Three-Dimensional Free Vibration Analysis of a Viscothermoelastic Hollow Sphere
Affiliation(s)
Department of Mathematics, National Institute of Technology, Hamirpur (HP), India. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Sangrur, India.
Department of Mathematics, National Institute of Technology, Hamirpur (HP), India. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Sangrur, India.
ABSTRACT
This paper concentrates on the study of the three-dimensional free vibrations in a homogenous isotropic, viscothermoelastic hollow sphere whose surfaces are subjected to stress free, thermally insulated or isothermal boundary conditions. The use of governing partial differential equations is solved into a coupled system of ordinary differential equations. The equation for toroidal motion gets decoupled from rest of the motion and remains unaffected due to thermal variations. Matrix Fr?benious method of extended power series is employed to obtain the solution. The secular equations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived in the com- pact form. The special cases of spheroidal and toroidal modes of vibrations of a hollow sphere have also been deduced and discussed. In order to explore the characteristics of vibrations the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software tools. The computer simulated results have been presented graphically for copper material.
This paper concentrates on the study of the three-dimensional free vibrations in a homogenous isotropic, viscothermoelastic hollow sphere whose surfaces are subjected to stress free, thermally insulated or isothermal boundary conditions. The use of governing partial differential equations is solved into a coupled system of ordinary differential equations. The equation for toroidal motion gets decoupled from rest of the motion and remains unaffected due to thermal variations. Matrix Fr?benious method of extended power series is employed to obtain the solution. The secular equations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived in the com- pact form. The special cases of spheroidal and toroidal modes of vibrations of a hollow sphere have also been deduced and discussed. In order to explore the characteristics of vibrations the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software tools. The computer simulated results have been presented graphically for copper material.
Cite this paper
J. Sharma, D. Sharma and S. Dhaliwal, "Three-Dimensional Free Vibration Analysis of a Viscothermoelastic Hollow Sphere," Open Journal of Acoustics, Vol. 2 No. 1, 2012, pp. 12-24. doi: 10.4236/oja.2012.21002.
J. Sharma, D. Sharma and S. Dhaliwal, "Three-Dimensional Free Vibration Analysis of a Viscothermoelastic Hollow Sphere," Open Journal of Acoustics, Vol. 2 No. 1, 2012, pp. 12-24. doi: 10.4236/oja.2012.21002.
References
[1] W. Q. Chen, J. B. Cai, G. R. Ye and H. J. Ding, “On Eigen Frequencies of an Anisotropic Sphere,” Journal of Applied Mechanics, Vol. 67, No. 2, 2000, pp. 422-424. doi:10.1115/1.1303803 [2] A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity,” Dover Publications, New York, 1944. [3] H. Cohen, A. H. Shah and C. V Ramakrishan, “Free Vibrations of a Spherically Isotropic Hollow Sphere,” Acustica, Vol. 26, 1972, pp. 329-333. [4] H. J. Ding, C. W. Qiu and L. Z. Hong, “Solutions to Equations of Vibrations of Spherical and Cylindrical Shells,” Applied Mathematics and Mechanics, Vol. 16, No. 1, 1995, pp. 1-15. doi:10.1007/BF02453770 [5] H. W. Lord and Y. Shulman, “A Generalization of Dynamical Theory of Thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5 [6] E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 77, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689 [7] R. B. Hetnarski and J. Ignaczac, “Generalized Thermoelasticity: Closed Form-Solutions,” Journal of Thermal Stresses, Vol. 16, No. 4, 1993, pp. 473-498. doi:10.1080/01495739308946241 [8] G. R. Buchanan and G. R. Ramirez, “A Note on the Vibration of Transversely Isotropic Solid Spheres,” Journal of Sound and Vibration, Vol. 253, No. 3, 2002, pp. 724- 732. doi:10.1006/jsvi.2001.4054 [9] J. N Sharma and N. Sharma, “Three-Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere,” Journal of Applied Mechanics, Vol. 77, No. 2, 2010, pp. 1-9. doi:10.1115/1.3172141 [10] J. L. Neuringer, “The Fr?benius Method for Complex Roots of the Indicial Equation,” International Journal of Mathematical Education in Science and Technology, Vol. 9, No. 1, 1978, pp. 71-72. doi:10.1080/0020739780090110 [11] C. Hunter, “Visco-Elastic Waves in Progress in Solid Mechanics,” Wiley Inter-Science, New York, 1960. [12] W. Flugge, “Visco-Elasticity,” Blasdell, London, 1960. [13] S. Mukhopadhyay, “Effect of Thermal Relaxation on Thermoviscoelastic Interactions in Unbounded Body with Spherical Cavity Subjected Periodic Load on the Boundary,” Journal of Thermal Stresses, Vol. 23, No. 7, 2000, pp. 675-684. doi:10.1080/01495730050130057 [14] J. N. Sharma, “Some Considerations on the Rayleigh-Lamb Wave Propagation in Visco-Thermoelastic Plates,” Journal of Vibration and Control, Vol. 11, No. 10, 2005, pp. 1311-1335. doi:10.1177/1077546305058267 [15] R. S. Dhaliwal and A. Singh, “Dynamic Coupled Thermoelasticity,” Hindustan Publishing Corporation, New Delhi, 1980. [16] C. G. Cullen, “Matrices and Linear Transformation,” Addison-Wesley Publishing Company, Reading, 1966. [17] H. Ding, W. Chen and L. Zhang, “Elasticity of Transversely Isotropic Materials,” Springer, Amsterdam, 2006.
[1] W. Q. Chen, J. B. Cai, G. R. Ye and H. J. Ding, “On Eigen Frequencies of an Anisotropic Sphere,” Journal of Applied Mechanics, Vol. 67, No. 2, 2000, pp. 422-424. doi:10.1115/1.1303803 [2] A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity,” Dover Publications, New York, 1944. [3] H. Cohen, A. H. Shah and C. V Ramakrishan, “Free Vibrations of a Spherically Isotropic Hollow Sphere,” Acustica, Vol. 26, 1972, pp. 329-333. [4] H. J. Ding, C. W. Qiu and L. Z. Hong, “Solutions to Equations of Vibrations of Spherical and Cylindrical Shells,” Applied Mathematics and Mechanics, Vol. 16, No. 1, 1995, pp. 1-15. doi:10.1007/BF02453770 [5] H. W. Lord and Y. Shulman, “A Generalization of Dynamical Theory of Thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5 [6] E. Green and K. A. Lindsay, “Thermoelasticity,” Journal of Elasticity, Vol. 77, No. 1, 1972, pp. 1-7. doi:10.1007/BF00045689 [7] R. B. Hetnarski and J. Ignaczac, “Generalized Thermoelasticity: Closed Form-Solutions,” Journal of Thermal Stresses, Vol. 16, No. 4, 1993, pp. 473-498. doi:10.1080/01495739308946241 [8] G. R. Buchanan and G. R. Ramirez, “A Note on the Vibration of Transversely Isotropic Solid Spheres,” Journal of Sound and Vibration, Vol. 253, No. 3, 2002, pp. 724- 732. doi:10.1006/jsvi.2001.4054 [9] J. N Sharma and N. Sharma, “Three-Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere,” Journal of Applied Mechanics, Vol. 77, No. 2, 2010, pp. 1-9. doi:10.1115/1.3172141 [10] J. L. Neuringer, “The Fr?benius Method for Complex Roots of the Indicial Equation,” International Journal of Mathematical Education in Science and Technology, Vol. 9, No. 1, 1978, pp. 71-72. doi:10.1080/0020739780090110 [11] C. Hunter, “Visco-Elastic Waves in Progress in Solid Mechanics,” Wiley Inter-Science, New York, 1960. [12] W. Flugge, “Visco-Elasticity,” Blasdell, London, 1960. [13] S. Mukhopadhyay, “Effect of Thermal Relaxation on Thermoviscoelastic Interactions in Unbounded Body with Spherical Cavity Subjected Periodic Load on the Boundary,” Journal of Thermal Stresses, Vol. 23, No. 7, 2000, pp. 675-684. doi:10.1080/01495730050130057 [14] J. N. Sharma, “Some Considerations on the Rayleigh-Lamb Wave Propagation in Visco-Thermoelastic Plates,” Journal of Vibration and Control, Vol. 11, No. 10, 2005, pp. 1311-1335. doi:10.1177/1077546305058267 [15] R. S. Dhaliwal and A. Singh, “Dynamic Coupled Thermoelasticity,” Hindustan Publishing Corporation, New Delhi, 1980. [16] C. G. Cullen, “Matrices and Linear Transformation,” Addison-Wesley Publishing Company, Reading, 1966. [17] H. Ding, W. Chen and L. Zhang, “Elasticity of Transversely Isotropic Materials,” Springer, Amsterdam, 2006.