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On the fundamental frequency of point supported plates
Journal of Sound and Vibration (1975) 40(4), 561-562
LETTERS
TO THE
EDITOR
ON THE FUNDAMENTAL FREQUENCY OF POINT SUPPORTED PLATES
In reference [! ], most accurate upper bounds for the natural frequencies of point supported square plates symmetrically supported along diagonals have been presented, as obtained by a finite element analysis in which a high precision triangular plate bending element [2] has been used. Also, it has been pointed out that the finite element results [1] for the fundamental frequency agree very well with the results of Dowell [3], except for the point ~[a = 0.2, where ct is the distance of the point support from an edge and a is the length of the plate. Later, in reference [4], Dowell has commented that this discrepancy is due to the fact that the lowest frequency mode changes at ~/a = 0.2, but gives the impression that the result presented in reference [3] is for the fundamental mode, which is not true. On the contrary, the result presented in reference [3] is for the first symmetric-symmetric mode of vibration of the point supported plate. We wish to point out that there are frequency cross-overs with support locations in the problem under study and we emphasize (at the risk of annoying the reader) that "fundamental" implies the lowest. The purpose of the present note is to give a most accurate upper bound for the fundamental frequency of a point supported square plate symmetrically supported along diagonals. The high precision triangular plate bending element of reference [2] is used to obtain the natural frequencies. Due to symmetries, only a quarter of the plate is considered and a 5 x 5 uniform finite element mesh is used for the analysis. Table I gives the frequency parameter (defined as ). = pto92aa/D, where p is the mass density, t is the thickness, o9 is the angular frequency, a is the length of the plate and D is the plate flexural rigidity) for the first three modes of a point supported plate for various support positions. Further, in Figure 1 of reference [5], it has been stated that, for the first symmetric-symmetric mode and first symmetric-antisymmetric mode, the frequencies when ct/a = 0.5 are the same. We have computed the frequency parameter for the first symmetric-antisymmetrie mode with the same finite element idealization and found that the frequency associated with the first symmetric-antisymmetric mode does not coincide with that of the first symmetric mode for ~[a = 0.5. This also is confirmed by the results of references [6] and [7]. TABLE I
Frequencies of a pohlt supported plate (2) ct/a 0"0 (Corner supports) 0"1 0-2 0"3 0'4 0"5 (One centre support)
First mode
Second mode
Third mode
7"11088 (SSI) 15-7702(SAI) 19"5961(SS2) 12-7745 (SSI) 19"5961 (SS2) 19-!234 (SS1) 13"2810 (SSI) 11-2303 (SSI)
19-5961(SS2) 23"7366(SAI) 23-0038(SSI) 32"2923(SAI) 19-5961(SS2) 23-5831 (SAI) 13"5426(SAI) ----
SSI : First symmetric-symmetric mode SS2: Second symmetric-symmetric mode SAI : First symmetric-antisymmetric mode 561
562
LETTERSTO TIlE EDITOR
It may be pointed out here that the results presented in an experimental study [8] are not for the fundamental mode of vibration but are for the first symmetric-symmetric mode of vibration.
The authors wish to thank Dr I. S. Raju for the discussions they had during the course ofthis work. Structural Enghzeering Dicision, G. VENKATESWARARAO Vikram Sarabhai Space Centre, C . L . AMBA-RAO Trit'andrum--695022, T. V. G. K. MURTIIY
btdia (ReceiL'ed 4 February 1975) REFERENCES
I. G. VENKATESWARARAO,I. S. RAJu and C. L. AMBA-RAo1973 Journal of Sound and Vibration 29, 387-391. Vibrations of point supported plates. 2. G. R. COWPER,E. KOSKO,G. M. LINDBERGand M. D. Or.SON 1968 NationalResearch Councilor Canada, Aeronautical Report LR-514. A high precision triangular plate bending element. 3. E. H. DOWELL1971 Journal of Applied Mechanics 38, 595-600. Free vibrations of a linear structure with arbitrary support conditions. 4. E. H. DOWELL1974 Journal of Sound and Vibration 32, 524. Comments on "Vibrations of point supported plates". 5. E. H. DOWELL1973 Journal of Spacecraft and Rockets 10, 389-395. Theoretical vibration and flutter studies of point suppoi'ted plates. 6. M. PErvTand W. H. MIRZA1972JournalofSoundand Vibration21,355-364. Vibration ofcolumnsupported floor slabs. 7. D.J. Joxxss and R. NATARAJA1972JournalofSoundand Vibration25, 75-82. Vibration of a square plate symmetrically supported at four points. 8. Y. V. K. SADASIVARAO, G. VENKATESWARARAO and C. L. AMBA-RAO1974 Journal of Sound and Vibration 32, 286-288. Experimental study of a four-point supported square plate.
LETTERS
TO THE
EDITOR
ON THE FUNDAMENTAL FREQUENCY OF POINT SUPPORTED PLATES
In reference [! ], most accurate upper bounds for the natural frequencies of point supported square plates symmetrically supported along diagonals have been presented, as obtained by a finite element analysis in which a high precision triangular plate bending element [2] has been used. Also, it has been pointed out that the finite element results [1] for the fundamental frequency agree very well with the results of Dowell [3], except for the point ~[a = 0.2, where ct is the distance of the point support from an edge and a is the length of the plate. Later, in reference [4], Dowell has commented that this discrepancy is due to the fact that the lowest frequency mode changes at ~/a = 0.2, but gives the impression that the result presented in reference [3] is for the fundamental mode, which is not true. On the contrary, the result presented in reference [3] is for the first symmetric-symmetric mode of vibration of the point supported plate. We wish to point out that there are frequency cross-overs with support locations in the problem under study and we emphasize (at the risk of annoying the reader) that "fundamental" implies the lowest. The purpose of the present note is to give a most accurate upper bound for the fundamental frequency of a point supported square plate symmetrically supported along diagonals. The high precision triangular plate bending element of reference [2] is used to obtain the natural frequencies. Due to symmetries, only a quarter of the plate is considered and a 5 x 5 uniform finite element mesh is used for the analysis. Table I gives the frequency parameter (defined as ). = pto92aa/D, where p is the mass density, t is the thickness, o9 is the angular frequency, a is the length of the plate and D is the plate flexural rigidity) for the first three modes of a point supported plate for various support positions. Further, in Figure 1 of reference [5], it has been stated that, for the first symmetric-symmetric mode and first symmetric-antisymmetric mode, the frequencies when ct/a = 0.5 are the same. We have computed the frequency parameter for the first symmetric-antisymmetrie mode with the same finite element idealization and found that the frequency associated with the first symmetric-antisymmetric mode does not coincide with that of the first symmetric mode for ~[a = 0.5. This also is confirmed by the results of references [6] and [7]. TABLE I
Frequencies of a pohlt supported plate (2) ct/a 0"0 (Corner supports) 0"1 0-2 0"3 0'4 0"5 (One centre support)
First mode
Second mode
Third mode
7"11088 (SSI) 15-7702(SAI) 19"5961(SS2) 12-7745 (SSI) 19"5961 (SS2) 19-!234 (SS1) 13"2810 (SSI) 11-2303 (SSI)
19-5961(SS2) 23"7366(SAI) 23-0038(SSI) 32"2923(SAI) 19-5961(SS2) 23-5831 (SAI) 13"5426(SAI) ----
SSI : First symmetric-symmetric mode SS2: Second symmetric-symmetric mode SAI : First symmetric-antisymmetric mode 561
562
LETTERSTO TIlE EDITOR
It may be pointed out here that the results presented in an experimental study [8] are not for the fundamental mode of vibration but are for the first symmetric-symmetric mode of vibration.
The authors wish to thank Dr I. S. Raju for the discussions they had during the course ofthis work. Structural Enghzeering Dicision, G. VENKATESWARARAO Vikram Sarabhai Space Centre, C . L . AMBA-RAO Trit'andrum--695022, T. V. G. K. MURTIIY
btdia (ReceiL'ed 4 February 1975) REFERENCES
I. G. VENKATESWARARAO,I. S. RAJu and C. L. AMBA-RAo1973 Journal of Sound and Vibration 29, 387-391. Vibrations of point supported plates. 2. G. R. COWPER,E. KOSKO,G. M. LINDBERGand M. D. Or.SON 1968 NationalResearch Councilor Canada, Aeronautical Report LR-514. A high precision triangular plate bending element. 3. E. H. DOWELL1971 Journal of Applied Mechanics 38, 595-600. Free vibrations of a linear structure with arbitrary support conditions. 4. E. H. DOWELL1974 Journal of Sound and Vibration 32, 524. Comments on "Vibrations of point supported plates". 5. E. H. DOWELL1973 Journal of Spacecraft and Rockets 10, 389-395. Theoretical vibration and flutter studies of point suppoi'ted plates. 6. M. PErvTand W. H. MIRZA1972JournalofSoundand Vibration21,355-364. Vibration ofcolumnsupported floor slabs. 7. D.J. Joxxss and R. NATARAJA1972JournalofSoundand Vibration25, 75-82. Vibration of a square plate symmetrically supported at four points. 8. Y. V. K. SADASIVARAO, G. VENKATESWARARAO and C. L. AMBA-RAO1974 Journal of Sound and Vibration 32, 286-288. Experimental study of a four-point supported square plate.