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Prediction of natural frequencies of rectangular plates with rectangular cutouts
Computers & Structures Vol. 36, No. 5. pp. 861469, Printed in Great Britain.
1990
lw5-7949/90 53.00 + 0.00 Q 1990 Pergamon Press plc
PREDICTION OF NATURAL FREQUENCIES OF RECTANGULAR PLATES WITH RECTANGULAR CUTOUTS H. P. LEE, S. P. LIM and S. T. CHOW Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge, Singapore 0511, Republic of Singapore (Received 25 April 1989)
Abstract-A
simple numerical method based on the Rayleigh quotient is presented for predicting the natural frequencies of the fundamental and some higher modes of a rectangular plate with an arbitrarily located rectangular cutout. The edges of the cutout are parallel to the edges of the complete plate. The nodal patterns of these higher modes are pre-selected based on the symmetry of the mode shapes about the geometrical axes. The method is applied to the study of linear vibration of rectangular plates simply supported along one pair of opposite edges with any other boundary conditions at the remaining edges. The predicted results compare favourably with the corresponding results produced by other methods whenever available.
NOTATION
b c, d 0,
h rlv 5 Sl.S2
D T V w W” A P w
dimensions of the plate dimensions of the cutout thickness of the plate x coordinates for the corners of cutout y coordinates for the corners of cutout flexural rigidity kinetic energy potential energy transverse displacement of the plate nondimensional frequency parameter Poisson’s ratio mass density angular frequency I. INTRODIJCI-ION
Rectangular plates with cutouts are frequently found in engineering structures. The cutouts are made for saving weight, for venting, for altering the natural frequencies and for providing accessibility to other parts of the structures. Past investigations on the vibration of these plates have been confined to the numerical analysis of rectangular plates with rectangular openings or circular holes. The numerical methods for computing the natural frequencies of rectangular plates with cutouts can be broadly classified into three categories, namely the finite element and finite difference methods, the series-type analytical method and the semianalytical approach based on the Rayleigh-Ritz principle. Numerical results obtained by the finite element method have been reported by Ali and Atwal [I], Shastry and Venkateswara Rao [2], Reddy [3], Laura et al. [4] and Tham et al. [S]. Paramasivam [6] and Aksu and Ali [7] analysed the problems using finite difference analysis.
The problems have also been solved by a number of authors using the series-type analytical method. Basdekas and Chi [8] investigated the forced and free dynamic response of a rectangular orthotropic plate by expressing the effect of a cutout as an external loading function on the plate. The assumed deflection function was taken to be a series of eigenfunctions for the complete plate without the cutout. Rajamani and Prabhakaran [9, lo] generalised the method by approximating the eigenfunctions by the products of beam functions for the given end conditions. Nagaya [l 1, 121 obtained the natural frequencies of square plates with central circular holes by conformal mapping and the external boundary conditions were satisfied by a Fourier expansion collocation method. Hegarty and Ariman [ 131solved the same problem by satisfying the external boundary conditions using a least square point matching method. Internal boundary conditions along the circular hole were satisfied by the assumed Bessel functions. Rayleigh-Ritz method has been used widely as an alternative to the aforementioned methods for predicting the linear frequencies of rectangular plates with cutouts. There are two different approaches in applying this method. The first approach is a combination of the point matching and the Rayleigh-Ritz method. The external boundaries and the internal free edge conditions are satisfied by the assumed deflection functions only at some discrete points, usually at the corners of the cutout and the plate. This approach is restricted to rectangular plates with centrally located cutouts or circular holes as the deflection functions are invariably symmetrical about the centres of the plates. Numerical results were reported by Kumai [14] and Joga Rao and Pickett [ 151. 861
H. P. LEE et al.
862
A different approach was reported by Takahasi [ 161 for predicting the natural frequencies of a rectangular plate with a circular hole which need not be located at the centre of the plate. The energy of a plate with a hole is approximated by deducting the energy of the hole from the energy of the whole plate without the hole. The assumed deflection functions for computing the energy of the plate and the hole only satisfy the external boundary conditions and disregard the presence of the hole. Ali and Atwal [1] used the same approach to predict the natural frequencies of a simply supported square plate with a central rectangular cutout. The results obtained for the fundamental mode need to be multiplied by a correction factor depending on the cutout size in order to reduce the percentage discrepancies from the corresponding finite element results to below 5%. All the works mentioned above, with the exception of [7J, deal with a rectangular plate with a single cutout. The same method was also used by Mizusawa et al. [17] and Laura et al. [4, 181, based on B-spline functions, polynomial functions and combinations of polynomial and trigonometric functions which satisfied only the external boundary conditions of the complete plates, for predicting the natural frequencies of rectangular plates with central or corner cutouts. A similar method, using the Hamilton principle instead of the Rayleigh quotient, was reported by Ganesan and Nagaraja Rao [ 191. In these two approaches, which are based on the Rayleigh-Ritz method, the assumed deflection functions are continuous throughout the whole plate. The internal free edge conditions of a rectangular cutout are disregarded or are satisfied only at the four comers of the cutout. In [7] Aksu and Ali used finite difference method to deal with the problem of rectangular plates with multiple rectangular cutouts. In the present analysis, the assumed deflection functions are no longer continuous throughout the whole plate. The deflection functions are, however, made to satisfy all or part of the internal free edge conditions along the four edges of the cutout. 2. ANALYSIS
2.1. Classical method According to the classical theory of plates [20], the linear vibration of a thin, homogeneous, isotropic plate is governed by the following partial differential equation:
An exact solution of eqn (1) is known only for simply supported rectangular plates and rectangular plates which are simply supported along one pair of opposite edges with any conditions at the remaining edges [20]. It is necessary to resort to numerical methods for plates with internal boundaries or other combinations of boundary conditions. The procedure developed by Ritz [21] has been found to be useful for solving such problems. For a rectangular plate which is vibrating harmonically with amplitude Wand natural frequency o, the maximum potential energy (V) and kinetic energy (T) are given by
-2(1-u)($-($)3]dxdy
T =;phw*
(2)
W= dx dy.
(3)
The above integrations are to be performed over the domain of the plate. Using Rayleigh quotient, the maximum values of the potential and kinetic energies are equated to determine the natural frequency. w*=_
2 ph
V (4) W’dxdy
Ritz method assumes the deflection function W as a linear series of admissible functions and adjusts the coefficients in the series so as to minimise eqn (4). It is usually written as w=C&Ln~m(~)y”(Y).
m ”
(5)
The functions X,,, and Y, for a rectangular plate without any cutout are usually beam functions]211 or degenerated beam functions [22] that satisfy the external boundary conditions. For a rectangular plate with a rectangular cutout, there is hitherto no reported function which is continuous throughout the whole domain of the plate and satisfies all, or part, of the internal free edge conditions besides the external boundary conditions. As a means of overcoming this difficulty a modified Rayleigh method is presented. 2.2. The present method
DA’W+ph$y=O.
The deflection W is assumed to be small compared with the plate thickness h. The effects of shear deformation and rotary inertia have been neglected.
The plate under consideration is a thin rectangular plate of size a x b having a rectangular cutout of size c x d as shown in Fig. 1. The plate is divided into smaller sub-domains based on the mode shapes and the locations of the cutouts. The eight sub-domains for the fundamental mode are presented in Fig. 1.
863
Natural frequencies of rectangular plates
Yf
1 ’
b 92
_-----
8
1 6 I--me
d
3 SI -----
I 7
5 C
---_
Ici
2
I
fl
0
i 4 t 12 0
x
In order to minimize ~mpu~tion time, a singleterm deflection function W, is assumed instead of a series for each sub-domain after the corresponding modified boundary conditions have been determined. The assumed functions will be either products of beam functions, degenerated beam functions or the single-term analytical solutions for the given boundary conditions. For the beam functions, only the essential boundary conditions are satisfied. The functions are assumed to be of the form
Fig. 1. A rectangular plate with a rectangular cutout.
wn=&w)YnCV) The four basic modes of vibration to be considered are the SX-SY, AX-SY, SX-AY and the AX-AY 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 anti-symmetric modes of vibration about x = a/2 and y = b/2. The SX-SY mode is the fundamental mode presented in Fig. 2. The other three higher modes will only be considered for a rectangular plate with a centrally located rectangular cutout. As a result of symmetry, it is necessary to consider the sub-domains for only the lower left hand portion of the plate. The sub-domains of these higher modes are presented in Fig. 2. For a rectangular plate with a single cutout, the sub-domains at the four comers, namely 1, 4, 6 and 8, are considered to be parts of a complete plate having the given external boundary conditions. The sub-domains between the external boundary and the internal free edge, such as sub-domains 2, 3, 5 and 7, are, however, taken to be parts of a rectangular plate with three sides having the same boundary conditions as the original plate and the remaining side being free.
for the nth sub-domain. All the deflection functions are expressed in the same global coordinate axes (x, y). Adjacent subdomains will have one of the two functions, X, and Y., in common. Relative magnitudes of coefficients A, are determined by matching the maximum deflections at the comers of the cutout. The potential energy and kinetic energy of each sub-domain are evaluated using eqns (2) and (3). The total potential and kinetic energies of the plate are taken to be the sum of the kinetic and potential energy of all the sub-domains. The frequency of every mode is obtained by equating the total potential energy to the total kinetic energy. The frequency is expressed in terms of a non~imensio~l frequency parameter d which is a function of only the Poisson’s ratio and the aspect ratio of the plate for a given cutout configuration. The parameter is defined as
A=;(-&
1
[17 2
I
:
AX-SY
sx-SY
3 ----I
~
0$ i’2.
3
3 ----I
I
_---__
_--_
2
I
I
SX-AY ----Sub-domain -Nodal
(6)
I’121
I
I
1 AX-AY
I
boundory line
Fig. 2. Sub-domains for a rectangular plate with a central rectangular cutout.
(7)
H. P. LEE et al.
864
The present method is used to calculate the natural frequencies of rectangular plates having two opposite edges simply supported with any conditions at the remaining edges. A single-term deflection function in the form of a product of beam functions does not produce numerical results of reasonable accuracy for rectangular plates with other boundary conditions [20]. The deflection functions for the sub-domains of such plates will need to be approximated by series of admissible functions. The amount of computation will, however, make the method less attractive compared with the finite element and the finite difference methods and will not be considered in the present study. For a rectangular plate having two opposite edges simply supported and the remaining edges clamped or free, the sub-domains between the simply supported boundaries and the internal free edges do not possess single-term analytical solutions if they are viewed as rectangular plates with the modified boundary conditions. Assumed deflection functions for these sub-domains are approximated by products of single-term beam functions or degenerated beam functions. The assumed deflection functions for the remaining corner sub-domains which do possess analytical solutions are also replaced by products of the corresponding beam functions. Adjacent subdomains will still have one of the two functions X, and Y,,in common. The predicted results are expected to be better for rectangular plates with small cutouts where the energy is predominantly contributed by the corner domains. There are six combinations of boundary conditions for which two opposite sides are simply supported (SS). The remaining sides are either clamped (C) or free (F). Boundary conditions will be expressed in the order of y = 0, x = a, y = b and x = 0. For example, an SS-C-SS-C rectangular plate is simply supported along y = 0, b and clamped along x = 0, a. Application of the present method will be illustrated only for the SS-SS-SS-SS and the C-SS-C-SS rectangular plates with Poisson’s ratio of 0.3. Finite elements results for comparison with the predicted results are generated by a computer software package [23,24]. A mesh of between 80 and 100 eight-node plate elements, depending on the cutout size, is used in the finite element analysis. 2.2.1. SS-SS-SS-SS rectangular plates. A rectangular plate with an arbitrarily located rectangular cutout is divided into eight sub-domains as shown in Fig. 1 for the fundamental mode. The assumed deflection functions for the sub-domains are as follows:
W,= W,= W,= W,=A,sinEsiny a
I
W,=A,rsiny, rl W, = A,
usin !!f! a’ S!
A,=A,siny
a-x. ny -a - rz s1n T ’
W,=ATzsinz,
a
2
A,=A,sink
A,=A,sinc.
a
a
(8)
A,, A,, A, and A, are determined by matching the deflections of adjacent sub-domains at the corners of the cutout. For example, A2 is determined by matching the deflections of sub-domains 1 and 2 at the corner defined by x = r, and y = s, . For a rectangular plate with a central rectangular cutout, it is not necessary to consider all eight subdomains because of symmetry. In the case of the SX-SY mode, only sub-domains 1, 2 and 3 are needed, as shown in Fig. 2. The natural frequencies for the AX-SY, SX-AY and AX-AY modes of vibration can also be obtained by the present method. The assumed deflection functions for the sub-domains as shown in Fig. 2 are as follows. AX-SY mode:
W,=A,sinGsiny
W2=A2isinf&,
A,=A,siny
W,=A,rsiny, rI
A,=A,sin$$.
(9)
SX-AY mode:
W2 = A 2
x sin !?a’
A,= A, sin2
Sl
W,=A,:sin$,
A,=A,sinT.
(10)
AX-AY mode: W, = A, sin --(1;) sin (T2)
W 2 = A 2- ‘sin SI
W2= A2
A,=A,sinK
7cx
(a/2)
W,= A3Esin&,
A,=A,sin$ ’
A,= A,sins.
(11)
Natural frequencies of rectangular plates 2.2.2. C-SS-C-SSrectangularplates. Theassumed deflection functions for the sub-domains of a C-SSC-SS rectangular plate with a central rectangular cutout are as follows.
865
Table 1. Frequency parameters for the SX-SY mode of an SS-SS-SS-SS
square plate with a central
Cutout size cxd
Finite element
rectangular
Present
cutout
Percentage
results
c/d=
1 O.Oa x 0.0~
W,=A,[cosi(y-q)+Fcoshi(y-i)]siny W, = A, [cash ay - cos ay ILX
+
G(sinh ay - sin ay)] sin a
,=,,[,s;(y-;)+Fcosh;(y-$1;
[cash CW,- cos as, + G(sinh W, - sin a.~)]
(12)
a
Part of the assumed deflection functions in y for W, and W, are the beam function and degenerated beam function, respectively [21,22]. G is determined by satisfying the vanishing bending moment condition at the internal free edge. a is the same as the value for the corresponding beam function for a clamped-free end condition. y equal to sin xx/a is replaced by sin nx/(a/2) for the AX-SY mode of vibration. For the SX-AY mode, only part of the plate is considered. W, =A,[sin
y(y
W,=A,[cosha(b
-i)+Fsinh
-y)-cosa(b
O.la 0.2a 0.3a Oh 0.5a 0.60 0.7a 0.80 0.90
19.752 19.357 19.120 19.357 20.732 23.235 28.241 37.579 57.452 120.392
19.739 19.176 18.901 19.245 20.556 23.329 28.49 1 38.186 58.847 122.663
-0.06 -0.93 -1.15 -0.58 -0.85 0.40 0.89
x x x x x
0.05a 0.2a 0.2% 0.3a 0.4a
19.487 19.008 19.570 20.501 23.583
19.297 18.980 19.556 20.576 23.807
-0.98 -0.14 -0.07 0.36 0.95
c/d = 3 0.15a x 0.24a x 0.3a x 0.6a x 0.90 x
0.05a 0.08~ O.la 0.2~ 0.30
19.341 18.948 18.396 18.981 20.827
19.146 18.856 18.721 19.117 21.001
-1.01 -0.49 1.77 0.72 0.83
c/d=2 O.la o&l 0.50 0.6~ 0.80
A,=A,[cos;(s,-;)+Fcosh;(s,-;)]I
A,=A,sin?.
x x x x x x x x x
0.10 0.2a 0.30 0.4a 0.5a 0.6a 0.7a 0.8a 0.9a
y(y -i)]sinf
-y)
+ G(sinh a@ - y) - sin a(b - y))] sin F
1.62 2.43 1.89
3. RESULTS AND DISCUSSION
3.1. SS-SS-SS-SS
rectangular plates
The computed frequencies and the finite element results for the fundamental mode of a square plate with a central rectangular cutout of various sizes and aspect ratios are presented in Table 1. The discrepancies of the predicted results from the finite element results are less than 2.5%. The predicted changes in frequency agree with the finite element results in showing an initial decreasing trend with increased cutout size. This trend was not reflected in the unmodified results of Ali and Atwal [l]. The results for the AX-SY mode of square plates having rectangular cutouts with c/d = 2 and 3 are presented in Table 2. The discrepancies are of the Table 2. Frequency parameters for the AX-SY mode of an SS-SS-SS-SS square plate with a central rectangular cutout Cutout size cxd
x [cash a(b - sJ - cos a(b - sJ + G(sinh a@ - sJ - sin a(b - sJ)]
A 3= A, sin 5. a
(13)
The assumed deflection function W, for the SX-AY mode is the same as the analytical solution for the fundamental mode of a rectangular plate with three sides simply supported and the remaining side clamped [20]. sin nx/a in W, and W, is replaced by sin nx/(a/Z) for the AX-AY mode of vibration.
Finite element
results
Present method
Percentage discrepancies
c/d=2 O.Oa x O.la x 0.4a x 0.5~ x 0.6a x 0.8~ x
o.oa 0.05a 0.2a 0.25~ 0.3a 0.4~
49.482 49.436 46.583 46.224 47.352 55.641
49.348 49.622 46.910 46.427 47.562 55.834
-0.27 0.38 0.70 0.44 0.44 0.38
c/d=3 0.15a 0.24a 0.3a 0.6a 0.9a
0.05a 0.08a O.Ia Q2a 0.3a
49.377 49.08 1 47.868 47.8 12 53.377
49.378 48.917 48.425 47.812 53.935
0.00 0.33 1.16 0.00 I .05
x x x x x
H. P. LEE et al.
866
Table 3. Frequency parameters for the SX-AY and AX-AY modes of an SS-SS-SS-SS rectangular cutout SX-AY mode Cutout size cxd
cxd=l O.la x O.la
Finite element results
49.239
0.2a x
0.2a
47.773
0.3a
x
0.3a
44.207
0.4a
x 0.4a
41.101
0.5a
x 0.5a
39.712
0.6a
x
42.576
0.7a
x 0.7a
0.6a
50.483
0.8~ x 0.8~
69.817
0.9a x 0.90
133.153
square plate with a central
AX-AY mode
Present method (a) (b)
Finite element results
50.185 (1.92) 49.648 (3.93) 47.029 (6.38) 43.924 (6.87) 42.638 (7.37) 45.120 (5.97) 54.179 (7.32) 77.834 (11.48) 156.645 (17.64)
78.767 76.806 73.906 71.550 69.868 74.988 89.325 124.205 246.022
Present method (a) (c) 74.197 (-5.86) 71.720 (-6.62) 70.975 (-3.97) 70.696 (- 1.19) 71.263 (2.00) 75.558 (0.76) 88.987 (-0.38) 124.998 (0.64) 248.324 (0.94)
78.416 (-0.45) 77.086 (0.37) 75.686 (2.41) 75.047 (4.89)
75.201 (-5.00) 72.283 (;24&) (2:22) 72.791 (6.14) 69.482 (6.60)
79.900 (0.93) 75.246 (1.54) 72.746 (2.05) 70.836 (3.29) 70.019 (7.42)
74.258 (- 5.99) 72.828 (-7.08) 72.575 (- 5.46) 76.843 (11.10) 63.570 (9.72)
80.707 (2.18) 80.206 (2.34) 79.203 (3.17) 72.280 (4.50) 70.544 (21.75)
c/d=2
c/d
O.la x O.OSa
49.403
0.40 x 0.20
41.429
OSa
36.541
x 0.25a
0.6~ x 0.3~
32.611
0.8a
28.260
x 0.4a
= 3 O.l5a x 0.050
0.24~ x
0.08a
49.301 48.428
0.3a x O.la
46.620
0.60 x 0.2~
32.532
0.9a x 0.30
23.529
50.513 (2.25) 49.949 (20.57) 45.979 (25.83) 40.750 (24.96) 31.668 (12.06)
50.420 (2.06) 44.085 (6.41) 37.907 (3.74) 32.292 (-0.98) 26.43 1 (-6.47)
51.333 (4.12) 52.716 (8.86) 53.496 (14.75) 47.535 (46.12) 27.017 (14.82)
51.019 (3.48) 51.212 (5.75) 50.352 (8.00) 33.172 (1.97) 22.132 (- 5.94)
Figures in parentheses denote percentage discrepancies. (a) Unmodified results. (b) W: is multiplied by 2 for c/d = 2. W: is multiplied by 3 for (c) W, and W, are replaced by eqns (14).
order of 1%. For square plates with square cutouts, the predicted frequencies for the AX-SY mode are the same as the values for the SX-AY mode. The discrepancies are higher than the corresponding values for the SX-SY mode. Results are presented in Table 3. For the SX-AY mode of square plates having central rectangular cutouts with c/d > 1, the nodal lines are parallel to the longer sides of the cutouts. The energy contribution from sub-domain 2 is underestimated as the point of matching for the maximum deflections of adjacent sub-domains is close to the nodal line. The results could be improved by multiplying the energy contribution from sub-domain 2 by
c/d
79.160 74.104 71.285 68.579 65.183
78.987 78.375 76.771 69.168 57.940
=
3.
the factor c/d, namely the aspect ratio of the cutout. However, this adjustment does not have a concrete logical base and will not be employed in subsequent analyses. The results are presented in Table 3. The predicted results, in general, agree with the finite element results for the AX-AY mode of vibration. The results for small cutouts with c, d < 0.4~ could be improved by replacing W, and W, by the following functions: W,=A,[E+FsinT]sin& W,=A,[~+Gsin~]sin&.
(14)
-
sx -SY 0 SX-SY --AX-W V AX-AY
Natural frequencies of rectangular plates
Present F.E.M. and SX-AY
Present
8.5 t
F.E.M.
0 SX-SY --AX-SY
0: 0
0.1
867
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
\
F.E.M. Present
\
\
!
C/O
Fig. 3. Frequency parameters for an SS-SASS-SS plate with a square cutout.
square
Fig. 5. Frequency parameters for an SS-SS-SS-SS square plate with a rectangular cutout (c/d = 3).
t, and s, are the x and y coordinates of the lower left hand corner of the cutout. F and G are determined by minimising the integral of the square of the differences in deflections along the boundary of adjacent sub-domains. The additional terms F sin xv/s, and G sin rrr/r, reduce the differences in deflections between adjacent sub-domains. The predicted results for the AX-AY mode are also presented in Table 3. The improved
results for all four modes are presented in Figs 3-5. The method is also applicable for the prediction of the natural frequencies of rectangular plates having central rectangular cutouts. Results for a rectangular plate of aspect ratio 2 with a central rectangular cutout of aspect ratios 1, 2 and 3 are presented in Figs 6-8. Results for some of the higher modes have been modified in the same way as that
200
-a-___4 --\_
-
sx - SY 7o -0 SX-SY -AX -SY 4 AX-SY 6. _-- SX -BY V SX-AY --AX-BY 0
AX-AY
Present F.E.M. Present F.E.M. Present F.E.M. Present
-\
$
-_--__ a
$ cl
5 z $
-I? ‘\ v
-a/’
Z .jis
‘1
2 E 7 ‘c:
30-
s 2
‘. ‘IO_
20()-
to 0
--+---A.
/’ F‘.,
B 0
iA ‘T-
40-
,/fw ,60
1 _-sx-SY 0 SX -SY
Present F.E.M.
--AX-SY ,20 _ A AX-SY -- SX -AY V SX-AY 100 _---AX -AY 0 AX-AY
Present F.E.M. Present F.E.M. Present F.E.M.
2
F.E.M.
50+q-=w
_-o__---,------~
180
,40
8o,F-*-_
l/V
-__--V
-Ii-----__ A
60
---
-
h
1
0.1
, 0.2
i 0.3
I
0.4
I 0.5
t 0.6
t 0.7
)
t
0.8
0.9
C/0
4. Frequency parameters for an SS-SS-SS-SS square plate with a rectangular cutout (c/d = 2).
201
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
c/a
Fig. 6. Frequency parameters for an SS-SS-SS-SS rectangular plate (u/b = 2) with a square cutout.
H. P. Lxs et al.
868
225
_I”” -
250
sx - SY
0 SX-SY --AX-SY A AX-W - -- sx-AY 0 SX-AY ---AX-AY 0 AX-AY
Present
- sx - SY
F.E.M. Pressni F.E.M. ~reaeni F.E.M. Present F.E.M.
0 SX-SY --AX-SY 200
_-P
A AX-SY
--.
-SX-AY V SX-AY ---AX -AY 0 AX-AY
?i\
c
‘\ ‘\ Cl
Present F.E.M. Present F.E.M. Present F.E.M. Present F.E.M.
‘\ .\
\
8 $
125
\ \
= s ‘$
P
V’\ \ \
100
/’ ‘\
E a
P /
V’
i
0
--6-F--a-y
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
c/a
Fig. 7. Frequency parameters for an SS-SS-SS-SS rectangular plate (a/b = 2) with a rectangular cutout (c/d = 2).
Fig. 8. Frequency parameters for an SS-SS-SS-SS rectangular plate (a/b = 2) with a rectangular cutout (c/d = 3).
required for a square plate with a central rectangular cutout.
results agree with the FEM results to within 3% if the longer internal free edge is perpendicular to the nodal line and c and d < 0.6~. The results are not very encouraging if the nodal line is parallel to the longer internal free edge. The error could top 28% even for a modest cutout. The error incurred by the AX-AY mode is generally higher than that for the three lower modes. This is expected as the consequences of the Rayleigh quotient approach. The present method is also applicable for the prediction of the natural frequencies of F-SS-F-SS, C-SS-F-SS, SS-SS-C-SS and SS-SS-F-SS rectangular plates with rectangular cutouts by the use of
3.2. C-SS-C-SS
rectangular plates
The computed results and the finite element results of a square plate with a central rectangular cutout of various sizes and aspect ratios are given in Table 4. For the SX-SY mode, the discrepancies are less than 1% for cutouts with c/d = 2. Discrepancies for cutouts with d/c = 2 are, in general, higher, in particular for large cutouts with d > 0.5~. This is because the energies from sub-domains 2 and 7 are overestimated. For the AX-SY and SX-AY modes, the computed
Table 4. Frequency parameters for a C-SS-C-SS square plate with a central rectangular cutout AX-SY mode SX-AY mode AX-AY method SX-SY mode Finite element results
Finite element
Present method
Finite element
Present method
O.la x o.osa
28.600
28.400
58.839
69.498
0.4a x 0.2a
29.753
51.975
0.6a x 0.3a
56.025 (2.16) 53.437 (2.80)
35.569
54.713
55.308
48.048
70.216
(1.09) 72.044 (2.W
Cutout size c/d=2
0.8~ x 0.4a d/c =
2 0.05a x O.la
(1.W 28.690
28.704 54.883 59.285 (0.03) (4.38) 0.2n x 0.4a 30.569 31.277 45.606 58.320 (2.31) (27.88) 0.3~ x 0.6~ 34.416 36.614 37.354 46.448 (6.39) (24.34) 0.40 x 0.80 34.841 41.131 34.854 39.443 (18.05) (13.17) Figures in parentheses denote percentage discrepancies.
56.004
51.067
Present method 70.229 (1.05) 65.383 (16.66)
Finite element results
Present method
94.999
90.796
81.929
(6.51) 87.223
54.641
(13.72) 53.045
(6.46)
(3.87)
69.582
69.415
95.319
90.333
66.873
(---9ob2;b’ (d.09, 77.026 (1.20) 96.442 (17.94)
88.842
(&313~ (0.98) 93.210 (11.21) 92.335 (12.81)
76.109 81.769
83.815 81.846
Natural frequencies of rectangular plates
suitable assumed deflection functions. The results are presented in [25]. Frequencies obtained for rectangular plates with multiple cutouts which compare favourably with other methods are also presented in [25].
5. L. G. Tham, A. H. C. Chan and Y. K. Cheung,
Free vibration and buckling of plates by the negative stiffness method. Comput. SWUCI.22, 687-692 (1986). 6. P. Paramasivam, Free vibration of square plates with square openings. J. Sound Vibr. 30, 173-198 (1973). 7. G. Aksu and R. Ali, Determination of dynamic charac-
4. CONCLUSIONS 8.
The present method serves as a simple alternative to the finite element and the finite difference methods for predicting the natural frequencies of rectangular plates with cutouts. The computation only involves the evaluation of standard integrals for a single-term assumed deflection function. The application has been demonstrated for rectangular plates simply supported along two opposite edges with the remaining edges clamped, free or simply supported. The predicted results by the present method, in general, compare favourably with the finite element results, except for those for square plates with large cutouts and mode shapes with nodal lines parallel to the longer sides of the cutouts. The predicted results for these cases can be improved by modifying the energy contribution from some of the sub-domains. The results can also be improved by using a RayleighRitz multiple terms approach. However, this will increase the computation time, which is not the intention of this work. In respect of the fundamental mode of a simply supported square plate with a central rectangular cutout, the results predicted by the present method differ only slightly from the finite element results although single term deflection functions are used. The present results also reflect the initial decreasing trend in frequency with increased cutout size and are dependent on the Poisson ratio of the plate. The method had also been extended to sheardeformable plate [25] and laminated plate [26].
REFERENCES
1. R. Ali and S. J. Atwal, Prediction of natural frequencies of vibration of rectangular plates with cutouts. Compur. Strucr. 12, 819-823 (1980). 2. B. P. Shastry and G. Venkateswara Rao, Vibrations of thin rectangular plates with arbitrary oriented stiffeners. Comput. Struct. 7, 627-629 (1977). 3. N. Reddy, Large amplitude flexural vibration of layered composite plates with cutouts. J. Sound Vibr. 83, l-10 (1982). 4. P. P. A. Laura, J. C. Utjes and V. H. Palluzzi, On the effect of free, rectangular cut-outs along the edge on the transverse vibrations of rectangular plates. Appl. Acousr. 19, 139-151 (1986).
869
9.
10.
11.
teristics of rectangular plates with cutouts using a finite difference formulation. J. Sound Vibr. 44. 147-158 (1976). N. L. Basdekas and M. Chi, Dynamic response of plates with cutouts. Shock Vibr. Bull. 29-35 (1970). A. Rajamani and R. Prabhakaran, Dynamic response of composite plates with cut-outs, part I: simplysupported plates. J. Sound. Vibr. 54, 549-564 (1977). A. Rajamani and R. Prabhakaran, Dynamic response of composite plates with cut-outs, part II: clamped plates. J. Sound Vibr. 54, 565-576 (1977). K. Nagaya, Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape, part I: derivation of frequency equation. J. Sound Vibr. 74, 543-551 (1981).
12. K. Nagaya, Simplified method of solving problems of vibrating plates of doubly connected arbitrary shape, part II: applications and experiments. J. Sound Vibr. 74, 553-564 (1981).
13. R. F. Hegarty and T. Ariman, Elasto-dynamic analysis of rectangular plates with circular holes. Inr. J. Solids Srruct. 11, 895-906 (1975). 14. T. Kumai, The flexural vibrations of a square plate with a central circular hole. In Proceedings of the 2nd Japan National Congress for Applied Mechanics, pp. 339-342 (1957). 15. C. Joga Rao and G. Picket& Vibration of plates with irregular shapes and plates with holes. J. aeronaut. Sot. India 13, 83-88 (1961).
16. S. Takahasi, Vibration of rectangular plates with circular holes. Bull. Jap. Sot. hfech. Engs. 1,380-385 (1958). 17. T. Mizusawa. T. Kaiita and M. Naruoka. Vibration and buckling analysis ofplates of abruptly varying stiffness. Comput. Struct. 12, 689-693
(1980).
18. P. P. A. Laura, P. Vemiere De Irassar, L. Ercoli and R. Gelos, Fundamental frequency of vibrations of a rectangular plate with a free, straight corner cut-out. J. Sound. Vibr. 78, 489-493
(1981).
19. G. Ganesan and S. Nagaraja Rao, Influence of location of cut-outs on natural frequencies of thin square plates. J. Sound. Vibr. 101, 598-600 (1985). 20. A. W. Leissa, Vibration of plates. NASA SP160 (1969). 21. D. Young, Vibration of rectangular plates by the Ritz method. J. appl. Mesh. 17, 448-453 (1950). 22. S. F. Bassily and S. M. Dickinson, On the use of beam functions for problems of plates involving free edges. J. aool. Mech. 42. 858-864 11975). 23. PAI%C Data Preparation, PAFEC. Ltd, Strelley Hall, Strelley, Nottingham, U.K. 24. PAFEC Theory, Results, PAFEC. Ltd, Strelley Hall, Strehey, Nottingham, U.K. 25. H. P. Lee, Vibration study of rectangular plates with cutouts. M. Engng thesis, National University of Singapore, Singapore (1986). 26. H. P. Lee, S. P. Lim and S. T. Chow, Free vibration of composite rectangular plates with rectangular cutouts. Camp. Strucr. 8, 63-81 (1987).
1990
lw5-7949/90 53.00 + 0.00 Q 1990 Pergamon Press plc
PREDICTION OF NATURAL FREQUENCIES OF RECTANGULAR PLATES WITH RECTANGULAR CUTOUTS H. P. LEE, S. P. LIM and S. T. CHOW Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge, Singapore 0511, Republic of Singapore (Received 25 April 1989)
Abstract-A
simple numerical method based on the Rayleigh quotient is presented for predicting the natural frequencies of the fundamental and some higher modes of a rectangular plate with an arbitrarily located rectangular cutout. The edges of the cutout are parallel to the edges of the complete plate. The nodal patterns of these higher modes are pre-selected based on the symmetry of the mode shapes about the geometrical axes. The method is applied to the study of linear vibration of rectangular plates simply supported along one pair of opposite edges with any other boundary conditions at the remaining edges. The predicted results compare favourably with the corresponding results produced by other methods whenever available.
NOTATION
b c, d 0,
h rlv 5 Sl.S2
D T V w W” A P w
dimensions of the plate dimensions of the cutout thickness of the plate x coordinates for the corners of cutout y coordinates for the corners of cutout flexural rigidity kinetic energy potential energy transverse displacement of the plate nondimensional frequency parameter Poisson’s ratio mass density angular frequency I. INTRODIJCI-ION
Rectangular plates with cutouts are frequently found in engineering structures. The cutouts are made for saving weight, for venting, for altering the natural frequencies and for providing accessibility to other parts of the structures. Past investigations on the vibration of these plates have been confined to the numerical analysis of rectangular plates with rectangular openings or circular holes. The numerical methods for computing the natural frequencies of rectangular plates with cutouts can be broadly classified into three categories, namely the finite element and finite difference methods, the series-type analytical method and the semianalytical approach based on the Rayleigh-Ritz principle. Numerical results obtained by the finite element method have been reported by Ali and Atwal [I], Shastry and Venkateswara Rao [2], Reddy [3], Laura et al. [4] and Tham et al. [S]. Paramasivam [6] and Aksu and Ali [7] analysed the problems using finite difference analysis.
The problems have also been solved by a number of authors using the series-type analytical method. Basdekas and Chi [8] investigated the forced and free dynamic response of a rectangular orthotropic plate by expressing the effect of a cutout as an external loading function on the plate. The assumed deflection function was taken to be a series of eigenfunctions for the complete plate without the cutout. Rajamani and Prabhakaran [9, lo] generalised the method by approximating the eigenfunctions by the products of beam functions for the given end conditions. Nagaya [l 1, 121 obtained the natural frequencies of square plates with central circular holes by conformal mapping and the external boundary conditions were satisfied by a Fourier expansion collocation method. Hegarty and Ariman [ 131solved the same problem by satisfying the external boundary conditions using a least square point matching method. Internal boundary conditions along the circular hole were satisfied by the assumed Bessel functions. Rayleigh-Ritz method has been used widely as an alternative to the aforementioned methods for predicting the linear frequencies of rectangular plates with cutouts. There are two different approaches in applying this method. The first approach is a combination of the point matching and the Rayleigh-Ritz method. The external boundaries and the internal free edge conditions are satisfied by the assumed deflection functions only at some discrete points, usually at the corners of the cutout and the plate. This approach is restricted to rectangular plates with centrally located cutouts or circular holes as the deflection functions are invariably symmetrical about the centres of the plates. Numerical results were reported by Kumai [14] and Joga Rao and Pickett [ 151. 861
H. P. LEE et al.
862
A different approach was reported by Takahasi [ 161 for predicting the natural frequencies of a rectangular plate with a circular hole which need not be located at the centre of the plate. The energy of a plate with a hole is approximated by deducting the energy of the hole from the energy of the whole plate without the hole. The assumed deflection functions for computing the energy of the plate and the hole only satisfy the external boundary conditions and disregard the presence of the hole. Ali and Atwal [1] used the same approach to predict the natural frequencies of a simply supported square plate with a central rectangular cutout. The results obtained for the fundamental mode need to be multiplied by a correction factor depending on the cutout size in order to reduce the percentage discrepancies from the corresponding finite element results to below 5%. All the works mentioned above, with the exception of [7J, deal with a rectangular plate with a single cutout. The same method was also used by Mizusawa et al. [17] and Laura et al. [4, 181, based on B-spline functions, polynomial functions and combinations of polynomial and trigonometric functions which satisfied only the external boundary conditions of the complete plates, for predicting the natural frequencies of rectangular plates with central or corner cutouts. A similar method, using the Hamilton principle instead of the Rayleigh quotient, was reported by Ganesan and Nagaraja Rao [ 191. In these two approaches, which are based on the Rayleigh-Ritz method, the assumed deflection functions are continuous throughout the whole plate. The internal free edge conditions of a rectangular cutout are disregarded or are satisfied only at the four comers of the cutout. In [7] Aksu and Ali used finite difference method to deal with the problem of rectangular plates with multiple rectangular cutouts. In the present analysis, the assumed deflection functions are no longer continuous throughout the whole plate. The deflection functions are, however, made to satisfy all or part of the internal free edge conditions along the four edges of the cutout. 2. ANALYSIS
2.1. Classical method According to the classical theory of plates [20], the linear vibration of a thin, homogeneous, isotropic plate is governed by the following partial differential equation:
An exact solution of eqn (1) is known only for simply supported rectangular plates and rectangular plates which are simply supported along one pair of opposite edges with any conditions at the remaining edges [20]. It is necessary to resort to numerical methods for plates with internal boundaries or other combinations of boundary conditions. The procedure developed by Ritz [21] has been found to be useful for solving such problems. For a rectangular plate which is vibrating harmonically with amplitude Wand natural frequency o, the maximum potential energy (V) and kinetic energy (T) are given by
-2(1-u)($-($)3]dxdy
T =;phw*
(2)
W= dx dy.
(3)
The above integrations are to be performed over the domain of the plate. Using Rayleigh quotient, the maximum values of the potential and kinetic energies are equated to determine the natural frequency. w*=_
2 ph
V (4) W’dxdy
Ritz method assumes the deflection function W as a linear series of admissible functions and adjusts the coefficients in the series so as to minimise eqn (4). It is usually written as w=C&Ln~m(~)y”(Y).
m ”
(5)
The functions X,,, and Y, for a rectangular plate without any cutout are usually beam functions]211 or degenerated beam functions [22] that satisfy the external boundary conditions. For a rectangular plate with a rectangular cutout, there is hitherto no reported function which is continuous throughout the whole domain of the plate and satisfies all, or part, of the internal free edge conditions besides the external boundary conditions. As a means of overcoming this difficulty a modified Rayleigh method is presented. 2.2. The present method
DA’W+ph$y=O.
The deflection W is assumed to be small compared with the plate thickness h. The effects of shear deformation and rotary inertia have been neglected.
The plate under consideration is a thin rectangular plate of size a x b having a rectangular cutout of size c x d as shown in Fig. 1. The plate is divided into smaller sub-domains based on the mode shapes and the locations of the cutouts. The eight sub-domains for the fundamental mode are presented in Fig. 1.
863
Natural frequencies of rectangular plates
Yf
1 ’
b 92
_-----
8
1 6 I--me
d
3 SI -----
I 7
5 C
---_
Ici
2
I
fl
0
i 4 t 12 0
x
In order to minimize ~mpu~tion time, a singleterm deflection function W, is assumed instead of a series for each sub-domain after the corresponding modified boundary conditions have been determined. The assumed functions will be either products of beam functions, degenerated beam functions or the single-term analytical solutions for the given boundary conditions. For the beam functions, only the essential boundary conditions are satisfied. The functions are assumed to be of the form
Fig. 1. A rectangular plate with a rectangular cutout.
wn=&w)YnCV) The four basic modes of vibration to be considered are the SX-SY, AX-SY, SX-AY and the AX-AY 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 anti-symmetric modes of vibration about x = a/2 and y = b/2. The SX-SY mode is the fundamental mode presented in Fig. 2. The other three higher modes will only be considered for a rectangular plate with a centrally located rectangular cutout. As a result of symmetry, it is necessary to consider the sub-domains for only the lower left hand portion of the plate. The sub-domains of these higher modes are presented in Fig. 2. For a rectangular plate with a single cutout, the sub-domains at the four comers, namely 1, 4, 6 and 8, are considered to be parts of a complete plate having the given external boundary conditions. The sub-domains between the external boundary and the internal free edge, such as sub-domains 2, 3, 5 and 7, are, however, taken to be parts of a rectangular plate with three sides having the same boundary conditions as the original plate and the remaining side being free.
for the nth sub-domain. All the deflection functions are expressed in the same global coordinate axes (x, y). Adjacent subdomains will have one of the two functions, X, and Y., in common. Relative magnitudes of coefficients A, are determined by matching the maximum deflections at the comers of the cutout. The potential energy and kinetic energy of each sub-domain are evaluated using eqns (2) and (3). The total potential and kinetic energies of the plate are taken to be the sum of the kinetic and potential energy of all the sub-domains. The frequency of every mode is obtained by equating the total potential energy to the total kinetic energy. The frequency is expressed in terms of a non~imensio~l frequency parameter d which is a function of only the Poisson’s ratio and the aspect ratio of the plate for a given cutout configuration. The parameter is defined as
A=;(-&
1
[17 2
I
:
AX-SY
sx-SY
3 ----I
~
0$ i’2.
3
3 ----I
I
_---__
_--_
2
I
I
SX-AY ----Sub-domain -Nodal
(6)
I’121
I
I
1 AX-AY
I
boundory line
Fig. 2. Sub-domains for a rectangular plate with a central rectangular cutout.
(7)
H. P. LEE et al.
864
The present method is used to calculate the natural frequencies of rectangular plates having two opposite edges simply supported with any conditions at the remaining edges. A single-term deflection function in the form of a product of beam functions does not produce numerical results of reasonable accuracy for rectangular plates with other boundary conditions [20]. The deflection functions for the sub-domains of such plates will need to be approximated by series of admissible functions. The amount of computation will, however, make the method less attractive compared with the finite element and the finite difference methods and will not be considered in the present study. For a rectangular plate having two opposite edges simply supported and the remaining edges clamped or free, the sub-domains between the simply supported boundaries and the internal free edges do not possess single-term analytical solutions if they are viewed as rectangular plates with the modified boundary conditions. Assumed deflection functions for these sub-domains are approximated by products of single-term beam functions or degenerated beam functions. The assumed deflection functions for the remaining corner sub-domains which do possess analytical solutions are also replaced by products of the corresponding beam functions. Adjacent subdomains will still have one of the two functions X, and Y,,in common. The predicted results are expected to be better for rectangular plates with small cutouts where the energy is predominantly contributed by the corner domains. There are six combinations of boundary conditions for which two opposite sides are simply supported (SS). The remaining sides are either clamped (C) or free (F). Boundary conditions will be expressed in the order of y = 0, x = a, y = b and x = 0. For example, an SS-C-SS-C rectangular plate is simply supported along y = 0, b and clamped along x = 0, a. Application of the present method will be illustrated only for the SS-SS-SS-SS and the C-SS-C-SS rectangular plates with Poisson’s ratio of 0.3. Finite elements results for comparison with the predicted results are generated by a computer software package [23,24]. A mesh of between 80 and 100 eight-node plate elements, depending on the cutout size, is used in the finite element analysis. 2.2.1. SS-SS-SS-SS rectangular plates. A rectangular plate with an arbitrarily located rectangular cutout is divided into eight sub-domains as shown in Fig. 1 for the fundamental mode. The assumed deflection functions for the sub-domains are as follows:
W,= W,= W,= W,=A,sinEsiny a
I
W,=A,rsiny, rl W, = A,
usin !!f! a’ S!
A,=A,siny
a-x. ny -a - rz s1n T ’
W,=ATzsinz,
a
2
A,=A,sink
A,=A,sinc.
a
a
(8)
A,, A,, A, and A, are determined by matching the deflections of adjacent sub-domains at the corners of the cutout. For example, A2 is determined by matching the deflections of sub-domains 1 and 2 at the corner defined by x = r, and y = s, . For a rectangular plate with a central rectangular cutout, it is not necessary to consider all eight subdomains because of symmetry. In the case of the SX-SY mode, only sub-domains 1, 2 and 3 are needed, as shown in Fig. 2. The natural frequencies for the AX-SY, SX-AY and AX-AY modes of vibration can also be obtained by the present method. The assumed deflection functions for the sub-domains as shown in Fig. 2 are as follows. AX-SY mode:
W,=A,sinGsiny
W2=A2isinf&,
A,=A,siny
W,=A,rsiny, rI
A,=A,sin$$.
(9)
SX-AY mode:
W2 = A 2
x sin !?a’
A,= A, sin2
Sl
W,=A,:sin$,
A,=A,sinT.
(10)
AX-AY mode: W, = A, sin --(1;) sin (T2)
W 2 = A 2- ‘sin SI
W2= A2
A,=A,sinK
7cx
(a/2)
W,= A3Esin&,
A,=A,sin$ ’
A,= A,sins.
(11)
Natural frequencies of rectangular plates 2.2.2. C-SS-C-SSrectangularplates. Theassumed deflection functions for the sub-domains of a C-SSC-SS rectangular plate with a central rectangular cutout are as follows.
865
Table 1. Frequency parameters for the SX-SY mode of an SS-SS-SS-SS
square plate with a central
Cutout size cxd
Finite element
rectangular
Present
cutout
Percentage
results
c/d=
1 O.Oa x 0.0~
W,=A,[cosi(y-q)+Fcoshi(y-i)]siny W, = A, [cash ay - cos ay ILX
+
G(sinh ay - sin ay)] sin a
,=,,[,s;(y-;)+Fcosh;(y-$1;
[cash CW,- cos as, + G(sinh W, - sin a.~)]
(12)
a
Part of the assumed deflection functions in y for W, and W, are the beam function and degenerated beam function, respectively [21,22]. G is determined by satisfying the vanishing bending moment condition at the internal free edge. a is the same as the value for the corresponding beam function for a clamped-free end condition. y equal to sin xx/a is replaced by sin nx/(a/2) for the AX-SY mode of vibration. For the SX-AY mode, only part of the plate is considered. W, =A,[sin
y(y
W,=A,[cosha(b
-i)+Fsinh
-y)-cosa(b
O.la 0.2a 0.3a Oh 0.5a 0.60 0.7a 0.80 0.90
19.752 19.357 19.120 19.357 20.732 23.235 28.241 37.579 57.452 120.392
19.739 19.176 18.901 19.245 20.556 23.329 28.49 1 38.186 58.847 122.663
-0.06 -0.93 -1.15 -0.58 -0.85 0.40 0.89
x x x x x
0.05a 0.2a 0.2% 0.3a 0.4a
19.487 19.008 19.570 20.501 23.583
19.297 18.980 19.556 20.576 23.807
-0.98 -0.14 -0.07 0.36 0.95
c/d = 3 0.15a x 0.24a x 0.3a x 0.6a x 0.90 x
0.05a 0.08~ O.la 0.2~ 0.30
19.341 18.948 18.396 18.981 20.827
19.146 18.856 18.721 19.117 21.001
-1.01 -0.49 1.77 0.72 0.83
c/d=2 O.la o&l 0.50 0.6~ 0.80
A,=A,[cos;(s,-;)+Fcosh;(s,-;)]I
A,=A,sin?.
x x x x x x x x x
0.10 0.2a 0.30 0.4a 0.5a 0.6a 0.7a 0.8a 0.9a
y(y -i)]sinf
-y)
+ G(sinh a@ - y) - sin a(b - y))] sin F
1.62 2.43 1.89
3. RESULTS AND DISCUSSION
3.1. SS-SS-SS-SS
rectangular plates
The computed frequencies and the finite element results for the fundamental mode of a square plate with a central rectangular cutout of various sizes and aspect ratios are presented in Table 1. The discrepancies of the predicted results from the finite element results are less than 2.5%. The predicted changes in frequency agree with the finite element results in showing an initial decreasing trend with increased cutout size. This trend was not reflected in the unmodified results of Ali and Atwal [l]. The results for the AX-SY mode of square plates having rectangular cutouts with c/d = 2 and 3 are presented in Table 2. The discrepancies are of the Table 2. Frequency parameters for the AX-SY mode of an SS-SS-SS-SS square plate with a central rectangular cutout Cutout size cxd
x [cash a(b - sJ - cos a(b - sJ + G(sinh a@ - sJ - sin a(b - sJ)]
A 3= A, sin 5. a
(13)
The assumed deflection function W, for the SX-AY mode is the same as the analytical solution for the fundamental mode of a rectangular plate with three sides simply supported and the remaining side clamped [20]. sin nx/a in W, and W, is replaced by sin nx/(a/Z) for the AX-AY mode of vibration.
Finite element
results
Present method
Percentage discrepancies
c/d=2 O.Oa x O.la x 0.4a x 0.5~ x 0.6a x 0.8~ x
o.oa 0.05a 0.2a 0.25~ 0.3a 0.4~
49.482 49.436 46.583 46.224 47.352 55.641
49.348 49.622 46.910 46.427 47.562 55.834
-0.27 0.38 0.70 0.44 0.44 0.38
c/d=3 0.15a 0.24a 0.3a 0.6a 0.9a
0.05a 0.08a O.Ia Q2a 0.3a
49.377 49.08 1 47.868 47.8 12 53.377
49.378 48.917 48.425 47.812 53.935
0.00 0.33 1.16 0.00 I .05
x x x x x
H. P. LEE et al.
866
Table 3. Frequency parameters for the SX-AY and AX-AY modes of an SS-SS-SS-SS rectangular cutout SX-AY mode Cutout size cxd
cxd=l O.la x O.la
Finite element results
49.239
0.2a x
0.2a
47.773
0.3a
x
0.3a
44.207
0.4a
x 0.4a
41.101
0.5a
x 0.5a
39.712
0.6a
x
42.576
0.7a
x 0.7a
0.6a
50.483
0.8~ x 0.8~
69.817
0.9a x 0.90
133.153
square plate with a central
AX-AY mode
Present method (a) (b)
Finite element results
50.185 (1.92) 49.648 (3.93) 47.029 (6.38) 43.924 (6.87) 42.638 (7.37) 45.120 (5.97) 54.179 (7.32) 77.834 (11.48) 156.645 (17.64)
78.767 76.806 73.906 71.550 69.868 74.988 89.325 124.205 246.022
Present method (a) (c) 74.197 (-5.86) 71.720 (-6.62) 70.975 (-3.97) 70.696 (- 1.19) 71.263 (2.00) 75.558 (0.76) 88.987 (-0.38) 124.998 (0.64) 248.324 (0.94)
78.416 (-0.45) 77.086 (0.37) 75.686 (2.41) 75.047 (4.89)
75.201 (-5.00) 72.283 (;24&) (2:22) 72.791 (6.14) 69.482 (6.60)
79.900 (0.93) 75.246 (1.54) 72.746 (2.05) 70.836 (3.29) 70.019 (7.42)
74.258 (- 5.99) 72.828 (-7.08) 72.575 (- 5.46) 76.843 (11.10) 63.570 (9.72)
80.707 (2.18) 80.206 (2.34) 79.203 (3.17) 72.280 (4.50) 70.544 (21.75)
c/d=2
c/d
O.la x O.OSa
49.403
0.40 x 0.20
41.429
OSa
36.541
x 0.25a
0.6~ x 0.3~
32.611
0.8a
28.260
x 0.4a
= 3 O.l5a x 0.050
0.24~ x
0.08a
49.301 48.428
0.3a x O.la
46.620
0.60 x 0.2~
32.532
0.9a x 0.30
23.529
50.513 (2.25) 49.949 (20.57) 45.979 (25.83) 40.750 (24.96) 31.668 (12.06)
50.420 (2.06) 44.085 (6.41) 37.907 (3.74) 32.292 (-0.98) 26.43 1 (-6.47)
51.333 (4.12) 52.716 (8.86) 53.496 (14.75) 47.535 (46.12) 27.017 (14.82)
51.019 (3.48) 51.212 (5.75) 50.352 (8.00) 33.172 (1.97) 22.132 (- 5.94)
Figures in parentheses denote percentage discrepancies. (a) Unmodified results. (b) W: is multiplied by 2 for c/d = 2. W: is multiplied by 3 for (c) W, and W, are replaced by eqns (14).
order of 1%. For square plates with square cutouts, the predicted frequencies for the AX-SY mode are the same as the values for the SX-AY mode. The discrepancies are higher than the corresponding values for the SX-SY mode. Results are presented in Table 3. For the SX-AY mode of square plates having central rectangular cutouts with c/d > 1, the nodal lines are parallel to the longer sides of the cutouts. The energy contribution from sub-domain 2 is underestimated as the point of matching for the maximum deflections of adjacent sub-domains is close to the nodal line. The results could be improved by multiplying the energy contribution from sub-domain 2 by
c/d
79.160 74.104 71.285 68.579 65.183
78.987 78.375 76.771 69.168 57.940
=
3.
the factor c/d, namely the aspect ratio of the cutout. However, this adjustment does not have a concrete logical base and will not be employed in subsequent analyses. The results are presented in Table 3. The predicted results, in general, agree with the finite element results for the AX-AY mode of vibration. The results for small cutouts with c, d < 0.4~ could be improved by replacing W, and W, by the following functions: W,=A,[E+FsinT]sin& W,=A,[~+Gsin~]sin&.
(14)
-
sx -SY 0 SX-SY --AX-W V AX-AY
Natural frequencies of rectangular plates
Present F.E.M. and SX-AY
Present
8.5 t
F.E.M.
0 SX-SY --AX-SY
0: 0
0.1
867
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
\
F.E.M. Present
\
\
!
C/O
Fig. 3. Frequency parameters for an SS-SASS-SS plate with a square cutout.
square
Fig. 5. Frequency parameters for an SS-SS-SS-SS square plate with a rectangular cutout (c/d = 3).
t, and s, are the x and y coordinates of the lower left hand corner of the cutout. F and G are determined by minimising the integral of the square of the differences in deflections along the boundary of adjacent sub-domains. The additional terms F sin xv/s, and G sin rrr/r, reduce the differences in deflections between adjacent sub-domains. The predicted results for the AX-AY mode are also presented in Table 3. The improved
results for all four modes are presented in Figs 3-5. The method is also applicable for the prediction of the natural frequencies of rectangular plates having central rectangular cutouts. Results for a rectangular plate of aspect ratio 2 with a central rectangular cutout of aspect ratios 1, 2 and 3 are presented in Figs 6-8. Results for some of the higher modes have been modified in the same way as that
200
-a-___4 --\_
-
sx - SY 7o -0 SX-SY -AX -SY 4 AX-SY 6. _-- SX -BY V SX-AY --AX-BY 0
AX-AY
Present F.E.M. Present F.E.M. Present F.E.M. Present
-\
$
-_--__ a
$ cl
5 z $
-I? ‘\ v
-a/’
Z .jis
‘1
2 E 7 ‘c:
30-
s 2
‘. ‘IO_
20()-
to 0
--+---A.
/’ F‘.,
B 0
iA ‘T-
40-
,/fw ,60
1 _-sx-SY 0 SX -SY
Present F.E.M.
--AX-SY ,20 _ A AX-SY -- SX -AY V SX-AY 100 _---AX -AY 0 AX-AY
Present F.E.M. Present F.E.M. Present F.E.M.
2
F.E.M.
50+q-=w
_-o__---,------~
180
,40
8o,F-*-_
l/V
-__--V
-Ii-----__ A
60
---
-
h
1
0.1
, 0.2
i 0.3
I
0.4
I 0.5
t 0.6
t 0.7
)
t
0.8
0.9
C/0
4. Frequency parameters for an SS-SS-SS-SS square plate with a rectangular cutout (c/d = 2).
201
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
c/a
Fig. 6. Frequency parameters for an SS-SS-SS-SS rectangular plate (u/b = 2) with a square cutout.
H. P. Lxs et al.
868
225
_I”” -
250
sx - SY
0 SX-SY --AX-SY A AX-W - -- sx-AY 0 SX-AY ---AX-AY 0 AX-AY
Present
- sx - SY
F.E.M. Pressni F.E.M. ~reaeni F.E.M. Present F.E.M.
0 SX-SY --AX-SY 200
_-P
A AX-SY
--.
-SX-AY V SX-AY ---AX -AY 0 AX-AY
?i\
c
‘\ ‘\ Cl
Present F.E.M. Present F.E.M. Present F.E.M. Present F.E.M.
‘\ .\
\
8 $
125
\ \
= s ‘$
P
V’\ \ \
100
/’ ‘\
E a
P /
V’
i
0
--6-F--a-y
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
c/a
Fig. 7. Frequency parameters for an SS-SS-SS-SS rectangular plate (a/b = 2) with a rectangular cutout (c/d = 2).
Fig. 8. Frequency parameters for an SS-SS-SS-SS rectangular plate (a/b = 2) with a rectangular cutout (c/d = 3).
required for a square plate with a central rectangular cutout.
results agree with the FEM results to within 3% if the longer internal free edge is perpendicular to the nodal line and c and d < 0.6~. The results are not very encouraging if the nodal line is parallel to the longer internal free edge. The error could top 28% even for a modest cutout. The error incurred by the AX-AY mode is generally higher than that for the three lower modes. This is expected as the consequences of the Rayleigh quotient approach. The present method is also applicable for the prediction of the natural frequencies of F-SS-F-SS, C-SS-F-SS, SS-SS-C-SS and SS-SS-F-SS rectangular plates with rectangular cutouts by the use of
3.2. C-SS-C-SS
rectangular plates
The computed results and the finite element results of a square plate with a central rectangular cutout of various sizes and aspect ratios are given in Table 4. For the SX-SY mode, the discrepancies are less than 1% for cutouts with c/d = 2. Discrepancies for cutouts with d/c = 2 are, in general, higher, in particular for large cutouts with d > 0.5~. This is because the energies from sub-domains 2 and 7 are overestimated. For the AX-SY and SX-AY modes, the computed
Table 4. Frequency parameters for a C-SS-C-SS square plate with a central rectangular cutout AX-SY mode SX-AY mode AX-AY method SX-SY mode Finite element results
Finite element
Present method
Finite element
Present method
O.la x o.osa
28.600
28.400
58.839
69.498
0.4a x 0.2a
29.753
51.975
0.6a x 0.3a
56.025 (2.16) 53.437 (2.80)
35.569
54.713
55.308
48.048
70.216
(1.09) 72.044 (2.W
Cutout size c/d=2
0.8~ x 0.4a d/c =
2 0.05a x O.la
(1.W 28.690
28.704 54.883 59.285 (0.03) (4.38) 0.2n x 0.4a 30.569 31.277 45.606 58.320 (2.31) (27.88) 0.3~ x 0.6~ 34.416 36.614 37.354 46.448 (6.39) (24.34) 0.40 x 0.80 34.841 41.131 34.854 39.443 (18.05) (13.17) Figures in parentheses denote percentage discrepancies.
56.004
51.067
Present method 70.229 (1.05) 65.383 (16.66)
Finite element results
Present method
94.999
90.796
81.929
(6.51) 87.223
54.641
(13.72) 53.045
(6.46)
(3.87)
69.582
69.415
95.319
90.333
66.873
(---9ob2;b’ (d.09, 77.026 (1.20) 96.442 (17.94)
88.842
(&313~ (0.98) 93.210 (11.21) 92.335 (12.81)
76.109 81.769
83.815 81.846
Natural frequencies of rectangular plates
suitable assumed deflection functions. The results are presented in [25]. Frequencies obtained for rectangular plates with multiple cutouts which compare favourably with other methods are also presented in [25].
5. L. G. Tham, A. H. C. Chan and Y. K. Cheung,
Free vibration and buckling of plates by the negative stiffness method. Comput. SWUCI.22, 687-692 (1986). 6. P. Paramasivam, Free vibration of square plates with square openings. J. Sound Vibr. 30, 173-198 (1973). 7. G. Aksu and R. Ali, Determination of dynamic charac-
4. CONCLUSIONS 8.
The present method serves as a simple alternative to the finite element and the finite difference methods for predicting the natural frequencies of rectangular plates with cutouts. The computation only involves the evaluation of standard integrals for a single-term assumed deflection function. The application has been demonstrated for rectangular plates simply supported along two opposite edges with the remaining edges clamped, free or simply supported. The predicted results by the present method, in general, compare favourably with the finite element results, except for those for square plates with large cutouts and mode shapes with nodal lines parallel to the longer sides of the cutouts. The predicted results for these cases can be improved by modifying the energy contribution from some of the sub-domains. The results can also be improved by using a RayleighRitz multiple terms approach. However, this will increase the computation time, which is not the intention of this work. In respect of the fundamental mode of a simply supported square plate with a central rectangular cutout, the results predicted by the present method differ only slightly from the finite element results although single term deflection functions are used. The present results also reflect the initial decreasing trend in frequency with increased cutout size and are dependent on the Poisson ratio of the plate. The method had also been extended to sheardeformable plate [25] and laminated plate [26].
REFERENCES
1. R. Ali and S. J. Atwal, Prediction of natural frequencies of vibration of rectangular plates with cutouts. Compur. Strucr. 12, 819-823 (1980). 2. B. P. Shastry and G. Venkateswara Rao, Vibrations of thin rectangular plates with arbitrary oriented stiffeners. Comput. Struct. 7, 627-629 (1977). 3. N. Reddy, Large amplitude flexural vibration of layered composite plates with cutouts. J. Sound Vibr. 83, l-10 (1982). 4. P. P. A. Laura, J. C. Utjes and V. H. Palluzzi, On the effect of free, rectangular cut-outs along the edge on the transverse vibrations of rectangular plates. Appl. Acousr. 19, 139-151 (1986).
869
9.
10.
11.
teristics of rectangular plates with cutouts using a finite difference formulation. J. Sound Vibr. 44. 147-158 (1976). N. L. Basdekas and M. Chi, Dynamic response of plates with cutouts. Shock Vibr. Bull. 29-35 (1970). A. Rajamani and R. Prabhakaran, Dynamic response of composite plates with cut-outs, part I: simplysupported plates. J. Sound. Vibr. 54, 549-564 (1977). A. Rajamani and R. Prabhakaran, Dynamic response of composite plates with cut-outs, part II: clamped plates. J. Sound Vibr. 54, 565-576 (1977). K. Nagaya, Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape, part I: derivation of frequency equation. J. Sound Vibr. 74, 543-551 (1981).
12. K. Nagaya, Simplified method of solving problems of vibrating plates of doubly connected arbitrary shape, part II: applications and experiments. J. Sound Vibr. 74, 553-564 (1981).
13. R. F. Hegarty and T. Ariman, Elasto-dynamic analysis of rectangular plates with circular holes. Inr. J. Solids Srruct. 11, 895-906 (1975). 14. T. Kumai, The flexural vibrations of a square plate with a central circular hole. In Proceedings of the 2nd Japan National Congress for Applied Mechanics, pp. 339-342 (1957). 15. C. Joga Rao and G. Picket& Vibration of plates with irregular shapes and plates with holes. J. aeronaut. Sot. India 13, 83-88 (1961).
16. S. Takahasi, Vibration of rectangular plates with circular holes. Bull. Jap. Sot. hfech. Engs. 1,380-385 (1958). 17. T. Mizusawa. T. Kaiita and M. Naruoka. Vibration and buckling analysis ofplates of abruptly varying stiffness. Comput. Struct. 12, 689-693
(1980).
18. P. P. A. Laura, P. Vemiere De Irassar, L. Ercoli and R. Gelos, Fundamental frequency of vibrations of a rectangular plate with a free, straight corner cut-out. J. Sound. Vibr. 78, 489-493
(1981).
19. G. Ganesan and S. Nagaraja Rao, Influence of location of cut-outs on natural frequencies of thin square plates. J. Sound. Vibr. 101, 598-600 (1985). 20. A. W. Leissa, Vibration of plates. NASA SP160 (1969). 21. D. Young, Vibration of rectangular plates by the Ritz method. J. appl. Mesh. 17, 448-453 (1950). 22. S. F. Bassily and S. M. Dickinson, On the use of beam functions for problems of plates involving free edges. J. aool. Mech. 42. 858-864 11975). 23. PAI%C Data Preparation, PAFEC. Ltd, Strelley Hall, Strelley, Nottingham, U.K. 24. PAFEC Theory, Results, PAFEC. Ltd, Strelley Hall, Strehey, Nottingham, U.K. 25. H. P. Lee, Vibration study of rectangular plates with cutouts. M. Engng thesis, National University of Singapore, Singapore (1986). 26. H. P. Lee, S. P. Lim and S. T. Chow, Free vibration of composite rectangular plates with rectangular cutouts. Camp. Strucr. 8, 63-81 (1987).