Appfied Acoustics 35 (1992) 91104
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Free Vibration of lsotropic and Orthotropic Rectangular Plates with Partially Clamped Edges H. P. Lee & S. P. Lim Department of Mechanical & Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 (Received 5 April 1991; revised version received 20 August 1991; accepted 22 August 1991)
A BS TRA C T A simple numerical method based on the energy principle is presented.for predicting the natural frequencies of a simply supported rectangular plate with an edge or two opposite edges partially clamped. The computation only #wolves the integration of simple assumed deflection functions. Numerical results are presented for isotropic and orthotropic plates. The predicted fi'equencies for isotropic and specially orthotropic rectangular plates, with the two principal axes of orthotropy parallel to the two pairs of opposite plate edges, are in good agreement with the reported analytical results.
NOTATION a,b c
D Dij
E
E ,E, G Gxr
h Qu T U
w,w
x, y
Dimensions of plate Length of clamped edge Bending stiffness of isotropic plates Bending stiffness of laminates Young's modulus of isotropic plates Young's moduli of orthotropic plates Shear modulus of isotropic plates Shear moduli of orthotropic plates Thickness of the plate Bending stiffness of a lamina Kinetic energy Potential energy Transverse deflection Rectangular cartesian coordinates 91
Applied Acoustics 0003682X/92/$05.00 © Elsevier Science Publishers Ltd, England. Printed in Great Britain
92
v Vxy
P CO
H. P. Lee, S. P. Lira
Nondimensional frequency parameter Poisson's ratio of isotropic plates Poisson's ratio of orthotropic plates Mass density Angular frequency
1 INTRODUCTION The vibration of a simply supported rectangular plate partially clamped on its edges has been examined by various seriestype analytical methods. The fundamental frequencies were obtained by the principle of stationary energy. 1'2 The problem was analysed by the use of Freholm Integrals a and by the finite element method. 4 An analytical solution was obtained by enforcing the mixed boundary conditions.5 These reported works have been restricted to isotropic rectangular plates. Numerical results for specially orthotropic rectangular plates, with the two principal axes of orthotropy parallel to the two pairs of opposite edges of the plate, have been obtained using a seriestype method. 6 These methods entail intensive computations and extensive settingup for the problem. To the authors' knowledge, the Rayleigh method, which is often used as a simple alternative for predicting the natural frequencies, has not been applied to solving this problem. A simple numerical method, based on the Rayleigh principle, is presented for predicting the fundamental and some selected higher modes of a simply supported rectangular plate with an edge or two opposite edges partially clamped. The method is illustrated for isotropic and orthotropic plates with special and general orthotropy.
2 ANALYSIS The rectangular plates under consideration are shown in Fig. 1. Figure l(a) is a rectangular plate clamped along the central portion of two opposite edges. The four modes to be considered are the SXSY, AXSY, SXAY and the A X  A Y modes. SX and SY represent the first symmetric modes of vibration about x = a/2 and y = b/2, respectively. Similarly, AX and AY are the respective first antisymmetric modes of vibration about x = a/2 and y = b/2. For a rectangular plate which is clamped at the end of an edge shown in Fig. l(b), only the fundamental mode is considered. The mode shape is equivalent to the A X  A Y mode of the plate in Fig. l(a). The rectangular plate is divided into smaller subdomains, as shown in Fig. 2. A deflection function Wn which satisfies all the essential boundary
Vibration of rectangular plates with partially clamped edges Y
L.
C
ii
L.
c
b Ia
93
..j
a
o
(a)
x
(b)
Fig. 1. Simply supported plates with partially clamped edges.
conditions is assumed for each subdomain. All the assumed deflection functions are expressed in the same global coordinate axes (x,y). The relative magnitudes of coefficients d, are determined by matching the maximum deflection at a suitable point along the boundary of adjacent subdomains. The assumed deflection functions for the mode shapes are as follows. For the SXSY mode, W 1 = W 3 = `41
sin 7tx 7ty a
b
(i)
W2= A2ICOSb(Y~)+ F cOShb(Y b ) l sin ? For the AXSY mode, zty a/2 b
W I = W 3 = ,41 s i n ~ x
W2=A 2 cos~ y~
1
2
3
(2) y
+Fcosh
1
2
SXSY 2
AXSY 1
',
2
I
i
AXAY Fig. 2. Subdomains for the plates.
SXAY
Sina/2
94
H. P. Lee, S. P. Lira
The part of the deflection function in y for W2 is the shape function of a beam clamped at y = 0 and y = b. The value ofy and F a r e determined from the clamped boundary conditions ( W = d W/dy = 0) for a beam at y = 0 or y = b since the assumed shape function is symmetric about y = b/2. A 2 is determined by matching the m a x i m u m deflection at y = b/2. 1 .42= A x(i~ff)
(3)
For the S X  A Y mode, WI = W3 = A 1 sin nx ny
a b/2
W2 = A 2 sin
y
(4)
+Fsinh
y
sin a
For the A X  A Y mode, W~ = W a = A 1 sin nx lty
a/2 b/2
I412= A 2 [ s i n b ( Y  ~ ) + F
(5) s i n h b ( Y  ~ ) ] Sina~2
The part of the deflection function in y for W2 is the shape function of a beam clamped at y = 0 and simply supported at y = b/2. For this assumed shape function of a beam, the zero displacement ( W = 0) and zero bending m o m e n t (d 2 W/dy2= 0) conditions for a beam are satisfied inherently. The values ofv and F a r e determined from the clamped boundary conditions for a beam at y = 0. A 2 is estimated by m a t c h i n g the maximum deflection of adjacent subdomains at a distance of b/4 from the nodal line, or at y = ¼b, 1
A2
=
a l sin ~(b/4) + Fsinh y(b/4)
(6)
The potential and kinetic energy of each subdomain are then evaluated:
u=2 JJLkax=
( 02 W \ 2 \ ]
+07)
2(1v)t~x2
ay 2
(7)
T=½ph o2f f W2dxdy
(8)
Vibration of rectangular plates with partially clamped edges
95
F o r a generally orthotropic plate, the potential energy is given by 1 f fl
u=]
~82W~ 2
//02 W~ 2 ~2WO2W + D 2 2 ~ y2) + +2o12 ox2 dy2
//t~ 2 W'~ 2 4D66~d~y )
a w8xw] dyd dx dy
+ 4 D ' 6 ~x 2 ~X ~~"3L4D26 ~
(9)
where D 11, D22, Dr2, D66, D16 and D26 are related to the principal bending stiffnesses Q I~, Q22, Q I 2, Q66 along the principal axes of o r t h o t r o p y by the following expressions: 7 D ll = Q II c°s4 0 + 2(Q 12 + 2Q66)sin2 0 cos 2 0 + Q22 sin4 0 D22 = Q I 1 sin4 0 + 2(Q12 4 2Q66) sin 2 0 cos 2 0 + Q22 cos4 0 DI 2 = (QI 1 + Q22  4Q66) sin 2 0 cos 2 0 + QI 2(sin4 0 + cos 4 0) D 16 = (Q 11  Q 12  2Q66) sin 0 cos 3 0 + (Q 12  Q22 + 2Q66) sin3 0 cos 0 (10) D26 = (Q11  Ql2  2Q66) sin a 0cos 0 + (QI2  Q22 + 2Q66) sin 0 cos a 0 D66 = (QI 1 + Q22  2Q 12  2Q66) sin 2 0 cos 2 0 + Q66(sin 4 0 + cos 4 0) where 0 is the angle between the x axis and the principal axis of o r t h o t r o p y , and D u are equal to Qij when 0 = 0 °. F o r a specially orthotropic plate, the principal axes of o r t h o t r o p y are in the x and y directions o f the plate. E q u a t i o n (3) can be simplified as
1 f /" [
[t~ 2 W~ 2
02 W t~2 W
+ = " ~x~
/t32 W\21
d y ~ + D 2 2 ~ y2 )
]dxdy
(11)
where H is given by vI2D22 I2D12. The total potential and kinetic energy of the plate is assumed to be the respective sums o f the potential and kinetic energy o f each subdomain. The frequency of a m o d e shape is obtained by equating the total potential energy to the total kinetic energy. The frequency is expressed in terms of a n o n d i m e n s i o n a l frequency p a r a m e t e r given by
2 = oga2 /r,.
(12)
where D is replaced by H for specially orthotropic plates and by D22 for generally orthotropic plates.
H. P. Lee, S. P. Lim
96
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< E II
N
0
r~
II
0
I
II
E II
0 0 0
Vibration of rectangular plates with partially clamped edges
97
3 RESULTS AND DISCUSSION
3.1 Isotropic plates The first problem considered is the vibration of an is0tropic simply supported plate clamped along a segment at the end of an edge shown in Fig. l(b). The fundamental frequencies are computed for different values of aspect ratios of the plate and various proportions of clamped edge. The predicted results and the analytical results by Gorman 5 are presented in Tables 1 and 2. The discrepancies are less than 4% for rectangular plates with b/a > 1 and a/b < 2"5. The highest discrepancy is of the order of 7% for a rectangular plate with a/b = 3 and clamped edge equal to 0.5a. The predicted trend of frequency variation is in good agreement with the reported results for all the plates under consideration. In the second problem, the vibration of a simply supported isotropic square plate partially clamped along the central portions of two opposite sides, shown in Fig. l(a), is considered. The predicted results and the reported results are presented in Table 3. The reported results for the AXAY mode by Gorman 5 for lengths of clamped edge equal to Xaaand aZaare interpolated from the other data presented. 5 The discrepancies of the predicted frequencies from the reported frequencies 6 are, in general, higher for a short length of clamped edge. The reported frequencies 6 are, however, higher than the corresponding values presented in Ref. 5 for the AXAY mode. The predicted frequencies, on the other hand, are in good agreement with those reported by Gorman 5 for this higher mode. The discrepancies in the reported results are probably caused by the differences in the assumed deflection functions and the number of terms for the series of deflection functions in their computations.
3.20rthotropic plates The effects of orthotropy are examined for the plate shown in Fig. l(a). Numerical results are presented in Figs 3 and 4 for three different combinations of specially orthotropic parameters. The predicted results and the previously reported results, 6 using a seriestype method with 10 to 15 terms, are appended in Table 4. The predicted frequencies are, in general, lower than the corresponding reported frequencies as in the case ofisotropic plates. Numerical results are also generated for the nondimensional frequency parameters for the orthotropic materials, with material properties given in Table 5. The variations of the frequencies with the length of the clamped edge are shown in Figs 58.
H. P. Lee, S. P. Lim
98
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II
t~
I ° 0~
II _o
< e,
U e.,
I
.<
II
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I
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I
I
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~J
II
E
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Vibration of rectangular plates with partially clamped edges
99
70 AXSY 60 50 E
40 U
e"
.o ?
30 0
015
40
SXSY......'"..........•................................
30
g
z
20 10
°o
o~s c/a
Fig. 3.
Frequency parameters for a simply supported orthotropic square clamped along two segments o f opposite edges.
120 AXAY
......................
100 .............
~
80 
•~
60
=~
40 .0
.~
100
.E ?
so
l0
z


°.°.*° ~
... ...........
t
0.5
SXAY. . . . , . . . . ' " ' .................................... ....."" ..
°. .................
60 40
200
015 cla
Fig. 4.
Frequency parameters for a simply supported orthotropic square clamped along two segments o f opposite edges.
100
H. P. Lee, S. P. Lira
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0
¢o q~
<
.< b,
"0
e.
Vibration of rectangular plates with partially clamped edges
t01
160 140 ~
120
[
100
..'"....................
8(1 •g
60
Z
. 20
.i
0 0
0.5 c/a
Fig. 5. Frequency parameters for the SXSY mode of a simply supported square plate clamped along two segments of opposite edges.
300
250
J
...................................
200 150
.o.o.o'°'" .o. . . . . . .o.o'°" °
100
,.........'"
5O
00
0[5 rJa
Fig. 6. Frequency parameters for the AXSY mode of a simply supported square plate clamped along two segments of opposite edges.
102
H. P. Lee, S. P. Lira 400 350 /"
E
300 ...,"
a., 250
.g 200 e~ to
"~
150
.e ~O
too
.... •;=:::::::::
.................
z
50
O0
I
0.5 c/a
Fig. 7.
Frequency parameters for the S X  A Y mode of a simply supported square plate clamped along two segments of opposite edges.
400
/
...°" .....
350 ,/
u
300 .... :.':::: .......................
250 200 tO
=o
150 100
Z
50 I
0
0.5 cla
Fig. 8.
Frequency parameters for the A X  A Y mode of a simply supported square plate clamped along two segments of opposite edges.
103
Vibration of rectangular plates with partially clamped edges TABLE 5
Material Properties for the Orthotropic Plates Case number
1 2
Typical material
Ex/Ey
Gxy/Ey
v~r
3 40
0"5 0'5
0"25 0"25
Glassepoxy Graphiteepoxy
The present method provides a quick and simple alternative for estimating the natural frequencies of a rectangular plate with partially clamped edges. The computation only involves the integration of simply assumed shape functions. The errors between the present predicted results and the corresponding analytical results are probably caused by the discontinuity in the assumed shape functions at boundaries between the subdomains. The results could be improved by using a higher order assumed shape function or a series of simple functions at the expense of simplicity. The unknown parameters built into the higherorder shape function or the series of simple functions could then be used for matching the maximum deflections at more points along the boundary of adjacent subdomains.
4 CONCLUSION A simple method, based on the Rayleigh principle, has been presented for predicting the natural frequencies of simply supported rectangular plates with partially clamped edges. The predicted frequencies for isotropic and specially orthotropic plates are in good agreement with the reported results. The method is also capable of predicting the natural frequencies of generally orthotropic plates and simply supported plates with multiple clamped segments, without increased complexity in the computation.
REFERENCES I. Kurata, M. & Okamura, H., Natural vibration of partially clamped plate. Proc. ASCE, Eng. Mech., 3 (1963) 16986. 2. Ota, T. & Hamada, M., Fundamental frequencies of simply supported but partially clamped square plates. Bull. JSME, 6 (1963) 397403. 3. Keer, L. M. & Stahl, B., Eigenvalue problem of rectangular plates with mixed boundary conditions. J. Appl. Mech., 39 (1972) 51320. 4. Venkateswara Rao, G., Raju, I. S. & Mrurthy, T. V. G. K., Vibration of rectangular plates with mixed boundary conditions. J. Sound Vih., 30 (1973) 25760.
104
H. P. Lee, S. P. Lira
5. Gorman, D. J., An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions. J. Sound Vib., 93 (1984) 23547. 6. Narita, Y., Application of a seriestype method to vibration of orthotropic rectangular plates with mixed boundary conditions. J. Sound Vib., 77 (1981) 34555. 7. Jones, R. M., Mechanics of Composite Materials. McGrawHill, New York, 1975.