Three-dimensional Free Vibration Analysis of a Transversely Isotropic Thermoelastic Diffusive Cylindrical Panel

Document Type: Research Paper

Authors

Department of Mathematics, Kurukshetra University

Abstract

The present paper is aimed to study an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic, cylindrical panel based on three-dimensional generalized theories of thermoelastic diffusion. After applying the displacement potential functions in the basic governing equations of generalized thermoelastic diffusion, it is noticed that a purely transverse mode is independent of thermal and concentration fields and gets decoupled from the rest of motion. The equations for free  vibration problem are reduced to four equations of second-order and one fourth-order ordinary differential equation after expanding the displacement potential, temperature change and concentration functions with an orthogonal series. The formal solution of this system of equations can be expressed by using modified Bessel function with complex arguments.  The numerical results for lowest frequency have been obtained and presented graphically. The effect of diffusion on lowest frequency has also been presented graphically. Some special cases of secular equation are also discussed.

Keywords


[1] Aouadi M., 2007, Uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion, Journal of Thermal Stresses 30: 665-678.

[2] Aouadi M., 2008, Generalized theory of thermoelastic diffusion for anisotropic media, Journal of Thermal Stresses 31: 270-285.

[3] Bert C.W., Baker J.K., Egle D.M., 1969, Free vibrations of multilayer anisotropic cylindrical shells, Journal of Composite Materials 3: 1173-1186.

[4] Buchwald V.P., 1961, Rayleigh waves in transversely isotropic media, Quarterly Journal of Mechanics and Applied Mathematics, 14: 193-304.

[5] Chau K.T., 1994, Vibrations of transversely isotropic circular cylinders, ASME Journal of Applied Mechanics 61: 964-970.

[6] Fan J.R.., Ding K.W., 1993, Analytical solution for thick closed laminated cylindrical shells, International Journal of Mechanical Sciences 35: 657-668.

[7] Gawinecki J.A.., Kacprzyk P., Bar-Yoseph P., 2000, Initial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body, J. Anal. Appl, 19: 121-130.

[8] Gawinecki J. A., Szymaniec A., 2002, Global solution of the cauchy problem in nonlinear thermoelastic diffusion in solid body, Proceedings in Applied Mathematics and Mechanics 1: 446-447.

[9] Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.

[10] Ip K.H., Chan W.K., Tse P.C., Lai T.C., 1996, Vibration analysis of orthotropic cylindrical shells with free ends by Rayleigh-Ritz method, Journal of Sound and Vibration 195: 117-135.

[11] Jiang X.Y.,1997, 3-D vibration analysis of fiber reinforced composite laminated cylindrical shells, Journal of Vibration and Acoustics 119: 46-51.

[12] Jones R.M., Margan H., 1975, Buckling and vibration of cross ply laminated circular cylindrical shells, AIAA Journal 13: 664-671.

[13] Kumar R., Kansal T., 2008, Propagation of lamb waves in transversely isotropic thermoelastic diffusive plate, International Journal of Solids and Structures 45: 5890-5913.

[14] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299-309.

[15] Mirsky I., 1965, Wave propagation in transversely isotropic circular cylinders (part I and II), Journal of the Acoustical Society of America 37: 1016-1026.

[16] Nowacki W., 1974, Dynamical problems of thermodiffusion in solids-I, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 55- 64.

[17] Nowacki W., 1974, Dynamical problems of thermodiffusion in solids-II, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 129-135.

[18] Nowacki W., 1974,Dynamical problems of thermodiffusion in solids-III, Bulletin of Polish Academy of Sciences Series, Science and Technology 22: 257-266.

[19] Nowacki W., 1976, Dynamical problems of thermodiffusion in solids, Engineering Fracture Mechanics 8: 261-266.

[20] Sharma J.N., Sharma P.K., 2002, Free vibration analysis of homogeneous transversely isotropic thermoelastic cylindrical panel, Journal of Thermal Stresses 25: 169-182.

[21] Sherief H.H., Hamza F., Saleh H., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591- 608.

[22] Sherief H.H., Saleh H., 2005, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 4484-4493.

[23] So J.Y., Leissa A.W., 1997, Free vibrations of thick hollow cylinders from three dimensional analysis, Journal of Vibration and Acoustics 119: 89-95.

[24] Soedel W., 1983, Simplified equations and solutions for the vibration of orthotropic cylindrical shells, Journal of Sound and Vibration 87: 555-566.

[25] Soldatos K.P., 1987, Influence of thickness shear deformation on free vibrations of rectangular plates and cylinders of antisymmetric angle ply construction, Journal of Sound and Vibration 119: 111-137.

[26] Soldatos K.P., Hadhgeorgian V.P., Three-dimensional solutions of the free vibration problem of homogeneous isotropic cylindrical shells and panels, Journal of Sound and Vibration 137: 369-384.

[27] Ye J.Q., Soldatos K.P., 1994, 3-D vibrations of laminated cylinders and cylindrical panels with symmetric and antisymmetric cross-ply layup, Composites part B: Engineering 4: 429-444.