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Vibration of cylindrically orthotropic circular plates
Journal of Sound and Vibration (1974) 36(3), 433-434
LETTERS TO THE EDITOR VIBRATION OF CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATES Free vibration of a fixed circular plate possessing cylindrical orthotropy has been investigated by Akasaka and Takagishi [1 ], Mossakowski and Borsuk [2], Minkarah and H o p p m a n n [3], Pandalai and Patel [4] and Kirmser, Huang and Woo [5]. Woo, Kirmser and Huang [6] also investigated free vibration of a cylindrically orthotropic circular plate with concentric isotropic core for clamped and simply supported boundary conditions. The calculations of the results presented in references [1]-[4], which show some differences, were verified in reference [5]. In this note the results obtained by using a finite element method are presented. For the results presented in reference [5] it was assumed that there was an TABLE 1
Axisymmetrie frequency parameters 21/2[= (phD,)l/2 a2tn]for a clamped orthotropic circular plate with a concentric isotropic core 2 '/z for values ofbla A
.B 0"2 0"4 0"6 0"8 I-0 1-2 1"4 1"6 1"8 2"0
0.00
0-2
8-2120 8"8359 9"3437 9"7964 10"2162 10"6140 10-9977 11-3710 11-7378 12"1004
8"8670 9"2107 9"5378 9"8842 10"2162 10-5465 10"8758 11"2048 11"5352 11"8667
0-5 9-8248 9"9221 10"0198 10"1178 10-2162 10-3150 10"4142 10"5137 10"6136 10"7139
TABLE 2
AMsymnletric frequency parameter 21/2 [= (phD,)t/2 a2 og]for a simply supported orthotropic circular plate with concentric isotropic core 2 ~t2 for values ofb/a
[I
0.00
0"2
0.5
0-2 0"4 0'6 0"8 I'0 1.2 1"4 1"6 1"8 2"0
2"6769 3"4182 3"9925 4"4872 4-9352 5"3521 5"7473 6"1271 6-4954 6"8556
3"1239 3"6383 4"1018 4"5307 4"9352 5"3213 5"6939 6-0561 6"4105 6-7591
3"8790 4"1638 4"4330 4.6894 4-9352 5"1719 5-4007 5"6228 5"8390 6-0500
433
434
LETTERS TO THE EDITOR
isotropic core at the centre, of a radius b which was 0.025 of the radius a of the orthotropic plate. This assumption is not necessary in the finite element approach. The frequency parameters presented here were obtained by using a shell finite element [7] with 6 degrees of freedom at each node. Five elements were used and to avoid a singularity at the centre a hole with a radius of 0.001 of the outer radius was assumed. Tables 1 and 2 give the frequency parameters for various ratios of radius of isotropic core to the outer radius (namely 0"5, 0-2 and 0.0) for different values offl(=Ee/E,) for clamped and simply supported boundary conditions, respectively. It is seen from the results that for fl = I (isotropic case) the values obtained are exact for both the cases. However there is little discrepancy between the results presented here and those of reference [6] for other values of ft.
Structural Engineering Division, Vikram Sarabhai Space Centre, Trivandrum-69 5022, lndia
K. SINGA RAO K. GANAPATHI G. VENKATESWARARAO
(Received 23 April 1974)
REFERENCES 1. T. AKASAKAand T. TAKAGISHI1958 lhdletin of the Japanese Society of Mechanical Engineers 1,215-221. Vibration of corrugated diaphragm. 2. J. MOSSAKOWSKIand K. BORSUK1960 in .4ppliedMechanics (edited by A. Rolla and N. Koiter), 266-269. Buckling and vibrations of circular plates with cylindrical orthotropy. Amsterdam: Elsevier. 3. L A. MINKARAHand W. H. HOPPMANN1964 Journal of the Acoustical Society of America 36, 470475. Flexural vibrations of cylindrically aeolotropic circular plates. 4. K. A. V. PANDALAIand S. A. PATEL 1965 American Institute of Aeronautics and Astronautics Journal 3, 780-781. Natural frequencies of orthotropic circular plates. 5. P. G. K1RMSER,C. L. HUANGand H. K. Woo 1972 American blstitute of Aeronautics attdAstronautics Journal 10, 1690--1691. Vibration of cylindrically orthotropic circular plates. 6. H. K. Woo, P. G. KIRMSERand C. L. HUANG 1973 American Institute of Aeronautics andAstronauties Journal 11, 1421-1422. Vibration of orthotropic circular plates with concentric isotopic core.
7. H. M. AOELMAN,D. S. CATHERINESand W. C. WALTON,JR. 1969 NASA TN D-4972. A method for computation of vibration modes and frequencies of orthotropie thin shells of revolution having general meridional curvature.
LETTERS TO THE EDITOR VIBRATION OF CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATES Free vibration of a fixed circular plate possessing cylindrical orthotropy has been investigated by Akasaka and Takagishi [1 ], Mossakowski and Borsuk [2], Minkarah and H o p p m a n n [3], Pandalai and Patel [4] and Kirmser, Huang and Woo [5]. Woo, Kirmser and Huang [6] also investigated free vibration of a cylindrically orthotropic circular plate with concentric isotropic core for clamped and simply supported boundary conditions. The calculations of the results presented in references [1]-[4], which show some differences, were verified in reference [5]. In this note the results obtained by using a finite element method are presented. For the results presented in reference [5] it was assumed that there was an TABLE 1
Axisymmetrie frequency parameters 21/2[= (phD,)l/2 a2tn]for a clamped orthotropic circular plate with a concentric isotropic core 2 '/z for values ofbla A
.B 0"2 0"4 0"6 0"8 I-0 1-2 1"4 1"6 1"8 2"0
0.00
0-2
8-2120 8"8359 9"3437 9"7964 10"2162 10"6140 10-9977 11-3710 11-7378 12"1004
8"8670 9"2107 9"5378 9"8842 10"2162 10-5465 10"8758 11"2048 11"5352 11"8667
0-5 9-8248 9"9221 10"0198 10"1178 10-2162 10-3150 10"4142 10"5137 10"6136 10"7139
TABLE 2
AMsymnletric frequency parameter 21/2 [= (phD,)t/2 a2 og]for a simply supported orthotropic circular plate with concentric isotropic core 2 ~t2 for values ofb/a
[I
0.00
0"2
0.5
0-2 0"4 0'6 0"8 I'0 1.2 1"4 1"6 1"8 2"0
2"6769 3"4182 3"9925 4"4872 4-9352 5"3521 5"7473 6"1271 6-4954 6"8556
3"1239 3"6383 4"1018 4"5307 4"9352 5"3213 5"6939 6-0561 6"4105 6-7591
3"8790 4"1638 4"4330 4.6894 4-9352 5"1719 5-4007 5"6228 5"8390 6-0500
433
434
LETTERS TO THE EDITOR
isotropic core at the centre, of a radius b which was 0.025 of the radius a of the orthotropic plate. This assumption is not necessary in the finite element approach. The frequency parameters presented here were obtained by using a shell finite element [7] with 6 degrees of freedom at each node. Five elements were used and to avoid a singularity at the centre a hole with a radius of 0.001 of the outer radius was assumed. Tables 1 and 2 give the frequency parameters for various ratios of radius of isotropic core to the outer radius (namely 0"5, 0-2 and 0.0) for different values offl(=Ee/E,) for clamped and simply supported boundary conditions, respectively. It is seen from the results that for fl = I (isotropic case) the values obtained are exact for both the cases. However there is little discrepancy between the results presented here and those of reference [6] for other values of ft.
Structural Engineering Division, Vikram Sarabhai Space Centre, Trivandrum-69 5022, lndia
K. SINGA RAO K. GANAPATHI G. VENKATESWARARAO
(Received 23 April 1974)
REFERENCES 1. T. AKASAKAand T. TAKAGISHI1958 lhdletin of the Japanese Society of Mechanical Engineers 1,215-221. Vibration of corrugated diaphragm. 2. J. MOSSAKOWSKIand K. BORSUK1960 in .4ppliedMechanics (edited by A. Rolla and N. Koiter), 266-269. Buckling and vibrations of circular plates with cylindrical orthotropy. Amsterdam: Elsevier. 3. L A. MINKARAHand W. H. HOPPMANN1964 Journal of the Acoustical Society of America 36, 470475. Flexural vibrations of cylindrically aeolotropic circular plates. 4. K. A. V. PANDALAIand S. A. PATEL 1965 American Institute of Aeronautics and Astronautics Journal 3, 780-781. Natural frequencies of orthotropic circular plates. 5. P. G. K1RMSER,C. L. HUANGand H. K. Woo 1972 American blstitute of Aeronautics attdAstronautics Journal 10, 1690--1691. Vibration of cylindrically orthotropic circular plates. 6. H. K. Woo, P. G. KIRMSERand C. L. HUANG 1973 American Institute of Aeronautics andAstronauties Journal 11, 1421-1422. Vibration of orthotropic circular plates with concentric isotopic core.
7. H. M. AOELMAN,D. S. CATHERINESand W. C. WALTON,JR. 1969 NASA TN D-4972. A method for computation of vibration modes and frequencies of orthotropie thin shells of revolution having general meridional curvature.