Composite Structures 8 (1987) 6381
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 midsurface of the plate Transverse deflection of the midsurface of the plate at time t Rectangular Cartesian coordinates Orientation of the principal axis of orthotropy Nondimensional frequency parameter Poisson's ratio 63
Composite Structures 02638223/87/$0350 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 RayleighRitz 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 simplysupported 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~za+ph_7v= 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~"2x 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 preselected 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 singleterm 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
=
DI2TTDz2TT2026
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 KelvinKirchhoff 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 SXSY, 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.
SXSY 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 ~
AXSY 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
SXAYmode:
=
=
A
lsin 7rsl b
(9)
A3 ~x try ¢rrl rl sinff,A3 = Alsin (a/2'~
W1 = Alsin 7rx sin Try a (b/2) W2 = A2Ysin rrx ,A2 = Alsin 7rsl
s,
W 3 :
a
A 3x sin rl
AXAYmode:
(8)
try ~
(b/2)
(10)
, A 3 : A t sin ~'rSl a
7rx
Wt = Alsin(aa~sin try (b/2) W2 = A2 ysin
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 lefthand 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 = A3smzz~.~.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/2ff
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 ~sL1 a (b/2)
(14) W4
Try
=
A4Xrxsin (   ~ 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 = Alsin77~,_.sin (a/z)
7rx
W2 = A2 Y( a sin ~,Sl
A
Try
(b/2) 7rs~
2
=
Aysin (b/2ff
(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 51232 for thickness over a span ratio of 001. 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 57. 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 higherorder modes.
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Z
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ZI
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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 graphiteepoxy square plate with a rectangular cutout.
TABLE 1
Material Properties for the Orthotropic Plates Case number l 2 3
Typical material Balanced bidirectional Glassepoxy Graphiteepoxy
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 911. 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
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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 glassepoxy and graphiteepoxy composites. The predicted results for principal axis orientations of 0 ° and 90 ° are presented in Figs 1215. 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 SXSY modes of the graphiteepoxy rectangular plates. The phenomenon was examined further in Figs 1619 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
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Fig. 11. Fundamental frequency parameters for a graphiteepoxy square plate with a square cutout.
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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 SXSY modes for the rectangular plates having the principal axis of orthotropy along the x axis with Ex > Ey. The energy for the SXSY and the SXAY 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 SXSY and the SXAY 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 SXAY 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 SXSY and A X  S Y modes. This phenomenon will not occur in the case of rectangular plates without cutouts since higherorder modes will always have higher frequencies in comparison with the lowerorder modes.
REFERENCES 1. Reddy, J. N., Large amplitude flexural vibration of layered composite plates with cutouts, J. Sound and Vibration, 83 (1982) 110. 2. Rajamani, A. and Prabhakaran, R., Dynamic response of composite plates with cutouts, Part I: Simplysupported plates, J. Sound and Vibration, 54 (1977) 54964. 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: Clampedclamped plates, J. Sound and Vibration, 54 (1977) 56576. 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. 3338. Hearman, R. F. S., The frequency of flexural vibration of rectangular orthotropic plates with clamped or simplysupported edges, J. Applied Mechanics, 26 (1959) 53740. Dickinson, S. M., The flexural vibration of rectangular orthotropic plates, J. Applied Mechanics, 36 (1976) 1016. 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) 18. Jones, R. M., Mechanics of composite materials, McGrawHill, 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) 2669. Leissa, A. W., Vibration ofplates, NASA SP160, 1969. Timoshenko, S. and WoinowskyKreiger, S., Theory of plates and shells, McGrawHill, New York, 1959.