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Free vibration of square plates with square openings
Jottntal of Sound and Vibration (1973) 30(2), 173-178
FREE VIBRATION
OF SQUARE
PLATES WITH
SQUARE
OPENINGS
P. PARAMASIVAM
Department of Cicil Enghteering, Unicersity of Singapore, Shtgapore
(Receired 12 March 1973, attd ht revisedform 17 May 1973)
This paper describes a method of determining the effects of openings on the fundamental frequencies in plates with different types of boundary conditions. By extending the grid framework model, the finite difference operators can be developed for the re-entrant corner, junction point, etc., without involving fictitious points. Numerical examples are presented and the results are discussed in comparison with those of other methods.
1. INTRODUCTION Knowledge of the natural frequencies of square plates with square or rectangular openings is an important factor in the design of many types of aeronautical, mechanical and civil structures. The investigation described here was concerned with extending the discrete model analogue proposed by Witteveen [1] to include situations involving different types of boundary conditions. Previously, the experimental as well as theoretical values of fundamental frequencies of plates with square opening had not been available. The exact solution to the free vibration problem of such plates is difficult, except for the cases where circular openings are considered. The widely used Rayleigh-Ritz method is not suitable in this case as it is difficult to assume a reasonable initial wave form in the case of plates with openings. When there is a square opening, the problem becomes more complicated because of the presence of re-entrant corners. It is also not possible to use the finite difference method for re-entrant corners of the opening because it involves fictitious points which are difficult to determine. This can be easily overcome by the use of a physical model of the plate from which the same difference operators can be derived directly without involving fictitious points. The physical model used here was first suggested by Witteveen [1] for bending of plates of abruptly varying stiffnesses. However, the effect of lateral deformations was neglected. It has been found that, based on the same principles, one can develop mathematically consistent finite difference equations in which lateral deformation effects are taken into consideration. The problems of stability, vibration and bending of such plates have been solved by the author and Sridhar Rao [2, 3, 4]. In this paper, concerned with the free vibration problem, suitable operators are developed for the points on the re-entrant corners, junction points and points adjacent to the re-entrant corners of the openings, by using the grid framework model. Numerical examples are presented to illustrate the convergence and the effects of lateral deformations (10, and size of the openings, on the values of fundamental frequencies, higher order frequencies, etc. 173
174
P. P A R A M A S I V A M
2. ANALYSIS The governing differential equation for plate vibration problems can be Written as a 2 iv
DV 4 w = pd at 2,
(1)
where D is the flexural rigidity, w the deflection, p the mass density, d the thickness of the plate and t the time. In accord with the proposed method, the left-hand side of equation (1) is to be replaced by finite difference operators, by using the concept of the discrete physical model of Witteveen [1] and extending it to take care of lateral deformations also in the formulation of the model.
<
ac
I}
,,i(D
|
.......
|
-| (a)
=q
L
L
{b)
Figure I. Grid framework model for the (a) model at re-entrant corner, (b) problem.
The discrete model (Figure l(a)) consists of an orthogonal grid of flexurally stiff beams of zero torsional stiffness and rectangular torsional panels connected to the beams by hinges which transfer the reaction due to twisting to the beams. Since the beams are assumed to be torsionless, they are considered as two beams disposed freely side by side. Each is assumed to have a stiffness value equal to one-half of the plate stiffness of the adjacent panel of the plate. The connections and hinges are considered to undergo negative curvature (i.e., ll times) in perpendicular direction. The sum of the shear forces due to bending of beams and of the reactions due to torsional panels is equated to the inertia load in order to obtain the operator at the re-entrant corner o (see Figure l(a)). Then, these moments are related to curvatures by introducing the approximation that the curvature has a constant value over the adjacent beams and that the curvatures can be expressed in terms of displacements. The appropriate operators, which include the effects of lateral deformations, can be derived for re-entrant corner points, junction points, and points adjacent to the re-entrant corner by using the principles as described here, and from the author's previous work [2, 3, 4]. They are presented in Figure 2(a), (b) and (c), respectively. For a plate vibrating harmonically with an amplitude ~(x,y), then w(x,y, t) = ~(x,y) cos (cot),
(2)
where w is the angular frequency. For simplicity of notation, ff(x,y) is replaced by w(x,y). Then the right-hand side of equation (1) becomes p~o2dw(x,y). In dynamic problems, the inertia force can be expressed as pto2dw(x,) ,) by assuming that one-quarter of the plate
VIBRATION OF PLATES WITH OPENINGS
175
element is associated with each grid point. Then equation (1) is reduced to the following matrix form: [A] {w} = fl[B] {w),
(3)
where fl = pdm2h4/D. [A] is the matrix of the terms on the left-hand side of equation (1) and [B] is the diagonal mass matrix corresponding to the inertia load. Equation (3) is a standard eigenvalue problem and is modified suitably in order to apply the iterative technique as explained in reference [5] to obtain the minimum value o f ft. i t ~--/a~ t
(-4+3
2 + 5/.z2/2),,,~ \
2
(i/2_=2/2)/- 7
il
(- 4 + 2/z + 2~z)
(z-,,) _ ;-~+2~ I._
~2-#)
(1/2-~z/2)
'~
--
(-7§247
1
,~)
2
-I_
(8_ 4/z_3,,~21 ~
/-"
\
"
_1
(-4"F2/'z+2/"2)
(2-/.,)
'(,3-4/,-3J).
(-4 + 3/Z/2+ 5/~/2),~,
(112-/2/2)
c.l
2
-
t (a)(I/2-/2/21
-
(-4+2/z+2/'g)
( B_ 4/z _ i i/z ~./41: ~) (- 4+3/z/2 § 3~2/2)\ ~ " (/~/4) " "
-N--
(b)
_ (z-/,) -6+2~ I-
~
(2-3,./4) {~/2 ~1'/4)
(c)
Figure 2. Operators by grid framework analogue: (a) re-entrant corner; (b) junction point; (c) point adjacent to re-entrant corner. 3. ILLUSTRATIVE EXAMPLES The square plate with square opening (2L x 2L) at its centre as shown in Figure l(b) is considered for the application o f the method. Table 1 shows the convergence o f fundamental frequencies for the different grid sizes n = 8, 12 and 16 with 2 = 0.5, for simply supported and clamped boundaries. The values o f the fundamental frequencies for ll = 0 and 0.3 with 2 = 0.5 and n = 12 are presented in Table 2, and also the results are c o m p a r e d with results obtained by using the N e w m a r k plate analogue [6]. Table 3 and Figure 3 show the effect TABLE 1
Vahtes of fimdamental frequency for a square plate with square openhzg L/2 x L/2 Grid n xn
Fundamental frequency ^ Simply supported Clamper
8x8
25"45
12 • 12 16 • 16
26.05 26"38
57"25 ) 62"37 I 66-05
Multiplier 1
L--s~ p d
176
p. PARAMASIVAM
TABLE 2 Vahtes offim(lamental freqttenc), for a shnply supported square plate with square opening L/2 x L/2 for d(fferent values of Poisson's ratio (It) with n = 12
F u n d a m e n t a l frequency by A.
t
It 0'0 0"3
9
Grid framework model
Newmark plate analogue
26"05 24"25
25.85 22"98
Multiplier
}
1 __ -~ ~/D/pd
TABLE 3 Values offundamental frequencies for d(fferent sizes of square opening (2) with n = 12
Fundamental frequency for simply supported F u n d a m e n t a l frequency for clamped
70
i
,
0
L/6
L[3
19"63
19'48
21"45
26-05 ~
34-85
35"80
43"25
62"40 /
i
I
I
L/2
Multiplier 1 ~-i v/-~Pd
I
60
40
--
3s
g E ~c
20
, Simply supported plate with square opentng
IO
0
I 0'1
I 0 Z
1 0 3
I 0-4
l 0-5
Size of lhe opening (A)
Figure 3. Variation of fundamental frequency with size of the opening (2), for n = 12, It = 0. zx, Grid framework model; O, finite element method.
177
V I B R A T I O N OF PLATES W I T H O P E N I N G S
TABLE 4 Higher frequencies for a sqvare plate with square openhzg L/2 x L/2 Fundamental frequency A-...
Mode shape Simply supported t 2 3 4
25.45 42.15 70-50 78'65
Clampe~t 57-25 68.55 98-52 143-95
Multiplier /
i -~ V ~ p d
of the size of the opening (2) on the fundamental frequencies. The frequencies for assumed, corresponding higher modes are presented in Table 4.
4. DISCUSSIONS AND CONCLUSIONS The method used is an extension of the Witteveen grid framework model to vibration problems of plates with openings, in which effects of lateral deformations are also incorporated in the model. The main purpose of the investigation was to provide a general indication of the changes in fundamental frequencies that occur in plates when different sizes of square openings are introduced. It can be seen from Table 1 that the convergence for the clamped boundary condition is slower than that for the simply supported case. This behaviour of the rate of convergence is similar to that encountered in buckling and bending problems of plates without openings when finite difference methods or the grid framework model are used [2, 4]. Better results can be obtained by decreasing the grid size or by using Richardson's extrapolation technique. The error in the values of the fundamental frequencies obtained by finite difference methods is always on the safe side in comparison with those obtained by the Rayleigh-Ritz method. The results for square plates with circular openings, as obtained by Anderson et aL [7] by using finite element methods, are compared in Figure 3 with results obtained in this study for square openings, to establish trends and to verify the method of approach used herein, when different sizes of the openings are introduced. Table 2 shows that the values of fundamental frequencies obtained by using the grid framework model and the Newmark plate analogue [6], respectively, are very close to each other. The effect of Poisson's ratio (I0 is found to be about 7 to l0 ~ for ll = 0.3. The formulation is simpler and the number of equations handled are far less in comparison with the finite element method. However, this method is suitable only for square or rectangular openings and fails in the case of irregular boundaries. The grid framework model as developed in this study has an advantage over the Newmark plate analogue, in that the abrupt variation of stiffness in the plate can also be easily incorporated. ACKNOWLEDGMENT The author would like to thank Dr J. K. Sridhar Rao, Department of Civil Engineering, Indian Institute of Technology, Kanpur, for his valuable guidance. REFERENCES I. I.J. WIT'rEVEEN1966 Heron (English) 4, 1-20. The analysis of plates of abruptly varying thickness with the aid of the method of differences.
178
P. PARAMASIVAM
2. P. PARAMASWAMand J. K. SRIOHAR RAO 1969 Journal of the Strtwtures Division of the American Society of Civil Enghwers 95, 1314-1337. Buckling of plates of abruptly varying stiffnesses. 3. P. PARAMASlVAMand J. K. SRIDllAR R^O 1969 htternational Journal of Mechanical Science 11, 885-895. Free vibrations of rectangular plates of abruptly varying stiffnesses. 4. P. PARAMASIVAMand J. K. SRIDHAR RAO 1972 Journal of the Aeronautical Society of btdia 24, 383-389. Discrete flexural analyses of rectangular plates of abruptly varying stiffnesses. 5. S. H. CRANDALL 1956 Enghteerh~g Analysis. New York: McGraw-Hill Book Company, Inc. 6. W. E. FLUrtR, A. ANO and C. P. SEISS 1961 Strttctttral Research Series 228, Unirersity of Illinois. Theoretical analysis of effects of openings on the bending moments in square plates with fixed edges. 7. R. G. ANDERSON, B. M. IRONS and O. C. ZIENKIEXVICZ1968 htternationalJottrnal of Solids and Structures 4, 1031-1055. Vibration and stability of plates using finite element method.
FREE VIBRATION
OF SQUARE
PLATES WITH
SQUARE
OPENINGS
P. PARAMASIVAM
Department of Cicil Enghteering, Unicersity of Singapore, Shtgapore
(Receired 12 March 1973, attd ht revisedform 17 May 1973)
This paper describes a method of determining the effects of openings on the fundamental frequencies in plates with different types of boundary conditions. By extending the grid framework model, the finite difference operators can be developed for the re-entrant corner, junction point, etc., without involving fictitious points. Numerical examples are presented and the results are discussed in comparison with those of other methods.
1. INTRODUCTION Knowledge of the natural frequencies of square plates with square or rectangular openings is an important factor in the design of many types of aeronautical, mechanical and civil structures. The investigation described here was concerned with extending the discrete model analogue proposed by Witteveen [1] to include situations involving different types of boundary conditions. Previously, the experimental as well as theoretical values of fundamental frequencies of plates with square opening had not been available. The exact solution to the free vibration problem of such plates is difficult, except for the cases where circular openings are considered. The widely used Rayleigh-Ritz method is not suitable in this case as it is difficult to assume a reasonable initial wave form in the case of plates with openings. When there is a square opening, the problem becomes more complicated because of the presence of re-entrant corners. It is also not possible to use the finite difference method for re-entrant corners of the opening because it involves fictitious points which are difficult to determine. This can be easily overcome by the use of a physical model of the plate from which the same difference operators can be derived directly without involving fictitious points. The physical model used here was first suggested by Witteveen [1] for bending of plates of abruptly varying stiffnesses. However, the effect of lateral deformations was neglected. It has been found that, based on the same principles, one can develop mathematically consistent finite difference equations in which lateral deformation effects are taken into consideration. The problems of stability, vibration and bending of such plates have been solved by the author and Sridhar Rao [2, 3, 4]. In this paper, concerned with the free vibration problem, suitable operators are developed for the points on the re-entrant corners, junction points and points adjacent to the re-entrant corners of the openings, by using the grid framework model. Numerical examples are presented to illustrate the convergence and the effects of lateral deformations (10, and size of the openings, on the values of fundamental frequencies, higher order frequencies, etc. 173
174
P. P A R A M A S I V A M
2. ANALYSIS The governing differential equation for plate vibration problems can be Written as a 2 iv
DV 4 w = pd at 2,
(1)
where D is the flexural rigidity, w the deflection, p the mass density, d the thickness of the plate and t the time. In accord with the proposed method, the left-hand side of equation (1) is to be replaced by finite difference operators, by using the concept of the discrete physical model of Witteveen [1] and extending it to take care of lateral deformations also in the formulation of the model.
<
ac
I}
,,i(D
|
.......
|
-| (a)
=q
L
L
{b)
Figure I. Grid framework model for the (a) model at re-entrant corner, (b) problem.
The discrete model (Figure l(a)) consists of an orthogonal grid of flexurally stiff beams of zero torsional stiffness and rectangular torsional panels connected to the beams by hinges which transfer the reaction due to twisting to the beams. Since the beams are assumed to be torsionless, they are considered as two beams disposed freely side by side. Each is assumed to have a stiffness value equal to one-half of the plate stiffness of the adjacent panel of the plate. The connections and hinges are considered to undergo negative curvature (i.e., ll times) in perpendicular direction. The sum of the shear forces due to bending of beams and of the reactions due to torsional panels is equated to the inertia load in order to obtain the operator at the re-entrant corner o (see Figure l(a)). Then, these moments are related to curvatures by introducing the approximation that the curvature has a constant value over the adjacent beams and that the curvatures can be expressed in terms of displacements. The appropriate operators, which include the effects of lateral deformations, can be derived for re-entrant corner points, junction points, and points adjacent to the re-entrant corner by using the principles as described here, and from the author's previous work [2, 3, 4]. They are presented in Figure 2(a), (b) and (c), respectively. For a plate vibrating harmonically with an amplitude ~(x,y), then w(x,y, t) = ~(x,y) cos (cot),
(2)
where w is the angular frequency. For simplicity of notation, ff(x,y) is replaced by w(x,y). Then the right-hand side of equation (1) becomes p~o2dw(x,y). In dynamic problems, the inertia force can be expressed as pto2dw(x,) ,) by assuming that one-quarter of the plate
VIBRATION OF PLATES WITH OPENINGS
175
element is associated with each grid point. Then equation (1) is reduced to the following matrix form: [A] {w} = fl[B] {w),
(3)
where fl = pdm2h4/D. [A] is the matrix of the terms on the left-hand side of equation (1) and [B] is the diagonal mass matrix corresponding to the inertia load. Equation (3) is a standard eigenvalue problem and is modified suitably in order to apply the iterative technique as explained in reference [5] to obtain the minimum value o f ft. i t ~--/a~ t
(-4+3
2 + 5/.z2/2),,,~ \
2
(i/2_=2/2)/- 7
il
(- 4 + 2/z + 2~z)
(z-,,) _ ;-~+2~ I._
~2-#)
(1/2-~z/2)
'~
--
(-7§247
1
,~)
2
-I_
(8_ 4/z_3,,~21 ~
/-"
\
"
_1
(-4"F2/'z+2/"2)
(2-/.,)
'(,3-4/,-3J).
(-4 + 3/Z/2+ 5/~/2),~,
(112-/2/2)
c.l
2
-
t (a)(I/2-/2/21
-
(-4+2/z+2/'g)
( B_ 4/z _ i i/z ~./41: ~) (- 4+3/z/2 § 3~2/2)\ ~ " (/~/4) " "
-N--
(b)
_ (z-/,) -6+2~ I-
~
(2-3,./4) {~/2 ~1'/4)
(c)
Figure 2. Operators by grid framework analogue: (a) re-entrant corner; (b) junction point; (c) point adjacent to re-entrant corner. 3. ILLUSTRATIVE EXAMPLES The square plate with square opening (2L x 2L) at its centre as shown in Figure l(b) is considered for the application o f the method. Table 1 shows the convergence o f fundamental frequencies for the different grid sizes n = 8, 12 and 16 with 2 = 0.5, for simply supported and clamped boundaries. The values o f the fundamental frequencies for ll = 0 and 0.3 with 2 = 0.5 and n = 12 are presented in Table 2, and also the results are c o m p a r e d with results obtained by using the N e w m a r k plate analogue [6]. Table 3 and Figure 3 show the effect TABLE 1
Vahtes of fimdamental frequency for a square plate with square openhzg L/2 x L/2 Grid n xn
Fundamental frequency ^ Simply supported Clamper
8x8
25"45
12 • 12 16 • 16
26.05 26"38
57"25 ) 62"37 I 66-05
Multiplier 1
L--s~ p d
176
p. PARAMASIVAM
TABLE 2 Vahtes offim(lamental freqttenc), for a shnply supported square plate with square opening L/2 x L/2 for d(fferent values of Poisson's ratio (It) with n = 12
F u n d a m e n t a l frequency by A.
t
It 0'0 0"3
9
Grid framework model
Newmark plate analogue
26"05 24"25
25.85 22"98
Multiplier
}
1 __ -~ ~/D/pd
TABLE 3 Values offundamental frequencies for d(fferent sizes of square opening (2) with n = 12
Fundamental frequency for simply supported F u n d a m e n t a l frequency for clamped
70
i
,
0
L/6
L[3
19"63
19'48
21"45
26-05 ~
34-85
35"80
43"25
62"40 /
i
I
I
L/2
Multiplier 1 ~-i v/-~Pd
I
60
40
--
3s
g E ~c
20
, Simply supported plate with square opentng
IO
0
I 0'1
I 0 Z
1 0 3
I 0-4
l 0-5
Size of lhe opening (A)
Figure 3. Variation of fundamental frequency with size of the opening (2), for n = 12, It = 0. zx, Grid framework model; O, finite element method.
177
V I B R A T I O N OF PLATES W I T H O P E N I N G S
TABLE 4 Higher frequencies for a sqvare plate with square openhzg L/2 x L/2 Fundamental frequency A-...
Mode shape Simply supported t 2 3 4
25.45 42.15 70-50 78'65
Clampe~t 57-25 68.55 98-52 143-95
Multiplier /
i -~ V ~ p d
of the size of the opening (2) on the fundamental frequencies. The frequencies for assumed, corresponding higher modes are presented in Table 4.
4. DISCUSSIONS AND CONCLUSIONS The method used is an extension of the Witteveen grid framework model to vibration problems of plates with openings, in which effects of lateral deformations are also incorporated in the model. The main purpose of the investigation was to provide a general indication of the changes in fundamental frequencies that occur in plates when different sizes of square openings are introduced. It can be seen from Table 1 that the convergence for the clamped boundary condition is slower than that for the simply supported case. This behaviour of the rate of convergence is similar to that encountered in buckling and bending problems of plates without openings when finite difference methods or the grid framework model are used [2, 4]. Better results can be obtained by decreasing the grid size or by using Richardson's extrapolation technique. The error in the values of the fundamental frequencies obtained by finite difference methods is always on the safe side in comparison with those obtained by the Rayleigh-Ritz method. The results for square plates with circular openings, as obtained by Anderson et aL [7] by using finite element methods, are compared in Figure 3 with results obtained in this study for square openings, to establish trends and to verify the method of approach used herein, when different sizes of the openings are introduced. Table 2 shows that the values of fundamental frequencies obtained by using the grid framework model and the Newmark plate analogue [6], respectively, are very close to each other. The effect of Poisson's ratio (I0 is found to be about 7 to l0 ~ for ll = 0.3. The formulation is simpler and the number of equations handled are far less in comparison with the finite element method. However, this method is suitable only for square or rectangular openings and fails in the case of irregular boundaries. The grid framework model as developed in this study has an advantage over the Newmark plate analogue, in that the abrupt variation of stiffness in the plate can also be easily incorporated. ACKNOWLEDGMENT The author would like to thank Dr J. K. Sridhar Rao, Department of Civil Engineering, Indian Institute of Technology, Kanpur, for his valuable guidance. REFERENCES I. I.J. WIT'rEVEEN1966 Heron (English) 4, 1-20. The analysis of plates of abruptly varying thickness with the aid of the method of differences.
178
P. PARAMASIVAM
2. P. PARAMASWAMand J. K. SRIOHAR RAO 1969 Journal of the Strtwtures Division of the American Society of Civil Enghwers 95, 1314-1337. Buckling of plates of abruptly varying stiffnesses. 3. P. PARAMASlVAMand J. K. SRIDllAR R^O 1969 htternational Journal of Mechanical Science 11, 885-895. Free vibrations of rectangular plates of abruptly varying stiffnesses. 4. P. PARAMASIVAMand J. K. SRIDHAR RAO 1972 Journal of the Aeronautical Society of btdia 24, 383-389. Discrete flexural analyses of rectangular plates of abruptly varying stiffnesses. 5. S. H. CRANDALL 1956 Enghteerh~g Analysis. New York: McGraw-Hill Book Company, Inc. 6. W. E. FLUrtR, A. ANO and C. P. SEISS 1961 Strttctttral Research Series 228, Unirersity of Illinois. Theoretical analysis of effects of openings on the bending moments in square plates with fixed edges. 7. R. G. ANDERSON, B. M. IRONS and O. C. ZIENKIEXVICZ1968 htternationalJottrnal of Solids and Structures 4, 1031-1055. Vibration and stability of plates using finite element method.