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Vibration analysis of plates with cutouts by the modified Rayleigh-Ritz method
Applied Acoustics 28 (1989) 49-60
Vibration Analysis of Plates with Cutouts by the Modified Rayleigh-Ritz Method K. Y. L a m , K. C. H u n g & S. T. C h o w Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 051 ! (Received 19 December 1988; accepted 23 February 1989)
A BS TRA C T An efficient and accurate numerical method in the study of the vibration of rectangular plates with cutouts and non-homogeneity is presented. By dividing the problem domain into appropriate rectangular segments, the deflection function for the originally complex domain can easily befound. The method is simple and versatile by virtue of the use of a newly developed characteristic polynomial function in the Rayleigh-Ritz procedures. Numerical results obtainedfor isotropic and orthotropic plates are compared with the open literature and the agreement is found to be very good.
INTRODUCTION The Rayleigh-Ritz method is widely used in the study of the vibration of rectangular plates. The stability and accuracy of the method is dependent upon the deflection functions. The functions assumed for a continuous plate domain are usually the characteristic beam functions t'2 or the degenerate beam functions? Recently, Bhat 4'5 proposed the use of the orthogonally generated polynomial functions in the study of the vibration of rectangular plates. Modifications to the sets of orthogonal polynomial functions were presented by Dickinson & Blasio 6 in the vibration and buckling analysis of isotropic and orthotropic plates. Geannakakes 7 worked out a set of normalized characteristic orthogonal polynomials which can be used to analyse plates of various geometries defined by natural co-ordinates. However, no detailed results were published in his paper. 49 Applied Acoustics 0003-682X/89/$03.50 O 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
50
K. Y. Lam, K. C. Hung, S. T. Chow
For a rectangular plate with cutouts, which is commonly encountered in marine and aerospace structures, the normal Rayleigh-Ritz method will require a beam function that is continuous over the plate domain while satisfying the inner and external boundary requirements. No such function has been reported in the open literature, and the analysis of such problem using the Rayleigh-Ritz scheme will require some modifications to the numerical procedures. This paper describes an extension of the Rayleigh-Ritz method to vibration problems of a plate with cutouts and plates with internal non-homogeneity. By dividing the plate domain into smaller rectangular segments, appropriate deflection functions for each segment can easily be found. Continuity in the transverse displacement is enforced at evenly spaced discrete points along the interconnecting lines, and the coupled deflection functions of each segment are substituted into the energy functional to arrive at the eigenvalue equation. Flexural vibrations of isotropic and orthotropic plates with centrally located cutouts are analysed and the results verified by comparison with the reported numerical, analytical and experimental results, whenever possible. Isotropic rectangular plates with rectangular inhomogeneity have also been considered. METHOD OF ANALYSIS To demonstrate the numerical procedures, a rectangular plate with a centrally located rectangular cutout is considered. The geometry and dimensions of the plate are shown in Fig. 1. Taking the advantage of geometrical symmetry, vibration modes symmetric and antisymmetric about the axes can be evaluated by considering only the lower quadrant of
b/2 i
2
i i
Q
=
X
Fig. 1. Geometryand dimensionsof plate with centrally located rectangularcutout.
Vibration of plates with cutouts
51
the plate domain. The lower quadrant is further divided into smaller rectangular segments. Depending on the mode of vibration considered, the edges along the axes of symmetry, x = a/2 and y = b/2, will be subjected to different restraints. Solutions for the symmetric modes will be obtained from the condition that the slope and the shear force are zero and, for the antisymmetric modes, that the displacement and moment are zero. The deflection function assumed for each segment is in the following form: (e) W(x,y ) =
22 m
m~) ¢m(X)(e)¢n( y)(e)
(1)
n
where 4),. and ~h. are polynomial functions satisfying at least the geometric boundary conditions of the segment. The superscript (e) represents the segment number. The polynomial functions ~b,.(x) (and ~h.(y)) assumed in the deflection are the characteristic orthogonal polynomial functions generated using the Gram-Schmidt process. Starting with an initial polynomial ~bl(x) satisfying the geometric and natural boundary conditions of the equivalent beam function, the subsequent terms are generated using the recursive relationship, ~b2(x) = [ f ( x ) - B1]q~l(x) ~bk(x)= [f(x) -- Bk- a]~bk- l(x) - Ck- 2q~k-2(x),
(2) K> 2
where i 2 Bk-x = [~'0f(x)¢k - ,(x)dx]/[~'ox ~b~_x(x)dx]
and Ck - 2 = [I~ f ( x ) dpk_ x(x)dp k_ 2(x) dx]/[j'~ 4~_ ,(x) dx]
The starting polynomial ~bl(x) and the generating function f ( x ) corresponding to each boundary condition are given in Table 1. The deflection function of the adjacent segments will have one of the two polynomials, 45.(x) or 0.(y), in common. By enforcing continuity of transverse deflection at evenly spaced discrete points along the interconnecting boundaries, the undetermined coefficients I')Am., e = 1,2,3, of the deflection function for each segment will be coupled together. Taking segments 1 and 2 as an illustration, the following condition is imposed: ~l)W(x,y) = (2)W(x,y)
where
at evenly spaced discrete points P,, (3)
t = 1,2 ..... N
K. Y. Lam, K. C. Hung, S. T. Chow
52
TABLE 1 Starting Polynomial and the Corresponding Generating Function
Boundary conditions
Startingpolynomial, d~l(x)
Generating.function, fix)
x - 2x 3 "4-x a x 3x 2 - 5x 3 + 2x 4 x2 x 2 - 2x 3 + X 4 8x - 4x 3 + X 4 4x 2 - 4x 3 + x 4
x2 x x x
SS-SS SS-F C-SS C-F C-C SS-Sym C-Sym ~
X
2x + x 2 - 2x + x 2 --
a Sym denotes the boundary condition at the line of symmetry during the symmetric mode of vibration. U p o n s u b s t i t u t i o n o f t h e c o - o r d i n a t e v a l u e s o f P , i n t o e q n (3), t h e f o l l o w i n g m a t r i x e x p r e s s i o n is f o r m u l a t e d : [CONT(1)]~)A,,. = [CONT(2)]~2)A,.,
(4)
Mx
M2 where [CONT(e)] =
A)/,
_MN and
M, = {cpl(Xt)lp1(Yt)
dpl(Xt)~z(Yt)
P o s t - m u l t i p l y i n g the inverse of [ C O N T ( 2 ) ]
...
dpu(Xt)lpN(yt)}
o n e q n (4) g i v e s
[ C O N T ( 2 ) ] - 1[ C O N T ( 1 ) ] ~ t ) A , . , =
~2)Amn
or
~2)A,,. = L I N K ~ I ) A m .
(5)
2~1
where [ L I N K ] (~) = [ C O N T ( 2 ) ] - a [ C O N T ( 1 ) ] 2~1
T h e deflection f u n c t i o n for s e g m e n t 2 c a n n o w be expressed in the f o l l o w i n g form:
(2)W~x'r) = ~ ~ WI
n
[LINK] l
(6)
Vibration of plates with cutouts
53
Likewise. the deflection functions of other segments can also be expressed in terms of coefficients tl)A,..: ~)Wo,,y) -
~ ~ [LINK] ~-.1 {(X)A,.} m
(7)
/t
where [LINK]= [LINK] e'-* 1
x.-.
e-~e - 1
[LINK] x...[LINK]
e-j'-*e-j- 1
2-'* 1
The coupled deflection function for each segment is substituted into the strain and kinetic energy expression: U=~f[DllW~x2-4-O22W;2
T= phi2 2 ff
+ 2 D 1 2 W x x W , r-4-4966W~x2]dxdy
W 2 dx dy
(8)
(9)
where the integrations are to be carried out over the domain of the segment considered, and
Elh 3
E2 ha
D11 - 12(1 -v12v21 )'
D22
D12 = v21D22 ,
D66 =
12(1 -
v12v21) Gha/12
The total strain and kinetic energies of the entire domain are assumed to be the sum of the contributions from each segment. Minimizing the resulting functional (UTotaI - TTotal) with respect to the undetermined coefficients of segment 1, tl)A,.., leads to the governing eigenvalue equation:
~[Kmnij-~2Mmnij](1)Amn=O //I
(10)
?1
where ~ = (phoFa4/Dl 1)1/2 and K,..ij, M,..~¢ are the total stiffness and mass matrices of the plate considered. Solving eqn (10) yields the natural frequency of the plate at the specific mode of vibration considered. In the above expression,
Kmnij
=
C(1),.n~j+ [ L I N K ] w C(2)mnij [ L I N K ] + . . . 2-.1
2-'1
+ [ L I N K ] T C(e)m.,j[ L I N K ] + " " [ L I N K ] T C(n),.. o [ L I N K ] e~l
Mranij
e~l
__ F.(Hto,o)k-lH(o,o) - _,_,,.~ _ ,_,.~ + [LINK]
n~l
(11)
~-*1
T E(2),,, (0,0) F ( 2 ) .(0,0) j [LINK] 2--.I
....
+ [ L I N K ] "rE(e),.~F(e).~[ L I N K ] + ' - " + [ L I N K ] TE(n)m~F(n).j[ L I N K ] e'-*l
e-*l
n-*l
n--*l
(12)
K. Y. Lain, K. C. Hung, S. T. Chow
54
where e is the segment number and ~z is the total number of segments. C(e)mnij = E(e),.i (2,2) ~-~(0,0) ), gTt . , ~ ( 0 , 0 ) F(ej.j + /L.,22x.,~Zjm i
L-If.A(2,2 )
a ~¢lnj
L-y~'I( 2, O) K:'t~'L(O, 2 )) + 2D12{E(e),,,(0,2) i F(e),j(2,0) + "-,~¢J,,i ,~J,j s ,
71_
r (1,1) 4D66E(e),. i P~e).j(1,1)
(13)
and E ( e ) ~ ~1 =
f?
F(e)~}'~ =
(d'~bm/dx')(d~4~Jdx ~)dx ( d ' O d d J X d ~ J / j / d y ~) d y
D'12 = D12/D11,
D'22 = D22/D11,
D'66 = D 6 6 / D l l
NUMERICAL RESULTS AND DISCUSSION The numerical method developed is used to study the free vibration of rectangular plates with cutouts and internal non-homogeneities. For a simply supported square plate with a free square opening under the fundamental mode of vibration, the starting polynomials q~l(x) and ~kI(Y) for each segment, for an SS-SS-SS-SS square plate, are given in Table 2. TABLE Polynomial
Segment
Functions
2
for an SS-SS-SS-SS
4at(x)
,fix)
Square
$l(Y)
Plate
.tI y)
no.
1
8x -- 4x 3 + x 4
- - 2.~ -4- x 2
y
|'
E
8 x - - 4 x 3 --~ x 4
- 2 x -{- x 2
8].' - 41 '3 -;v 3 ,4
- - 2 y + ) ,2
3
x
x
83' - 43 '3 + ) ,4
- 23' + y 2
Convergence study The convergence of the fundamental mode for a square plate with a clamped and simply supported opening is shown in Table 3. Satisfactory convergence is achieved with 4 x 4 terms in the deflection function. The computed results are in good agreement with those obtained by Nagaya. 8
Free vibration of rectangular plates with cutouts To check the validity of the present method, the experimental results obtained by Aksu et aL 9 for an isotropic rectangular plate with single or double cutouts are used for comparison. The geometries and dimensions of
55
Vibration of plates with cutouts
TABLE 3 Fundamental Frequency Parameter Q(= [phtn2a4/D] 1/2) with Number of Terms in DeflectionFunction (Doubly Connected Isotropic Square Plate, Inner Opening= 0.2a) Convergence
of
Boundary condition
Number of terms 2x 2
3x 3
4x 4
5x 5
Nagaya 8
S ~
55"95
57.27
60"62
60'87
60-93
C ~
112"83
116-29 119"38 119"73 117"98
the plates studied, and the number of segments considered in the analysis, are shown in Fig. 2. The numerical results obtained are compared in Tables 4 and 5. The agreement is good in the case of single and double cutouts. It is noted that the present method yields a slightly higher frequency as compared to the experimental results. This may be due to the difficulty in imposing a perfectly clamped boundary condition in the experiment. The free vibration of orthotropic square plates with a center cutout has I_
a:gin
--I
-7
I--
t o=13in
b=7in
Fig. 2.
I
I a/5 I ~= a/5 =I Geometry and dimensions of plates with rectangular cutouts.
56
K. }'. Lam, K. C. Hung, S. T. Chow TABLE 4 Frequency Parameters of a C-SS-C-SS lsotropic Rectangular Plate with a Single Cutout (v = 0"3) Mode shape
1 2 3 4
f2
Frequency (Hz)
Aksu et al.
Present
Aksu 9
Present
E.vperiment 9
33"22 53'01 61.91 91.87
34.04 54.57 65'05 95'38
179.68 286"73 334.89 496"95
184'12 295"16 351.84 515"89
183-0 292'0 338'0 514.0
TABLE 5 Frequency Parameters of a C-SS-C-SS Isotropic Rectangular Plate with Double Cutouts (v = 0-3) Mode shape
1 2 3 4
f~
Frequency (Hz)
Aksu et al.
Present
Aksu 9
Present
Experiment 9
47-86 93-62 122'04 151"95
49"26 97"97 130-82 160'35
124-08 242"71 316"41 393"92
127"71 253-99 339"17 415"70
127 246 324 398
B0 70-
O~ tad t---
,.- 50
oo
20~° II0 , , , , =I 0
0.0
01
0.2
0.3
0./.,
0.5
0.5
0.7
0£~
C/A Fig. 3.
Fundamental frequency parameter for a simply supported orthotropic square plate ©, Rajamani; t° , present method.
57
Vibration o f plates with cutouts 270
250 230 7t0
190
o
o
150, 130 11o
390.o
Fig. 4.
Ot
o.2
o.3
O.t,
0.5
0.6
0.7
o.8
C/A Fundamental frequency parameter for a fully clamped orthotropic square plate. ©, Rajamani; 11 , present method.
also been studied. The material properties are given in Table 6. The effect of cutout size and modulus ratio upon the fundamental frequency are depicted in Figs 3 and 4. The curves obtained agree closely with the reported results of Rajamani & Prabhakaran ~°'11 for both simply supported and clamped boundary conditions.
Free vibration of plate with abrupt change in thickness The configuration of a square plate with an abrupt change in thickness is shown in Fig. 5. The lower quadrant of the plate is divided into four segments as shown. DEand D, are the flexural rigidities ofthe outer and inner domains respectively. The results for simply supported and clamped plates with various inner and outer side ratios, c/a, are presented in Tables 7 and 8. The results are in Material
TABLE 6 Properties of Typical Composites Material
Balanced bidirectional Glass-epoxy Boron-epoxy Graphite-epoxy
Unidirectional
E1/E2
G I 2/E2
vl2
1 3 10 40
0.200 0"500 0"333 0.500
0-10 0-25 0'30 0"25
TABLE 7 F u n d a m e n t a l Frequency Parameter for a Simply Supported Square Plate with D I = 4D, DII = D and v = 0.0;
f2 = [ph~o2a4/D] 1/2 Finite difference ~2
0"0 0-25 0"50 0'75 1.00
n=4
n=8
n=12
--23-39 -18'75
31'14 30"46 29-58 26-74 19"52
--29"80 -19-63
Negative stiffness 13
Present method
31"34 30"90 30"22 27'61 19'74
31"34 31-04 30-18 27"65 19"74
TABLE 8 F u n d a m e n t a l F r e q u e n c y P a r a m e t e r for a C l a m p e d Square Plate with D r = 4 D , D n = D, v = 0"0; D = [phto2a4/D] 1/2
Finite difference i z
0"0 0-25 0'50 0-75 1"00
n=4
n=8
-. . 46-20 -28"78
-. . 54-01 -33"63
Negative stiffness 13
Present method
n=12 -56-14 -34"85
57.07 57-28 58-50 52.11 35"98
57"15 57"46 58"40 51"22 35"99
TABLE 9 Material Properties for O r t h o t r o p i c Plates with Rectangular H o m o g e n e i t y
Subdomain
D11
D22
Dl 2
D66
P
DI Dn
1-432 1'00
1' 131 0"502
1"084 0"776
0"383 0-279
2"95 1.00
TABLE 10 F u n d a m e n t a l F r e q u e n c y P a r a m e t e r for Square Plate with Square N o n - h o m o g e n e i t y ; D = p2ho921a4/D~
Boundary condition
Side ratio, c/a
Laura TM
Present method
SS-SS-SS-SS
0"2 0"3 0"4
18"19 16-74 15'57
18"23 16-82 15"71
C-C-C-C
0"2 0"3 0-4
29.26 26.51 24.55
29'30 26"60 24"66
Vibration of plates with cutouts
59
11
C
I:i I
t
Fig. 5. Simply supported square plate with abrupt change in thickness.
good agreement with those obtained by finite difference formulation 12 and by the negative stiffness method) 3
Free vibration of orthotropic plates with rectangular non-homogeneity The problem of transverse vibration of orthotropic, non-homogeneous rectangular plates is first solved by Laura et al. 14 The geometry of the plates is depicted in Fig. 6. Following the conventions of Laura, D I and D n are subdomains of different orthotropic characteristics. The problem domain is again divided into four segments, in order to apply the present method. Simply supported and clamped square plate with the material properties listed in Table 9 are investigated. The fundamental frequency parameter computed is presented in Table 10, and is found to agree well with that calculated by Laura. I1
C
I-
ii!i 71
2
,I
Ilr
Fig. 6. Square plate with centrally located square inhomogeneity.
60
K.Y. Lam, K. C. Hung, S. T. Chow CONCLUSION
A modification of the Rayleigh-Ritz method to study the free vibration of rectangular plates with a cutout or other form o f inhomogeneity has been presented herein. The method is simple and versatile and, with the use of the orthogonal polynomial function generated via the G r a m - S c h m i d t process, a great variety o f plates problems can be readily analysed. Numerical calculations are performed for isotropic and orthotropic plates, and comparison with the results in the existing literature shows good agreement.
REFERENCES t. Leissa, A. W., Vibration of plates. NASA SPI60, 1969. 2. Young, D., Vibration of rectangular plates by the Ritz method. J. Appl. Mech., 17 (1950) 448-53. 3. Bassily, S. F. & Dickinson, S. M., On the use of beam functions for problems of plates involving free edges. J. Appl. Mech., Trans. A S M E , 42 (1975) 858-64. 4. Bhat, R. B., Vibration of structures using characteristic orthogonal polynomials in Rayleigh-Ritz method. Proceedings of the Tenth Canadian Congress of Applied Mechanics, London, Ontario, Canada, A129-A130, 1985. 5. Bhat, R. B., Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. J. Sound Vibr., 102 (1985) 493-99. 6. Dickinson, S. M. & Blasio, A. D., On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexurai vibration and buckling of isotropic and orthotropic rectangular plates. J. Sound Vibr., 108 (1988) 51-62. 7. Geannakakes, G. N., A theoretical formulation for the natural frequency, buckling, and deflection analyses of plates using natural co-ordinates and characteristic orthogonal polynomials. J. Sound Vibr., 124 (1988) 385-7. 8. Nagaya, K., Simplified method for solving problems of vibrating plates of double connected arbitrary shape, Part II: Application and experiments. J. Sound Vibr., 74 (1981) 553-64. 9. Aksu, G. & Ali, R., Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation. J. Sound Vibr., 44(1) (1976) 147-58. 10. Rajamani, A. & Prabhakaran, R., Dynamics response of composite plates with cutouts. Part I: Simply supported plates. J. Sound Vibr., 54(4) (1977) 549-64. 11. Rajamani, A. & Prabhakaran, R., Dynamics response of composite plates with cutouts. Part II: Clamped-clamped plates. J. Sound Vibr., 54(4) (1977) 565-76. 12. Paramasivam, P. & Rao, J. K. S., Free vibration of rectangular plates of abruptly varying stiffness. Int. J. Mech. Sci., 11 (1969) 885-95. 13. Tham, L. C., Chan, A. H. C. & Cheung, Y. K., Free vibration and buckling analysis of plates by the negative stiffness method. Comput. Struct., 22(4) (1986) 687-92. 14. Laura, P. A. A. & Gutierrez, R. H., Transverse vibrations of orthotropic, nonhomogeneous rectangular plates. Fiber Sci. Technol., 21 (1984) 125-33.
Vibration Analysis of Plates with Cutouts by the Modified Rayleigh-Ritz Method K. Y. L a m , K. C. H u n g & S. T. C h o w Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 051 ! (Received 19 December 1988; accepted 23 February 1989)
A BS TRA C T An efficient and accurate numerical method in the study of the vibration of rectangular plates with cutouts and non-homogeneity is presented. By dividing the problem domain into appropriate rectangular segments, the deflection function for the originally complex domain can easily befound. The method is simple and versatile by virtue of the use of a newly developed characteristic polynomial function in the Rayleigh-Ritz procedures. Numerical results obtainedfor isotropic and orthotropic plates are compared with the open literature and the agreement is found to be very good.
INTRODUCTION The Rayleigh-Ritz method is widely used in the study of the vibration of rectangular plates. The stability and accuracy of the method is dependent upon the deflection functions. The functions assumed for a continuous plate domain are usually the characteristic beam functions t'2 or the degenerate beam functions? Recently, Bhat 4'5 proposed the use of the orthogonally generated polynomial functions in the study of the vibration of rectangular plates. Modifications to the sets of orthogonal polynomial functions were presented by Dickinson & Blasio 6 in the vibration and buckling analysis of isotropic and orthotropic plates. Geannakakes 7 worked out a set of normalized characteristic orthogonal polynomials which can be used to analyse plates of various geometries defined by natural co-ordinates. However, no detailed results were published in his paper. 49 Applied Acoustics 0003-682X/89/$03.50 O 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
50
K. Y. Lam, K. C. Hung, S. T. Chow
For a rectangular plate with cutouts, which is commonly encountered in marine and aerospace structures, the normal Rayleigh-Ritz method will require a beam function that is continuous over the plate domain while satisfying the inner and external boundary requirements. No such function has been reported in the open literature, and the analysis of such problem using the Rayleigh-Ritz scheme will require some modifications to the numerical procedures. This paper describes an extension of the Rayleigh-Ritz method to vibration problems of a plate with cutouts and plates with internal non-homogeneity. By dividing the plate domain into smaller rectangular segments, appropriate deflection functions for each segment can easily be found. Continuity in the transverse displacement is enforced at evenly spaced discrete points along the interconnecting lines, and the coupled deflection functions of each segment are substituted into the energy functional to arrive at the eigenvalue equation. Flexural vibrations of isotropic and orthotropic plates with centrally located cutouts are analysed and the results verified by comparison with the reported numerical, analytical and experimental results, whenever possible. Isotropic rectangular plates with rectangular inhomogeneity have also been considered. METHOD OF ANALYSIS To demonstrate the numerical procedures, a rectangular plate with a centrally located rectangular cutout is considered. The geometry and dimensions of the plate are shown in Fig. 1. Taking the advantage of geometrical symmetry, vibration modes symmetric and antisymmetric about the axes can be evaluated by considering only the lower quadrant of
b/2 i
2
i i
Q
=
X
Fig. 1. Geometryand dimensionsof plate with centrally located rectangularcutout.
Vibration of plates with cutouts
51
the plate domain. The lower quadrant is further divided into smaller rectangular segments. Depending on the mode of vibration considered, the edges along the axes of symmetry, x = a/2 and y = b/2, will be subjected to different restraints. Solutions for the symmetric modes will be obtained from the condition that the slope and the shear force are zero and, for the antisymmetric modes, that the displacement and moment are zero. The deflection function assumed for each segment is in the following form: (e) W(x,y ) =
22 m
m~) ¢m(X)(e)¢n( y)(e)
(1)
n
where 4),. and ~h. are polynomial functions satisfying at least the geometric boundary conditions of the segment. The superscript (e) represents the segment number. The polynomial functions ~b,.(x) (and ~h.(y)) assumed in the deflection are the characteristic orthogonal polynomial functions generated using the Gram-Schmidt process. Starting with an initial polynomial ~bl(x) satisfying the geometric and natural boundary conditions of the equivalent beam function, the subsequent terms are generated using the recursive relationship, ~b2(x) = [ f ( x ) - B1]q~l(x) ~bk(x)= [f(x) -- Bk- a]~bk- l(x) - Ck- 2q~k-2(x),
(2) K> 2
where i 2 Bk-x = [~'0f(x)¢k - ,(x)dx]/[~'ox ~b~_x(x)dx]
and Ck - 2 = [I~ f ( x ) dpk_ x(x)dp k_ 2(x) dx]/[j'~ 4~_ ,(x) dx]
The starting polynomial ~bl(x) and the generating function f ( x ) corresponding to each boundary condition are given in Table 1. The deflection function of the adjacent segments will have one of the two polynomials, 45.(x) or 0.(y), in common. By enforcing continuity of transverse deflection at evenly spaced discrete points along the interconnecting boundaries, the undetermined coefficients I')Am., e = 1,2,3, of the deflection function for each segment will be coupled together. Taking segments 1 and 2 as an illustration, the following condition is imposed: ~l)W(x,y) = (2)W(x,y)
where
at evenly spaced discrete points P,, (3)
t = 1,2 ..... N
K. Y. Lam, K. C. Hung, S. T. Chow
52
TABLE 1 Starting Polynomial and the Corresponding Generating Function
Boundary conditions
Startingpolynomial, d~l(x)
Generating.function, fix)
x - 2x 3 "4-x a x 3x 2 - 5x 3 + 2x 4 x2 x 2 - 2x 3 + X 4 8x - 4x 3 + X 4 4x 2 - 4x 3 + x 4
x2 x x x
SS-SS SS-F C-SS C-F C-C SS-Sym C-Sym ~
X
2x + x 2 - 2x + x 2 --
a Sym denotes the boundary condition at the line of symmetry during the symmetric mode of vibration. U p o n s u b s t i t u t i o n o f t h e c o - o r d i n a t e v a l u e s o f P , i n t o e q n (3), t h e f o l l o w i n g m a t r i x e x p r e s s i o n is f o r m u l a t e d : [CONT(1)]~)A,,. = [CONT(2)]~2)A,.,
(4)
Mx
M2 where [CONT(e)] =
A)/,
_MN and
M, = {cpl(Xt)lp1(Yt)
dpl(Xt)~z(Yt)
P o s t - m u l t i p l y i n g the inverse of [ C O N T ( 2 ) ]
...
dpu(Xt)lpN(yt)}
o n e q n (4) g i v e s
[ C O N T ( 2 ) ] - 1[ C O N T ( 1 ) ] ~ t ) A , . , =
~2)Amn
or
~2)A,,. = L I N K ~ I ) A m .
(5)
2~1
where [ L I N K ] (~) = [ C O N T ( 2 ) ] - a [ C O N T ( 1 ) ] 2~1
T h e deflection f u n c t i o n for s e g m e n t 2 c a n n o w be expressed in the f o l l o w i n g form:
(2)W~x'r) = ~ ~ WI
n
[LINK] l
(6)
Vibration of plates with cutouts
53
Likewise. the deflection functions of other segments can also be expressed in terms of coefficients tl)A,..: ~)Wo,,y) -
~ ~ [LINK] ~-.1 {(X)A,.} m
(7)
/t
where [LINK]= [LINK] e'-* 1
x.-.
e-~e - 1
[LINK] x...[LINK]
e-j'-*e-j- 1
2-'* 1
The coupled deflection function for each segment is substituted into the strain and kinetic energy expression: U=~f[DllW~x2-4-O22W;2
T= phi2 2 ff
+ 2 D 1 2 W x x W , r-4-4966W~x2]dxdy
W 2 dx dy
(8)
(9)
where the integrations are to be carried out over the domain of the segment considered, and
Elh 3
E2 ha
D11 - 12(1 -v12v21 )'
D22
D12 = v21D22 ,
D66 =
12(1 -
v12v21) Gha/12
The total strain and kinetic energies of the entire domain are assumed to be the sum of the contributions from each segment. Minimizing the resulting functional (UTotaI - TTotal) with respect to the undetermined coefficients of segment 1, tl)A,.., leads to the governing eigenvalue equation:
~[Kmnij-~2Mmnij](1)Amn=O //I
(10)
?1
where ~ = (phoFa4/Dl 1)1/2 and K,..ij, M,..~¢ are the total stiffness and mass matrices of the plate considered. Solving eqn (10) yields the natural frequency of the plate at the specific mode of vibration considered. In the above expression,
Kmnij
=
C(1),.n~j+ [ L I N K ] w C(2)mnij [ L I N K ] + . . . 2-.1
2-'1
+ [ L I N K ] T C(e)m.,j[ L I N K ] + " " [ L I N K ] T C(n),.. o [ L I N K ] e~l
Mranij
e~l
__ F.(Hto,o)k-lH(o,o) - _,_,,.~ _ ,_,.~ + [LINK]
n~l
(11)
~-*1
T E(2),,, (0,0) F ( 2 ) .(0,0) j [LINK] 2--.I
....
+ [ L I N K ] "rE(e),.~F(e).~[ L I N K ] + ' - " + [ L I N K ] TE(n)m~F(n).j[ L I N K ] e'-*l
e-*l
n-*l
n--*l
(12)
K. Y. Lain, K. C. Hung, S. T. Chow
54
where e is the segment number and ~z is the total number of segments. C(e)mnij = E(e),.i (2,2) ~-~(0,0) ), gTt . , ~ ( 0 , 0 ) F(ej.j + /L.,22x.,~Zjm i
L-If.A(2,2 )
a ~¢lnj
L-y~'I( 2, O) K:'t~'L(O, 2 )) + 2D12{E(e),,,(0,2) i F(e),j(2,0) + "-,~¢J,,i ,~J,j s ,
71_
r (1,1) 4D66E(e),. i P~e).j(1,1)
(13)
and E ( e ) ~ ~1 =
f?
F(e)~}'~ =
(d'~bm/dx')(d~4~Jdx ~)dx ( d ' O d d J X d ~ J / j / d y ~) d y
D'12 = D12/D11,
D'22 = D22/D11,
D'66 = D 6 6 / D l l
NUMERICAL RESULTS AND DISCUSSION The numerical method developed is used to study the free vibration of rectangular plates with cutouts and internal non-homogeneities. For a simply supported square plate with a free square opening under the fundamental mode of vibration, the starting polynomials q~l(x) and ~kI(Y) for each segment, for an SS-SS-SS-SS square plate, are given in Table 2. TABLE Polynomial
Segment
Functions
2
for an SS-SS-SS-SS
4at(x)
,fix)
Square
$l(Y)
Plate
.tI y)
no.
1
8x -- 4x 3 + x 4
- - 2.~ -4- x 2
y
|'
E
8 x - - 4 x 3 --~ x 4
- 2 x -{- x 2
8].' - 41 '3 -;v 3 ,4
- - 2 y + ) ,2
3
x
x
83' - 43 '3 + ) ,4
- 23' + y 2
Convergence study The convergence of the fundamental mode for a square plate with a clamped and simply supported opening is shown in Table 3. Satisfactory convergence is achieved with 4 x 4 terms in the deflection function. The computed results are in good agreement with those obtained by Nagaya. 8
Free vibration of rectangular plates with cutouts To check the validity of the present method, the experimental results obtained by Aksu et aL 9 for an isotropic rectangular plate with single or double cutouts are used for comparison. The geometries and dimensions of
55
Vibration of plates with cutouts
TABLE 3 Fundamental Frequency Parameter Q(= [phtn2a4/D] 1/2) with Number of Terms in DeflectionFunction (Doubly Connected Isotropic Square Plate, Inner Opening= 0.2a) Convergence
of
Boundary condition
Number of terms 2x 2
3x 3
4x 4
5x 5
Nagaya 8
S ~
55"95
57.27
60"62
60'87
60-93
C ~
112"83
116-29 119"38 119"73 117"98
the plates studied, and the number of segments considered in the analysis, are shown in Fig. 2. The numerical results obtained are compared in Tables 4 and 5. The agreement is good in the case of single and double cutouts. It is noted that the present method yields a slightly higher frequency as compared to the experimental results. This may be due to the difficulty in imposing a perfectly clamped boundary condition in the experiment. The free vibration of orthotropic square plates with a center cutout has I_
a:gin
--I
-7
I--
t o=13in
b=7in
Fig. 2.
I
I a/5 I ~= a/5 =I Geometry and dimensions of plates with rectangular cutouts.
56
K. }'. Lam, K. C. Hung, S. T. Chow TABLE 4 Frequency Parameters of a C-SS-C-SS lsotropic Rectangular Plate with a Single Cutout (v = 0"3) Mode shape
1 2 3 4
f2
Frequency (Hz)
Aksu et al.
Present
Aksu 9
Present
E.vperiment 9
33"22 53'01 61.91 91.87
34.04 54.57 65'05 95'38
179.68 286"73 334.89 496"95
184'12 295"16 351.84 515"89
183-0 292'0 338'0 514.0
TABLE 5 Frequency Parameters of a C-SS-C-SS Isotropic Rectangular Plate with Double Cutouts (v = 0-3) Mode shape
1 2 3 4
f~
Frequency (Hz)
Aksu et al.
Present
Aksu 9
Present
Experiment 9
47-86 93-62 122'04 151"95
49"26 97"97 130-82 160'35
124-08 242"71 316"41 393"92
127"71 253-99 339"17 415"70
127 246 324 398
B0 70-
O~ tad t---
,.- 50
oo
20~° II0 , , , , =I 0
0.0
01
0.2
0.3
0./.,
0.5
0.5
0.7
0£~
C/A Fig. 3.
Fundamental frequency parameter for a simply supported orthotropic square plate ©, Rajamani; t° , present method.
57
Vibration o f plates with cutouts 270
250 230 7t0
190
o
o
150, 130 11o
390.o
Fig. 4.
Ot
o.2
o.3
O.t,
0.5
0.6
0.7
o.8
C/A Fundamental frequency parameter for a fully clamped orthotropic square plate. ©, Rajamani; 11 , present method.
also been studied. The material properties are given in Table 6. The effect of cutout size and modulus ratio upon the fundamental frequency are depicted in Figs 3 and 4. The curves obtained agree closely with the reported results of Rajamani & Prabhakaran ~°'11 for both simply supported and clamped boundary conditions.
Free vibration of plate with abrupt change in thickness The configuration of a square plate with an abrupt change in thickness is shown in Fig. 5. The lower quadrant of the plate is divided into four segments as shown. DEand D, are the flexural rigidities ofthe outer and inner domains respectively. The results for simply supported and clamped plates with various inner and outer side ratios, c/a, are presented in Tables 7 and 8. The results are in Material
TABLE 6 Properties of Typical Composites Material
Balanced bidirectional Glass-epoxy Boron-epoxy Graphite-epoxy
Unidirectional
E1/E2
G I 2/E2
vl2
1 3 10 40
0.200 0"500 0"333 0.500
0-10 0-25 0'30 0"25
TABLE 7 F u n d a m e n t a l Frequency Parameter for a Simply Supported Square Plate with D I = 4D, DII = D and v = 0.0;
f2 = [ph~o2a4/D] 1/2 Finite difference ~2
0"0 0-25 0"50 0'75 1.00
n=4
n=8
n=12
--23-39 -18'75
31'14 30"46 29-58 26-74 19"52
--29"80 -19-63
Negative stiffness 13
Present method
31"34 30"90 30"22 27'61 19'74
31"34 31-04 30-18 27"65 19"74
TABLE 8 F u n d a m e n t a l F r e q u e n c y P a r a m e t e r for a C l a m p e d Square Plate with D r = 4 D , D n = D, v = 0"0; D = [phto2a4/D] 1/2
Finite difference i z
0"0 0-25 0'50 0-75 1"00
n=4
n=8
-. . 46-20 -28"78
-. . 54-01 -33"63
Negative stiffness 13
Present method
n=12 -56-14 -34"85
57.07 57-28 58-50 52.11 35"98
57"15 57"46 58"40 51"22 35"99
TABLE 9 Material Properties for O r t h o t r o p i c Plates with Rectangular H o m o g e n e i t y
Subdomain
D11
D22
Dl 2
D66
P
DI Dn
1-432 1'00
1' 131 0"502
1"084 0"776
0"383 0-279
2"95 1.00
TABLE 10 F u n d a m e n t a l F r e q u e n c y P a r a m e t e r for Square Plate with Square N o n - h o m o g e n e i t y ; D = p2ho921a4/D~
Boundary condition
Side ratio, c/a
Laura TM
Present method
SS-SS-SS-SS
0"2 0"3 0"4
18"19 16-74 15'57
18"23 16-82 15"71
C-C-C-C
0"2 0"3 0-4
29.26 26.51 24.55
29'30 26"60 24"66
Vibration of plates with cutouts
59
11
C
I:i I
t
Fig. 5. Simply supported square plate with abrupt change in thickness.
good agreement with those obtained by finite difference formulation 12 and by the negative stiffness method) 3
Free vibration of orthotropic plates with rectangular non-homogeneity The problem of transverse vibration of orthotropic, non-homogeneous rectangular plates is first solved by Laura et al. 14 The geometry of the plates is depicted in Fig. 6. Following the conventions of Laura, D I and D n are subdomains of different orthotropic characteristics. The problem domain is again divided into four segments, in order to apply the present method. Simply supported and clamped square plate with the material properties listed in Table 9 are investigated. The fundamental frequency parameter computed is presented in Table 10, and is found to agree well with that calculated by Laura. I1
C
I-
ii!i 71
2
,I
Ilr
Fig. 6. Square plate with centrally located square inhomogeneity.
60
K.Y. Lam, K. C. Hung, S. T. Chow CONCLUSION
A modification of the Rayleigh-Ritz method to study the free vibration of rectangular plates with a cutout or other form o f inhomogeneity has been presented herein. The method is simple and versatile and, with the use of the orthogonal polynomial function generated via the G r a m - S c h m i d t process, a great variety o f plates problems can be readily analysed. Numerical calculations are performed for isotropic and orthotropic plates, and comparison with the results in the existing literature shows good agreement.
REFERENCES t. Leissa, A. W., Vibration of plates. NASA SPI60, 1969. 2. Young, D., Vibration of rectangular plates by the Ritz method. J. Appl. Mech., 17 (1950) 448-53. 3. Bassily, S. F. & Dickinson, S. M., On the use of beam functions for problems of plates involving free edges. J. Appl. Mech., Trans. A S M E , 42 (1975) 858-64. 4. Bhat, R. B., Vibration of structures using characteristic orthogonal polynomials in Rayleigh-Ritz method. Proceedings of the Tenth Canadian Congress of Applied Mechanics, London, Ontario, Canada, A129-A130, 1985. 5. Bhat, R. B., Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. J. Sound Vibr., 102 (1985) 493-99. 6. Dickinson, S. M. & Blasio, A. D., On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexurai vibration and buckling of isotropic and orthotropic rectangular plates. J. Sound Vibr., 108 (1988) 51-62. 7. Geannakakes, G. N., A theoretical formulation for the natural frequency, buckling, and deflection analyses of plates using natural co-ordinates and characteristic orthogonal polynomials. J. Sound Vibr., 124 (1988) 385-7. 8. Nagaya, K., Simplified method for solving problems of vibrating plates of double connected arbitrary shape, Part II: Application and experiments. J. Sound Vibr., 74 (1981) 553-64. 9. Aksu, G. & Ali, R., Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation. J. Sound Vibr., 44(1) (1976) 147-58. 10. Rajamani, A. & Prabhakaran, R., Dynamics response of composite plates with cutouts. Part I: Simply supported plates. J. Sound Vibr., 54(4) (1977) 549-64. 11. Rajamani, A. & Prabhakaran, R., Dynamics response of composite plates with cutouts. Part II: Clamped-clamped plates. J. Sound Vibr., 54(4) (1977) 565-76. 12. Paramasivam, P. & Rao, J. K. S., Free vibration of rectangular plates of abruptly varying stiffness. Int. J. Mech. Sci., 11 (1969) 885-95. 13. Tham, L. C., Chan, A. H. C. & Cheung, Y. K., Free vibration and buckling analysis of plates by the negative stiffness method. Comput. Struct., 22(4) (1986) 687-92. 14. Laura, P. A. A. & Gutierrez, R. H., Transverse vibrations of orthotropic, nonhomogeneous rectangular plates. Fiber Sci. Technol., 21 (1984) 125-33.