Thermal Effect on Vibration of Parallelogram Plate of Bi-Direction Linearly Varying Thickness

ABSTRACT

In this paper, the effect of thermal gradient on the vibration of parallelogram plate with linearly varying thickness in both direction having clamped boundary conditions on all the four edges is analyzed. Thermal effect on vibration of such plate has been taken as one-dimensional distribution in linear form only. An approximate but quiet convenient frequency equation is derived using Rayleigh-Ritz technique with a two-term deflection function. The frequencies corresponding to the first two modes of vibration of a clamped parallelogram plate have been computed for different values of aspect ratio, thermal gradient, taper constants and skew angle. The results have been presented in tabular forms. The results obtained in this study are reduced to that of unheated parallelogram plates of uniform thickness and have generally been compared with the published one.

In this paper, the effect of thermal gradient on the vibration of parallelogram plate with linearly varying thickness in both direction having clamped boundary conditions on all the four edges is analyzed. Thermal effect on vibration of such plate has been taken as one-dimensional distribution in linear form only. An approximate but quiet convenient frequency equation is derived using Rayleigh-Ritz technique with a two-term deflection function. The frequencies corresponding to the first two modes of vibration of a clamped parallelogram plate have been computed for different values of aspect ratio, thermal gradient, taper constants and skew angle. The results have been presented in tabular forms. The results obtained in this study are reduced to that of unheated parallelogram plates of uniform thickness and have generally been compared with the published one.

Cite this paper

nullA. Gupta, M. Kumar, S. Kumar and A. Khanna, "Thermal Effect on Vibration of Parallelogram Plate of Bi-Direction Linearly Varying Thickness,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 33-38. doi: 10.4236/am.2011.21004.

nullA. Gupta, M. Kumar, S. Kumar and A. Khanna, "Thermal Effect on Vibration of Parallelogram Plate of Bi-Direction Linearly Varying Thickness,"

References

[1] A. W. Leissa, “Vibration of Plates,” NASA SP-160, 1969.

[2] A. W. Leissa, “Recent Studies in Plate Vibration 1981-1985, Part-II Complicating Effect,” The Shock and Vibration Digest, Vol. 19, No. 3, 1987, pp. 10-24. doi: 10.1177/058310248701900304

[3] A. W. Leissa, “Recent Studies in Plate Vibration 1982-1985, Part-I Classical Theory,” The Shock and Vibration Digest, Vol. 19, 1987, pp. 11-18. doi:10.1177/ 058310248701900204

[4] B. Singh and S. Chakraverty, “Flexural Vibration of Skew Plates Using Boundary Characteristic Orthogonal Polynomials in Two Variables,” Journal of Sound and Vibration, Vol. 173, No. 2, 1994, pp. 157-178. doi: 10.1006/jsvi.1994.1224

[5] P. S. Nair and S. Durvasula, “Vibration of Skew Plates,” Journal of Sound and Vibration, Vol. 26, 1973, pp. 1-20. doi:10.1016/S0022-460X(73)80201-9

[6] R. M. Orris and M. Petyt, “A Finite Element Study of Vibration of Trapezoidal Plates,” Journal of Sound and Vibration, Vol. 27, No. 3, 1973, pp. 325-344. doi: 10.1016/S0022-460X(73)80349-9

[7] P. A. A. Laura, R. H. Gutierrez and R. B. Bhat, “Transverse Vibration of a Trapezoidal Cantilever Plate of Variable Thickness,” AIAA Journal, Vol. 27, No. 7, 1989, pp. 921-922. doi:10.2514/3.10201

[8] K. M. Liew and M. K. Lim, “Transverse Vibration of Trapezoidal Plates of Variable Thickness: Symmetric Trapezoids,” Journal of Sound and Vibration, Vol. 165, No. 1, 1993, pp. 45-67. doi:10.1006/jsvi.1993.1242

[9] K. Y. Lam, K. M. Liew and S. T. Chow, “Free Vibration Analysis of Isotropic and Orthotropic Triangular Plates,” International Journal of Mechanical Sciences, Vol. 32, No. 5, 1990, pp. 455-464. doi:10.1016/0020-7403(90) 90172-F

[10] T. Sakiyama and M. Huang, “Free Vibration Analysis of Right Triangular Plates with Variable Thickness,” Journal of Sound and Vibration, Vol. 234, No. 5, 2000, pp. 841-858. doi:10.1006/jsvi.2000.2903

[11] N. J. Hoff, “High Temperature Effect in Aircraft Structures,” Pergamon Press, New York, p.62.

[12] J. S. Tomar and A. K. Gupta “Thermal Effect of Frequencies of an Orthotropic Rectangular Plate of Linearly Varying Thickness,” Journal of Sound and Vibration, Vol. 90, No. 3, 1983, pp. 325-331. doi:10.1016/0022-460X(83) 90715-0

[13] J. S. Tomar and A. K. Gupta, “Thermal Effect of Axis Symmetric Vibration of an Orthotropic Circular Plate of Variable Thickness,” American Institute of Aeronautics and Astronautics Journal, Vol. 22, No. 7, 1984, pp. 1015-1017.

[14] N. S. Bhatnagar and A. K. Gupta, “Thermal Effect on Vibration of Visco-Elastic Elliptic Plate of Variable Thickness,” Proceedings of International Conference on Modeling and Simulation, Melbourne, 1987, pp. 424-429.

[15] N. S. Bhatnagar and A. K. Gupta, “Vibration Analysis of Visco-Elastic Circular Plate Subjected to Thermal Gradient,” Modeling, Simulation and Control, B, AMSE Press 15, Cairo, 1988, pp. 17-31.

[16] B. Singh and V. Saxena, “Transverse Vibration of Skew Plates with Variable Thickness,” Journal of Sound and Vibration, Vol. 206. No. 1, 1997, pp. 1-13. doi:10.1006/ jsvi.1997.1032

[17] A. K. Gupta and A. Khanna, “Vibration of Visco-Elastic Rectangular Plate with Linearly Thickness Variation in both Directions,” Journal of Sound and Vibration, Vol. 301, No. 3-5, 2007, pp. 450-457. doi:10.1016/j.jsv.2006. 01.074

[18] A. K. Gupta, Anuj Kumar and Y.K. Gupta, “Vibration of Visco Elastic Parallelogram Plate with Parabolic Thickness Variation,” Applied Mathematics, Vol. 1, No. 2, 2010, pp. 128-136. doi:10.4236/am.2010.12017

[19] D. V. Bambill, S. Maize and R. E. Rossi “Free Vibration of Super Elliptical Plates with Constant and Variable Thickness,” Journal of Sound and Vibration, Vol. 329, No. 21, 2010, pp. 4578-4580. doi:10.1016/j.jsv.2010.05. 007

[20] A. K. Gupta and Lalit Kumar “Effect of Thermal Gradient on Vibration of Non-Homogeneous Visco-Elastic Elliptic Plate of Variable Thickness,” Meccanica, Vol. 44, No. 5, 2009, pp. 507-518. doi:10.1007/s11012-008-9184-9

[21] W. L. Li, “Vibration Analysis of Rectangular Plate with General Elastic Boundary Supports,” Journal of Sound and Vibration, Vol. 273, No. 3, 2004, pp. 619-635. doi: 10.1016/S0022-460X(03)00562-5

[22] T. Sakiyama, M. Haung, H. Matuda and C. Morita, “Free Vibration of Orthotropic Square Plate with a Square Hole,” Journal of Sound and Vibration, Vol. 259, No. 1, 2003, pp. 66-80. doi:10.1006/jsvi.2002.5181

[1] A. W. Leissa, “Vibration of Plates,” NASA SP-160, 1969.

[2] A. W. Leissa, “Recent Studies in Plate Vibration 1981-1985, Part-II Complicating Effect,” The Shock and Vibration Digest, Vol. 19, No. 3, 1987, pp. 10-24. doi: 10.1177/058310248701900304

[3] A. W. Leissa, “Recent Studies in Plate Vibration 1982-1985, Part-I Classical Theory,” The Shock and Vibration Digest, Vol. 19, 1987, pp. 11-18. doi:10.1177/ 058310248701900204

[4] B. Singh and S. Chakraverty, “Flexural Vibration of Skew Plates Using Boundary Characteristic Orthogonal Polynomials in Two Variables,” Journal of Sound and Vibration, Vol. 173, No. 2, 1994, pp. 157-178. doi: 10.1006/jsvi.1994.1224

[5] P. S. Nair and S. Durvasula, “Vibration of Skew Plates,” Journal of Sound and Vibration, Vol. 26, 1973, pp. 1-20. doi:10.1016/S0022-460X(73)80201-9

[6] R. M. Orris and M. Petyt, “A Finite Element Study of Vibration of Trapezoidal Plates,” Journal of Sound and Vibration, Vol. 27, No. 3, 1973, pp. 325-344. doi: 10.1016/S0022-460X(73)80349-9

[7] P. A. A. Laura, R. H. Gutierrez and R. B. Bhat, “Transverse Vibration of a Trapezoidal Cantilever Plate of Variable Thickness,” AIAA Journal, Vol. 27, No. 7, 1989, pp. 921-922. doi:10.2514/3.10201

[8] K. M. Liew and M. K. Lim, “Transverse Vibration of Trapezoidal Plates of Variable Thickness: Symmetric Trapezoids,” Journal of Sound and Vibration, Vol. 165, No. 1, 1993, pp. 45-67. doi:10.1006/jsvi.1993.1242

[9] K. Y. Lam, K. M. Liew and S. T. Chow, “Free Vibration Analysis of Isotropic and Orthotropic Triangular Plates,” International Journal of Mechanical Sciences, Vol. 32, No. 5, 1990, pp. 455-464. doi:10.1016/0020-7403(90) 90172-F

[10] T. Sakiyama and M. Huang, “Free Vibration Analysis of Right Triangular Plates with Variable Thickness,” Journal of Sound and Vibration, Vol. 234, No. 5, 2000, pp. 841-858. doi:10.1006/jsvi.2000.2903

[11] N. J. Hoff, “High Temperature Effect in Aircraft Structures,” Pergamon Press, New York, p.62.

[12] J. S. Tomar and A. K. Gupta “Thermal Effect of Frequencies of an Orthotropic Rectangular Plate of Linearly Varying Thickness,” Journal of Sound and Vibration, Vol. 90, No. 3, 1983, pp. 325-331. doi:10.1016/0022-460X(83) 90715-0

[13] J. S. Tomar and A. K. Gupta, “Thermal Effect of Axis Symmetric Vibration of an Orthotropic Circular Plate of Variable Thickness,” American Institute of Aeronautics and Astronautics Journal, Vol. 22, No. 7, 1984, pp. 1015-1017.

[14] N. S. Bhatnagar and A. K. Gupta, “Thermal Effect on Vibration of Visco-Elastic Elliptic Plate of Variable Thickness,” Proceedings of International Conference on Modeling and Simulation, Melbourne, 1987, pp. 424-429.

[15] N. S. Bhatnagar and A. K. Gupta, “Vibration Analysis of Visco-Elastic Circular Plate Subjected to Thermal Gradient,” Modeling, Simulation and Control, B, AMSE Press 15, Cairo, 1988, pp. 17-31.

[16] B. Singh and V. Saxena, “Transverse Vibration of Skew Plates with Variable Thickness,” Journal of Sound and Vibration, Vol. 206. No. 1, 1997, pp. 1-13. doi:10.1006/ jsvi.1997.1032

[17] A. K. Gupta and A. Khanna, “Vibration of Visco-Elastic Rectangular Plate with Linearly Thickness Variation in both Directions,” Journal of Sound and Vibration, Vol. 301, No. 3-5, 2007, pp. 450-457. doi:10.1016/j.jsv.2006. 01.074

[18] A. K. Gupta, Anuj Kumar and Y.K. Gupta, “Vibration of Visco Elastic Parallelogram Plate with Parabolic Thickness Variation,” Applied Mathematics, Vol. 1, No. 2, 2010, pp. 128-136. doi:10.4236/am.2010.12017

[19] D. V. Bambill, S. Maize and R. E. Rossi “Free Vibration of Super Elliptical Plates with Constant and Variable Thickness,” Journal of Sound and Vibration, Vol. 329, No. 21, 2010, pp. 4578-4580. doi:10.1016/j.jsv.2010.05. 007

[20] A. K. Gupta and Lalit Kumar “Effect of Thermal Gradient on Vibration of Non-Homogeneous Visco-Elastic Elliptic Plate of Variable Thickness,” Meccanica, Vol. 44, No. 5, 2009, pp. 507-518. doi:10.1007/s11012-008-9184-9

[21] W. L. Li, “Vibration Analysis of Rectangular Plate with General Elastic Boundary Supports,” Journal of Sound and Vibration, Vol. 273, No. 3, 2004, pp. 619-635. doi: 10.1016/S0022-460X(03)00562-5

[22] T. Sakiyama, M. Haung, H. Matuda and C. Morita, “Free Vibration of Orthotropic Square Plate with a Square Hole,” Journal of Sound and Vibration, Vol. 259, No. 1, 2003, pp. 66-80. doi:10.1006/jsvi.2002.5181