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A note on transverse vibrations of a rectangular plate with a free, rectangular, corner cut-out
Journal
qf Sound and Vibration
A NOTE
(1986) W(2)
ON
RECTANGULAR
187-192
TRANSVERSE PLATE
WITH
CORNER
VIBRATIONS A FREE,
OF A
RECTANGULAR,
CUT-OUT
P. A. A. LAURA Institute
of Applied Mechanics,
Puerto Belgrano
Naval
Base, 8111 -Argentina
AND
P. A. LAURA AND V. H. CORTINEZ Universidad
(Received
National
14 July
de1 Sur, Bahia Blanca,
Argentina
and in revised form 9 April 1985)
1984,
An approximate solution lo the title problem is presented, obtained by using the Rayleigh-Ritz method. The analysis is presented for the case of simply supported and clamped plates. For the case of a rigidly clamped plate results are presented of numerical experiments on minimizing the calculated value of the fundamental frequency coefficient by using Schmidt’s approach. An experimental investigation is described on a clamped square plate with a free square, comer cut-out, which has led to the conclusion that the fundamental frequency coefficient remains practically invariant with respect to size when compared with the frequency coefficient of the fully clamped plate. A similar conclusion is arrived at by means of the mathematical model. The problem under consideration is important from a practical viewpoint since cut-outs of the type considered here are quite common in engineering practice.
1. INTRODUCTION Plates with cut-outs constitute quite common realistic structural elements in a wide variety of engineering systems; from aerospace, civil and naval configurations to electronic packages, nuclear reactor components, etc., and applied mechanicists have dedicated a very substantial portion of specialized literature to their static and dynamic analysis?. Recent contributions on the subject matter can be found in references [4-61. Treatments of corner edge cut-outs, on the other hand, are rather scarce in the open literature even though this is a very common engineering situation. Exceptions to this are the treatments of straight corner cut-outs published rather recently [7,8]. The present paper deals with rectangular plates with free, rectangular corner, cut-outs (see Figure l), and with simply supported and clamped edges. An experimental investigation has also been performed on a clamped, square, plate.
2. APPROXIMATE Let the fundamental supported
or clamped
mode
of vibration
edges
be represented Wr,(x,
t No attempt
SOLUTION
to review the literature
OF THE PROBLEM
of the “complete”, by a functional
rectangular relation
plate
of simply
of the type (1)
Y) = A,,.!-(x)g(y),
will be performed
here since thorough
surveys
are available
[l-d].
187 0022-460X/86/080187+06
$03.00/O
@ 1986 Academic Press Inc. (London)
Limited
188
P. A. A. LAURA,
Figure
P. A. LAURA
1. Rectangular
AND V. H. CORTINEZ
plate with a rectangular
free, corner
cut-out.
where it is assumed that W,,(x, y) may be either the exact or a valid approximation the mode shape of the “complete” plate R, (see Figure 1). The Rayleigh-Ritz method requires that the maximum strain energy
a2w,,a2w,, 2
-+-
ax2
be equal to the maximum
kinetic
energy
ay2
of the structural
T:g, = 4 phw’
>
to
(2)
dx dy
system
W:, dx dy. Rl
Once and (3) original, mation,
the “cut-out” process has taken place one must subtract from expressions (2) the corresponding energies of the sub-plate R, which has been cut from the “complete” structural element (see Figure 1). Accordingly, and as a first approxione may write
a2w,* a2w,, 2
-+-
ax2
ay2 >
dx dx
-2(1-/J)
(4)
Tit&= $phm2 Consequently,
and by the principle
W:, dx dy.
(5)
R2
of conservation
of energy,
one has
(6) and the approximate value of the fundamental frequency coefficient m w,,u2 can then be obtained. Two types of boundary conditions of the original plate two cases are to be considered here I, simply supported, and II, clamped, edges. 2.1. SIMPLY SUPPORTED Equation
(1) is taken
EDGES
in the form W,,(x, y) =A,,
sin (TX/~)
sin (ry/b).
(7)
RECTANGULAR
Substituting fashion,
2.2.
RIGIDLY
expression
PLATE WITH CORNER
(7) in expressions
CLAMPED
(2)-(6)
one finally obtains,
W,,,(x, where in order to satisfy 2-Y
ff4=y_3a
Substituting
in a straightforward
EDGES
Since the exact, fundamental mode shape for the complete extremely complicated functional relation [l] it is considerably an approximate expression of the form
1
189
CUT-OUT
Y) = A,,(~,xY+~(y3x3+x2)(P4yY+P3y3+yZ), identically a3=--a
’
expression
plate is, in this case, an more convenient to use
the boundary y-2 y-3
-1 ’
(9) in expressions
&,=fi,,(?)=[{U,-
(9)
conditions
P4=- ,l,b-
(2)-(6)
results
27
I
1 y-2. P3=-;v_3.
in an expression
u,+2(I-~)u,}/{T,-T,}]“2,
of the type (10)
where U,, U,, U,, Tl and T2 are functions of y, a/b, al/a and b,/ b. The parameter in equation (9) is determined in such a manner as to minimize expression (10) [8,9]:
an,,/ay=o. Since the Rayleigh-Ritz expression (11) optimizes,
y
(11)
method provides an upper bound to the exact eigenvalue, in general, the value of the fundamental frequency coefficient.
3. NUMERICAL
RESULTS
Figure 2 depicts the variation of the fundamental frequency coefficients of simply supported and clamped square plates? with square, corner cut-outs (a = b; a, = b,). The scale of the frequency ordinate has been considerably exaggerated in order to obtain a better appreciation of its variation but from a practical viewpoint one can conclude that a,, remain practically invariant as the cut-out dimensions increase. It is observed that for both types of boundary conditions the frequency coefficient tends to decrease because the loss of rigidity of the edge “boundary layer” is considerably more important than the loss of plate mass due to the presence of the cut-out. Some numerical experiments were performed in order to test Schmidt’s optimization procedure [8] in the case of the clamped square plate. It was found that for small cut-outs the minimum value of ai, was attained for y = 4 while for al/a = b,/ b = 0.40 a value of 3.50 allowed for a minimized value of ai, which was 0.4% lower than the value shown in Figure 2 (a,, = 34.915). i The lower graph of Figure 2 has been obtained by making y = 4 in equation (9).
190
P. A. A. LAURA, I
I
I
I
P. A. LAURA
AND V. H. CORTINEZ
I
I
(0) 36 -
35
0
I
I
1
I
(b)
I 0.1
I 0.2
1 0.3
I 0.4
I 0.5
3:& o
?? -0
0
1
-
0
I 0.1
I 0.2
I 0.3
I 0.4
I 0.5
b
Figure 2. Variation of the fundamental frequency coefficient Simply supported square plate; (b) clamped square plate.
as a function
of the cut-out
dimensions.
(a)
4. EXPERIMENTAL DETERMINATIONS The experimental set-up and procedure used in the investigation described in the present paper are similar to those described in reference [7] (see Figure 3, where the plate configuration resulting when the cut-out is q/a = b,/ b = 0.50, is shown).
Figure
3. Experimental
set-up.
RECTANGULAR
PLATE
WITH
CORNER
191
CUT-OUT
Experiments were performed on a stainless steel, clamped square plate with the following geometrical and mechanical parameters: a = 19.2 cm; a* = 20.7 cm; h = 0.3 cm; p = 7.84 g/cm”, E = 19.6 x 10” (g cm/s2)/cm2, I_L= 0.28. The dimension a is the distance between inner edges of the clamp while the parameter a * is the value of the side of the square plate’with the “root effect” taken into account; it is obtained once the fundamental frequency of the complete square plate have been measured [7] and with the use of the expression for the fundamental frequency coefficient.
I
1
0.1
I
0.2
1
II
0.4
0.5
I
0.3 0, 0
Figure 4. Comparison
Figure 4 quite good validity of applied to
of experimental
(0) and analytical
(-_)
results:
case of a clamped
square
plate
shows a comparison of analytical and experimental results. The agreement is from an engineering viewpoint since the experimental results confirm the the simple mathematical model proposed in the present investigation and a particular configuration (square domain with a corner square cut-out).
5. CONCLUSIONS A simple analytical solution has been proposed in order to predict some of the vibrational characteristics of a rather complex and very realistic problem of structural dynamics. Experimental and analytical values are in very good agreement for the particular plate-“cut-out” configuration chosen, and investigated in some detail, and the results lead to the interesting conclusion that in the case of a clamped square plate with a square corner cut-out the frequency coefficient remains practically invariant with respect to size: in other words, the decrease of mass and stiffness are approximately the same on a percentage basis.
ACKNOWLEDGMENTS The authors are indebted to research engineers J. L. Pombo, G. Ficcadenti, V. H. Palluzzi and P. Verniere for their valuable aid in performing some of the preliminary calculations and during the experimental stage of the project.
192
P. A.A.LAURA,P.A.LAURAANDV.H.CORTINEZ
The present investigation has been partially Cientificas (Buenos Aires Province).
sponsored
by Comisi6n
de Investigaciones
REFERENCES 1. A. W. LEISSA 1969 NASA SP 160. Vibration of plates. 2. A. W. LEISSA 1977 The Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations: classical theory. 3. A. W. LEISSA 1978 The Shock and Vibration Digest 10,21-35. Recent research in plate vibrations 1973-1976: Complicating effects. 4. F. E. EASTEP and F. G. HEMMIG 1978 Journal ofSound and Vibration 56, 155-165. Estimation of fundamental frequency of non-circular plates with free, circular cutouts. 5. K. NAGAYA 1981 Journal of Sound and Vibration 74, 543-551. Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape, Part I: Derivation of the frequency equation. 6. K. NAGAYA 1981 Journal ofSound and Vibration 74, 553-564. Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape, Part II: Application and experiments. 7. P.A.A. LAURA,P.VERNIERE DE IRASSAR,L.ERCOLI~~~R. G~~0s1981 JournalofSound and Vibration 78, 489-493. Fundamental frequency of vibrations of a rectangular plate with a free, straight corner cut-out. 8. P. A. A. LAURA, P. L. VERNIERE DE IRASSAR and G. M. FICCADENTI 1982 Journalofthe Acoustical Society of America 71, 501-502. Vibrations of a clamped, rectangular plate of generalized orthotropy with a free, straight corner cut-out. 9. R. SCHMIDT 1982 Journal of Applied Mechanics 49,639-640. Estimation of buckling loads and other eigenvalues via a modification of the Rayleigh-Ritz method. 10. L. ERCOLI, L. C. NAVA and P. A. A. LAURA 1984 Institute of Applied Mechanics Publication No 84-25. Vibrations of plates of arbitrary shape by a modified Galerkin method.
qf Sound and Vibration
A NOTE
(1986) W(2)
ON
RECTANGULAR
187-192
TRANSVERSE PLATE
WITH
CORNER
VIBRATIONS A FREE,
OF A
RECTANGULAR,
CUT-OUT
P. A. A. LAURA Institute
of Applied Mechanics,
Puerto Belgrano
Naval
Base, 8111 -Argentina
AND
P. A. LAURA AND V. H. CORTINEZ Universidad
(Received
National
14 July
de1 Sur, Bahia Blanca,
Argentina
and in revised form 9 April 1985)
1984,
An approximate solution lo the title problem is presented, obtained by using the Rayleigh-Ritz method. The analysis is presented for the case of simply supported and clamped plates. For the case of a rigidly clamped plate results are presented of numerical experiments on minimizing the calculated value of the fundamental frequency coefficient by using Schmidt’s approach. An experimental investigation is described on a clamped square plate with a free square, comer cut-out, which has led to the conclusion that the fundamental frequency coefficient remains practically invariant with respect to size when compared with the frequency coefficient of the fully clamped plate. A similar conclusion is arrived at by means of the mathematical model. The problem under consideration is important from a practical viewpoint since cut-outs of the type considered here are quite common in engineering practice.
1. INTRODUCTION Plates with cut-outs constitute quite common realistic structural elements in a wide variety of engineering systems; from aerospace, civil and naval configurations to electronic packages, nuclear reactor components, etc., and applied mechanicists have dedicated a very substantial portion of specialized literature to their static and dynamic analysis?. Recent contributions on the subject matter can be found in references [4-61. Treatments of corner edge cut-outs, on the other hand, are rather scarce in the open literature even though this is a very common engineering situation. Exceptions to this are the treatments of straight corner cut-outs published rather recently [7,8]. The present paper deals with rectangular plates with free, rectangular corner, cut-outs (see Figure l), and with simply supported and clamped edges. An experimental investigation has also been performed on a clamped, square, plate.
2. APPROXIMATE Let the fundamental supported
or clamped
mode
of vibration
edges
be represented Wr,(x,
t No attempt
SOLUTION
to review the literature
OF THE PROBLEM
of the “complete”, by a functional
rectangular relation
plate
of simply
of the type (1)
Y) = A,,.!-(x)g(y),
will be performed
here since thorough
surveys
are available
[l-d].
187 0022-460X/86/080187+06
$03.00/O
@ 1986 Academic Press Inc. (London)
Limited
188
P. A. A. LAURA,
Figure
P. A. LAURA
1. Rectangular
AND V. H. CORTINEZ
plate with a rectangular
free, corner
cut-out.
where it is assumed that W,,(x, y) may be either the exact or a valid approximation the mode shape of the “complete” plate R, (see Figure 1). The Rayleigh-Ritz method requires that the maximum strain energy
a2w,,a2w,, 2
-+-
ax2
be equal to the maximum
kinetic
energy
ay2
of the structural
T:g, = 4 phw’
>
to
(2)
dx dy
system
W:, dx dy. Rl
Once and (3) original, mation,
the “cut-out” process has taken place one must subtract from expressions (2) the corresponding energies of the sub-plate R, which has been cut from the “complete” structural element (see Figure 1). Accordingly, and as a first approxione may write
a2w,* a2w,, 2
-+-
ax2
ay2 >
dx dx
-2(1-/J)
(4)
Tit&= $phm2 Consequently,
and by the principle
W:, dx dy.
(5)
R2
of conservation
of energy,
one has
(6) and the approximate value of the fundamental frequency coefficient m w,,u2 can then be obtained. Two types of boundary conditions of the original plate two cases are to be considered here I, simply supported, and II, clamped, edges. 2.1. SIMPLY SUPPORTED Equation
(1) is taken
EDGES
in the form W,,(x, y) =A,,
sin (TX/~)
sin (ry/b).
(7)
RECTANGULAR
Substituting fashion,
2.2.
RIGIDLY
expression
PLATE WITH CORNER
(7) in expressions
CLAMPED
(2)-(6)
one finally obtains,
W,,,(x, where in order to satisfy 2-Y
ff4=y_3a
Substituting
in a straightforward
EDGES
Since the exact, fundamental mode shape for the complete extremely complicated functional relation [l] it is considerably an approximate expression of the form
1
189
CUT-OUT
Y) = A,,(~,xY+~(y3x3+x2)(P4yY+P3y3+yZ), identically a3=--a
’
expression
plate is, in this case, an more convenient to use
the boundary y-2 y-3
-1 ’
(9) in expressions
&,=fi,,(?)=[{U,-
(9)
conditions
P4=- ,l,b-
(2)-(6)
results
27
I
1 y-2. P3=-;v_3.
in an expression
u,+2(I-~)u,}/{T,-T,}]“2,
of the type (10)
where U,, U,, U,, Tl and T2 are functions of y, a/b, al/a and b,/ b. The parameter in equation (9) is determined in such a manner as to minimize expression (10) [8,9]:
an,,/ay=o. Since the Rayleigh-Ritz expression (11) optimizes,
y
(11)
method provides an upper bound to the exact eigenvalue, in general, the value of the fundamental frequency coefficient.
3. NUMERICAL
RESULTS
Figure 2 depicts the variation of the fundamental frequency coefficients of simply supported and clamped square plates? with square, corner cut-outs (a = b; a, = b,). The scale of the frequency ordinate has been considerably exaggerated in order to obtain a better appreciation of its variation but from a practical viewpoint one can conclude that a,, remain practically invariant as the cut-out dimensions increase. It is observed that for both types of boundary conditions the frequency coefficient tends to decrease because the loss of rigidity of the edge “boundary layer” is considerably more important than the loss of plate mass due to the presence of the cut-out. Some numerical experiments were performed in order to test Schmidt’s optimization procedure [8] in the case of the clamped square plate. It was found that for small cut-outs the minimum value of ai, was attained for y = 4 while for al/a = b,/ b = 0.40 a value of 3.50 allowed for a minimized value of ai, which was 0.4% lower than the value shown in Figure 2 (a,, = 34.915). i The lower graph of Figure 2 has been obtained by making y = 4 in equation (9).
190
P. A. A. LAURA, I
I
I
I
P. A. LAURA
AND V. H. CORTINEZ
I
I
(0) 36 -
35
0
I
I
1
I
(b)
I 0.1
I 0.2
1 0.3
I 0.4
I 0.5
3:& o
?? -0
0
1
-
0
I 0.1
I 0.2
I 0.3
I 0.4
I 0.5
b
Figure 2. Variation of the fundamental frequency coefficient Simply supported square plate; (b) clamped square plate.
as a function
of the cut-out
dimensions.
(a)
4. EXPERIMENTAL DETERMINATIONS The experimental set-up and procedure used in the investigation described in the present paper are similar to those described in reference [7] (see Figure 3, where the plate configuration resulting when the cut-out is q/a = b,/ b = 0.50, is shown).
Figure
3. Experimental
set-up.
RECTANGULAR
PLATE
WITH
CORNER
191
CUT-OUT
Experiments were performed on a stainless steel, clamped square plate with the following geometrical and mechanical parameters: a = 19.2 cm; a* = 20.7 cm; h = 0.3 cm; p = 7.84 g/cm”, E = 19.6 x 10” (g cm/s2)/cm2, I_L= 0.28. The dimension a is the distance between inner edges of the clamp while the parameter a * is the value of the side of the square plate’with the “root effect” taken into account; it is obtained once the fundamental frequency of the complete square plate have been measured [7] and with the use of the expression for the fundamental frequency coefficient.
I
1
0.1
I
0.2
1
II
0.4
0.5
I
0.3 0, 0
Figure 4. Comparison
Figure 4 quite good validity of applied to
of experimental
(0) and analytical
(-_)
results:
case of a clamped
square
plate
shows a comparison of analytical and experimental results. The agreement is from an engineering viewpoint since the experimental results confirm the the simple mathematical model proposed in the present investigation and a particular configuration (square domain with a corner square cut-out).
5. CONCLUSIONS A simple analytical solution has been proposed in order to predict some of the vibrational characteristics of a rather complex and very realistic problem of structural dynamics. Experimental and analytical values are in very good agreement for the particular plate-“cut-out” configuration chosen, and investigated in some detail, and the results lead to the interesting conclusion that in the case of a clamped square plate with a square corner cut-out the frequency coefficient remains practically invariant with respect to size: in other words, the decrease of mass and stiffness are approximately the same on a percentage basis.
ACKNOWLEDGMENTS The authors are indebted to research engineers J. L. Pombo, G. Ficcadenti, V. H. Palluzzi and P. Verniere for their valuable aid in performing some of the preliminary calculations and during the experimental stage of the project.
192
P. A.A.LAURA,P.A.LAURAANDV.H.CORTINEZ
The present investigation has been partially Cientificas (Buenos Aires Province).
sponsored
by Comisi6n
de Investigaciones
REFERENCES 1. A. W. LEISSA 1969 NASA SP 160. Vibration of plates. 2. A. W. LEISSA 1977 The Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations: classical theory. 3. A. W. LEISSA 1978 The Shock and Vibration Digest 10,21-35. Recent research in plate vibrations 1973-1976: Complicating effects. 4. F. E. EASTEP and F. G. HEMMIG 1978 Journal ofSound and Vibration 56, 155-165. Estimation of fundamental frequency of non-circular plates with free, circular cutouts. 5. K. NAGAYA 1981 Journal of Sound and Vibration 74, 543-551. Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape, Part I: Derivation of the frequency equation. 6. K. NAGAYA 1981 Journal ofSound and Vibration 74, 553-564. Simplified method for solving problems of vibrating plates of doubly connected arbitrary shape, Part II: Application and experiments. 7. P.A.A. LAURA,P.VERNIERE DE IRASSAR,L.ERCOLI~~~R. G~~0s1981 JournalofSound and Vibration 78, 489-493. Fundamental frequency of vibrations of a rectangular plate with a free, straight corner cut-out. 8. P. A. A. LAURA, P. L. VERNIERE DE IRASSAR and G. M. FICCADENTI 1982 Journalofthe Acoustical Society of America 71, 501-502. Vibrations of a clamped, rectangular plate of generalized orthotropy with a free, straight corner cut-out. 9. R. SCHMIDT 1982 Journal of Applied Mechanics 49,639-640. Estimation of buckling loads and other eigenvalues via a modification of the Rayleigh-Ritz method. 10. L. ERCOLI, L. C. NAVA and P. A. A. LAURA 1984 Institute of Applied Mechanics Publication No 84-25. Vibrations of plates of arbitrary shape by a modified Galerkin method.