Free vibration of composite rectangular plates with rectangular cutouts

Free vibration of composite rectangular plates with rectangular cutouts

Composite Structures 8 (1987) 63-81 Free Vibration of Composite Rectangular Plates with Rectangular Cutouts H. P. Lee, S. P. L i m a n d S. T. C h o...

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Composite Structures 8 (1987) 63-81

Free Vibration of Composite Rectangular Plates with Rectangular Cutouts

H. P. Lee, S. P. L i m a n d S. T. C h o w Department of Mechanical and Production Engineering, National University of Singapore, Singapore 0511

ABSTRACT A simple numerical method based on the Rayleigh principle is presented for predicting the natural frequencies of composite rectangular plates which exhibit special and general orthotropy. The method is illustrated for simplysupported rectangular plates having central rectangular cutouts and double square cutouts. The results are compared with the reported finite element and analytical results.

NOMENCLATURE

A , An

Dij Ex,

Ey

Gxy h Q,; SI~ $2

W, W. W

x, y , Z

0 h i~xy

Amplitude of vibration Bending stiffness of the plate Young's moduli Shear modulus Thickness of the plate Bending stiffness of a layer of a laminate along the principal axes of orthotropy x coordinates of the comers of cutouts y coordinates of the comers of cutouts Maximum deflection of the mid-surface of the plate Transverse deflection of the mid-surface of the plate at time t Rectangular Cartesian coordinates Orientation of the principal axis of orthotropy Non-dimensional frequency parameter Poisson's ratio 63

Composite Structures 0263-8223/87/$03-50 O Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

H. P. Lee, S. P. Lim, S. 7". Chow

64

Mass density of the plate Angular frequency

P O3

1 INTRODUCTION The linear vibration of rectangular orthotropic plates with cutouts have been studied by several authors by extending the established methods for the isotropic plates to the orthotropic plates. Reddy ~presented the natural frequencies of simply supported and clamped specially orthotropic square plates with central rectangular cutouts by the finite element method. Rajamani and Prabhakaran z'3 reported the natural frequencies of specially and generally orthotropic square plates with central square cutouts by expressing the effect of a cutout as an external loading function on the plates. They assumed the deflection functions to be products of characteristic beam functions for the given boundary conditions. The method in conjunction with the Lagrange's equations of motion led to an infinite system of differential equations which were then truncated based on the required accuracy. The other contributions based on the Rayleigh-Ritz principle or the Rayleigh method were made by Kanazawa and Kawai, 4 Hearman 5 and Dickinson. 6,7 All these works were confined to rectangular orthotropic plates without any cutout. The present method is based on the Rayleigh principle and the classical plate theory for predicting the fundamental and selected higher modes of composite rectangular plates with rectangular cutouts. The method is illustrated by applying it to simply-supported rectangular plates with either special or general orthotropy.

2 ANALYSIS 2.1 Theory The plates under consideration are thin rectangular plates of size a x b having central cutouts and double cutouts as shown in Figs 1 and 2. The transverse vibration of the plates is governed by the following partial differential equation 8 D

04W + 4D 04w O4w 04w n c)X---'a16~-ff~-~y+Z(D12+2D66)~x~-~+gD26ax-~~y O4W

~2 w

+D22~-z---a-+ph_--7-v-= oy " ot"

0

(1)

Free vibration of composite rectangularplates

65

T h e m a x i m u m potential energy and kinetic energy are given by 2 +2D12-~"2-x 7 + D 2 2

\ Oy ]

O2W ~ 2 02W 02W 02W 02W -] + 4016-'~x OxOy + 4026--~-y 0--~y ] dx dy + 4D66 ~

)

K E = (½)Phto2 S

~ W2dxdy

(2) (3)

T h e above integrations are to be performed over the domain of the plate. T h e bending stiffnesses Dijof the plate in terms of the bending stiffnesses Qi~ along the principal axis of orthotropy are as follows. D n = Q . cos40 + 2(Qu + 2Qos) sinE0cos20 + Q22sin40 D22 = Qusin40 + 2(Qn + 2Qss) sin2Ocos2O + Q22cos40

D 12 = (Q n + Q2z - 4Q66)sinZ0cos20 + Qlz(sin40 + cos40) D 16

=

(Qn

-

(4)

Q 12- 2Q66) sin 0 cos30 + (Q 12- Q22 + 2Q66) sin30 cos 0

D26 = (Q t t - Q u - 2Q66) sin30 COS0 + (Q 12 Q22 d- 2Q66) sin 0 cos30 - -

D66 = (Qn + Q22 - 2Q12 - 2Q66) sin20cosZO + Q66(sin40 + cos40) The orientation of the principal axis of orthotropy is measured anticlockwise from the x axis. D 16 and D26 are equal to zero for a specially orthotropic laminate where the orientation of the principal axis of orthotropy is equal to 0 ° or 90 °. The rectangular plates are divided into smaller subdomains depending on the m o d e shapes and the locations of the cutouts. The m o d e shapes to be considered are pre-selected based on the symmetry of the cutouts about the geometrical axes. The four modes of vibration under consideration are the S X - S Y , A X - S Y , S X - A Y and the A X - A Y modes. SX and SY represent the first symmetric modes of vibration. Similarly, A X and A Y are the respective first antisymmetric modes of vibration about the lines defined by x = a/2 and y = b/2. The subdomains for the four modes of vibrations are presented in Figs 1 and 2. A single-term deflection function W, is assumed for each subdomain. The deflection function is taken to be of the form

W. = A . X . ( x ) Y.(y)

(5)

X . ( n ) and Y.(y) are beam functions which satisfy prescribed boundary conditions. The adjacent subdomains will have one of the two functions, X .

H. P. Lee, S. P. Lira, S. T. Chow

66

!

!

8

!

6

I I

3

Si

!

!I

6

I

0

l

r~

r2

x

(1

Fig. 1. A rectangular plate with a central rectangular cutout.

b

!

131

12

[9

11

I

Sz

#

Si

I

1 t !

2

3

j j

!

r3

r4

m

0

rl

r2

a

x

Fig. 2. A rectangular plate with double square cutouts.

and Yn in common. The relative magnitudes of coefficients An between adjacent subdomains are determined by matching the maximum deflections at the corners of the cutouts. The potential energy and kinetic energy are evaluated for each subdomain using eqns (2) and (3). The total potential energy and kinetic energy of the plate are assumed to be the sum of the potential and kinetic energy from all the subdomains. The frequency of the mode shape is obtained by equating the total potential energy to the total kinetic energy. The frequency is expressed in terms of a nondimensional frequency parameter given by 2[ oh \ 1/2

67

Free vibration of composite rectangularplates ORIENTATION OF FIBRE

/ /

2 a)

I

f

.-~1

×

GENERAL ORTHOTROPIC PLATE

b)

SPECIAL ORTHOTROPIC PLATE

Fig. 3. Geometry of orthotropic plate. h is related to the material properties as well as the aspect ratio of the plate for a given cutout configuration.

2.2 Simply supported rectangular plates The simply supported boundary conditions of a generally orthotropic plate are as follows 8 -

02W

02W

02W

a2W

aZW

02W

x = 0, a: W = 0, Mx = - Dlr-~-rx - Dl2---~y -2D16 axc)-----~

-

0

(7) y = O, b: W = O, My

=

--DI2---T--T--Dz2--T-T---2026

ox

oy

OxOy

-

0

The presence of D t6 and D26 in the governing differential equation and the b o u n d a r y conditions renders a closed form solution impossible. The deflection functions are assumed to be the same for specially orthotropic, generally orthotropic and isotropic plates. However, the assumed deflection functions which satisfy both the zero deflection and vanishing bending m o m e n t conditions at the external edges for specially orthotropic plates will only satisfy the essential boundary condition, namely zero deflection along the external boundaries for generally orthotropic plates. For these cases, the natural boundary conditions in respect of bending m o m e n t at both the external and internal boundaries and the Kelvin-Kirchhoff edge reaction at the internal free edges are not satisfied.

2.3 Mode shapes under consideration For specially orthotropic plates, the four basic modes of vibration namely the SX-SY, A X - S Y , S X - A Y and the A X - A Y modes are the same as that of

68

H. P. Lee, S. P. Lira, S. T. Chow

isotropic rectangular plates and are amenable to the present m e t h o d . T h e n o d a l lines for the higher m o d e s of generally orthotropic plates, as p r e s e n t e d by M o h a n and Kinsbury, 9 are not parallel to the plate edges. T h e p r e s e n t m e t h o d is t h e r e f o r e restricted to predicting the frequencies of only the f u n d a m e n t a l m o d e s for generally orthotropic plates. 2 . 4 T h e a s s u m e d deflection functions

T h e a s s u m e d deflection functions for the subdomains of a simply s u p p o r t e d s q u a r e plate with a central rectangular cutout depicted in Fig. I are the s t a n d a r d b e a m functions. 10,11These are listed below.

SX-SY mode:

rrx Try W~ = A ~ s i n - - s i n - a b W2 = A 2 Y s i n __rrx,Az = Atsin 7rslsl a b W 3 ~-

AX-SY mode:

A3 xrl sin ---b--, ~'y A 3 = A, sin -rrrl -a

7rx try W1 = A l s i n ~ sin (a/z) b W2 = A2 ySl sm " - ~rrx - ~ - , A2 W3

SX-AYmode:

=

=

A

lsin 7rsl b

(9)

A3 ~x try ¢rrl rl sin---ff-,A3 = Alsin (a/2-'--~

W1 = Alsin 7rx sin Try a (b/2) W2 = A2Y---sin rrx ,A2 = Alsin 7rsl

s,

W 3 :

a

A 3-x sin rl

AX-AYmode:

(8)

try ~

(b/2------)

(10)

, A 3 : A t sin ~'r-----Sl a

7rx

Wt = Alsin(a--a-~-sin try (b/2) W2 = A2 y---sin

s~

7rx ~'S 1 (a--~-,A2 = Alsin (b/2-----)

W3 = A3 xrl sin (-~-~-, ~'Y A3 = A1 sin (a/2------) rrrl

(11)

Free vibration o f composite rectangular plates

69

A 2 a n d A 3 a r e d e t e r m i n e d b y m a t c h i n g t h e m a x i m u m deflections at t h e l o w e r left-hand corner of the cutout. T h e a s s u m e d d e f l e c t i o n functions f o r the s u b d o m a i n s in Fig. 2 a r e as follows S X - S Y mode:

W~

=

W 3 =

A~sin 7rx sin 7r__y_y a b •

T/'S 1

W2 = A 2 Y s i n ~-x ,Az = ,~lsm b Sl a

(12) W 4 =

x zry ~rrl A 4 - - sin , A 4 ~--- A 1s i n - rl --ifa Try

W5 = Assin--~-, A5 = Alsin rrr2a

A X - S Y mode:

• zrx try W1 --- W3 = A3sm-z---z~-.~.sin (a/z)

I W2 = AEySl s i' n -(-a]-~rrx ' m 2 = A l s i n 7TS -b

(13) • zry ~rl W4 = A4 x sm --~-, A4 = A l s i n - (a/2) El

W5 =

S X - A Y mode:

A

a/2 - x

.

~ry

7rr2

5~sm-~--,A5

= Atsin (a/2----ff

W1 = W3 = A~sin ~x sin try a (b/2)

W2 = A z Y s i n sl

~x ,A2 = A l s i n ~s-----L-1 a (b/2)

(14) W4

Try

=

A4X--rxsin ( - - ~ y , A4

=

Alsin rrrla

• (--~3-, try A5 = Alsin rcrz W5 = Assln a

H. P. Lee, S. P. Lim, S. T. Chow

70

rrx

A X - A Y mode: W1 = W3 = Alsin-7--7~-,_.sin (a/z)

7rx

W2 = A2 Y--( a -sin -~-,Sl

A

Try

(b/2) 7rs~

2

=

Aysin (b/2----ff

(15) x ~-y W4 = A 4-- sin , A4 rl ~ a/2 - x W 5 = A 5 a / 2 - rl

.

7rr 1

:

A ~s i n - (a/2)

Try

rrr2

sm (-~-~, As = A tsin (a/2---~

A 2, A 4 a n d A 1 are matched at x = r 1 a n d y = x 4. A 5 a n d A 1 are matched at x = r 2 a n d y = s l.

3 RESULTS AND DISCUSSION 3.1

Rectangular

plates with central cutouts

The predicted fundamental frequencies for simply supported square plates having central rectangular cutouts were compared with the reported finite element results by R e d d y I and shown in Fig. 4. The orthotropic material properties were taken to be as follows: E x / E y = 40, G x y / E y = 0.5 and Uxy = 0.25. The computed frequencies by the present method were in good agreement with the reported results for thin plates. For a square plate with a square cutout of size 0.5a × 0.5a the computed frequency was 53.226 c o m p a r e d with the reported value of 51-232 for thickness over a span ratio of 0-01. The agreement is expected to be better if the reported result is for a very thin plate. Numerical results were generated for a simply supported square plate having a central square cutout for the materials given in Table 1. The c o m p u t e d results for specially orthotropic plates are presented in Figs 5-7. The curves in general agreed with the reported results of Rajamani and P r a b h a k a r a n 2 except for the A X - S Y and A X - A Y modes of a graphiteepoxy square plate as shown in Fig. 7. For these two cases, the present m e t h o d predicted a more logical trend since the computed frequencies for the A X - A Y m o d e must be higher than the corresponding frequencies for the A X - S Y m o d e for all sizes of cutouts. These figures also confirm that the Rayleigh quotient is getting less accurate for higher-order modes.

90 * d/c=l

[...,, .,~ >.

80

7O

~

0

d/c=l - Ref I



d/c=2.-....~AUTH.....~0R...SS

n

d/c=2 - Ref 1



d / e = 3 - AUTHORS

~o

Z

- AUTHORS

b / /

/

d/c=3 - Ref 1

/

60-

,-1 .< 5OZ 0

Z

4O.

ZI

30-

O Z

200

0.'1 0.'2 0.'3 0.~4 0.'5 0.'6 0.'7 0.;8 0.~9

Fig. 4. Fundamental frequency parameters for a graphite-epoxy square plate with a rectangular cutout.

TABLE 1

Material Properties for the Orthotropic Plates Case number l 2 3

Typical material Balanced bidirectional Glass--epoxy Graphite-epoxy

Ex/E r

Gxy/Ey

Vxy

l 3 40

0.2 0.5 0.5

0.10 0.25 0.25

For generally orthotropic plates with 0 = 45 °, the computed and reported results are very close as shown in Fig. 8. The variations of the computed fundamental frequencies with the orientation of the principal axis of orthotropy 0 were examined further in Figs 9-11. Computed frequencies and trends agree with the reported results 2 for all the three orthotropic materials. This indicates that although the assumed deflection functions do not satisfy the vanishing bending moment at the simply supported edges they are still reasonably good approximations for square plates with increased orthotropy.

72

H. P. Lee, S. P. Lim, S. T. Chow

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Free vibration of composite rectangularplates

73

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Free vibration of composite rectangularplates

75

3.2 Rectangular plates with double cutouts The present m e t h o d is also used to compute for simply supported rectangular plates with a/b = 2 having double square cutouts with centres located at 1/4a and 3/4a. No results for comparison are available to the authors' knowledge. Numerical results were generated for glass--epoxy and graphite--epoxy composites. The predicted results for principal axis orientations of 0 ° and 90 ° are presented in Figs 12-15. For/9 = 90 °, the predicted curves for the A X - A Y mode dipped below the curves for the S X - A Y mode at some cutout sizes. A similar p h e n o m e n o n was observed for the A X - S Y and the SX-SY modes of the graphite-epoxy rectangular plates. The phenomenon was examined further in Figs 16-19 by varying the Ex/Ey ratio for a rectangular plate with Poisson's ratio 0.25. The p h e n o m e n o n only occurred for rectangular plates with the principal axis of orthotropy perpendicular to the line joining the centre of the two cutouts

I

<: 250-1

p,

Z;

r-~

200"



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©/*=.8 - Ae! 2

~.

c/*=.?

- Re! 2



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0

©/*,,.e

- Ref 2



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c/a--'.4 - I l e f 2



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1,50

2: 100 I

Z 0 Z

50

o

;

l'o

1;

2'o 15 0

3'o 3'5 4'o ,'5

50

(DEGREES)

Fig. 11. Fundamental frequency parameters for a graphite-epoxy square plate with a square cutout.

H. P. Lee, S. P. Lim, S. T. Chow

76

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Free vibration of composite rectangularplates

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80

H. P. Lee, S. P. Lim, S. T. Chow

with Ey > Ex. The computed frequencies for the A X - A Y mode and the A X - S Y modes were always higher than the corresponding values for the S X - A Y and SX-SY modes for the rectangular plates having the principal axis of orthotropy along the x axis with Ex > Ey. The energy for the SX-SY and the SX-AY modes was found to be dominated by the subdomains between the cutouts when the principal axis of orthotropy was along the thin strip. An increase of Ey/Ex ratio would thus result in higher frequencies for the SX-SY and the SX-AY modes. These frequencies could be higher than the corresponding frequencies for the A X - S Y and the A X - A Y modes, respectively, for a rectangular plate with a high degree of orthotropy having a long and narrow strip between the cutouts. This conclusion has been verified by the foregoing predicted results.

4 CONCLUSION A simple numerical method based on the Rayleigh principle and the classical plate theory has been presented for predicting the natural frequencies of composite rectangular plates with rectangular cutouts. The predicted results are in good agreement with the reported results for a simply supported square plate having a central rectangular or a square cutout. There are some surprising results for orthotropic rectangular plates with double square cutouts. For a rectangular plate having the principal axis of orthotropy perpendicular to the line joining the centres of the two cutouts with Ey > Ex, the predicted frequencies for the SX-AY mode, which is of lower order than the A X - A Y mode, are higher than the corresponding frequencies for the A X - A Y mode for some cutout sizes. The same phenomenon occurs for the predicted frequencies for the SX-SY and A X - S Y modes. This phenomenon will not occur in the case of rectangular plates without cutouts since higher-order modes will always have higher frequencies in comparison with the lower-order modes.

REFERENCES 1. Reddy, J. N., Large amplitude flexural vibration of layered composite plates with cutouts, J. Sound and Vibration, 83 (1982) 1-10. 2. Rajamani, A. and Prabhakaran, R., Dynamic response of composite plates with cutouts, Part I: Simply-supported plates, J. Sound and Vibration, 54 (1977) 549-64. 3. Rajamani, A. and Prabhakaran, R., Dynamic response of composite plates

Free vibration of composite rectangularplates

4. 5. 6. 7. 8. 9. 10. I I.

81

with cutouts, Part II: Clamped-clamped plates, J. Sound and Vibration, 54 (1977) 565-76. Kanazawa, T. and Kawai, T., On the lateral vibration of anisotropic rectangular plates, Proceedings of the 2nd Japan National Congress for Applied Mechanics, 1952, pp. 333--8. Hearman, R. F. S., The frequency of flexural vibration of rectangular orthotropic plates with clamped or simply-supported edges, J. Applied Mechanics, 26 (1959) 537--40. Dickinson, S. M., The flexural vibration of rectangular orthotropic plates, J. Applied Mechanics, 36 (1976) 101--6. Dickinson, S. M., The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh method, J. Sound and Vibration, 61 (1978) 1-8. Jones, R. M., Mechanics of composite materials, McGraw-Hill, New York, 1975. Mohan, D. and Kinsbury, H. B., Free vibration of generally orthotropic plates, The Journal of the Acoustical Society of America, 50 (1971) 266-9. Leissa, A. W., Vibration ofplates, NASA SP-160, 1969. Timoshenko, S. and Woinowsky-Kreiger, S., Theory of plates and shells, McGraw-Hill, New York, 1959.