The study of free vibrations in a generalized elastic solid has been a subject of extensive investigation in the literature. It is of great importance of variety applications in many engineering fields like Aerospace, Civil, Mechanical, Navel, Chemical and Nuclear Engineering. The generalized theory of elasticity has drawn widespread attention because it removes the physically unacceptable situation of the classical theory of elasticity. Some of spherical or a part of spherical shape structures are saturated soil, osseous tissues, sedimentary rocks and human body. The generalized theory of elasticity such as thermo elasticity was developed by Lord-Shulman ^{1},^{ }Green –Lindsay ^{2}. The effect of non-locality on free vibrations in a thermo-elastic hollow cylinder with diffusion was studied by Dinesh Kumar Verma et.al. ^{3}. Forced axisymmetric vibrations in an inhomogeneous piezoceramic hollow sphere are investigated by Grigorenko and Loza ^{4}. Hamdy ^{5} studied the effects of mechanical damage, radial distance, diffusion on Lord-Shulman^{'}s thermo elastic sphere. Eman ^{6} presented vibration analysis of a nanobeam due to a ramp type heating under Moore-Gibson-Thompson theory of thermo-elasticity. Free vibrations in the visco-thermo elastic hollow sphere are presented by Dinesh Kumar sharma and Himani Mittal ^{7} .

Some of authors like Somaiah ^{8}^{ }studied the rotation effect on radial vibrations, and inhomogeneous and attenuation waves respectively.

This present paper is arranged in the following manner. Governing equations are presented and solved in section 2, results and discussion are given with the help of MATLAB software in section 3 and overall conclusion in section 4.

With the usual notations of Eringen

where

We consider a homogeneous rotating elastic hollow solid sphere of inner radius

In this case,

With the help of equation (3) and fundamental vector calculus, the equations (1) and (2) reduces in the directions of

where ;

To study the propagation of harmonic waves, we seek the solution of the form

where A and B are arbitrary constants, q is the wave number,

Because of equation (9), the solutions (8) reduces of the form

where A_{1}, A_{2}, B_{1}, B_{2} are arbitrary constants and

The solutions of the hollow sphere with different boundary conditions are performed, the mixed boundary conditions which consist of two kinds of boundary conditions, the inner surface fixed and the outer surface free i.e.,

Inserting equations (11) and (12) in equation (14), we obtain the following system of homogeneous equations in A_{1}, A_{2}, B_{1}, B_{2}

The system (15) has non-trivial solutions if and only if

where

The equation (16) is the coupled (radial and tangential) dispersion relation of free vibrations in a rotating hollow sphere.

(i) When m_{1} vanishes i.e.,

(ii) When m_{2} vanishes, i.e.,_{ },

Under the theoretical illustrations presented in the previous sections, we now present some numerical results. The physical data of the magnesium crystal - like material which modelled as generalized elastic solid is given by Sharma et.al ^{11};

The values of angle

The coupled frequency (CF) curves for non rotation and rotations I, II and III in the given range of angle

Tangential frequencies are shown in figure (2). Also tangential waves in non-rotating material are faster than in rotating material. Coupled and tangential waves are rapidly jumped at 210^{0}

The comparative CF and TF curves are shown for non-rotation and the rotations-I, II, III in figures (3) to (6). All the frequency curves are compared in figure (7).

From these figures we observed that tangential waves are faster than coupled waves in rotating and non-rotating solid materials.

From the above graphical study we observed that CF and TF are inverse proportional to the angular rotation of the solid. TF is faster than to the CF in the non-rotating and rotating solid for free vibrations. All these frequency curves are rapidly jumped at

For deriving free vibrations in a hollow solid sphere, the basic equations are converted into spherical coordinates and solved by the method of plane harmonic solution. Three types of vibrations named as radial, tangential and coupled (radial and tangential) are derived for hollow solid sphere. The effect of rotations on free vibrations also discussed. Also we observed that:

Tangential and coupled vibrations in a non-rotating solid are faster than in rotating solid

Tangential vibrations are faster than coupled vibrations

Tangential and coupled vibrations are inverse proportional to the angular rotations of the solid

The frequencies are rapidly jumped at 210^{0}