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Research Papers

# Three-Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere

[+] Author and Article Information
J. N. Sharma

Department of Applied Sciences, National Institute of Technology, Hamirpur 177005, Indiajns@nitham.ac.in

N. Sharma

Department of Applied Sciences, National Institute of Technology, Hamirpur 177005, Indianivi2j@gmail.com

J. Appl. Mech 77(2), 021004 (Dec 09, 2009) (9 pages) doi:10.1115/1.3172141 History: Received August 30, 2008; Revised February 19, 2009; Published December 09, 2009; Online December 09, 2009

## Abstract

In the present paper, an exact three-dimensional vibration analysis of a transradially isotropic, thermoelastic solid sphere subjected to stress-free, thermally insulated, or isothermal boundary conditions has been carried out. Nondimensional basic governing equations of motion and heat conduction for the considered thermoelastic sphere are uncoupled and simplified by using Helmholtz decomposition theorem. By using a spherical wave solution, a system of governing partial differential equations is further reduced to a coupled system of three ordinary differential equations in radial coordinate in addition to uncoupled equation for toroidal motion. Matrix Fröbenious method of extended power series is used to investigate motion along radial coordinate from the coupled system of equations. Secular equations for the existence of various types of possible modes of vibrations in the sphere are derived in the compact form by employing boundary conditions. Special cases of spheroidal and toroidal modes of vibrations of a solid sphere have also been deduced and discussed. It is observed that the toroidal motion remains independent of thermal variations as expected and spheroidal modes are in general affected by thermal variations. Finally, the numerical solution of the secular equation for spheroidal motion ($S$-modes) is carried out to compute lowest frequency and dissipation factor of different modes with MATLAB programming for zinc and cobalt materials. Computer simulated results have been presented graphically. The analyses may find applications in aerospace, navigation, and other industries where spherical structures are in frequent use.

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## Figures

Figure 3

Variation in lowest frequency (Ω) versus spherical harmonics (n) in cobalt material for different values of radius (R)

Figure 2

Variation in dissipation versus spherical harmonics (n) in zinc material for different values of radius (R)

Figure 1

Variation in lowest frequency (Ω) versus spherical harmonics (n) in zinc material for different values of radius (R)

Figure 5

Variation in lowest frequency (Ω) versus thermal conductivity ratio (K¯) for different spherical harmonics (n) in zinc material for radius R=1

Figure 6

Variation in dissipation (D) versus thermal conductivity ratio (K¯) for different spherical harmonics (n) in zinc material for radius R=1

Figure 7

Variation in lowest frequency (Ω) versus thermal conductivity ratio (K¯) for different spherical harmonics (n) in cobalt material for radius R=1

Figure 8

Variation in dissipation (D) versus thermal conductivity ratio (K¯) for different spherical harmonics (n) in cobalt material for radius R=1

Figure 4

Variation in dissipation (D) versus spherical harmonics (n) in cobalt material for different values of radius (R)

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