an iteration method for the large amplitude flexural vibration of antisymmetric cross-ply...

Composite Structures 18 ( 1991 ) 263-282

An Iteration Method for the Large Amplitude Fiexural Vibration of Antisymmetric Cross-Ply Rectangular Plates

Bharat Bhushan, a Gajbir Singh b & G. Venkateswara R a o b

~Quality Assurance Division, System Reliability Group, bStructural Design and Analysis Division, Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum

695022, India

ABSTRA CT

An analytical sohttion to the large amplitude free-vibration problem of antisymmetric cross-ply rectangular composite plates, having an additional quadratic nonlinear term in the modal equation of equilibrium is contained herein. It is shown that the classical two-term perturbation solution and further extension of the same for five-term fail to yield any meaningful results when the coefficients of nonlinear terms in the modal equation are large. Hence, an iteration method, used to solve the Duffing's equation for isotropic plates, has been extended to solve the present modal equation of equilibrium corresponding to the antisymmetric cross-ply rectangular plates with simply supported edges. It is observed that the Duffing's assumption of considering the first Fourier coefficient as amplitude results in erroneous estimation of nonlinear free- vibration behaviour of such plates. The nonlinear frequencies obtained from second iteration of the proposed iteration method compare well with those obtained from the direct numerical integration method which is found to yield accurate results. It is seen that the presence of an additional quadratic term in the modal equation results in different amplitudes in the positive and negative half cycles of vibration.

NOTATION

a, b Plate length and width along the x- and y-directions A, B Maximum deflections of the plate for the positive and

negative half cycles of vibration Ai/, Bi/, Di/ Extensional, bending-extensional coupling and bending

stiffness coefficients of plate 263

Composite Structures 0263-8223/91/S03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

264

AI,A~ t) EL, Et

GLT KE l m, r/

SE t ti II O, U 0, W

u , v , w x, y, z

B. Bhushan, G. Singh, G. Venkateswara Rao

First Fourier coefficients Tensile moduli of lamina in filament and transverse directions Shear modulus of lamina Kinetic energy of the plate Number of layers in plate Half sine waves in x- and y-directions Reduced stiffness coefficients of a layer Strain energy of plate Plate thickness in z-direction Thickness of ith layer Mid-plane displacements in x-, y- and z-directions Maximum displacements at any instant of time Cartesian coordinates

a Linear stiffness coefficient fl Quadratic nonlinearity coefficient y Cubic nonlinearity coefficient ex °, ey,° exy° Mid-surface strains R;x, Ky, K'x.y Curva tu re s VLT Poisson's ratio p~ Density of ith layer r Time o Frequency o~ 0 Linear frequency (= ,~a) () d( )/dr

INTRODUCTION

Fibre-reinforced composite materials are widely used in aerospace structures due to their high strength and stiffness. Because of the aniso- tropi.c properties of these materials the analysis of composite structures becomes a complex task compared with the conventional metallic structures. The high strength of these materials results in the requirement of low thicknesses of laminates in these structures which are often subjected to severe dynamic environments. Thus, these structures may vibrate with large amplitudes, where linear flee-vibration analysis becomes insufficient. Nonlinear dynamic analysis of composite laminates have attracted several researchers. Chia ! has summed up the information on nonlinear response of plates in his classic book. A comprehensive

An iteration method for large amplitude flexural vibration 265

survey on the large amplitude vibration of plates has been presented by Sathyamoorthy. 2 Antisymmetric angle-ply laminates and all symmetric laminates have only cubic nonlinear terms in the modal equation of equilibrium and have been widely analysed by researchers. Wu and Vinson 3'~ evaluated the nonlinear frequencies of orthotropic and symmetric laminates based on Berger's hypothesis. 5 Bennett 6 and Chandra and Raju 7 presented the large amplitude free-vibration behaviour of antisymmetric angle-ply plates.

Whitney and Leissa s derived the nonlinear equations of motion for unsymmetrically laminated plates and presented solutions for the linearised equations. The second-order, ordinary, differential equation obtained after substituting the spatial distribution satis~ing the boundary conditions for such plates has a quadratic nonlinear term in addition to a cubic nonlinear term, because of the bending-extension coupling. The presence of this quadratic nonlinear term, causes the plate to vibrate with different amplitudes in positive and negative half cycles of vibration. Bert 9 employed the Ritz-Galerkin technique to study the nonlinear vibrations of clamped-clamped unsymmetrically laminated plates. In order to simplify the solution, nonlinear terms from the inplane equilibrium equations are dropped, on the grounds that these terms vanish in the first stage of application of the perturbation technique (cf. the Appendix in Ref. 10).

Chandra and Raju ~'12 studied the nonlinear vibrations of simply- supported cross-ply rectangular plates, wherein the two-term perturbation method is employed to solve the nonlinear governing equation having both quadratic and cubic nonlinear terms. Recently, Singh e t al. ~3

demonstrated that the two-term perturbation method used by Chandra and Raju and many other researchers, fails to yield any meaningful results and proposed a 'direct numerical integration method' to predict nonlinear frequencies accurately.

In the present paper an analytical solution to study the nonlinear vibration behaviour of antisymmeric cross-ply plates is obtained. It is shown that the two-term or even five-term perturbation solutions fails to yield any meaningful results. An iteration method used to solve the Duffing's equation for isotropic plates is extended to solve the present modal equation of equilibrium containing quadratic and cubic nonlinear terms. It is demonstrated that the Duffing's assumption of considering the first Fourier coefficient as amplitude of vibration leads to erroneous results when the coefficient of the quadratic nonlinear term is significant. Hence, in the present study, an iteration method is proposed wherein the first Fourier coefficient is used to derive a relationship between

266 B. Bhushan, G. Singh, G. Venkateswara Rao

frequency and amplitude. The present study indicates that the second iteration of this method leads to accurate results.

FORMULATION

A conventional x, y, z coordinate system is used to identify the geometry of rectangular plate of length, a, width, b and thickness, t as shown in Fig. 1.

Q

...... iiii ¥

Fig. 1. Geometry of the plate.

r

The strain energy (SE) of antisymmetric cross-ply plate having I layers 0 0 and 0 and can be expressed in terms of mid-plane strains, e~, ey exV,

bending curvatures, r X, ry and rxy as

l ( a ( b

+ 2A 12ex ey + + + 2B1 - ° r .~22~.y A66e~ lex x ~ J 0 J0

9 9 2 +2B22eOy~.y+Dllgx+2D12gxTcy+D227¢y+D667¢xy}dxdy (1)

where,

1 [h,_, , ~ d (Ai / ,Bi / ,Di / ) = Z - (~k/(1 Z,Z ~) Z ( i , ] = 1 , 2 , 6 ) (2)

k=l .,Ih k

a n d AI6 , A26, BI2, BI6, B26, B66, DI6, D26 = 0. The nonlinear strain-displacement relations (von-Karman type) can


be expressed as

o ex ax 2 k ax }

o or"+ l {Owl2 gy = ay 2 ~-0-fy]

o Ou°+Ov° OwOw

(3a)

Ky

Kxy

a w/ax z } = - 02 w/Of

202 w/Ox Oy (3b)

The kinetic energy (KE) of the system may be expressed as

f:f0 K E = ~ Piti w2 dx dy ) i=l

(4)

The admissible functions, satisfying simply supported conditions with immovable edges are assumed as

2m~r nzr u ° = U(r) sin - - x sin

a --flY

m~ 2n.zr v °= V(r)sin - - a xs in ---if- y (5)

m~ n~r w = W(r) sin - - x sin - - y

a b

Substituting eqns (5) and (3) in eqns (1) and (4) and integrating gives total strain energy (SE), and kinetic energy (KE). Applying Hamilton's principle, i.e. taking variation of (SE - KE) leads to two algebraic and a second-order, ordinary, differential equation in terms of U, V and W. Substituting U and V, obtained from the first two equations in terms of IV, in the third equation, gives a second-order, ordinary, differential


equation with quadratic and cubic nonlinear terms as follows:

#+aW+ flW2 + y W 3 "= 0 where,

( )/ a= Ts-I 2 T~ Ta Ts - T3 T-5 - T'~ T6. T~T6-T{. (Zpit')

( 3T2T4T~+T, T2Tv-T3TsTv-TiTaT6)/ fl = T 9 4 " - - - - ~ (Yp,t,)

T3 T6 - Tg_

y= (Tto+ 4T2T4Tv-2T3TeT-2T~T6)/ -r--Tg6:--f':,. <zp,,,)

(6)

(7)

Expressions for 'T ' are given in Appendix A as functions of plate stiffnesses, dimensions and wave numbers. In the following subsections a brief discussion of the perturbation method and iteration method is presented.

Perturbation method The differential equation of equilibrium (eqn 6) has been solved by Chandra and Raju ~ L ~2 using a two-term perturbation technique leading to the frequency-amplitude relationship as

t/2

(8)

This solution reveals that the frequency ratio is invariant with the sign of the amplitude, whereas the presence of the 'fl' term in the modal equation" of equilibrium does not suggest so. It could be because of the solution being limited to two terms, hence the authors have extended this solution to five terms. Assuming 'fl' and 'y ' as the perturbation parameters, the deflection and frequency can be expressed as

w= Wo+/~w, + yw2 +#2 w3 +~yw4 + y2 w5 (9)

o ~ = O,o ~ +/~o~, ~ + yo,~ + / ~ + ~yo,~ + y~a,i

Solving the differential equation (eqn 6) with these assumptions, the


frequency-amplitude relationship becomes

w l + _ 3 A , y 5 A 2 1 fly 3 a . . . . ----;+- A 3 - - A Y' (10) Wo 4 a 6 a- 2 a 2 128

and the deflection-time relationship is

IV= ' _ : cos 309r Co+ C', cos w r + C" cos 2wr + C'.

+ C ~ c o s 4 w r + C ' s cos 5wr (11)

Expressions for the coefficients C' are given in Appendix B. These equations reveal that the solution is unsymmetric about the

zero amplitude position. But the solution is valid for large amplitudes only if 'fl' and "y' are small compared to ' a ' as the series represented by eqn (10) diverge for large 'fl' and 'y'.

However, in general for unsymmetric composites, "fl' and 'y' are quite often greater than ' a ' so the perturbation method even with higher terms will fail to give any meaningful results. Hence, in this paper Duffing's iteration method is used with some modification and accurate solutions are obtained even for large 'fl' and '7'.

Iteration method

In order to apply the iteration method the differential equation (eqn 6)is rewrittenl 4 as

if '+ 092 W= (¢.02 - a ) W - ~ W 2 - y l / V 3 (12)

An approximate solution to eqn (12) is obtained by solving the linear differential equation by assuming 'fl' and 'y' to be zero. as

W 0 =A 1 cos wr (13)

In eqn (13) the sine term of the solution vanishes, as the coefficient of this term becomes zero for the initial condition I)¢= 0 at r = 0. In this solution, the first Fourier coefficient, A ~, becomes the amplitude for the first approximation.

Inserting W 0 in the right-hand side of eqn (12), we obtain a differential equation for the next approximation W~, as

14' l + w 2 W 1 =(w 2 - a - ] A ~ y ) A 1 cos w r - ½ A ~fl

- ½ A ~ f l c o s 2 w r - ¼ A 3 y c o s 3 w r (14)

2 7 0 B. Bhushan, G. Singh, G. Venkateswara Rao

Now applying the condition of periodicity for the solution, the coefficient of cos tot in the right-hand side of the above equation should vanish. Equating this coefficient to zero yields a constraint on frequency 'w' as follows:

0) 2 - a - ~ - A i y = 0 (15)

Satisfying this constraint, the solution of eqn (14) is

1 Wt =A~ll cos tot+ 2(-½A215+~AZttScos2tor+3-~A3)'cos3wr )

to

(16)

in which 'to' and 'A~' are related by the frequency constraint equation (eqn 15) and the term A~%os tor is the homogeneous solution of eqn (14) with an arbitrary coefficient A ~) (first Fourier coefficients of second approximation). In the second approximation, Wt(r) is not fixed until the value of coefficient A~ ~/ is prescribed. Thus, following Duffing's procedure,~4 coefficient A I~} is assumed as A~. Accordingly we have

1 W l = A l cos to r+ ..... , ( - ~ A 2 / 3 + ~ A 2 1 3 c o s 2 t o r + ~ A 3 ) , c o s 3 w r ) ( 1 7 )

to

Again it is clear from eqn (17) that the amplitude is unsymmetric about its zero amplitude position and the first Fourier coefficient of the oscillation ~4 I' is not the amplitude of oscillation, A, as assumed by Duffing. ~4 The coefficient ~4~' can be calculated from the frequency constraint equation (eqn 15) if 'to' is prescribed. The maximum deflection (amplitude) at r = 0 and r = x/to can be computed using eqn (17) as

1 A =A, +---5(-~ A ~/3+~ A 3),) (18a)

to

at r = 0 ,

1 B= -A~ +--7 ( -~ A ~ f l - ~ A ~ ) (lSb)

to

at r = ~r/to The other way to obtain the nonlinear frequency for a prescribed

initial amplitude is, the coefficient 7t !' is initially assumed as ~4' and 'to' is computed from eqn (15), then using these 'A~' and 'to ', amplitude :'A" is computed from eqn (18a). Based on this value of amplitude, a new value


of 'A,' is fixed and the procedure is repeated until the prescribed amplitude is obtained.

To obtain the second iteration results, the same procedure is repeated and the following expressions for the frequency and deflection are obtained:

/•2 2 , 2 + 5 2 3 4 Y 105 4fl2

°J'-a-3A'Y4 6 A I 022 128 A , o~2 --A,192 ~TY

~2 3 AI6 < + 11 AI6 Y6"=0 (19)

2048 a~ 1536 ~o

and

W 2 ---A, cos a~r+ Co + C2 cos 2 ~ r + C 3 cos 3oor+ C~ cos 4oJr

+ C 5 cos 5~or+ C 6cos 6o~r+ C 7 cos 709r+ C s cos 8oJr

+ C9 cos 9wr (20)

Expressions for the C coefficients are given in Appendix C. Again the amplitude at r = 0 and r = ~/a~ are

A=AI +Co +C2-FC3 WC4 +C5 WC6-FC7-FCs-FC9 (21a)

and

B= -A1-I-C0+C2-C3+C4-C5+C6-C7+C8-C9 (21b)

N U M E R I C A L RESULTS AND DISCUSSION

Based on eqns (8, 10, 15 and 17-21) the nonlinear frequency ratios and variation of deflection with time 'r ' are presented in the form of figures for simply supported, antisymmetric cross-ply rectangular plates. The following unidirectional lamina properties are used in the analysis:

EL/ E T = 40.0

GLT/E T = 0"5

liLT = 0"2 5

The coefficients 'a ' , 'fl' and 'y ' for a square and rectangular (a/b = 2), two-layered cross-ply plate are computed using eqn (7) for m = 1, n-- 1 and are presented in Table 1. It may be observed that the coefficient of


TABLE 1 Values of a, fl and y for a Two-Layered Cross-Ply (00/90 °)

Plate

a/b 1 2

a 19-5543 10-0318 fl 0"0 - 22"5889 )' 48"2833 25"5135

the nonlinear term y is large compared to linear stiffness coefficient a. Further, for a two-layered cross-ply rectangular plate, due to predominant bending-extension coupling the coefficient fl also is large.

The frequency ratio to/to 0 at various amplitudes are computed using two-term and five-term perturbation method, for two-layered cross-ply square and rectangular plates, and are presented in Figs 2 and 3 respect-

Fig. 2.

~2

0:3 iz :1 A - - - - ' ~

Ampli tude versus frequency ratio of a square plate - - perturbation method. a / b ffi 1, ( A ) two-term solution, ( o ) five-term solution, ( o ) Ref. 13.


3

T ~2

-3 -2 -1 0 1 2 3

A P

Fig. 3. Ampl i tude versus frequency ratio of a rectangular plate - - per turbat ion method, a/b = 2, (zx) two-term solution, (D) five-term solution, (o) Ref. 13.

ively. Direct numerical integration results of Singh et al. 13 a r e also included in these figures for comparison. It may be seen in Fig. 2 that in the case of a square plate (fl = 0.0) the two-term perturbation solution gives fairly correct results when compared to numerical integration results; whereas, the five-term perturbation solution differs from the numerical integration results as amplitude increases, since, the higher power terms in the series of eqn (10) become larger for higher amplitudes. Hence, it may be concluded that the perturbation solution will oscillate about the exact solution as the number of terms in the perturbation series increases and diverges for large amplitudes. Further, for two-layered cross-ply rectangular plates, wherein the nonlinear governing eqn (6) has non-zero fl and 7, the perturbation method (two- term/five-term) fails to yield any meaningful results for amplitudes of practical interest.

In the case of the iteration method, the variation of the difference in the amplitude and first Fourier coefficient (at the same frequency) with

274 B. Bhushan, G. Singh, G. I/enkateswara Rao

amplitude is shown in Fig. 4 to bring out the error in the Duffin~'s assumption of considering this Fourier coefficient, A m as amplitude, A ."It may be observed that this difference is within 4% for a square, two- layered cross-ply plate, where coefficient fl is zero. Whereas for a rectangular (a/b= 2), two-layered cross-ply plate, this difference is quite large (of the order of 24% at an amplitude of 0.8 and 35% at an amplitude of

Fig. 4.

T

.7

3° I I

20! !

10L

-10

-20

-30

-,.o "2 -'1 ~ A

Error in Duffing's assumption, a/b = 1, (zx) first iteration, (t~) second iteration; a/b = 2, (<>)first iteration, (o) second iteration.

- 0.5), and hence, may lead to erroneous estimation of the nonlinear frequencies. Further, the difference in the two iteration results is seen to be small.

The percentage difference between first and second iteration frequencies is plotted against amplitude in Fig. 5. The difference in the case of rectangular plate is seen to be as high as 35% for an amplitude of

A n iteration method for large amplitude flexural vibration 2 7 5

about 0"5 and - 0-3; whereas the difference is less for lower and higher amplitudes. For very small amplitudes the higher power terms in eqn (19) become negligible due to high powers of small A l and thus, the difference in two frequencies from first and second iteration is small. As the amplitude increases, initially up to around a value of 0"5 or - 0.3, the frequency, w, decreases (softening type of nonlinearity). 13 Since w is in

o

-- -5 ! .'K

-10

"~ <'c

'~ -20

i

~ -30

-35

Fig. 5.

~2 ' ~ -t,O -3 - -1 0 I

Improvement in frequency by second iteration. (o) a/b = 1, ( o ) a/b = 2.

the denominator of the additional higher power term (eqn 19), the effect of these terms becomes significant. Thus, the difference between two frequencies increases up to about 0.5 or -0-3 amplitude. But as the amplitude increases further the frequency also increases (hardening type of nonlinearity) and the higher power terms again become small, thus reducing the frequency difference. Whereas, for a square plate (fl = 0.0),


the frequency increases monotonically with amplitude, ~3 resulting in small, higher-power terms at any amplitude. Hence, the difference in the frequency obtained from the first iteration and the second iteration is small.

The amplitude-frequency ratio relationship by the proposed iteration method is shown in Figs 6 and 7 for square and rectangular ( a / b = 2)

t

-3 -Z -1 0 ! 2 3

A Ila

Fig. 6. Amplitude versus frequency ratio of a square plate - - proposed iteration method, a/b = 1,( A ) first iteration solution.(o) second iteration solution and Ref. i 3.

plates, respectively. An excellent agreement between proposed method and numerical integration method m3 is seen for the square plates. However, a very small difference is seen between first and second iteration results of the proposed method for the same problem. Whereas for rectangular plates, the second iteration of the proposed method is essential to obtain accurate results, as presented by Singh et at., 13 a little

An iteration method jbr large amplitude flexural vibration 277

difference is seen at the amplitude where the difference between the first and the second iteration frequencies is the highest, suggesting that the difference will be narrowed down if the next iteration is attempted. The frequencies obtained from the first and the second iteration are seen to match better at large amplitudes, because the difference in these solutions reduces, as shown in Fig. 5. It can be seen from Fig. 7 that the

l 3

6

2

0 i , I i r J - 3 - 2 -1 0 1 2

k . - - D -

Fig, 7. Amplitude versus frequency ratio of a rectangular plate -- proposed iteration method, a/b = 2, ( A ) first iteration solution, (cJ) second iteration solution, (o) Ref. 13.

variation of w/to 0 is not symmetric about the frequency ratio axis when fl is present.

The variation of deflection with time is shown in Fig. 8. It is clear from this figure that for the square plate the deflection is symmetric about the zero amplitude position, whereas for a rectangular plate, where quadratic nonlinear term fl is present, the deflection is unsymmetric about the zero amplitude position, as expected for this type of laminate.


1 ~ -7 -;

0.8

0.6

O.t.

0.2

I ° -0,2

-O.t.

-0.6

Fig. 8.

-0 .8

-1 0 0.2 O.g 0.6 O.B 1 1.2 1.g 1.6 1.8

T

Variation of deflect ion with time. ( ~ ) a / b = 1, (o) a /b = 2.

CONCLUSIONS

The following conclusions can be drawn from the present work:

(1) For an unsymmetric laminate wherein bending-extension coupling is present the perturbation method fails to yield any meaningful results.

(2) Duffmg's assumption of the first Fourier coefficient as amplitude will lead to erroneous estimation of nonlinear free-vibration behaviour of unsymmetricaUy laminated plates.

(3) The second iteration results of the proposed method agree well with the direct numerical integration results.

(4) The antisymmetric, cross-ply rectangular (a/b > 1) plates oscillate with different amplitudes in positive and negative half cycles of vibration.


(5) The proposed method is general and can be applied to plates with other boundary conditions, even though the efficacy of the method is demonstrated with respect to a simply supported plate.

REFERENCES

1. Chia, C. Y., Nonlinear Analysis of Plates. McGraw-Hill, New York, 1980. 2. Sathyamoorthy, M., Nonlinear vibration analysis of plates: A review and

survey of current developments. Appl. Mechan. Rev., 40 (1987) 1553-61. 3. Wu, C. I. & Vinson, J. R., On the nonlinear oscillations of plates composed

of composite materials. J. Comp. Mater., 3 (1969) 548-61. 4. Wu, C. I. & Vinson, J. R., Nonlinear oscillations of laminated specially

orthotropic plates with clamped and simply supported edges. J. Acoust. Soc. Amer., 49, no. 5(2), (1971) 1561-7.

5. Berger, H. M., A new approach to the analysis of large deflection of plates. J. Appl. Mechan., 22 (1955)465-72.

6. Bennett, J. A., Nonlinear vibrations of simply supported angle-ply laminated plates. AIAA J., 9 (10) ( 1971 ) 1997-2003.

7. Chandra, R. & Raju, B. B., Large deflection vibration of angle-ply laminated plates. J. Sound Vibrat., 40 (3) (1973) 393-408.

8. Whitney, J. M. & Leissa, A. W., Analysis of heterogeneous anisotropic plates. J. Appl. Mechan., 36 (1969) 261-6.

9. Bert, C. W., Nonlinear vibration of a rectangular plate arbitrarily laminated of anisotropic material. J. Appl. Mechan., 40 (1973) 452-8.

10. Mayberry, B. L. & Bert, C. W., Experimental investigation of nonlinear vibrations of laminated anisotropic panels. Shock Vibrat. Bull., 39 (3) (1969) 191-9.

11. Chandra, R. & Raju, B. B., Large amplitude flexural vibrations of cross-ply laminated composite plates. Fibre Sci. Technol., 8 (1975) 243-64.

12. Chandra, R., Large deflection vibration of cross-ply laminated plates with certain edge conditions. J. Sound Vibrat., 47 (1976) 509-14.

13. Singh, G., Venkateswara Rao, G. & Iyengar, N. G. R., Large amplitude free- vibration of simply supported antisymmetric cross-ply plates. AIAA J. 29 (5)(1991)784-90.

14. Stoker, J. J., Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers, New York.

280 B. Bhushan, G. Singh. G. Venkateswara Rao

APPENDIX A

where C, = 1 - ( - 1)"

C. = 1 - ( - 1)"

C.,. = C., C.

4 c T~ = 3net BY'

T 2 = ~ (AI2 +A66) 9mno'U

= 4 A22 + A66

T~=3m'----~ - k a ] k b ]

Ts= 3m:t Btl

- ?

T6=4 Au + ~ A66

T = Cn [2(~.~)3A _(m__~)(/¢)2(m,2_Z66)] 7 3n:t u

( 7 ) 4 + 2(m~12(n~t2 ( ~ ) a Ts= Du ~-~-i I l l (DI' +2D66)+ D22

4 C,,,,, Bu 4- B22 T 9 = 3mnzt2

1 (m~12(n~rl2 ( 7 ) a 1 lo 32 +-i--6 ~--a] k-if]


A P P E N D I X B

, . . . . . . 1 A 2 f l 1 A 3 f l 2 + 9 A 4 f l Y C o = 2 5 3 5" 32 5 2

2 _ _ _ _ f12 ~ 23 C ] = A + I A fl 1 A 3 Z + 29 A3 , 1 A4 + 3 5 32 5 144 5 48 a 1024

AsZ__ 5

f12 , 1 2 f l 1 3 1 4fl?' C2=-A - + - A 2 A

6 5 9 5 3 5

f12 ~, 2 1 A 3 Y + l A 3 = __ 1 4fl~' 3 5 C~ = - - ---~ + - - A , - - A

32 a 48 a" 16 5" 128 a -

1 fit, C '4=-~ A 4 --52

, y2 1 5

C 5 = 1024 A 2 5

A P P E ~ I X C

c - l ( a ) ' f l + 15 4 f ly 0)

f13 " 19 A 4 17 f l u 72 1 6 A6 ----g- w 2048 w

+ - - 21 A 6 3 8fl~ 3 t + A l s

144 to 4096 to

1 a 5 4 1 4 C 2 = i ~ + 2 A I - A l - A1 6

w 32 w 18 w 5 A 6 f l ~ '2

1 6 384 o)

+ 3 y3

3 7 A6 fl_.fl ~ 1 fl i + - - A S I 8 864 o~ 1 2 2 8 8 to


3 ~ - -

/32 3 ~ )'- 25 ~ fl: 1 + 7 A ~---~Y,+ 1 A 3 - - - 5 + - - A A- 256 o9" 48 1 l 4 l o9 512 o9 768 o9

+ n 57 A7/3-) ' - 3 9 74 + A t 1 S 8

18 432 o9 1 048 576 o9

/33 1 4 _ _ + . _ _ ~ 1 A4 1 6 flY:' 1 6 /33)/ C4 =~-~ A l o9 1080 1 (_D6 --384 A t co6 720-- A t ws

1 s/3)'3 + A t s

1 2 2 8 8 0 o9

2 1 " )'4 5 5 C s = - - A i - - + - - A t + - -

1024 o9 4608 o9

1 7 73 1 v/3272 A t 6 - - A t s

3 2 7 6 8 ~o 3 0 7 2 co

17 /3y2 1 f13)' 3 8 fly3 C 6 = _ _ A 6 . 6 A

71 680 co 30 240 o9 143 360 co

1 . t32 y2 1 7 C 7 = 6 5 536 A t 4 - - A i 8 co 7 3 7 2 8 o9

1 s/3), 3 C 8 = A 1 s

5 1 6 0 9 6 o9

1 A 9 ),4 C9 -- 10 485 760 t --qt.o

an iteration method for the large amplitude flexural vibration of antisymmetric cross-ply...

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