3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method

ABSTRACT

This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal), transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity. The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been uncoupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal motion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal relaxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT) and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where spherical structures are in frequent use.

This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal), transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity. The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been uncoupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal motion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal relaxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT) and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where spherical structures are in frequent use.

Cite this paper

nullJ. Sharma and N. Sharma, "3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method,"*World Journal of Mechanics*, Vol. 2 No. 2, 2012, pp. 98-112. doi: 10.4236/wjm.2012.22012.

nullJ. Sharma and N. Sharma, "3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method,"

References

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[2] H. W. Lord and Y. Shulman, “The Generalized Dynamical Theory of Thermoelasticity,” Journal of Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5

[3] A. E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689

[4] R. S. Dhaliwal and R. S. Sherief, “Generalized Thermoelasticity for Anisotropic Media,” Quarterly of Applied Mathematics, Vol. 38, No. 3, 1980, pp. 1-8.

[5] D. S. Chandrasekharaiah, “Thermoelasticity with Second Sound—A Review,” Applied Mechanics Review, Vol. 39, No. 3, 1986, pp. 355-376. doi:10.1115/1.3143705

[6] C. C. Ackerman, B. H. Bentman, A. Fairbank and R. A. Krumhansal, “Second sound in helium,” Physical Review Letters, Vol. 16, No. 18, 1966, pp. 789-791. doi:10.1103/PhysRevLett.16.789

[7] H. Singh and J. N. Sharma, “Generalized Thermoelastic Waves in Transversely Isotropic,” Journal of Acoustic Society of America, Vol. 77, No. 3, 1985, pp. 1046-1053. doi:10.1121/1.392391

[8] J. N. Sharma, “Three Dimensional Vibration Analysis of a Homogeneous Transversely Isotropic Cylindrical Panel,” Journal of Acoustical Society of America, Vol. 110, No. 1, 2001, pp. 254-259. doi:10.1121/1.1378350

[9] J. N. Sharma and P. K. Sharma, “Free Vibration Analysis of Homogeneous Transversely Isotropic Thermoelastic Cylindrical Panel,” Journal of Thermal Stresses, Vol. 25, No. 2, 2002, pp. 169-182. doi:10.1080/014957302753384405

[10] H. Lamb, “On the Vibrations of an Elastic Sphere,” Proceedings of the London Mathematical Society, Vol. 13, No. 1, 1882, pp. 189-212. doi:10.1112/plms/s1-13.1.189

[11] E. R. Lapwood and T. Usami, “Free Oscillations of the Earth,” Cambridge University Press, Cambridge, 1981.

[12] C. Chree, “The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Co-Ordinates,” Their Solution Trans- actions of the Cambridge Philosophical Society, Vol. 14, 1889, pp. 250-269.

[13] Y. Sato and T. Usami, “Basic Study on the Oscillation of Homogeneous Elastic Sphere-Part I,” Frequency of the Free Oscillations Geophysics Magazine, Vol. 31, No. 1, 1962, pp. 15-24.

[14] Y. Sato and T. Usami, “Basic Study on the Oscillation of Homogeneous Elastic Sphere-Part II Distribution of Displacement,” Geophysics Magazine, Vol. 31, No. 1, 1962, pp. 25-47.

[15] A. H. Shah, C. V. Ramkrishana and S. K. Datta, “Three Dimensional and Shell Theory Analysis of Elastic Waves in a Hollow Sphere: Part I—Analytical Foundation,” Jour- nal of Applied Mechanics, Vol. 36, No. 17, 1969, pp. 431- 439. doi:10.1115/1.3564698

[16] A. H. Shah, C. V. Ramkrishana and S. K. Datta, “Three Dimensional and Shell Theory Analysis of Elastic Waves in a Hollow Sphere: Part II—Numerical Results,” Journal of Applied Mechanics, Vol. 36, No. 3, 1969, pp. 440-444. doi:10.1115/1.3564699

[17] A. Gupta and S. J. Singh, “Toroidal Oscillations of a Transradially Isotropic Elastic Sphere,” Proceeding of Indian Academy of Science, Vol. 99, No. 3, 1990, pp. 383- 391.

[18] A. Bargi and M. R. Eslami, “Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory,” Journal of Thermal Stresses, Vol. 30, No. 12, 2007, pp. 1175-1193. doi:10.1080/01495730701519508

[19] J. N. Sharma and Sharma, N. (2011), “Vibration Analysis of Homogeneous Transradially Isotropic Generalized Thermoelastic Spheres”, Journal of Vibration and Acoustics, Vol. 133, No. 4, 2011, Article ID: 021004. doi:10.1115/1.4003396

[20] H. Cohen, A. H. Shah and C. V. Ramakrishna, “Free Vibrations of a Spherically Isotropic Hollow Sphere,” Acoustica, Vol. 26, 1972, pp. 329-333.

[21] J. L. Neuringer, “The Frobenius Method for Complex Roots of the Indicial Equation,” International Journal of Mathematics Education Science Technology, Vol. 9, No. 1, 1978, pp. 71-77. doi:10.1080/0020739780090110

[22] C. G. Cullen, “Matrices and Linear Transformations,” Addison-Wesley Publishing Company, London, 1966.

[23] H. J. Ding, W. Q. Chen and L. Zhang, “Elasticity of Transversely Isotropic Materials (Series: Solid Mechanics and Its Applications),” Springer, Dordrecht Netherland, 2006.

[24] A. E. |H. Love, “A Treatise on the Mathematical Theory of Elasticity,” Cambridge University Press, Cambridge, 1927.

[25] W. Q. Chen and H. J. Ding, “Free Vibration of Multi- Layered Spherically Isotropic Hollow Spheres,” Interna- tional Journal of Mechanical Sciences, Vol. 43, No. 3, 2001, pp. 667-680. doi:10.1016/S0020-7403(00)00044-8

[26] R. S. Dhaliwal and A. Singh, “Dynamic Coupled Thermoelasticity,” Hindustan, New Delhi, 2001.

[27] D. S. Chandrasekharaiah, “Hyperbolic Thermoelasticity, A Review of Recent Literature,” Applied Mechanics Re- view, Vol. 51, No. 12, 1998, pp. 705-729. doi:10.1115/1.3098984

[1] W. Nowacki, “Dynamic Problem of Thermoelasticity,” Noordhoff, Leyden, 1975.

[2] H. W. Lord and Y. Shulman, “The Generalized Dynamical Theory of Thermoelasticity,” Journal of Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5

[3] A. E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 2, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689

[4] R. S. Dhaliwal and R. S. Sherief, “Generalized Thermoelasticity for Anisotropic Media,” Quarterly of Applied Mathematics, Vol. 38, No. 3, 1980, pp. 1-8.

[5] D. S. Chandrasekharaiah, “Thermoelasticity with Second Sound—A Review,” Applied Mechanics Review, Vol. 39, No. 3, 1986, pp. 355-376. doi:10.1115/1.3143705

[6] C. C. Ackerman, B. H. Bentman, A. Fairbank and R. A. Krumhansal, “Second sound in helium,” Physical Review Letters, Vol. 16, No. 18, 1966, pp. 789-791. doi:10.1103/PhysRevLett.16.789

[7] H. Singh and J. N. Sharma, “Generalized Thermoelastic Waves in Transversely Isotropic,” Journal of Acoustic Society of America, Vol. 77, No. 3, 1985, pp. 1046-1053. doi:10.1121/1.392391

[8] J. N. Sharma, “Three Dimensional Vibration Analysis of a Homogeneous Transversely Isotropic Cylindrical Panel,” Journal of Acoustical Society of America, Vol. 110, No. 1, 2001, pp. 254-259. doi:10.1121/1.1378350

[9] J. N. Sharma and P. K. Sharma, “Free Vibration Analysis of Homogeneous Transversely Isotropic Thermoelastic Cylindrical Panel,” Journal of Thermal Stresses, Vol. 25, No. 2, 2002, pp. 169-182. doi:10.1080/014957302753384405

[10] H. Lamb, “On the Vibrations of an Elastic Sphere,” Proceedings of the London Mathematical Society, Vol. 13, No. 1, 1882, pp. 189-212. doi:10.1112/plms/s1-13.1.189

[11] E. R. Lapwood and T. Usami, “Free Oscillations of the Earth,” Cambridge University Press, Cambridge, 1981.

[12] C. Chree, “The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Co-Ordinates,” Their Solution Trans- actions of the Cambridge Philosophical Society, Vol. 14, 1889, pp. 250-269.

[13] Y. Sato and T. Usami, “Basic Study on the Oscillation of Homogeneous Elastic Sphere-Part I,” Frequency of the Free Oscillations Geophysics Magazine, Vol. 31, No. 1, 1962, pp. 15-24.

[14] Y. Sato and T. Usami, “Basic Study on the Oscillation of Homogeneous Elastic Sphere-Part II Distribution of Displacement,” Geophysics Magazine, Vol. 31, No. 1, 1962, pp. 25-47.

[15] A. H. Shah, C. V. Ramkrishana and S. K. Datta, “Three Dimensional and Shell Theory Analysis of Elastic Waves in a Hollow Sphere: Part I—Analytical Foundation,” Jour- nal of Applied Mechanics, Vol. 36, No. 17, 1969, pp. 431- 439. doi:10.1115/1.3564698

[16] A. H. Shah, C. V. Ramkrishana and S. K. Datta, “Three Dimensional and Shell Theory Analysis of Elastic Waves in a Hollow Sphere: Part II—Numerical Results,” Journal of Applied Mechanics, Vol. 36, No. 3, 1969, pp. 440-444. doi:10.1115/1.3564699

[17] A. Gupta and S. J. Singh, “Toroidal Oscillations of a Transradially Isotropic Elastic Sphere,” Proceeding of Indian Academy of Science, Vol. 99, No. 3, 1990, pp. 383- 391.

[18] A. Bargi and M. R. Eslami, “Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory,” Journal of Thermal Stresses, Vol. 30, No. 12, 2007, pp. 1175-1193. doi:10.1080/01495730701519508

[19] J. N. Sharma and Sharma, N. (2011), “Vibration Analysis of Homogeneous Transradially Isotropic Generalized Thermoelastic Spheres”, Journal of Vibration and Acoustics, Vol. 133, No. 4, 2011, Article ID: 021004. doi:10.1115/1.4003396

[20] H. Cohen, A. H. Shah and C. V. Ramakrishna, “Free Vibrations of a Spherically Isotropic Hollow Sphere,” Acoustica, Vol. 26, 1972, pp. 329-333.

[21] J. L. Neuringer, “The Frobenius Method for Complex Roots of the Indicial Equation,” International Journal of Mathematics Education Science Technology, Vol. 9, No. 1, 1978, pp. 71-77. doi:10.1080/0020739780090110

[22] C. G. Cullen, “Matrices and Linear Transformations,” Addison-Wesley Publishing Company, London, 1966.

[23] H. J. Ding, W. Q. Chen and L. Zhang, “Elasticity of Transversely Isotropic Materials (Series: Solid Mechanics and Its Applications),” Springer, Dordrecht Netherland, 2006.

[24] A. E. |H. Love, “A Treatise on the Mathematical Theory of Elasticity,” Cambridge University Press, Cambridge, 1927.

[25] W. Q. Chen and H. J. Ding, “Free Vibration of Multi- Layered Spherically Isotropic Hollow Spheres,” Interna- tional Journal of Mechanical Sciences, Vol. 43, No. 3, 2001, pp. 667-680. doi:10.1016/S0020-7403(00)00044-8

[26] R. S. Dhaliwal and A. Singh, “Dynamic Coupled Thermoelasticity,” Hindustan, New Delhi, 2001.

[27] D. S. Chandrasekharaiah, “Hyperbolic Thermoelasticity, A Review of Recent Literature,” Applied Mechanics Re- view, Vol. 51, No. 12, 1998, pp. 705-729. doi:10.1115/1.3098984