Free Vibrations of Homogenous Isotropic Viscothermoelastic Spherical Curved Plates

D. K.  Sharma

doi:10.6180/jase.2016.19.2.04

Free Vibrations of Homogenous Isotropic Viscothermoelastic Spherical Curved Plates

D. K. Sharma This email address is being protected from spambots. You need JavaScript enabled to view it.^1,2

¹Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Lingowal, Sangrur (Punjab) 148106, India
²Department of Applied Sciences and Humanities, Shiva Institute of Engineering and Technology, Bilaspur (HP) 174004, India

Received: July 9, 2015
Accepted: November 4, 2015
Publication Date: June 1, 2016

Download Citation: ||https://doi.org/10.6180/jase.2016.19.2.04

ABSTRACT

In this paper free vibration analysis of viscothermoelastic spherical curved plate has been presented to study the spheroidal and toroidal vibrations. The basic governing partial differential equations have been reduced to ordinary differential equations for time harmonic vibrations. Coupled systems of equations represent spheroidal vibrations, while uncoupled equation corresponds to toroidal vibrations. These uncoupled vibrations remain independent of temperature variations. Matrix Fröbenious method of extended power series has been applied to derive the formal solution of the coupled system of ordinary differential equations. The analytical results have been computed numerically for polymethyle methacrylate material. The fundamental iteration technique have been used to compute eigen values and corresponding eigen functions to represent field quantities with the help of MATLAB software. The numerical results in respect of lowest frequency, dissipation factor, stresses, displacements and temperature change have been presented graphically.

Keywords: Toroidal, Spheroidal, Stresses, Vibrations, Matrix Fröbenius Method, Spherical Structures

REFERENCES

[1] Liu, G. and Qu, J., “Guided Circumferential Waves in a Circular Annulus,” J. Appl. Mechs. ASME, Vol. 65, No. 2, pp. 424430 (1998). doi: 10.1115/1.2789071
[2] Towfighi, S., “Elastic Wave Propagation in Circumferential Direction in Anisotropic Pipes,” Ph.D. Dissertation, The University of Arizona, Tucson, AZ (2001).
[3] Towfighi, S., Kundu, T. and Ehsani, M., “Elastic Wave Propagation in Circumferential Direction in Anisotropic Cylindrical Curved Plates,” J. Appl. Mechs. ASME, Vol. 69, pp. 283291 (2002). doi: 10.1115/1.1464872
[4] Hunter, C., Sneddon, I. and Hill, R., Visco-elastic Waves in Progress in Solid Mechanics. Wiley Interscience, New York (1960).
[5] Flugge, W., Visco-elasticity, BLASDELL, London (1960).
[6] Chen, W. Q., Cai, J. B., Ye, G. R. and Ding, H. J., “On Eigen Frequencies of an Anisotropic Sphere,” J. Appl. Mechs. ASME., Vol. 67, No. 2, pp. 422424 (2000). doi: 10.1115/1.1303803
[7] Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Dover, New York (1994).
[8] Cohen, H., Shah, A. H. and Ramakrishan, C. V., “Free Vibrations of a Spherically Isotropic Hollow Sphere,” Acustica., Vol. 26, pp. 329333 (1972).
[9] Ding, H. J., Qiu, C. W. and Hong, L. Z., “Solutions to Equations of Vibrations of Spherical and Cylindrical Shells,” Appl. Maths Mechs. (Eng. Edi), Vol. 16, No. 1, pp. 115 (1995). doi: 10.1007/BF02453770
[10] Lord, H. W. and Shulman, Y. A., “A Generalization of Dynamical Theory of Thermoelasticity,” J. Mechs. Physics Solids., Vol. 15, No. 5, pp. 299309 (1967). doi: 10.1016/0022-5096(67)90024-5
[11] Green, A. E. and Lindsay, K. A., “Thermoelasticity,” J. Elasticity., Vol. 2, No. 1, pp. 17 (1972). doi: 10.1007/ BF00045689
[12] Hetnarski, R. B. and Ignaczac, J., “Generalized Thermoelasticity: Closed form Solutions,” J. Therm. Stresses., Vol. 16, No. 4, pp. 473498 (1993). doi: 10.1080/0149 5739308946241
[13] Othman, I. A. M., Ezzat, M. A., Zaki, S. A. and ElKaramany, A. S., “Generalized Thermo-viscoelasctic Plane Waves with Two Relaxation Times,” Int. J. Engig Scie., Vol. 40, No. 12, pp. 13291347 (2002). doi: 10. 1016/S0020-7225(02)00023-X
[14] Singh, R. P. and Jain, S. K., “Free Asymmetric Transverse Vibration of Parabolically Varying Thickness Polar Orthotropic Annular Plate with Flexible Edge Conditions,” Tamkang J. Science Engineering, Vol. 7, No. 1, pp. 4152 (2004). doi: 10.6180/jase.2004.7.1.07
[15] Sharma, J. N., “Some Considerations on the RayleighLamb Wave Propagation in Visco-thermoelastic Plates,” J. Vib. Cont., Vol. 11, No. 10, pp. 13111335 (2005). doi: 10.1177/1077546305058267
[16] Neuringer, J. L., “The Fröbenius Method for Complex Roots of the Indicial Equation,” Int. J. Math Edu. Sci Tech., Vol. 9, No. 1, pp. 7172 (1978). doi: 10.1080/00 20739780090110
[17] Towfighi, S. and Kundu, T., “Elastic Wave Propagation in Anisotropic Spherical Curved Plates,” Int. J. Solids Struct., Vol. 40, No. 20, pp. 54955510 (2003). doi: 10.1016/S0020-7683(03)00278-6
[18] Sharma, J. N. and Sharma, N., “Three Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermo-elastic Sphere,” J. Appl. Mech. ASME., Vol. 77, No. 2, pp. 02100419 (2009). doi: 10.1115/1.3172141
[19] Sharma, J. N. and Sharma, N., “3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method,” World J. Mechs., Vol. 2, No. 2, pp. 98112 (2012). doi: 10.4236/wjm.2012. 22012
[20] Sharma, J. N., Sharma, D. K. and Dhaliwal, S. S., “Free Vibration Analysis of a Viscothermoelastic Solid Sphere,” Int. J. Appl. Maths. Mechs., Vol. 8, No. 1, pp. 4568 (2012).
[21] Sharma, J. N., Sharma, D. K. and Dhaliwal, S. S., “Free Vibration Analysis of a Rigidly Fixed Viscothermoelastic Hollow Sphere,” Indian J. Pure Appl. Maths., Vol. 44, No. 5, pp. 559586 (2013). doi: 10.1007/s13 226-013-0030-y
[22] Sharma, J. N., Sharma, P. K. and Mishra, K. C., “Analysis of Free Vibrations in Axisymmetric Functionally Graded Thermoelastic Cylinder,” Acta Mechanica., Vol. 225, No. 6, pp. 15811594 (2014). doi: 10.1007/ s00707-013-1010-3
[23] Sharma, J. N., Sharma, D. K., Dhaliwal, S. S. and Walia, V., “Vibration Analysis of Axisymmetric Functionally Graded Viscothermoelastic Sphere,” Acta Mechanica Sinica., Vol. 30, No. 1, pp. 100111 (2014). doi: 10.1007/s10409-014-0016-y
[24] Hsu, M. H., “Vibration Analysis of Annular Plates,” Tamkang J. Science Engineering., Vol. 10, No. 3, pp. 193199 (2007). doi: 10.6180/jase.2007.10.3.02
[25] Sharma, S., Gupta, U. S. and Singhal, P., “Vibration Analysis of Non-Homogenous Orthotropic Rectangular Plates of Variable Thickness Resting on Winkler Foundation,” J. Appl. Science and Engineering., Vol. 15, No. 3, pp. 291300 (2012). doi: 10.6180/jase.2012. 15.3.10
[26] Khanna, A. and Sharma, A. K., “Natural Vibration of Visco-elastic Plate of Varying Thickness with Thermal Effects,” J. Appl. Science and Engineering., Vol. 16, No. 2, pp. 135140 (2013). doi: 10.6180/jase.2013.16. 2.04
[27] Dhaliwal, R. S. and Singh, A., Dynamic Coupled Thermo Elasticity, Hindustan Publication Corporation, New Delhi (1980).
[28] Watson, G. N., Theory of Bessel Functions, Cambredge University, Press London (1922).
[29] Cullen, C. G., Matrices and Linear Transformations, Addison-Wesley Publishing Company Reading Massachusetts, Palo Alto London (1966).
[30] Ding, H., Chen, W. and Zhang, L., Elasticity of Transversally Isotropic Materials, Springer, The Netherlands (2006).
[31] Richards, R. J. Principles of Solid Mechanics, CRC Press, Boca Raton London New York Washington, D.C. (2001).
[32] Ning, Y. U., Shoji, I. and Tatsuo, I., “Characteristics of Temperature Field due to Pulsed Heat Input Calculated by Non-Fourier Heat Conduction Hypothesis”, JSME, Int. J. Ser. A., Vol. 47, pp. 574580 (2004). doi: 10. 1299/jsmea.47.574