Study the Effect of Thermal Gradient on Transverse Vibration of Non-Homogeneous Orthotropic Trapezoidal Plate of Parabolically Varying Thickness

ABSTRACT

The present paper deals with the effect of linearly temperature on transverse vibration of non-homogeneous orthotropic trapezoidal plate of parabolically varying thickness. The deflection function is defined by the product of the equations of the prescribed continuous piecewise boundary shape. The non homogeneity of the plate is characterized by taking linear variation of the Young's modulus and parabolically variation of the density of the material. The non homogeneity is assumed to arise due to the variation in the density of the plate material and it is taken as parabolically. Rayleigh Ritz method is used to evaluate the fundamental frequencies. The equations of motion, governing the transverse vibrations of orthotropic trapezoidal plates, are derived with boundary condition clamped-simply supported-clamped-simply supported. Frequencies corresponding to first two modes of vibration are calculated for the trapezoidal plate for various combinations of the parameters of the non-homogeneity, thermal gradient, taper constant and for different values of the aspect ratios and shown by figures. All The results presented here are entirely new and are not found elsewhere. Comparison can only be made for homogeneous plates, and in that cases the results have been compared with those found in the existing literatures and are in excellent agreement.

The present paper deals with the effect of linearly temperature on transverse vibration of non-homogeneous orthotropic trapezoidal plate of parabolically varying thickness. The deflection function is defined by the product of the equations of the prescribed continuous piecewise boundary shape. The non homogeneity of the plate is characterized by taking linear variation of the Young's modulus and parabolically variation of the density of the material. The non homogeneity is assumed to arise due to the variation in the density of the plate material and it is taken as parabolically. Rayleigh Ritz method is used to evaluate the fundamental frequencies. The equations of motion, governing the transverse vibrations of orthotropic trapezoidal plates, are derived with boundary condition clamped-simply supported-clamped-simply supported. Frequencies corresponding to first two modes of vibration are calculated for the trapezoidal plate for various combinations of the parameters of the non-homogeneity, thermal gradient, taper constant and for different values of the aspect ratios and shown by figures. All The results presented here are entirely new and are not found elsewhere. Comparison can only be made for homogeneous plates, and in that cases the results have been compared with those found in the existing literatures and are in excellent agreement.

KEYWORDS

Thermal Gradient, Vibration, Orthotropic Trapezoidal Plate, Parabolically Thickness, Non-Homogeneity

Thermal Gradient, Vibration, Orthotropic Trapezoidal Plate, Parabolically Thickness, Non-Homogeneity

Cite this paper

nullA. Gupta and S. Sharma, "Study the Effect of Thermal Gradient on Transverse Vibration of Non-Homogeneous Orthotropic Trapezoidal Plate of Parabolically Varying Thickness,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 1-10. doi: 10.4236/am.2011.21001.

nullA. Gupta and S. Sharma, "Study the Effect of Thermal Gradient on Transverse Vibration of Non-Homogeneous Orthotropic Trapezoidal Plate of Parabolically Varying Thickness,"

References

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

[2] A. W. Leissa, “Recent Studies in Plate Vibration: 1981-1985. Part Classical Theory,” The Shock and Vibration Digest, Vol. 19, No. 2, 1987, pp. 11-18. doi:10.1177/ 058310248701900204

[3] A. W. Leissa, Vibration of Plates, NASA SP-160, 1969.

[4] I. Chopra and S. Durvasula, “Vibration of Simple-Supported Trapezoidal Plates. I: Symmetric Trapezoids,” Journal of Sound and Vibration, Vol. 19, No. 4, 1971, pp. 379-392. doi:10.1016/0022-460X(71)90609-2

[5] P. S. Nair and S. Durvasula, “Vibration of Skew Plate,” Journal of Sound Vibration, Vol. 26, No. 1, 1973, pp. 1-20. doi:10.1016/S0022-460X(73)80201-9

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

[7] 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

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

[9] Y. Narita, K. Maruyama and M. Sonada “Transverse Vibration of Clamped Trapezoidal Plates Having Rectangular Orthotropy,” Journal of Sound and Vibration, Vol. 85, No. 3, 1982, pp. 315-322. doi:10.1016/0022-460X(82)90257-7

[10] S. Mirza and M. Bijlani, “Vibration of Triangular Plates,” American Institute of Aeronautics and Astronautics, Vol. 21, No. 10, 1983, pp. 1472-1475.

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

[12] H. T. Sahba, “Transverse Free Vibration of Fully Clamped Symmetrical Trapezoidal Plates,” Journal of Sound and Vibration, Vol. 126, No. 2, 1988, pp. 237-247. doi: 10.1016/0022-460X(88)90238-6

[13] P. Laura, R. Gutierrez and R. Bhat, “Transverse Vibrations of a Trapezoidal Cantilever Plate of Various Thickness,” AIAA, Vol. 27, No. 7, 1989, pp. 921-922. doi: 10.2514/3.10201

[14] K. M. Liew and K. Y. Lam, “A Rayleigh-Ritz Approach to Transverse Vibration of Isotropic and Anisotropic Trapezoidal Plates Using Orthogonal Plate Functions,” International Journal of Solids and Structures, Vol. 27, No. 2, 1991, pp. 189-203. doi:10.1016/0020-7683(91) 90228-8

[15] K. M. Liew, “Variable of Symmetric Laminated Cantilever Trapezoidal Composite Plates,” Journal of Mechanical Sciences, Vol. 34, No. 4, 1992, pp. 299-308. doi: 10.1016/0020-7403(92)90037-H

[16] 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

[17] M. S. Qatu, N. A. Jaber and A. W. Leissa, “Natural Frequencies for Completely Free Trapezoidal Plates,” Journal of Sound and Vibration, Vol. 167, No. 1, 1993, pp. 183-191. doi:10.1006/jsvi.1993.1328

[18] C. W. Lim and K. M. Liew, “Vibration of Pretwisted Cantilever Trapezoidal Symmetric Laminates,” Acta Mechanica, Vol. 111, No. 3-4, 1995, pp. 193-208. doi: 10. 1007/BF01376930

[19] T. Sakiyama and M. Hung, “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

[20] R. Lal, “Transverse Vibration of Orthotropic Non Uniform Rectangular Plate with Continuously Varying Density,” Indian Journal of Pure and Applied Mathematics, Vol. 34, No. 4, 2003, pp. 587-606.

[21] A. W. Leissa, “The Historical Bases of the Rayleigh and Ritz Methods,” Journal of Sound and Vibration, Vol. 287, No. 4-5, 2005, pp. 961-978. doi:10.1016/j.jsv.2004.12.021

[22] C.-H. Huang, C.-H. Hsu and Y.-K. Lin, “Experimental and Numerical Investigations for the Free Vibration of Cantilever Trapezoidal Plates,” Journal of the Chinese Institute of Engineers, Vol. 29, No. 5, 2006, pp. 863-872.

[23] A. K. Gupta, Johri Tripti and R. P. Vats, “Thermal Effect on Vibration of Non Homogeneous Orthotropic Rectangular Plate Having Bi-directional Parabolically Varying Thickness,” Proceedings of International Conference on Engineering and Computer Science, San Francisco, 2007, pp. 784-787.

[24] A. K. Gupta, Johri Tripti and R. P. Vats, “Thermal Gradient Effect on Vibration of a Non-Homogeneous Orthotropic Rectangular Plate Having Bi-direction Linearly Thickness Variation,” Meccanica, Vol. 45, No. 3, 2010, pp. 393-400. doi:10.1007/s11012-009-9258-3

[25] G. Karami, S. A. Shahpari and P. Malekzadeh, “DQM Analysis of Skewed and Trapezoidal Laminated Plates,” Computer Structures, Vol. 59, No. 3, 2003, pp. 393-402. doi: 10.1016/S0263-8223(02)00188-5

[26] A. K. Gupta and S. Sharma, “Thermally Induced Vibration of Orthotropic Trapezoidal Plate of Linearly Varying Thickness,” Journal of Vibration and Control (Accepted for publication).

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

[2] A. W. Leissa, “Recent Studies in Plate Vibration: 1981-1985. Part Classical Theory,” The Shock and Vibration Digest, Vol. 19, No. 2, 1987, pp. 11-18. doi:10.1177/ 058310248701900204

[3] A. W. Leissa, Vibration of Plates, NASA SP-160, 1969.

[4] I. Chopra and S. Durvasula, “Vibration of Simple-Supported Trapezoidal Plates. I: Symmetric Trapezoids,” Journal of Sound and Vibration, Vol. 19, No. 4, 1971, pp. 379-392. doi:10.1016/0022-460X(71)90609-2

[5] P. S. Nair and S. Durvasula, “Vibration of Skew Plate,” Journal of Sound Vibration, Vol. 26, No. 1, 1973, pp. 1-20. doi:10.1016/S0022-460X(73)80201-9

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

[7] 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

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

[9] Y. Narita, K. Maruyama and M. Sonada “Transverse Vibration of Clamped Trapezoidal Plates Having Rectangular Orthotropy,” Journal of Sound and Vibration, Vol. 85, No. 3, 1982, pp. 315-322. doi:10.1016/0022-460X(82)90257-7

[10] S. Mirza and M. Bijlani, “Vibration of Triangular Plates,” American Institute of Aeronautics and Astronautics, Vol. 21, No. 10, 1983, pp. 1472-1475.

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

[12] H. T. Sahba, “Transverse Free Vibration of Fully Clamped Symmetrical Trapezoidal Plates,” Journal of Sound and Vibration, Vol. 126, No. 2, 1988, pp. 237-247. doi: 10.1016/0022-460X(88)90238-6

[13] P. Laura, R. Gutierrez and R. Bhat, “Transverse Vibrations of a Trapezoidal Cantilever Plate of Various Thickness,” AIAA, Vol. 27, No. 7, 1989, pp. 921-922. doi: 10.2514/3.10201

[14] K. M. Liew and K. Y. Lam, “A Rayleigh-Ritz Approach to Transverse Vibration of Isotropic and Anisotropic Trapezoidal Plates Using Orthogonal Plate Functions,” International Journal of Solids and Structures, Vol. 27, No. 2, 1991, pp. 189-203. doi:10.1016/0020-7683(91) 90228-8

[15] K. M. Liew, “Variable of Symmetric Laminated Cantilever Trapezoidal Composite Plates,” Journal of Mechanical Sciences, Vol. 34, No. 4, 1992, pp. 299-308. doi: 10.1016/0020-7403(92)90037-H

[16] 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

[17] M. S. Qatu, N. A. Jaber and A. W. Leissa, “Natural Frequencies for Completely Free Trapezoidal Plates,” Journal of Sound and Vibration, Vol. 167, No. 1, 1993, pp. 183-191. doi:10.1006/jsvi.1993.1328

[18] C. W. Lim and K. M. Liew, “Vibration of Pretwisted Cantilever Trapezoidal Symmetric Laminates,” Acta Mechanica, Vol. 111, No. 3-4, 1995, pp. 193-208. doi: 10. 1007/BF01376930

[19] T. Sakiyama and M. Hung, “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

[20] R. Lal, “Transverse Vibration of Orthotropic Non Uniform Rectangular Plate with Continuously Varying Density,” Indian Journal of Pure and Applied Mathematics, Vol. 34, No. 4, 2003, pp. 587-606.

[21] A. W. Leissa, “The Historical Bases of the Rayleigh and Ritz Methods,” Journal of Sound and Vibration, Vol. 287, No. 4-5, 2005, pp. 961-978. doi:10.1016/j.jsv.2004.12.021

[22] C.-H. Huang, C.-H. Hsu and Y.-K. Lin, “Experimental and Numerical Investigations for the Free Vibration of Cantilever Trapezoidal Plates,” Journal of the Chinese Institute of Engineers, Vol. 29, No. 5, 2006, pp. 863-872.

[23] A. K. Gupta, Johri Tripti and R. P. Vats, “Thermal Effect on Vibration of Non Homogeneous Orthotropic Rectangular Plate Having Bi-directional Parabolically Varying Thickness,” Proceedings of International Conference on Engineering and Computer Science, San Francisco, 2007, pp. 784-787.

[24] A. K. Gupta, Johri Tripti and R. P. Vats, “Thermal Gradient Effect on Vibration of a Non-Homogeneous Orthotropic Rectangular Plate Having Bi-direction Linearly Thickness Variation,” Meccanica, Vol. 45, No. 3, 2010, pp. 393-400. doi:10.1007/s11012-009-9258-3

[25] G. Karami, S. A. Shahpari and P. Malekzadeh, “DQM Analysis of Skewed and Trapezoidal Laminated Plates,” Computer Structures, Vol. 59, No. 3, 2003, pp. 393-402. doi: 10.1016/S0263-8223(02)00188-5

[26] A. K. Gupta and S. Sharma, “Thermally Induced Vibration of Orthotropic Trapezoidal Plate of Linearly Varying Thickness,” Journal of Vibration and Control (Accepted for publication).