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Vibrations of point supported plates
Jouraal of Sound wzd Vibration (1973) 29(3), 387-391
LETTERS TO THE EDITOR VIBRATIONS OF POINT SUPPORTED PLATES I. INTRODUCTION
The evaluation of the fundamental frequency of a square plate, symmetrically supported at four points on the diagonals has attracted the attention of many researchers [1-6] in recent years. They have used several approximate methods to solve this apparently difficult problem; namely, energy methods [1, 2], finite difference methods [2, 6], Lagrangian multipliers [3] and finite elements [4, 5]. Some experimental investigations on this problem have also been reported in references [1] and [2]. Nevertheless, considerable discrepancies exist among the results obtained by various authors. The purpose of the work reported in this letter was to use a finite element method to get a very accurate upper bound for the first few frequencies and to compare the results obtained with the earlier solutions. The results presented in this paper were obtained on an IBM 360 Model 44 computer with double precision arithmetic. 2. PLATE CONFIGURATION
Figure 1 shows the square plate of length a symmetrically supported on the diagonals at distance ~ from the free edges, o~/a--- 0 corresponds to a corner supported plate and ~[a = 0.5 corresponds to a single point support at the centre of the plate.
i
/
\
/
\,
Figure 1. Square plate symmetrically supported at four points on the diagonals. 3. FINIT~EELEMENT METHOD
The application of the finite element method for vibration problems is well established [7]. The use of this method with consistent and conforming elements gives upper bounds to frequencies. In the work reported in this paper, the high precision conforming triangular plate bending element [8] was used to solve the problem. Figure 2 shows the finite element mesh divisions of a quarter of a plate. Figure 2(a) shows a uniform mesh division and Figure 2(b) shows a non-uniform mesh division. The non-uniform mesh division was used in this work with the aim of getting more points near the centre of the plate. 387
388
LETTERSTO THE EDITOR I
IL
4/i/~----/I/1/i /1/1/
///,// ///// ///,// ////'/ (o)
L
(b)
Figure 2. A discrete model of a quarter of a plate--two alternate mesh configurations. (a) Uniform mesh, N = 10; (b) non-uniform mesh, N = 10. 4. RESULTS AND DISCUSSION To check the results obtained by the present method, the fundamental frequency of a simply supported square plate was calculated by using a 10 • 10 mesh (both uniform and non-uniform meshes are used). Table 1. gives the frequency parameter )2 (defined as )-2= pttoZa4[D, where p is the mass density, t is the thickness, to is the angular frequency, a is the length of the plate and D is the plate flexural rigidity) for the fundamental frequency o f a simply supported square plate. The results obtained by both the meshes differ from the exact value only in the sixth significant figure. TABLE
I
Fundamental eigem'alue )2 for simply supported square plate (see Figure 2), 22 = ptto2a4/D Uniform mesh Non-Uniform mesh N = 10
389.6364'
389.6379
Exact 389.63636
N = Number of elements per side. TABLE 2
Comparison offinite element sohttions for the fundamental eigenrahte 2 of a pohlt supported plate, 22 = pto92a 4[D, ct = 0 Element High precision triangular (reference [8])
ACM$
Number of elements per side 2 4 6 8 10 101" 10
Eigenvalue 7.1110 I0 7.110899 7.110888 7.110885 7.110884 7.110892 7.196
t Non-uniform mesh (Figure 2(b)). $ ACM. Rectangular plate bending element with 12 degrees of freedom (reference [9]).
389
0
0
o~E ~co
~111 .~- ~ ~
,..1 <
[.. 0
~'~
o o ~
~1
.~
11-
r.-
.~E~-_
-II
II
_~_
~
~
=~._o
o
I~_ .o
390
LETTERS TO THE EDITOR
Table 2 gives the results of the convergence study for the fundamental frequency of a point supported square plate supported at the corner for various mesh sizes. It can be seen from the table that the convergence is very rapid and the results are accurate up to six significant figures. In Table 3, the values for the fundamental frequency parameter of a square plate, symmetrically supported at four points on the diagonals, as obtained by various researchers and by using the present solution, for various values of or~a,varying from 0 to 0.5, are presented. In general, the agreement between the experimental and theoretical values is not very good except in the limiting cases of ct/a = 0 and 0.5. It is observed from this table that agreement between the present work and Dowell's [3] is very good except at ct/a = 0.2. Further, the three finite element solutions obtained with three different elements agree well with each other. It is noted that, in general, the present solution gives the best upper bound. The values obtained by Johns and Nataraja [6] appear to be lower bounds. Thus the present solution and the solution obtained in reference [6] bracket the exact solution. TABLE4
Comparison between finite difference and finite element sohttions for the first three frequency parameters )- of a corner-supported square plate (c~ = 0), )2 = pto~2a4/D
Mode
Finite difference [Johns & Nataraja]
1 2 3
7"111" 15"43 17.10
Finite element solutions , " , [Damle & [Petyt & Present Feeser] Mirza] work 7"196 15"840 19.630
7.143 15"590 19.599
7.111 15"770 19.596
t Values given are extrapolated values. TABLE 5
Fundamental frequency parameter )- of a shnply supported square plate with a central pohzt support, 22 -- pto~2a4/D Finite difference [Johns & Nataraja]
Energy solutions [Nowacki~]
Present solution
44.5 (53"4)t
49.3; 52.6
52.6277
Value given in the bracket is obtained by extrapolation.
These values are taken from reference [6]. Table 4 gives the first three frequency parameters for a corner supported square plate as obtained by finite difference and finite element methods. The finite element solutions agree very well with each other and, in general, the present solution yields the best upper bound. Johns and Nataraja [6], for e = 0, obtained a nodal line for the second mode along one of the diagonals but the present solution gave a nodal line along one of the coordinate axes, which agrees with the second and third mode shapes obtained in references [4] and [5]. Results have also been obtained for a simply supported square plate with a central point support. Table 5 gives the fundamental frequency parameter as obtained by finite difference, energy methods and the present solution. In view of the high precision of the element, the
LETTERSTO THE EDITOR
391
present solution is assumed to be m o s t accurate. In reference [6] it was stated that the fundamental m o d e has one diagonal o r one centre line as a nodal line. The present finite element solution, however, did not show any nodal line for the fundamental mode. ACKNOWLEDGMENT The authors gratefully acknowledge their indebtedness to D r Y. V. K. Sadasiva R a o and D r J. V e n k a t a r a m a n for valuable and stimulating discussions. G. VENKATESWARARAO I. S. RAJU C. L. AMBA-RAO
Structural Enghteering Division, Space Science and Technology Centre, Trivandrum-22, Kerala, htdia (Received5 February 1973, attd ht revisedform 4 April 1973)
REFERENCES 1. W. K. Tso 1966 American Institute of Aeronautics and Astronautics Journal 4, 733-735. On the fundamental frequency of a four point supported square elastic plate. 2. D. J. JOHNSand V. T. NAGARAJ1969 Journal of Sound and Vibration 10, 404--410. On the fundamental frequency of a square plate symmetrically supported at four points. 3. E. H. DOWELL 1971 Journal of Applied Mechanics 38, 595-600. Free vibrations of a linear structure with arbitrary support conditions. 4. S. K. DAMLE and L. J. FEESER 1972 Journal of the Aeronautical Society ofhzdia 24, 375-377. Vibration of four point supported plates by a finite element method. 5. M. PETYT and W. H. MmZA 1972 Journal of Sound and Vibration 21, 355-364. Vibration of column-supported floor slabs. 6. D. J. JOHNS and R. NATARAJA1972 Jottrnal of Sottnd and Vibration 25, 75-82. Vibration of a square plate symmetrically supported at four points. 7. O. C. ZIENKIEWICZ1971 The Finite Element Method in Engineerhrg Science. London: McGrawHill Book Company. 8. G. R. COWPER, E. KOSKO, G. M. L1NDBERGand M. D. OLSON 1968 National Research Cottncil of Canada, Aeronautical Report LR-514. A high precision triangular plate-bending element. 9. R..W. CLOUGH and J. L. TOCHER 1965 Proceedings of the First Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Dayton, U.S.A. Finite element stiffness matrices for analysis of plate bending. 10. H. L. Cox and J. BOXER 1960 Aeronautical Quarterly 11, 41-50. Vibration of rectangular plates point supported at the corners.
LETTERS TO THE EDITOR VIBRATIONS OF POINT SUPPORTED PLATES I. INTRODUCTION
The evaluation of the fundamental frequency of a square plate, symmetrically supported at four points on the diagonals has attracted the attention of many researchers [1-6] in recent years. They have used several approximate methods to solve this apparently difficult problem; namely, energy methods [1, 2], finite difference methods [2, 6], Lagrangian multipliers [3] and finite elements [4, 5]. Some experimental investigations on this problem have also been reported in references [1] and [2]. Nevertheless, considerable discrepancies exist among the results obtained by various authors. The purpose of the work reported in this letter was to use a finite element method to get a very accurate upper bound for the first few frequencies and to compare the results obtained with the earlier solutions. The results presented in this paper were obtained on an IBM 360 Model 44 computer with double precision arithmetic. 2. PLATE CONFIGURATION
Figure 1 shows the square plate of length a symmetrically supported on the diagonals at distance ~ from the free edges, o~/a--- 0 corresponds to a corner supported plate and ~[a = 0.5 corresponds to a single point support at the centre of the plate.
i
/
\
/
\,
Figure 1. Square plate symmetrically supported at four points on the diagonals. 3. FINIT~EELEMENT METHOD
The application of the finite element method for vibration problems is well established [7]. The use of this method with consistent and conforming elements gives upper bounds to frequencies. In the work reported in this paper, the high precision conforming triangular plate bending element [8] was used to solve the problem. Figure 2 shows the finite element mesh divisions of a quarter of a plate. Figure 2(a) shows a uniform mesh division and Figure 2(b) shows a non-uniform mesh division. The non-uniform mesh division was used in this work with the aim of getting more points near the centre of the plate. 387
388
LETTERSTO THE EDITOR I
IL
4/i/~----/I/1/i /1/1/
///,// ///// ///,// ////'/ (o)
L
(b)
Figure 2. A discrete model of a quarter of a plate--two alternate mesh configurations. (a) Uniform mesh, N = 10; (b) non-uniform mesh, N = 10. 4. RESULTS AND DISCUSSION To check the results obtained by the present method, the fundamental frequency of a simply supported square plate was calculated by using a 10 • 10 mesh (both uniform and non-uniform meshes are used). Table 1. gives the frequency parameter )2 (defined as )-2= pttoZa4[D, where p is the mass density, t is the thickness, to is the angular frequency, a is the length of the plate and D is the plate flexural rigidity) for the fundamental frequency o f a simply supported square plate. The results obtained by both the meshes differ from the exact value only in the sixth significant figure. TABLE
I
Fundamental eigem'alue )2 for simply supported square plate (see Figure 2), 22 = ptto2a4/D Uniform mesh Non-Uniform mesh N = 10
389.6364'
389.6379
Exact 389.63636
N = Number of elements per side. TABLE 2
Comparison offinite element sohttions for the fundamental eigenrahte 2 of a pohlt supported plate, 22 = pto92a 4[D, ct = 0 Element High precision triangular (reference [8])
ACM$
Number of elements per side 2 4 6 8 10 101" 10
Eigenvalue 7.1110 I0 7.110899 7.110888 7.110885 7.110884 7.110892 7.196
t Non-uniform mesh (Figure 2(b)). $ ACM. Rectangular plate bending element with 12 degrees of freedom (reference [9]).
389
0
0
o~E ~co
~111 .~- ~ ~
,..1 <
[.. 0
~'~
o o ~
~1
.~
11-
r.-
.~E~-_
-II
II
_~_
~
~
=~._o
o
I~_ .o
390
LETTERS TO THE EDITOR
Table 2 gives the results of the convergence study for the fundamental frequency of a point supported square plate supported at the corner for various mesh sizes. It can be seen from the table that the convergence is very rapid and the results are accurate up to six significant figures. In Table 3, the values for the fundamental frequency parameter of a square plate, symmetrically supported at four points on the diagonals, as obtained by various researchers and by using the present solution, for various values of or~a,varying from 0 to 0.5, are presented. In general, the agreement between the experimental and theoretical values is not very good except in the limiting cases of ct/a = 0 and 0.5. It is observed from this table that agreement between the present work and Dowell's [3] is very good except at ct/a = 0.2. Further, the three finite element solutions obtained with three different elements agree well with each other. It is noted that, in general, the present solution gives the best upper bound. The values obtained by Johns and Nataraja [6] appear to be lower bounds. Thus the present solution and the solution obtained in reference [6] bracket the exact solution. TABLE4
Comparison between finite difference and finite element sohttions for the first three frequency parameters )- of a corner-supported square plate (c~ = 0), )2 = pto~2a4/D
Mode
Finite difference [Johns & Nataraja]
1 2 3
7"111" 15"43 17.10
Finite element solutions , " , [Damle & [Petyt & Present Feeser] Mirza] work 7"196 15"840 19.630
7.143 15"590 19.599
7.111 15"770 19.596
t Values given are extrapolated values. TABLE 5
Fundamental frequency parameter )- of a shnply supported square plate with a central pohzt support, 22 -- pto~2a4/D Finite difference [Johns & Nataraja]
Energy solutions [Nowacki~]
Present solution
44.5 (53"4)t
49.3; 52.6
52.6277
Value given in the bracket is obtained by extrapolation.
These values are taken from reference [6]. Table 4 gives the first three frequency parameters for a corner supported square plate as obtained by finite difference and finite element methods. The finite element solutions agree very well with each other and, in general, the present solution yields the best upper bound. Johns and Nataraja [6], for e = 0, obtained a nodal line for the second mode along one of the diagonals but the present solution gave a nodal line along one of the coordinate axes, which agrees with the second and third mode shapes obtained in references [4] and [5]. Results have also been obtained for a simply supported square plate with a central point support. Table 5 gives the fundamental frequency parameter as obtained by finite difference, energy methods and the present solution. In view of the high precision of the element, the
LETTERSTO THE EDITOR
391
present solution is assumed to be m o s t accurate. In reference [6] it was stated that the fundamental m o d e has one diagonal o r one centre line as a nodal line. The present finite element solution, however, did not show any nodal line for the fundamental mode. ACKNOWLEDGMENT The authors gratefully acknowledge their indebtedness to D r Y. V. K. Sadasiva R a o and D r J. V e n k a t a r a m a n for valuable and stimulating discussions. G. VENKATESWARARAO I. S. RAJU C. L. AMBA-RAO
Structural Enghteering Division, Space Science and Technology Centre, Trivandrum-22, Kerala, htdia (Received5 February 1973, attd ht revisedform 4 April 1973)
REFERENCES 1. W. K. Tso 1966 American Institute of Aeronautics and Astronautics Journal 4, 733-735. On the fundamental frequency of a four point supported square elastic plate. 2. D. J. JOHNSand V. T. NAGARAJ1969 Journal of Sound and Vibration 10, 404--410. On the fundamental frequency of a square plate symmetrically supported at four points. 3. E. H. DOWELL 1971 Journal of Applied Mechanics 38, 595-600. Free vibrations of a linear structure with arbitrary support conditions. 4. S. K. DAMLE and L. J. FEESER 1972 Journal of the Aeronautical Society ofhzdia 24, 375-377. Vibration of four point supported plates by a finite element method. 5. M. PETYT and W. H. MmZA 1972 Journal of Sound and Vibration 21, 355-364. Vibration of column-supported floor slabs. 6. D. J. JOHNS and R. NATARAJA1972 Jottrnal of Sottnd and Vibration 25, 75-82. Vibration of a square plate symmetrically supported at four points. 7. O. C. ZIENKIEWICZ1971 The Finite Element Method in Engineerhrg Science. London: McGrawHill Book Company. 8. G. R. COWPER, E. KOSKO, G. M. L1NDBERGand M. D. OLSON 1968 National Research Cottncil of Canada, Aeronautical Report LR-514. A high precision triangular plate-bending element. 9. R..W. CLOUGH and J. L. TOCHER 1965 Proceedings of the First Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Dayton, U.S.A. Finite element stiffness matrices for analysis of plate bending. 10. H. L. Cox and J. BOXER 1960 Aeronautical Quarterly 11, 41-50. Vibration of rectangular plates point supported at the corners.