1990 g singh kk raju gv rao, ngr iyengar - non-linear vibrations of simply supported rectangular...

7/28/2019 1990 G Singh KK Raju GV Rao, NGR Iyengar - Non-linear vibrations of simply supported rectangular cross-ply plates - Journal of Sound and Vibration 142 21

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Journal of Sound and Vibration (1990) 142(2), 213-226

NON -LINEAR VIBRATIONS OF SIMPLY SUPPORTEDRECTANG ULAR CROSS-PLY PLATES

GAJBIR SINGH, K. KANAKA RAJUAND G. VENKATESWARA RAOStructural Design and Analysis Division, Structural Engineering Group, Vikram Sarabhai Space Centre,

Trivandrum 695 022, IndiaAND

N.G.R. IYENGARIndian Institute of Technology, Kanpur, India

(Received 15 May 1989, and in revised form 7 December 1989)

A metho d of direct numerical integration of the frequency-ratio expression is propo sedto study the non-linear free vibration behaviour of rectangular cross-ply laminates. Thepropo sed metho d, even with singie-term approximations for the admissillie functions,yields results that agree very well with the existing perturbation solutions. Non-linearbehaviour of the cross-ply lam inates is also studied with the harmonic oscillations assump-tion, by using the conservation of energy and the modal equation. The results are foundto be lower and upper bounds to those obtained from the direct numerical integrationmetho d. It is also observed that the arithmetic mean of the two solutions with the harmonicoscillations assumption match es very well with that of the direct numerical integrationmetho d. Non-linear vibration characteristics are obtained for several configurations ofcross-ply plates. Results for orthotropic and isotropic plates are also obtained as specialcases.

Com posite ma terials are finding increasing applications in modern aerospace andautomobile structures. This is primarily due to their high strength and stiffness to weightratio, and the lower machining and maintenance costs associated with composite ma terials.However, the analysis of composite structures is more complex when compared to metallicstructures, because composite structures are anisotropic and are characterized by bending-stretching coupling. Very often these structures are subjected to severe environmentalconditions, which necessitates the study of their vibration behaviour in the non-lineardomain. This topic has attracted many researchers and a number of approximate methodshave been developed, in view of the complexity of the problem, to study the non-linearvibration behaviour of cross-ply plates, wh ich are of comm on occurence in practicalstructures such as solar panels.

Sathyamoorthy [l] and Ch ia [2] have presented many references on approximateanalytical method s and numerical methods for large amp litude vibrations of plates.Am bartsumy an [3] and Hassert and Now inski [4] reported on non-linear vibrations ofsingle-layer orthotropic plates. Wu and Vinson [5] evaluated non-linear frequencies oforthotropic plates using Bergers [6] approxima tion. Furthermore, W u and Vinson [7]extended their earlier work to symm etric lamina tes.The lack of symmetry about the mid-plane of a laminate exhibits b ending-stretchingcoupling. The effect of this coupling in non-linear dynamic plate theory was recognized

213~~2pimxion/mn7l~+ 14 9nuwo__._, ._ ,__-___ _ __.__, @ 1990 cademic Press Limited


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21 4 G. SINGH ETAL.by Wh itney and Leissa [8]. Banne t [9] presented the non-linear free vibration responseof simply supported angle-ply plates. Chandra and Raju [lo-121 studied the large-amplitude flexural vibrations of cross-ply and angle-ply plates. Their study is based onthe two-term perturbation solution for non-symm etric laminates.The mod al equation for non-symm etric laminate usually has even and odd powers ofthe amplitude (see equation (14) of section 2). The perturbation method used by Chandraand R aju [lo-121 and many other researchers may fail when the coefficient of the evenpower of the amplitude is larger than the other two. In view of this, in the present paper,a direct numerical integration method which does not suffer from this problem is proposedto study the non-linear free flexural vibrations of antisymmetric cross-ply plates. Large-amplitude vibrations of these plates are also studied with the harmon ic oscillationsassum ption, yielding two solutions based on energy conservation and the equation ofmotion. The arithmetic mean of these two solutions matches very well with the resultfrom direct integration method. A number of samp le problems are presented to illustratethe accuracy and effectiveness of the method.

7 FnRMlll ATInN AND XII lITInN_. . .,~.I,~vy..~vI. .l.- yl..,..v..

Consider a thin rectangular plate of total thickness t composed of many orthotropiclayers, layered at 0 and 9 0 alternately. The origin of the Cartesian co-ordinate systemis located in the mid-plane with the Z-axis perpendicular to this plane as shown in Figure1. The material of each layer is assum ed to possess a plane of elastic symm etry parallelto the mid-plane (x-y plane). Since the plate is assum ed to have large-amp litude motion,the dynamic von K& -rniln type strain-displacement relations are used:

a~ 1 aw '( ) au au awdw. T2=- t - - )ay 2 ay &,5=2&*2=-+-+--,ay ax ax ayK! = -d2W/dX2, K 2= -d2W!ay2, ( 1)Kg=zK!z= -2 a2w/ axay.

Here u, v, w are the mid-plane displacements in the X, Y and 2 directions respectively.The stresses and mom ent resultants per unit length are defined a s

(Ni, Mi)=/2 (l,Z)Ui dz, (2)-r/2

where Vi (i = 1,2,6) are the in-plane stress compon ents ((rl = a,, u2 = a,,, o6 = axy).

Figure 1. Geometry of a cross-ply plate.


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CROSS-PLY PLATE NON-LINEAR VIBRATIONS 21 5For monoclinic behaviour in each layer, the constitutive equations for the mth layer,

in plate co-ordinates, are

(3 )

whe re Qi, are the reduced stiffness coefficients of the mth layer in the plate co-ordinates.Now , from equations (2) and (3), the plate constitutive equations are given by

(4 )where A,, B, and 0, are the respe ctive in-plane, bending-in-plane and bending ortwisting stiffness coefficients defined as

(A,,, B,,, Dij)=Cm I

I Q;T(l, z, z) dz, (5 )=,,Iwhe re z, is the distance from the mid-plane to the lower su rface of the mth layer.

The strain energy (U) and kinetic energy (T) of an antisymmetric cross-ply laminatescan be written as

(1 bUC ; II (A,,E:+~A,~E~E~+A~~E:+A~~E:,0 0

+2B,,&,K,+2B22E2K*+D,,K:+2D12K,K2+D22K:+D66K~)dxdy. (6)and

a b7-z; li (1 pfti)G2 dx d_v (7 )0 0

(with in-plane inertia neglected) whe re pi, ti are the density and thickness of the ith layerand u and b are the dimensions of the plate.

The non-linear free-vibration response of the plate can be obtained by introducing firstthe following two sets of admissible functions [13, 141, for simply supported boundaryconditions:

2mrxu= U(t)sin- n=ysin -, m7Tx 2nryz1= V(t)sin-sin- m7TXw= W(t)sin- n=ya b b sin -;a a b(8)

m7rxu= U(t) cos- sin %, mrx m7rx n7rya b ZI= V(t) sin - cos cz? w= W(t)sin- sin -.a b a b(9 )

Here m and n are the numbers of half sine waves in the x- and y-directions respectively.These sets of admissible functions (8), (9) satisfy the following displacement boundaryconditions, respectively:

for x = 0, a, u=fj=w=O; fo r y=O,b, u=u=w=O; (10)forx=O,a, v=w=O; for y = 0, b, u = w = 0. (11)

Substituting equations (l), (8) and (1) and (9) in the energy expressions (6) and (7) givesU =(ab/8)[T,W*+ T,U+ ~V2+2T5UW+2T,VW+2T7UW2+2T2UV

+2T4VW2+~T9W3+;T,oW4], (12)T=(ab/8)(C pit,)ti. (13)


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216 G. SINGH ET AL.The coefficients T,- T,, for bound ary conditions (IO) are

T = -4c, n7T -- 3nw b ) Tz=~(~)(~)(A1z+Ad,

T;&)2A66+(~)2A22,

T,=~[2(~)3A2r(~)2(~)~A~rAdb)lt-4c, 171773T5=- -3m7r a ) Te=(y)2A,,+(y)2A66,

T,=& ,2(~~A,,-(~)(~)2~A,*-A~~~,

T,=(~)4D,,+2(~)2(~)2~~*+2~~~~+(~)4~~*,

Tg=& [ (y)4B,,+(y)4B2z],

TIO=;[(~)~AH+(~)~A,,]+~(~)~(~)~(A,,+~A&and for boundary conditions (11) are

T,=- T 3B22,( )

Tz=(~)(~)(A,,+A,,).

T,=(y)2Aee+(y)2A,,, T4=&[A22(~)3+(~)2(~)CA,,-A,,)l,

T,=- = 3Bl,,( ) T,=(y)2A,,+(y)2A66,

T,=;[A,,(~)3+(~)(~)2tA~z-Ad].

T,=(54D,,+2[(92(52(D,,t2D,,)+(!f)4D2z],

T~=$$[(~)BH+(~)~Bzz],

T,o=~[(~)~A,,+(~)~A~z]+~(~)~(~)~~A,,+~A,J ,where c,=l-(-l)m, c,=l-(-l), and c,, =c,c,.

Based on the Ham iltons principle, the variation of the Lagrang ian L = U - T providesthe governing equation of motion as(Cpti)3+YW+PW2+yW3=0, (14)with

(Y= T,+(ZT,T,T,- T,T:- T:T,)/(T,T,- T;),/3= T,+3(T,T4T,+T,T2T,-T,T,T,--T,T,T,)/(T,T,-T:),

y = T,,+2(2T,T,T,- T,T:- T:T,)/( T3T6- T:).


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CROSS-PLY PLATE NON-LINEAR VIBRATIONS 21 7By multiplying equation (14) by 3 and integrating with respect to time, or by equating

U and T as given in equations (12) and (13), the following energy balance equation isobtained:

(1 piti)ti2+aw2+$w 3+$yw4= H =constant. (15)For a harmo nic solution w = w. cos wt, choice of equations (14) and (15) at w = w,)yields, respectively,

an d(%J%)2= l+(Pla)wo+(rla)wi, (16)

(WNL/w,)2= l+&?/(Y)Wg+f(Y/a)w:,. (17)Chandra and R aju [lo] have solved the equation of motion (14) by a perturbation

method for small finite values of amplitude to yield the solution(~NLI~L)*=(TL/~NL)~=~+[~~Y/~)-~(P/~~~Iw~. (18)

This solution sugges ts that the sign of the amplitude does n ot affect the frequencies,whereas the harmo nic solutions (16) and (17) reveal that the non-linear frequencies coulddiffer with amplitude sign. Secondly, the perturbation method fails when

[&/(Y)-&?/a)]w;< -1.0. (19)Hence, a direct numerical integration scheme is now used, to evaluate the non-linear

to linear frequency ratios as follows.From equation (15), the constant H can be compu ted for linear and non-linear vibrationsas

HL = (C pir;)C2+aw2, (20a)an d

HNL=(CP~~~)~~+(~W*+SPW~+~~W~ (20b)At w = w,,, (3 = 0), the constants H, an d HNL become

HL = aw:,, an d HNL = aw ;,,+$w;,,+~yw 4,,,. (20~ 4Substituting expressions (20~) and (20d) in equation (15), and rearranging the resultingequation suitably, leads to

2~ T,=4 Wmdii Jdw

OL 0 ( H~- aw' ) / ( E~i l t ) '2~ TNL=4 Wm*lI J

dwWNL 0 ( HNL- ( ~w* - _ ~w~- ~~w~) / ( CP~~~) '

Using equations (21a) and (21b) then gives@NL TL- =-WL TNL

(214

(2lb)

dw( H~- aw' ) I ( C~i f i )

dwJ ( H~~- aw' - Spw~- t yw~) l ( C~, t i )


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218 G. SINGH ET AL.The integrands in the numerator and denominator can be evaluated by using Gau ss

rule. In the present study a five-point Gau ss rule has been used to compute the frequencyratios.It can be noticed that no solution is assum ed in time when deriving the expressions

for the non-linear frequency (21b) and the frequency ratio (22). Further, the expressions(21b) and (22) by themselves show that the sign of the amplitude can alter the non-linearfrequency and frequency ratios, in contrast to the prediction from the perturbation solution(18).The linear frequency can be easily obtained from equations (14) and (15) as

ml.=-. (23)3. RESULTS AND DISCUSSIONS

On the basis of the results for the frequency ratios given by the above different method s,it is found that the present numerical integration scheme y ields results which match withthose of the perturbation method for the isotropic, orthotropic and cross-ply square plates,and solutions (16) and (17) obtained based on harmo nic oscillations assum ption arefound to yield results which form upper and lower bounds to those of the other twomethod s. An arithmetic mean of the solutions (16) and (17) gives

(WVLIW )* = (7-L/ TAX)*= 1 +SV~ )%+:(rl~)& (24)which is exactly the same as expression (18), in the absence of p (p =0 for squarecross-ply plates). It can be observed from the numerical results presented later that theratios of the non-linear frequency to linear frequency obtained from expression (24) arein close agreement with those of the numerical integration method.Linear frequencies for plates of various aspect ratios and different number of layershave been computed and on comp arison with those of Jones [14 ] they are found to beexactly the same; thus they are not presented here. As obtained from the expressions(16), (17), (18) and (22), calculated frequency ratios (wNL/wL) are presented for isotropic,orthotropic, symm etric a nd antisymmetric cross-ply plates. Effects of mod ulus ratio andaspect ratio on non-linear frequency are brought out. Boundary conditions of the typeshown in equation (10) are used in Tables l-4 and 11 for purposes of comp arison andthe results for boundary conditions (11) are presented in Tables 5- 10. The linear frequencyparameter AL= (cob/ r2)w for isotropic plates and hL = (cob*/ 7r2)J12 C piti/ ETt3for orthotropic and cross-ply plates are given in Tables l-10.The frequency ratios ( OJ~JOJ~) for a square isotropic plate are presented in Table 1.Resu lts are given for two m ode shapes, rn = 1, n = 1 and m = 2, n = 1. The frequency ratiocompu ted from expressions (18) and (22) are in good agreem ent with those of reference[15] and the one computed from equation (17) compare well with those of references[13,16-181 , wherein the effect of in-plane inertia also has been considered. Furthermore,the expressions (16) and (17) yield upper and lower bounds respectively on the frequencyratios, and their arithmetic mean matches very well with the direct integration solutionand coincides exactly with the perturbation solution because for this problem (p = 0). Asimilar trend in the behaviour of the frequencies also can be seen at higher modes (m = 2,n = 1). In all the cases, the effect of non-linearity increases with amplitude. Furthermore,from the table it can b e seen that for the mode m = 2, n = 1 the effect of non-linearity isgreater.

The frequency ratios for a rectangular plate of aspect ratio two are given in Table 2.The results comp uted by using the perturbation solution (18) and the numerical integrationscheme (22) compare well with those of reference [15] and those computed from


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220 G. SINGH E-J-AL.expression (17) match very well with those of references [ 13,16-18 1. From Tables 1 and2, it is clearly seen that as the aspect ratio increases, the non-linearity increases for thefirst fundam ental mode. Otherw ise, results for a rectangular plate show behaviour tenden-cies similar to those of a square plate.The non-linear frequency to linear frequency ratio values at different amp litudes for asquare orthotropic and two layered cross-ply are presented in Tables 3 and 4 along withvalues from references [lo, 121. It can be seen that a close agreement exists between thepresent values obtained by using equations (18) and (22) and the referenced values.From a ll these comp arisons, it can be seen that the proposed direct numerical integrationscheme with a one term approximation to the in-plane and lateral displacemen ts predictsthe non-linear free-vibration behaviour of isotropic, orthotropic and cross-ply plates veryaccurately. Further results for symm etric an d antisymmetric cross-ply plates based on thedirect numerical integration scheme are presented in Tables 5-10.

In Table 5 are given the frequency ratios for various amplitudes, for square plate havingmod ulus ratio 40. The table contains solutions for two- and four-layered antisymmetricand four-layered symm etric cross-ply plates, w ith the total thickness of the plate keptconstant, for rn = 1, n = 1 and m = 2, n = 1. It is clear from the table that a two-layeredplate has the largest oNL/wL for all amplitudes, whereas the symm etric lay-up(oO/90/900/oO) has the smallest (for m = 1, n = 1 and m = 2, n = 1). The two symmetriclay-ups presented in Table 5 have the same non-linear frequency for M = 1, n = 1 andhave a large variation in the other m ode: i.e., m = 2, n = 1. Furthermore, the frequencyratio uNL/uL reduces with increase in number of layers for an antisymmetric lay-up.

Frequency ratios mN L/wL at various amplitudes for a number of stack sequences of arectangular plate having modulus ratio 40 and aspect ratio two are presented in Table 6.As in Table 5, the two-layered plate sh ows the largest non-linearity for m = 1, n = 1 an dm = 2, n = 1 at all the amplitudes. The four-layered antisymmetric lay-up is found to havesma ll wNL/oL at all the amplitudes for m = 1, n = 1 when comp ared to a four-layeredsymm etric cross-ply (O a/900/900/0a ), whereas the four-layered symm etric stack(90/Oo/Oo /900) has the sma llest non-linearity. Furthermore, the two symm etric stackshave exactly the same non-linear frequency for the mode m = 2, n = 1. On comparisonwith results of Table 5, it can be seen that wNL/wL increases with aspect ratio for the firstthree lay-ups an d decreases for a symm etric lay-up (90 /Oo/Oo/9~). Resu lts for rectangularplates of aspect ratio four, which are found to be of a similar nature, are given in Table 7 .

In Tables 8-10 are given frequency ratios at various amplitudes for plates having a spectratios 1, 2 and 4, and different stack sequences. The modulus ratio considered in thesecomputations is 10. By comp aring the frequency ratios of Tables 5-7 with the correspond-ing ratios of Tables 8-10, it can be observed that the effect of non-linearity is greater forplates of higher modulus ratios. However, the symm etric lay-up (90/Oo/00 /900) showsthe reverse trend. The two symmetric lay-ups chosen, i.e., (O/90/900 /Oo) and(90/Oo/Oo /900), will yield the same frequency for a square plate w ith m = 1, n = 1 andfor rectangular plates say a/b = 2.0, with m = 2, n = 1 and so on.From the results presented in Tables 5-10 it can be seen that for antisymmetric cross-plyplates, the frequency ratio (wNL /wL) increases with increase of aspect ratio, modulusratio and for higher modes (e.g., m = 2, n = l), and reduces with increase in number oflayers. For the two symm etric lay-ups considered no general con clusion could be drawn,because wNL /wL ratios are mainly dependent on the bending stiffness distribution in thetwo directions, which differs appreciably in symm etric stacks.

The effect of sign and m agnitude of the amplitude (solution of equation (22)) on thenon-linear frequency ratio for a simply supported six layered antisymmetric cross-ply(0/90), rectangular plate (a/b = 2-O), with imm ovable edges, is shown in Table 11. It


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22 2 G.SINGH ETAL.TABLE 4

Compari son of frequency rati os of two layered cross-ply plate; AL (m = 1, n = 1) = 2.2022

Glass properties [lo]M odeshape w0/ t E q. (16) E q. (17) E q. (18) E q. (22) Ref. [lo]m=l o-3 1.0674 1.0343 1.0510 1.0480 1.050n=l 0.6 1.2481 1.1309 1.1909 1.1827 1.200

0.9 1.5016 1.2757 1.3932 1.3762 1.4001.2 1.7974 1.4544 1.6350 1.6069 1.6501.5 2.1179 1.6561 1.9011 1.8610 1.9001.8 2.4534 1.8734 2.1827 2.1304 2.2002.1 2.7985 1.1014 2.4746 2.4099 2.5002.4 3.1501 2.3370 2.7735 2.6965 2.8002.7 3.5062 2.5781 3.0773 2.9882 3.0503-o 3.8655 2.8233 3.3848 3.2836 3.333

TABLE 5Frequency ratios of a square cross-ply plate; EL / ET = 40; GLT/ ET = 0.5; uLT = 0.25

%w/~LM odeshape w,/ t 0/90 w/9cY /o/900 o/900/900/00 90/oo/00/900m=l 0.25 1.1433 1.0634 1.0535 1.0535n=l 0.50 1.5003 1.2388 1.2038 1.20380.75 1.9517 1.4832 1.4172 1.41721.00 24463 1.7679 1.6691 1.6691

1.50 3.4917 2.4000 2.2355 2.23552.00 4.5693 3.0729 2.8439 2.8439AL 3.9184 6.0425 6.6004 6.6004

m=2 0.25 1.1899 1.0820 1.0435 1.1669n=l 0.50 1.6411 1.3025 1.1678 1.57270.75 2.1922 1.6010 1.3480 2.07611.00 2.7852 1.9416 1.5643 2.62221.50 4.0229 2.6850 2.0584 3.76812.00 5.2894 3.4666 2.5955 4.9444

AL 10.9904 17.3056 24.0575 11.8038

can be seen that the frequency ratios change with the sign of the amplitude; in otherword s, the plate stiffness in the positive and negative z directions is different for thisconfiguration.

4. CONCLUDING REMARKSA general theory of antisymmetric cross-ply plates based on Kirchhoffs hypothesis

incorporating von Karman type strain-displacement relations has been used to study thelarge-amplitude vibrations of cross-ply plates. The bending and extension coupling isfound to increase the frequency ratios and reduce the linear frequencies. The arithmetic


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CROSS-PLY PLATE NON-LINEAR VIBRATIONS

TABLE 6

223

Frequency ratios of a rectangular (a/b = 2.0) cross-pl y plate; EJ ET = 40; GLT = 0.5;VL7-= 0.25

~NL/~LModeshape Walt 0/90 0/90/0/90 0/90/90/0 90/w/oo/900

m=l 0.25 1.1516 1.0645 1.1327 1.0340n= l 0.50 1.5254 1.2427 1.4674 1.13310.75 1.9949 1.4905 1.8946 1.27981.00 2.5074 1.7787 2.3652 I.45931.50 3.5876 2.4178 3.3634 1.87772.00 4.6994 3.0976 4.3949 2.3396Al. 2.7476 4.3264 2.9509 6.0144

m=2 0.25 1.1777 1.0795 1.0672 1.0672n= l 0.50 1.6049 1.2942 1.2519 I.25190.75 2.1310 1.5857 1.5077 1.50771.00 2.6993 1.9193 1.8042 1.80421.50 3.8888 2.6486 2.4600 2.46002.00 5.1079 3.4166 3.1561 3.1561AL 3.9184 6.0425 6.6004 6.6004

TABLE 7Frequency ratios of a rectangular (a/b = 4.0) cross-ply plate; EL / ET = 40; GLT/ ET = 0.5;

uLT = 0.25%I./%Mode

shape W0l t oO /90 0/900/00/900 0/900/900/oo 90 /0 /0 /90m=l 0.25 1.1575 1.0653 1.1707 1.0326n= l 0.50 1.5437 1.2454 I.5838 1.1275

0.75 2.0263 1.4956 2.0951 1.26881.00 2.5518 1.7863 2.6489 1.44231.50 3.6575 2.4303 3.8099 1.84792.00 4.7943 3.1148 5.0011 2.2971

2.5996 4.1552 2.4855 5.9464m=2 0.25 1.1899 1.0820 1.1669 1.0435n= l 0.50 1.6411 1.3025 1.5727 1.1678

0.75 2.1922 1.6010 2.0761 1.34801.00 2.7852 1.9416 2.6222 1.56421.50 4.0229 2.6849 3.7681 2.05842.00 5.2894 3.4666 4.9444 2.5955AL 2.7476 4.3264 2.95 10 6.0144


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22 4 G. SINGH E T A L .TABLE 8

Frequency rati os of a square cross-ply plate; EL / ET = 10; GLT/ ET = O -5; uLT = 0.25wwl WL

Modeshape WOl t o/ 90 0/900/oo/900 0/900/900/00 90/00/00/900m=l 0.25 1.0853 O-0556 1.0498 1.0498n=l 0.50 1.3136 1.2113 1.1907 1.19070.75 1.6212 1.4315 1.3922 1.39221.00 1.9713 1.6906 1.6314 1.63141.50 2.733 1 2.2715 2.1722 2.17222.00 3.5329 2.8941 2.7553 2.7553

AL 2.7839 3.4809 3.6841 3.6841m=2 0.25 1.1170 1.0729 1.0441 1.1224n=l 0.50 1.4176 1.2718 1.1701 1.43460.75 1.8073 1.5446 1.3524 1.83731.00 2.2404 1.8589 1.5711 2.28341.50 3.1654 2.5499 2.0699 3.23372.00 4.1250 3.2804 2.6118 4.2182

AL 7.4680 9.5926 12.4517 7.2909

TABLE 9Frequency rati os of a rectangular (a/b = 2) cross-pl y plate; EL / ET = 10; GLT/ ET = 0.5;

VLT.= 0.25wNL IWLMode

shape Wol 0/90 oQ/90/oo/900 o/900/900/oo 9o/oo/oo/900m=l 0.25 1.0955 1.0588 1.0986 1.0352n=l 0.50 1.3457 1.2222 1.3579 1.13750.75 1.6781 1.4516 1.7013 1.28851.00 2.0530 1.7204 2.0877 1.47291.50 2.8634 2.3206 2.9210 1.6795

2.00 3.7107 2.9620 3.7908 2.3730

m=2 0.25 1.1014 1.0663 1.0595 1.0595n=l 0.50 1.3669 1.2489 1 e2250 1.22500.75 1.7175 1.5022 1.4573 1.45731.00 2.1111 1.7961 1.7293 1.72931.50 2.9585 24466 2.3360 2.33602.00 3.8423 3.1375 2.9840 2.98402.7839 3.4809 3.6841 3.6841

1.8670 2.3982 1.8227 3.1129

mean of the two solutions obtained with the harmon ic oscillations assum ption leads toresults which are very close to those obtained from the direct numerical integrationsolution. An increase in the aspect ratio or mod ulus ratio increases the effect of non-linearity for antisymmetric cross-ply plates. The direct numerical integration methodproposed herein yields accurate frequency ratios and is more general.


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CROSS-PLY PLATE NON-LINEAR VIBRATIONS 225TABLE 10

Frequency ratios of a rectangular (a/b = 4.0) cross-ply plate; EL / ET = 10; GLT/ET = 0.5;VL T= 0.25

ONL IWL

Modeshape W 0l f 0 /90 0/900/oo/900 0/900/900/00 9o/00/oo/900 -m=l 0.25 1.1034 1.0613 1.1259 I.0342n=l 0.50 1.3722 1.2310 1.4459 1.13370.75 1.7259 1.4683 1.8572 1.2811

1.00 2.1225 1.7455 2.3117 1.4613I.50 2.9757 2.3624 3.2788 1.88112.00 3.8650 3.0204 4.2796 2.3444

AL 1.7044 2.2346 1.5263 3.0159m=2 0.25 1.1170 1.0729 1.1224 1.0441n=l 0.50 1.4176 1.2718 1.4346 I.1701

0.75 1.8073 1.5446 1.8373 1.35241.00 2.2404 1.8589 2.2834 I.57101.50 3.1654 2.5498 3.2337 2.06992.00 4.1250 3.2804 4.2182 2.6118

AL 1.8670 2.3982 1.8228 3.1129

TABLE 11Frequency ratio ( wNL/oL)variation with amplitude fora six-layered antisymmetriccross-ply plate; E,l ET = 10;

GLT/ ET = 0.5; VL T= 0.25-

2.0 2.47441.5 1.93101.0 1.44120. 5 1.07850 1.0

-0.5 1.2672-1.0 I.7201-1.5 2.2479-2.0 2.8099

REFERENCESM. SATHYAMOORTHY 1987 Applied M echanics Review 40, 1553 -156 1. Non-linear vibrationanalysis of plates: a review and survey of current development.C. Y. CHIA 1980 Non-linear Analysis of Plates. New York: McGraw-Hill.S. A. AMBARTSUMYAN 1970 Theory f AnisotropicPlates. Stanford, Connecticut: Technomic(English translation).J. E. HASSERT and J. . NOWINSKI 1962 Proceedings, 5th I nrernational Symposium on SpaceTechnology and Science, Toky o, 561-5 70. Nonlinear transverse vibration of a flat rectangularorthotropic plate supported by stiff rig.


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