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TRANSVERSE VIBRATIONS OF A THIN, ELASTIC CIRCULAR PLATE WITH MIXED BOUNDARY CONDITIONS
Journal of Sound and
LETTERS TO THE EDITOR TRANSVERSE VIBRATIONS OF A THIN, ELASTIC CIRCULAR PLATE WITH MIXED BOUNDARY CONDITIONS R. E. ROSSI
AND
P. A. A. LAURA
Department of Engineering, CONICE¹, ;niversidad Nacional del Sur, Bahn& a Blanca 8000, Argentina (Received 24 October 2001)
1. INTRODUCTION
The extremely important problem of vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder was approached by Bartlett four decades age [1]. In a very ingenious and rather pragmatic manner, he derived two variational principles for the lowest eigenvalue showing that the governing expressions can be determined by separation of variables. Upper and lower bounds for the fundamental frequency coe$cient were shown to be close together. Later, Noble [2, 3] obtained an asymptotic expression which yields very good agreement with Bartlett's results. The present research project arose from the need of knowledge of some higher natural frequencies. Furthermore, there is practically no information in the technical literature for the mixed boundary combination: clamped}free and simply supported}free. The lower natural frequency coe$cients are determined in the present study for the three combinations of edge conditions (Figure 1): simply supported}clamped, simply supported}free, clamped}free. Accurate values of the eigenvalues are obtained using a well-known "nite element code [4] and a dense net. The problem is of technological importance since it arises in the design of printed circuit boards, valve systems, transducers, etc.
2. FINITE ELEMENT RESULTS
The frequency coe$cients have been determined using a net of 20232 elements; see Figure 2. The Poisson ratio was taken equal to 0)25 for the con"guration shown in Figure 1(a) in order of verify the relative accuracy of the results obtained herein, as compared with the values presented in references [1, 2]. Very good agreement is observed when examining the results presented in Table 1 where additional frequency coe$cients obtained by means of the "nite element technique are shown. The "rst lower natural frequency coe$cients are again determined for "0)30 for the arrangement depicted in Figure 1(a) (see Table 2), while Tables 3 and 4 contain the natural frequency coe$cients for the simply supported}free and the clamped}free arrangements respectively (see Figures 1(b) and (c)). The agreement with the exact eigenvalues is excellent when classical boundary arrangements are considered. 0022-460X/02/$35.00
2002 Published by Elsevier Science Ltd.
984
LETTERS TO THE EDITOR
Figure 1. Vibrating structural element and di!erent arrangements of boundary conditions: (a) simply supported}clamped, (b) simply supported}free and (c) clamped}free.
TABLE 1
0
/8
/4
3/8
/2
5/8
3/4
7/8
4)868 13)842 13)842 25)557 25)564 29)669 39)907 39)907 48)432
5)859 14)070 15)939 26)331 27)054 31)585 41)338 41)523 48)653
6)351 14)748 16)904 27)128 28)122 33)051 42)109 42)965 49)989
6)878 15)886 17)404 28)004 29)829 33)771 43)223 44)554 52)206
7)511 17)485 17)550 30)217 30)277 34)504 45)490 45)705 54)153
8)272 17)590 19)493 30)690 31)603 36)621 46)536 46)895 54)664
9)118 18)276 20)833 31)704 32)873 38)134 47)552 48)639 56)264
9)880 19)980 21)232 32)930 34)697 38)902 49)009 50)485 58)202
10)216 21)262 21)262 34)878 34)882 39)775 51)037 51)037 60)837
985
LETTERS TO THE EDITOR
Figure 2. Finite element net used in the present investigation (20232 elements).
TABLE 2
0
/8
/4
3/8
/2
5/8
3/4
7/8
4)943 13)905 13)905 25)616 25)623 29)727 39)965 39)965 48)488
5)915 14)130 15)982 26)384 27)106 31)628 41)387 41)572 48)708
6)401 14)802 16)939 27)179 28)165 33)085 42)155 43)007 50)036
6)922 15)931 17)434 28)048 29)860 33)803 43)264 44)587 52)242
7)546 17)514 17)582 30)247 30)303 34)534 45)519 45)731 54)186
8)296 17)618 19)508 30)716 31)625 36)636 46)559 46)916 54)691
9)130 18)300 20)836 31)726 32)886 38)140 47)570 48)653 56)282
9)882 19)988 21)232 32)943 34)698 38)906 49)020 50)487 58)213
10)216 21)262 21)262 34)878 34)882 39)775 51)037 51)037 60)837
ACKNOWLEDGMENTS
The present study has been sponsored by CONICET Research and Development Program, by SecretarmH a General de Ciencia y TecnologmH a of Universidad Nacional del Sur and by FONCYT.
986
LETTERS TO THE EDITOR
TABLE 3
0
/8
/4
3/8
/2
5/8
3/4
7/8
4)943 13)905 13)905 25)616 25)623 29)727 39)965 39)965 48)488
4)924 13)574 13)903 23)782 25)588 28)811 34)491 39)973 43)207
4)592 8)675 13)672 16)649 22)512 27)318 29)966 31)688 41)304
3)047 6)455 10)672 15)715 16)456 23)379 27)996 28)530 33)322
1)782 6)309 6)555 13)948 15)488 17)325 24)004 27)672 30)836
1)032 4)195 6)221 9)733 13)908 17)064 18)983 22)482 30)039
0)584 2)834 5)458 8)149 10)525 15)873 17)903 18)669 23)661
0)261 1)890 4)190 7)869 7)978 12)294 16)561 17)668 20)654
0)000 0)000 0)000 5)358 5)358 9)002 12)437 12)437 20)470
TABLE 4
0
/8
/4
3/8
/2
5/8
3/4
7/8
10)216 21)262 21)262 34)878 34)883 39)775 51)037 51)037 60)837
9)871 19)682 21)230 29)720 34)687 37)306 41)219 50)387 52)295
8)008 10)978 20)584 21)198 26)356 35)047 36)036 39)050 49)899
4)225 9)775 12)455 20)573 21)169 25)659 34)272 34)974 39)345
2)455 7)196 9)320 14)865 19)850 20)939 25)411 32)777 35)201
1)600 4)573 8)443 10)400 15)354 20)783 21)021 23)586 32)665
1)143 3)091 6)422 9)354 11)090 16)804 20)026 20)853 24)007
0)855 2)092 4)784 8)249 8)946 12)723 17)256 20)084 20)740
0)000 0)000 0)000 5)358 5)358 9)002 12)437 12)437 20)470
REFERENCES 1. C. C. BARTLETT 1963 Quarterly Journal of Mechanics and Applied Mathematics 16, 431}440. The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder. 2. B. NOBLE 1965 Proceedings of the 9th Midwestern Conference on Solid and Fluid Mechanics, ;niversity of =isconsin, Madison, =isconsin, Vol. I, 540}546. The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder. 3. A. W. LEISSA 1969
LETTERS TO THE EDITOR TRANSVERSE VIBRATIONS OF A THIN, ELASTIC CIRCULAR PLATE WITH MIXED BOUNDARY CONDITIONS R. E. ROSSI
AND
P. A. A. LAURA
Department of Engineering, CONICE¹, ;niversidad Nacional del Sur, Bahn& a Blanca 8000, Argentina (Received 24 October 2001)
1. INTRODUCTION
The extremely important problem of vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder was approached by Bartlett four decades age [1]. In a very ingenious and rather pragmatic manner, he derived two variational principles for the lowest eigenvalue showing that the governing expressions can be determined by separation of variables. Upper and lower bounds for the fundamental frequency coe$cient were shown to be close together. Later, Noble [2, 3] obtained an asymptotic expression which yields very good agreement with Bartlett's results. The present research project arose from the need of knowledge of some higher natural frequencies. Furthermore, there is practically no information in the technical literature for the mixed boundary combination: clamped}free and simply supported}free. The lower natural frequency coe$cients are determined in the present study for the three combinations of edge conditions (Figure 1): simply supported}clamped, simply supported}free, clamped}free. Accurate values of the eigenvalues are obtained using a well-known "nite element code [4] and a dense net. The problem is of technological importance since it arises in the design of printed circuit boards, valve systems, transducers, etc.
2. FINITE ELEMENT RESULTS
The frequency coe$cients have been determined using a net of 20232 elements; see Figure 2. The Poisson ratio was taken equal to 0)25 for the con"guration shown in Figure 1(a) in order of verify the relative accuracy of the results obtained herein, as compared with the values presented in references [1, 2]. Very good agreement is observed when examining the results presented in Table 1 where additional frequency coe$cients obtained by means of the "nite element technique are shown. The "rst lower natural frequency coe$cients are again determined for "0)30 for the arrangement depicted in Figure 1(a) (see Table 2), while Tables 3 and 4 contain the natural frequency coe$cients for the simply supported}free and the clamped}free arrangements respectively (see Figures 1(b) and (c)). The agreement with the exact eigenvalues is excellent when classical boundary arrangements are considered. 0022-460X/02/$35.00
2002 Published by Elsevier Science Ltd.
984
LETTERS TO THE EDITOR
Figure 1. Vibrating structural element and di!erent arrangements of boundary conditions: (a) simply supported}clamped, (b) simply supported}free and (c) clamped}free.
TABLE 1
0
/8
/4
3/8
/2
5/8
3/4
7/8
4)868 13)842 13)842 25)557 25)564 29)669 39)907 39)907 48)432
5)859 14)070 15)939 26)331 27)054 31)585 41)338 41)523 48)653
6)351 14)748 16)904 27)128 28)122 33)051 42)109 42)965 49)989
6)878 15)886 17)404 28)004 29)829 33)771 43)223 44)554 52)206
7)511 17)485 17)550 30)217 30)277 34)504 45)490 45)705 54)153
8)272 17)590 19)493 30)690 31)603 36)621 46)536 46)895 54)664
9)118 18)276 20)833 31)704 32)873 38)134 47)552 48)639 56)264
9)880 19)980 21)232 32)930 34)697 38)902 49)009 50)485 58)202
10)216 21)262 21)262 34)878 34)882 39)775 51)037 51)037 60)837
985
LETTERS TO THE EDITOR
Figure 2. Finite element net used in the present investigation (20232 elements).
TABLE 2
0
/8
/4
3/8
/2
5/8
3/4
7/8
4)943 13)905 13)905 25)616 25)623 29)727 39)965 39)965 48)488
5)915 14)130 15)982 26)384 27)106 31)628 41)387 41)572 48)708
6)401 14)802 16)939 27)179 28)165 33)085 42)155 43)007 50)036
6)922 15)931 17)434 28)048 29)860 33)803 43)264 44)587 52)242
7)546 17)514 17)582 30)247 30)303 34)534 45)519 45)731 54)186
8)296 17)618 19)508 30)716 31)625 36)636 46)559 46)916 54)691
9)130 18)300 20)836 31)726 32)886 38)140 47)570 48)653 56)282
9)882 19)988 21)232 32)943 34)698 38)906 49)020 50)487 58)213
10)216 21)262 21)262 34)878 34)882 39)775 51)037 51)037 60)837
ACKNOWLEDGMENTS
The present study has been sponsored by CONICET Research and Development Program, by SecretarmH a General de Ciencia y TecnologmH a of Universidad Nacional del Sur and by FONCYT.
986
LETTERS TO THE EDITOR
TABLE 3
0
/8
/4
3/8
/2
5/8
3/4
7/8
4)943 13)905 13)905 25)616 25)623 29)727 39)965 39)965 48)488
4)924 13)574 13)903 23)782 25)588 28)811 34)491 39)973 43)207
4)592 8)675 13)672 16)649 22)512 27)318 29)966 31)688 41)304
3)047 6)455 10)672 15)715 16)456 23)379 27)996 28)530 33)322
1)782 6)309 6)555 13)948 15)488 17)325 24)004 27)672 30)836
1)032 4)195 6)221 9)733 13)908 17)064 18)983 22)482 30)039
0)584 2)834 5)458 8)149 10)525 15)873 17)903 18)669 23)661
0)261 1)890 4)190 7)869 7)978 12)294 16)561 17)668 20)654
0)000 0)000 0)000 5)358 5)358 9)002 12)437 12)437 20)470
TABLE 4
0
/8
/4
3/8
/2
5/8
3/4
7/8
10)216 21)262 21)262 34)878 34)883 39)775 51)037 51)037 60)837
9)871 19)682 21)230 29)720 34)687 37)306 41)219 50)387 52)295
8)008 10)978 20)584 21)198 26)356 35)047 36)036 39)050 49)899
4)225 9)775 12)455 20)573 21)169 25)659 34)272 34)974 39)345
2)455 7)196 9)320 14)865 19)850 20)939 25)411 32)777 35)201
1)600 4)573 8)443 10)400 15)354 20)783 21)021 23)586 32)665
1)143 3)091 6)422 9)354 11)090 16)804 20)026 20)853 24)007
0)855 2)092 4)784 8)249 8)946 12)723 17)256 20)084 20)740
0)000 0)000 0)000 5)358 5)358 9)002 12)437 12)437 20)470
REFERENCES 1. C. C. BARTLETT 1963 Quarterly Journal of Mechanics and Applied Mathematics 16, 431}440. The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder. 2. B. NOBLE 1965 Proceedings of the 9th Midwestern Conference on Solid and Fluid Mechanics, ;niversity of =isconsin, Madison, =isconsin, Vol. I, 540}546. The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder. 3. A. W. LEISSA 1969