0 (1 WdR T i u i ds S T R F i u i dr (2

3 In the abve expressin, W is the strain energy density functin, R, the vume f the eastic bdy, T i,thei th cmpnent f the surface tractin, u i,thei th cmpnent f dispacement, F i, the i th cmpnent f the bdy frce, and S T,the prtin f the bdy surface ver which the tractin are prescribed. The strain energy density functin, W, is defined as W = 1 2 σ ijɛ ij (3 The usua strain dispacement functins may be fund in the standard texts. The stress strain reatins fr the cmpsite amina is given by σ xx σ yy σ yz σ zx σ xy = Q 11 Q Q 16 Q 12 Q Q Q 44 2Q Q 45 2Q 55 0 Q 16 Q Q 66 ɛ xx ɛ yy ɛ yz ɛ zx ɛ xy (4 The stresses in the x-y-z system is btained by suitabe transfrmatin f the materia stiffness matrix [Q] frm the materia crdinate systems t the x-y-z reference crdinate system. The appearance f 2befre Q is due t the strain tensr cmpnent definitin f the shear strains. Assuming the functina frm f the dispacements fr the pate t be u(x, y, z = u (x, y + zα(x, y v(x, y, z = v (x, y + zβ(x, y w(x, y, z = w 0 (x, y (5 where u, v,andw 0 are the the midde surface dispacements and the secnd term in the first tw equatins are reated t the rtatins. In the cassica pate thery, α = w w and β =, y which negect the transverse shear defrmatin. On substituting the strain energy density functin fr each amina acrss the N aminae cmprising the pate, we btain V = 1 2 N k=1 A hk h k 1 {σ xx ɛ xx + σ yy ɛ yy +σ yz (2ɛ yz +σ zx (2ɛ zx +σ xy (2ɛ xy }dz da (6 We cnsider symmetric aminate cnstructin, which is a cmmn feature in many appicatins resuting in the absence f bendingstretching cuping. The stress resutants are given by N x N y N xy M x M y M xy = A 11 A 12 A A 12 A 22 A A 13 A 23 A D 11 D 12 D D 12 D 22 D D 13 D 23 D 33 where the curvatures are given by ɛ xx0 ɛ yy0 ɛ xy0 κ x κ y κ xy (7 κ x = α ; κ y = β ; κ y xy = α + β y (8 Cnsidering uncuped transverse vibratins f the pate ny, and using the definitins f strain dispacement reatins, the stresses, and the stress resutants, we can rewrite the equatin (6 as V = a D ( α α + D 16 β + D 26 y + D 66 2 ( 2 α α + D 12 y + β ( α y + β ( α y + β + D 22 2 β y ( β y 2 2 dx dy (9

4 where D ij are the bending stiffness cefficients, α and β are reated t the rtatins. The bending stiffness cefficients are cacuated using D ij = 1 3 N ( [ Qij h 3 k hk 1] 3 k=1 (10 where the summatin is carried ver the number f ayers, Q ij is the materia stiffness matrix f the k th ayer in the (x,y,z reference crdinate system, h k and h k 1 are the distances t the upper and wer surfaces f k th ayer frm the mid-pane f the aminate. The cntributin arising frm the transverse shear defrmatin, V s, t the ptentia enrgy expressin may be written as V s = 1 a ( b A 55 α + w ( + 2A 45 α + w ( β + w y ( + A 44 β + w 2 dx dy (11 y where 5 4 A ij = N ( Qij [h k h k (h3k h 3k 1 1 ] (12 h 2 k=1 k fri, j = 4, 5 ny. The maximum kinetic energy f the cmpsite pate is given by a T = ρhω2 w 2 da ( where ρ is the mass density per unit vume, h is the thickness, and ω is the natura frequency f the pate. In the Raeigh-Ritz prcedure, 7 the defrmatin is apprximated by the admissibe functins, as knwn as the kinematicay cnsistent functins, satisfying the gemetric bundary cnditins, as fws: M N α(x, y = A mn φ αm (xφ αn (y (14 m=1 n=1 M N β(x, y = B mn φ βm (xφ βn (y (15 m=1 n=1 M N w(x, y = C mn φ wm (xφ wn (y (16 m=1 n=1 where φ αm (x,φ βm (x and φ wm (x are the beam characteristic functins satisfying the apprpriate bundary cnditins in the x- directin. A mn, B mn,andc mn are as yet undetermined cefficients. The apprximatin functin in y-directins are btained by repacing x by y and interchanging α and β, andtheintegras ver the ength in y-directin. These cefficients are determined by invking the principe f minimum tta ptentia energy f the system, which resuts in δ(t V = 0 (17 (T V A ik = 0 (18 (T V = 0 B ik (19 (T V =0 C ik (20 fr i =1...M and k =1...N The resuting set f 3(MXN equatins may be sved t btain the eigenvaues and the crrespnding eigenvectrs. On substituting the equantins 14 and 15 int the ptentia energy expressin, we btain V = V a + V b + V c + V d + V e + V f (21 where the terms n the right hand crrespnd t the cntributin f the six terms n the right hand side f the ptentia energy expressin (6. Fr exampe, the first term is given as V a = D 11 2 a 0 0 ( Amn φ α m (xφ αn (y 2 dxdy (22

5 and the differentiatin f this expressin with respect t the undetermined cefficients may be written as V a = A ik a D 11 A mn φ α m (xφ α i (xdx 0 m n 0 φ αn (yφ αk (ydy (23 The transverse shear is as cacuated in a simiar fashin, V s = V g + V h + V i (24 By apprpriatey substituting the beam characteristic functins, the integras may be evauated. 8 We intrduce the ntatin fr the integras invving beam characteristic functins and their derivatives as fws: a i 02 im (αα = φ αi (xφ α m (xdx; i 02 mi(αα = j 02 a φ αm (xφ α i (xdx; (25 kn (αα = φ αk (yφ α n (ydy; j 02 nk (αα = φ αn (yφ α k (ydy; (26 i 11 im (αα = a j 11 φ α i (xφ α m (xdx; kn (αα = φ α k (yφ α n (ydy; (27 i 10 kn (αα = φ α k (yφ αn (ydy; j 10 nk(αα = i 01 kn(αα = j 01 φ α n (yφ αk (ydy; (28 φ αk (yφ α n (ydy; nk (αα = φ αn (yφ α k (ydy; (29 Simiar integras invving αβ, ββ, βα, αw, βw, wα, wβ, andww are defined with apprpriate substitutin f the φ α, φ β,andφ w functins in the integras n the right hand side. These equatins are nt given here in the interest f space. The rthgnaity reatins fr the beam characteristic functins are given by, a 0 φ i(xφ m (xdx = 0 i m (30 = a i = m 0 φ n(yφ k (ydy = 0 n k (31 = b n = k and a 0 φ m (xφ i (xdx = ɛ4 m a 3 m = i (32 = 0 m i 0 φ m (yφ i (ydy = ɛ4 m b 3 m = i (33 = 0 m = i On making use f these integras, the minimizatin f the tta ptentia energy resuts in a 3(M X N by 3(M X N system f hmgeneus agebraic equatins t be sved fr λs, A ik s, B ik s and C ik s. Substituting fr V and T, and perfrming the indicated differentiatins with respect t A ik s, B ik s and C ik s respectivey, we btain, the set f equatins V A ik = +D 12 Bmn i 10 mi j01 kn + M N Amn i 10 mi j01 nk +D 16 Amn i 10 mij 01 nk + +D 26 Bmn i 01 mi j10 nk + i01 im j10 kn +D 66 Bmn i 10 mij 01 nk + M N Amn i 10 imjkn 01 M N Cmn i 10 imjkn 01 +A 55 Amn i 00 mi j00 nk + M N Cmn i 10 mi j00 nk +A 45 Bmn i 00 im j00 nk + M N Cmn i 00 im j01 kn (34

6 ( V M N = D 12 Amni 10 B mijnk 01 ik + D 16 Amn i 11 mi j00 nk + D 22 Bmn i 00 mij 22 nk +D 26 Bmn i 01 mi j10 nk + M N Bmn i 10 mi j01 nk + D 26 Amn i 00 mi j11 nk +D 66 Bmn i 11 mi j00 nk + M N Amn i 01 mi j10 nk +A 45 Amn i 00 mi j00 nk + M N Cmn i 10 mi j00 nk +A 44 Bmn i 00 imj 00 nk + M N Cmn i 00 mijnk 10 (35 referenced withut expicity referring t them as i 10 mi (αα etc fr cnciseness. 3. BEAM CHARACTERISTIC FUNCTIONS: The beam characteristic functins that are used in this paper are f the fwing frms: (a a camped-free beam, which is camped at x =0andfreeatx = φ r =csh ɛ rx cs ɛ ( rx α r sinh ɛ rx sin ɛ rx (37 b a free-free beam, which is free at bth ends, φ 1 = 1 (38 φ 2 = 3(1 2x/ (39 V C ik = +A 55 Cmn i 11 mi j00 nk + M N Amn i 01 mi j00 nk +A 45 Amn i 00 mi j01 nk + M N Bmn i 01 mi j00 nk +A 45 Cmn i 10 mi j01 nk + M N Cmn i 01 mi j10 nk +A 44 Cmn i 00 mij 00 nk + M N Bmn i 00 mijnk 01 (36 The abve set f inear hmgeneus agebraic equatins are sved fr the frequency parameter and the crrespnding set f cefficients A mn, B mn, and C mn. These cefficients may be used t cmpute the mde shapes. It may be nted that the vaue f ɛ i is t be taken crrespnding t the characteristic functin φ m, whie the vaues fr ɛ k are taken crrespnding t the characteristic functin φ n. the equatins (34-36, the integras i 10 mi Further, in are φ r =csh ɛrx +cs ɛrx, ( α r sinh ɛ rx +sin ɛrx (r =3, 4, 5,... (40 and (c a camped-camped beam, which is fixed at bth x =0andx =. φ r =csh ɛ rx cs ɛ ( rx α r sinh ɛ rx sin ɛ rx (41 In the abve expressins, r =1, 2, 3,...;The numerica vaues fr ɛ are cacuated using characteristic equatins fr the respective beams. 7, 8 The vaues fr α are cacuated using : α r = ɛrx csh sinh ɛrx +cs ɛrx +sin ɛrx (42 4. INTEGRALS OF THE BEAM CHARACTERISTIC FUNCTIONS

7 In the evauatin f the equatins (34-36, varius integras invving the beam characteristic functins are required. These integras are evauated 8 using symbic math t bx f MATLAB where exact integras exist and Gauss quadratures fr evauating the integras numericay, where exact integras d nt exist. Sme f these vaues are as tabuated in a technica reprt RESULTS: The prcedure described here is used in evauating the free vibratins f sme seected exampe prbems. Sampe cacuatins have been carried ut fr graphite-epxy aminates, which are characteristic f typica aerspace and autmtive apicatins Simpy Supprted Laminated Cmpsite Pates: We present sme resuts fr a square simpy supprted cmpsite pate. Tabe 1 gives prperties f the unidirectina amina ut f which the aminates are cnstructed. Tabe 2ists the first six natura frequencies cmputed frm the present apprach fr a thin fur ayered cmpsite pate. As incuded in the tabe is a cmparisn f finite eement resuts frm MSC/NASTRAN. Figure 1 presents fundamenta frequency f crss-py and ange-py cmpsites fr varius side t thickness ratis. It may be nted that in the case f cmpsites, the effects f transverse shear defrmatin n the pate frequencies becmes imprtant fr side t thickness ratis as w as 10. Figure 2depicts the variatin f the fundamenta frequency with side t thickness rati. This figure as shws the effect n the natura frequency f the aspect rati a/b. Figures 3 and 4 shw the variatin f the fundamenta frequency as a functin f the side t thickness rati and aspect rati in a three dimensina surface pt. Such pts prve usefu in the design trade-ff studies f cmpsite panes. 6. CONCLUSIONS A frmuatin is deveped fr studying the dynamic behavir f shear defrmabe aminated cmpsite pates. The Raeigh-Ritz methd is used t derive the free vibratins estimates f the cmpsite aminates. The incusin f transverse shear significanty infuences the vibratin characterisitics and imprtant in studying deaminatin f aminated pates. The integras invving beam characterisitic functins are evauated exacty using symbic mathematics which give better estimates f the natura frequencies f the pates. ACKNOWLEDGMENTS The wrk reprted here is in part supprted by NASA Dryden Fight Research Center, Dryden, Caifrnia. REFERENCES 1. J. M.Whitney and N. Pagan, Shear defrmatin in hetergeneus anistrpic pates, J. f Appied Mechanics 37, p. 1031, J. Whitney and A. Leissa, Anaysis f hetergeneus anistrpic pates, J. f Appied Mechanics 36, p. 261, C. Wu and J. Vinsn, Infuence f arge ampitudes, transverse shear defrmatin, and rtary inertia n atera vibratins f transversey istrpic pates, J. f Appied Mechanics, p. 254, C. Wu and J. Vinsn, Nninear sciatins f aminated speciay rthtrpic pates with camped and simpy supprted edges, J. f the Acustica Sciety f America 49, p. 1561, C. Bert and T. Chen, Effect f shear defrmatin n vibratin f antisymmetric ange-py aminated rectanguar pates, Int. J. f Sids and Structures 414, p. 465, H. Nam, W. Hwang, and K. Han, Stacking sequence design f fiber-meta aminate fr maximum strength, J. f Cmpsite Materias 415, p. 1654, 2001.

8 Tabe 1. Prperties f the Unidirectina Lamina E 1 =20X10 6 ; E2 =2X10 6 ; G 12 =1X10 6 ; ν 12 =0.35 Tabe 2. Natura Frequencies fr a Square Simpy Supprted Pate, a/b = 1;h=0.0216;ρ =0.057 b/in 3 Mde Frequency, Hz, Present Frequency, Hz, MSC/NASTRAN e E e E e E e E e E e E D. Yung, Vibratin f rectanguar pates by the ritz methd, J. f Appied Mechanics 72, p. 481, R. Kar, Dynamics f cmpsite pates incuding shear defrmatin by raeigh ritz methd using beam characteristic functins, Department Of Aernautics and Astrnautics Reprt, p. In Preparatin, 2002.

9 Figure 1 Fundamenta Frequency Variatin with Side t Thickness Rati fr Ange-py and Crss-py Laminates Figure 2 Fundamenta Frequency Variatin fr Different Aspect Ratis

10 Figure 3 Fundamenta Frequency Variatins with Aspect Rati and Thickness Rati Figure 4 Fundamenta Frequency Variatins - Anther View