On Free Vibrations of Elastodynamic Problem in Rotating Non-Homogeneous Orthotropic Hollow Sphere

Abstract

The effect of non-homogenity and rotation on the free vibrations for elastodynamic problem of orthotropic hollow sphere is discussed. The free vibrations are studied on the basis of the linear elasticity. The determination is concerned with the eigenvalues of the natural frequency for mixed boundary conditions. The numerical results of the frequency equations are discussed in the presence and absence of non-homogenity and rotation. The computer simulated results indicate that the influence of non-homogenity and rotation in orthotropic material is pronounced.

1. Introduction

Hollow spheres are frequently encountered in engineering industries and the corresponding free vibration problem has become one of the basic problems in elastodynamics. The analyses for transient problems of spherical structures are important and interesting research fields for engineers and scientists. The applications for non-homogeneous orthotropic hollow sphere have continuously increased in some engineering areas, including aerospace, offshore, infrared detectors, frequency control filters, chemical vessels, information storage devices, and signal processing devices. Accidental failures of rotating sphere due to flexural vibrations have frequently occurred in rotodynamic machinery such as steam turbines and gas turbines. Free vibrations of elastodynamic have many applications in a micropolar porous cubic crystal, poroelastic material [1–3]. Many applications dealing with the elastic bodies and materials, we can only mention a few recent interesting investigations [4–8], the analysis of the dynamic problems of elastic bodies is an important and interesting research field for engineers and scientists. The hollow spheres are frequently used as structural components and their vibration characteristics are obviously important for practical design. Mahmoud et al. [1, 2] discussed the effect of the rotation on plane vibrations in a transversely isotropic infinite hollow cylinder and the effect of the rotation on wave motion through cylindrical bore in a micropolar porous cubic crystal. Mahmoud [3] studied wave propagation in cylindrical poroelastic dry bones. Abd-Alla and Mahmoud [8, 9] solved magnetothermoelastic problem in rotating non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model and investigated problem of radial vibrations in non-homogeneity isotropic cylinder under influence of initial stress and magnetic field. Influences of rotation, magnetic field, and gravity on Rayleigh waves in a homogeneous orthotropic elastic half space and the solution of electromechanical wave propagation are investigated by Abd-Alla et al. [10–13]. Marin et al. [14, 15] studied porous materials and nonsimple material problems addressed by the Lagrange's identity. Wang [16] studied the elastodynamic solution for an anisotropic hollow sphere. Ding et al. [17, 18] discussed elastodynamic solution of a non-homogeneous orthotropic hollow cylinder, a solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermo elastic problems. Inclusion of arbitrary shape in magnetoelectro-elastic composite materials has been investigated by Wang and Shen [19]. Ding et al. [20] obtained the analytical solution for the axisymmetric plane strain electroelastic dynamics of a non-homogeneous piezoelectric hollow cylinder. Hou and Leung [21] further study the corresponding problem of magnetoelectroelastic hollow cylinders. Buchanan and Liu [22] discussed an analysis of the free vibration of thick-walled isotropic toroidal shells. Yu et al. [23] investigated wave propagation in non-homogeneous magnetoelectroelastic hollow cylinders. Recently, Abd-Alla and Mahmoud [24] discussed analytical solution of wave propagation in non-homogeneous orthotropic rotating elastic media. Abd-Alla et al. [25] studied the effect of the rotation, magnetic field, and initial stress on peristaltic motion of micropolar fluid. Mahmoud [26] investigated wave propagation in piezoelectric hollow cylinder and influence of rotation and generalized magnetothermoelastic on Rayleigh waves in a granular medium under effect of initial stress and gravity field. Sharma et al. [27] studied free vibration analysis of a viscothermoelastic solid sphere. Abd-Alla et al. [28–33] investigated problem of radial and free vibrations in non-homogeneity cylinder under influence of initial stress rotation and magnetic field. Ozsahin and Taskner [34] investigated contact problem for an elastic layer on an elastic half plane. Daouadji et al. [35] studied the Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory. The present paper deals with the problem of free vibrationsof elastodynamic equations of rotating non-homogeneous and orthotropic hollow sphere. The effect of non-homogeneous and rotation in the equations of motion has been taken into account and the numerical results of the fundamental frequency equations are discussed. Comparisons are made with the result in the present and absence of non-homogeneous and rotation in cases of orthotropic hollow sphere.

2. Formulation of the Problem

Take the spherical coordinates (𝑟,𝜃,𝜑) and consider elastodynamic problem of non-homogeneous rotating hollow sphere of inner radius 𝑎 and outer radius 𝑏, as Figure 1. The stresses-strain relations for non-homogeneous spherically orthotropic material in two dimensions are in the form𝜎_𝑟𝑟=𝑟^2𝑚𝛼₁₁𝑒_𝑟𝑟+𝛼₁₂𝑒_𝜃𝜃+𝛼₁₃𝑒_𝜑𝜑,𝜎_𝜃𝜃=𝑟^2𝑚𝛼₁₂𝑒_𝑟𝑟+𝛼₂₂𝑒_𝜃𝜃+𝛼₂₃𝑒_𝜑𝜑,𝜎_𝜑𝜑=𝑟^2𝑚𝛼₁₃𝑒_𝑟𝑟+𝛼₂₃𝑒_𝜃𝜃+𝛼₃₃𝑒_𝜑𝜑,𝜏_𝑟𝜃=𝑟^2𝑚𝛼₄₄𝑒_𝑟𝜃,𝜏_𝑟𝜃=0,𝜏_𝜃𝜑=0. (1a)

Figure 1 

Problem geometry of non-homogeneous orthotropic material.

The strain-displacements relations in two dimensions are in the form 𝑒_𝑟𝑟=𝜕𝑢_𝑟𝜕𝑟,𝑒_𝜃𝜃=1𝑟𝜕𝑢_𝜃𝜕𝜃+𝑢_𝑟,𝑒_𝜑𝜑=1𝑢𝑟sin𝜃^𝑟sin𝜃+𝑢_𝜃,𝑒cos𝜃_𝑟𝜃=121𝑟𝜕𝑢_𝑟+𝜕𝜃𝜕𝑢_𝜃−𝑢𝜕𝑟𝑟^𝜃,𝑒_𝑟𝜑=0,𝑒_𝜃𝜑=0. (1b)

Substituting from (1b) into (1a) we obtain the stresses-displacements relations in two dimensions in the form 𝜎_𝑟𝑟=𝑟^1+2𝑚𝛼₁₁+𝛼₁₃𝑢_𝑟+𝛼₁₃𝑢_𝜃cot𝜃+𝛼₁₂𝜕𝑢_𝜃𝜕𝜃+𝑟𝛼₁₁𝜕𝑢_𝑟,𝜎𝜕𝑟_𝜃𝜃=𝑟^1+2𝑚𝛼₂₂+𝛼₂₂𝑢_𝑟+𝛼₂₃𝑢_𝜃cot𝜃+𝛼₂₂𝜕𝑢_𝜃𝜕𝜃+𝑟𝛼₁₂𝜕𝑢_𝑟,𝜎𝜕𝑟_𝜑𝜑=𝑟^1+2𝑚𝛼₂₃+𝛼₃₃𝑢_𝑟+𝛼₃₃𝑢_𝜃cot𝜃+𝛼₂₃𝜕𝑢_𝜃𝜕𝜃+𝑟𝛼₁₃𝜕𝑢_𝑟,𝜏𝜕𝜃_𝑟𝜃=12𝑟^1+2𝑚𝛼₄₄−𝑢_𝜃+𝜕𝑢_𝑟𝜕𝜃+𝑟𝜕𝑢_𝜃,𝜏𝜕𝑟_𝑟𝜃=0,𝜏_𝜃𝜑=0, (2) where 𝑢_𝑟 and 𝑢_𝜃 are, respectively, the components of displacement in the radial and tangential directions, 𝑒_𝑖𝑗 are the strain components, and 𝜎_𝑖𝑗 are the stress components. Where we have characterized the elastic constants 𝑐_𝑖𝑗 and the density 𝜌of non-homogeneous material in the form 𝑐_𝑖𝑗=𝛼_𝑖𝑗𝑟^2𝑚,𝜌=𝜌₀𝑟^2𝑚,𝑖,𝑗=1,2,3, (3) where 𝛼_𝑖𝑗 and 𝜌₀ are the values of 𝑐_𝑖𝑗 and 𝜌 in the homogeneous case, respectively, and 𝑚 is the non-homogeneous parameter. The displacement equations of motion in the rotating frame have two additional terms centripetal acceleration Ω^⇀×(Ω^⇀×^⇀𝑢)=(−Ω²𝑢_𝑟,−Ω²𝑢_𝜃,0), due to time varying motion only, where Ω^⇀=(0,0,Ω),  and^⇀𝑢=(𝑢_𝑟,𝑢_𝜃,0).

The elastodynamic equations of rotating non-homogeneous medium in two dimensions in the direction 𝑟,𝜃 are given by: 𝜕𝜎_𝑟𝑟+1𝜕𝑟𝑟𝜕𝜏_𝑟𝜃+1𝜕𝜃𝑟2𝜎_𝑟𝑟−𝜎_𝜃𝜃−𝜎_𝜑𝜑+𝜏_𝑟𝜃cot𝜃+𝜌₀𝑟^2𝑚Ω²𝑢_𝑟=𝜌₀𝑟^2𝑚𝜕²𝑢_𝑟𝜕𝑡²,𝜕𝜏_𝑟𝜃+1𝜕𝜃𝑟𝜕𝜎_𝜃𝜃+1𝜕𝜃𝑟𝜎_𝜃𝜃−𝜎_𝜑𝜑cot𝜃+3𝜏_𝑟𝜃+𝜌₀𝑟^2𝑚Ω²𝑢_𝜃=𝜌₀𝑟^2𝑚𝜕²𝑢_𝜃𝜕𝑡². (4)(5)

Substituting from (1a), (1b), and (2) into (4) and (5), we obtain: 𝑟^−1+𝑚2𝑎₀+𝑟²𝜌₀Ω²𝑢_𝑟+𝑎₁𝑢_𝜃cot𝜃+𝑎₂𝜕𝑢_𝜃𝜕𝜃+𝛼₄₄cot𝜃𝜕𝑢_𝑟−𝜕𝜃𝜕𝑢_𝜃+𝜕𝜕𝜃²𝑢_𝑟𝜕𝜃²+𝑟−2𝑟𝜌₀𝜕²𝑢_𝜃𝜕𝑡²+4(1+𝑚)𝛼₁₁𝜕𝑢_𝑟+𝜕𝑟2𝛼₁₃+𝛼₄₄cot𝜃𝜕𝑢_𝜃+𝜕𝑟2𝛼₁₂+𝛼₄₄𝜕²𝑢_𝜃𝜕𝑟𝜕𝜃+2𝛼₁₁𝑟𝜕²𝑢_𝑟𝜕𝑟²=0, (6) where 𝑎₀=𝛼₁₂+2𝑚𝛼₁₂+𝛼₁₃+2𝑚𝛼₁₃−𝛼₂₂−2𝛼₂₃−𝛼₃₃,  𝑎₁=((2+4𝑚)𝛼₁₃−2(𝛼₂₃+𝛼₃₃)−𝛼₄₄),  and 𝑎₂=2(𝛼₁₂+2𝑚𝛼₁₂−𝛼₂₂−𝛼₂₃).

One has 𝑟^−1+𝑚2𝛼₂₂−𝛼₃₃𝑢_𝑟𝑎cot𝜃−2₃−𝑟²𝜌₀Ω²+𝛼₃₃csc²𝜃𝑢_𝜃+𝑎₄𝜕𝑢_𝑟𝜕𝜃+2𝛼₂₂cot𝜃𝜕𝑢_𝜃+𝜕𝜕𝜃²𝑢_𝜃𝜕𝜃²+𝑟−2𝑟𝜌₀𝜕²𝑢_𝜃𝜕𝑡²+2𝛼₁₂−𝛼₁₃cot𝜃𝜕𝑢_𝑟()𝛼𝜕𝑟+21+𝑚₄₄𝜕𝑢_𝜃𝑎𝜕𝑟+𝑟₅𝜕²𝑢_𝑟𝜕𝑟𝜕𝜃+𝑟𝛼₄₄𝜕²𝑢_𝜃𝜕𝜃²=0, (7) where 𝑎₃=𝛼₂₃−𝛼₃₃+𝛼₄₄+𝑚𝛼₄₄,  𝑎₄=2(𝛼₂₂+𝛼₂₃+𝛼₄₄+𝑚𝛼₄₄),  and 𝑎₅=2𝛼₁₂+𝛼₄₄.

3. Solution of the Problem

By Helmohltz's theorem, the displacement vector 𝑢 can be written as 𝑢=∇Φ₁+∇∧Ψ, (8) where the two functions Φ₁and Ψ are known in the theory of elasticity, by Lame' potentials irrotational and rotatoinal parts of the displacement vector 𝑢, respectively. The displacement potentials are introduced for facilitating the solution of the field equations (5) and (6). It is possible to take only one components of the vector Ψ to be nonzero Ψ=(0,0,𝜓₁). From (8), we obtain 𝑢_𝑟=cot𝜃Ψ₁+(𝜕/𝜕𝜃)Ψ₁𝑟+𝜕Φ₁,𝑢𝜕𝑟_𝜃Ψ=−¹−𝜕Φ₁/𝜕𝜃+𝑟(𝜕/𝜕𝑟)Ψ₁𝑟. (9)

Substituting from (9) into (6) and (7) and regrouping them lead to the following equations for Φ₁ and Ψ₁: 𝑟^−2+𝑚2𝑎₁+𝑟²𝜌₀Ω²ℎ₁−𝑎₂cot𝜃𝜓₁−𝜕𝜙₁𝜕𝜃+𝑟𝜕𝜓₁𝜕𝑟−𝑎₃𝜕𝜓₁−𝜕𝜕𝜃²𝜙₁𝜕𝜃²𝜕+𝑟²𝜓₁𝜕𝑟𝜕𝜃+𝛼₄₄cot𝜃csc²𝜃𝜓₁−csc²𝜃𝜕𝜓₁−𝜕𝜕𝜃²𝜙₁𝜕𝜃²𝜕+2cot𝜃²𝜓₁𝜕𝜃²+𝜕³𝜓₁𝜕𝜃³+𝑟ℎ₂−2𝜌₀𝑟²𝜕cot𝜃²𝜓₁𝜕𝑡²+𝜕³𝜓₁𝜕𝜃𝜕𝑡²𝜕+𝑟³𝜙₁𝜕𝑟𝜕𝑡²+4(1+𝑚)𝛼₁₁𝑟²ℎ₃+2𝛼₁₃+𝛼₄₄×cot𝜃ℎ₃+2𝛼₁₂+𝛼₄₄ℎ₄+2𝛼₁₁2cot𝜃𝜓₁+2𝜕𝜓₁𝜕𝜃+𝑟ℎ₅=0, (10) where ℎ₁=cot𝜃𝜓₁+𝜕𝜓₁𝜕𝜃+𝑟𝜕𝜙₁,ℎ𝜕𝑟₂=𝜕cot𝜃²𝜙₁+𝜕𝜕𝑟𝜕𝜃²𝜓₁+𝜕𝜕𝑟𝜕𝜃³𝜙₁𝜕𝑟𝜕𝜃²,ℎ₃=−cot𝜃𝜓₁+𝜕𝜓₁𝜕𝜃−𝑟cot𝜃𝜕𝜓₁+𝜕𝜕𝑟²𝜓₁𝜕𝑟𝜕𝜃×𝑟⁻²+𝜕²𝜙₁𝜕𝑟²,ℎ₃=𝜓₁−𝜕𝜙₁𝜕𝜃−𝑟𝜕𝜓₁−𝜕𝜕𝑟²𝜙₁𝜕𝜕𝑟𝜕𝜃+𝑟²𝜓₁𝜕𝑟²,ℎ₄=𝜕𝜓₁−𝜕𝜕𝜃²𝜙₁𝜕𝜃²𝜕−𝑟²𝜓₁−𝜕𝜕𝑟𝜕𝜃³𝜙₁𝜕𝑟𝜕𝜃²𝜕+𝑟³𝜓₁𝜕𝑟²,ℎ𝜕𝜃₅=−2cot𝜃𝜕𝜓₁𝜕𝜕𝑟−2²𝜓₁𝜕𝜕𝑟𝜕𝜃+𝑟cot𝜃²𝜓₁𝜕𝑟²+𝜕³𝜓₁𝜕𝑟²𝜕𝜕𝜃+𝑟³𝜙₁𝜕𝑟³. (11) One has 𝑟^−2+𝑚2𝛼₂₂−𝛼₃₃cot𝜃cot𝜃𝜓₁+𝜕𝜓₁𝜕𝜃+𝑟𝜕𝜙₁𝑎𝜕𝑟+2₃−𝑟²𝜌₀Ω²+𝛼₃₃csc²𝜃𝜓₁−𝜕𝜙₁𝜕𝜃+𝑟𝜕𝜓₁𝑎𝜕𝑟+2₆−csc²𝜃𝜓₁+cot𝜃𝜕𝜓₁+𝜕𝜕𝜃²𝜓₁𝜕𝜃²𝜕+𝑟²𝜙₁𝜕𝑟𝜕𝜃−𝛼₂₂𝜕²𝜓₁𝜕𝜃²−𝜕³𝜙₁𝜕𝜃³+cot𝜃𝜕𝜓₁−𝜕𝜕𝜃²𝜙₁𝜕𝜃²𝜕+𝑟²𝜓₁𝜕𝜕𝑟𝜕𝜃+𝑟³𝜓₁𝜕𝑟𝜕𝜃²+𝑟²2𝜌₀𝜕²𝜓₁𝜕𝑡²−𝜕³𝜙₁𝜕𝜃𝜕𝑡²𝜕+𝑟³𝜓₁𝜕𝑟𝜕𝑡²−𝛼₄₄𝜕²𝜓₁𝜕𝜃²−𝜕³𝜙₁𝜕𝜃³𝜕+𝑟³𝜓₁𝜕𝑟𝜕𝜃²+2𝛼₁₂−𝛼₁₃−ℎ×cot𝜃⁸𝑟²+𝜕²𝜙₁𝜕𝑟²−2(1+𝑚)𝛼₄₄ℎ₇𝑟²+2𝛼₁₂+𝛼₄₄ℎ₆𝑟²=0, (12) where 𝑎₆=𝛼₂₂+𝛼₂₃+𝛼₄₄+𝑚𝛼₄₄,ℎ₆=−cot𝜃𝜕𝜓₁−𝜕𝜕𝜃²𝜓₁𝜕𝜃²+csc²𝜃𝜓₁−𝑟𝜕𝜓₁𝜕𝜕𝑟+𝑟cot𝜃²𝜓₁𝜕𝜕𝑟𝜕𝜃+𝑟³𝜓₁𝜕𝑟𝜕𝜃²𝑟𝜕³𝜙₁𝜕𝑟²,ℎ𝜕𝜃₇=−𝜓₁+𝜕𝜙₁𝜕𝜃+𝑟𝜕𝜓₁−𝜕𝜕𝑟²𝜙₁𝜕𝜕𝑟𝜕𝜃+𝑟²𝜓₁𝜕𝑟²,ℎ₈=cot𝜃𝜓₁+𝜕𝜓₁𝜕𝜃−𝑟cot𝜃𝜕𝜓₁+𝜕𝜕𝑟²𝜓₁.𝜕𝑟𝜕𝜃 (13)

To study the propagation of harmonic waves, we assume a solution in the form Φ₁(𝑟,𝜃,𝑡)=Φ₂(𝑟)𝑒^{𝑖(𝛾𝜃−𝜔𝑡)},Ψ₁(𝑟,𝜃,𝑡)=Ψ₂(𝑟)𝑒^{𝑖(𝛾𝜃−𝜔𝑡)}. (14)

Substituting from (14) into (10) and (12) and regrouping them lead to the following equations for Φ₂ and Ψ₂: 𝑒^{𝑖(𝛾𝜃−𝑡𝜔)}𝑟^−1+𝑚2𝛾𝑎₇−𝑎₈cot𝜃𝜙₂+−𝑖𝛾𝑎₉−2𝑟²𝜌₀𝜔²+Ω²+𝛼₄₄csc²𝜃+cot𝜃2𝑎₁₀+𝑟²𝜌₀𝜔²+Ω²+𝛼₄₄csc²𝜃Ψ₂+𝑟2𝑎₁₁+𝑟²𝜌₀𝜔²+Ω²+𝑖𝑎₁₃+𝛼₄₄𝛾cot𝜃𝑑Φ₂𝑎𝑑𝑟+2₁₂+𝑎₁₃cot𝜃𝑑Ψ₂4()𝑎𝑑𝑟+𝑟1+𝑚₁₁𝑑²Φ₂𝑑𝑟²+𝑎₁₄+𝑎₁₅𝑑cot𝜃²Ψ₂𝑑𝑟²+2𝑟𝛼₁₁𝑑³Φ₂𝑑𝑟³=0, (15) where 𝑎₇=−2𝑚𝛼₁₂+𝛼₂₂+𝛼₂₃+𝛼₄₄𝑎𝛾,₈=𝑖−2𝑚𝛼₁₃+𝛼₂₃+𝛼₃₃+𝛼₄₄,𝑎₉=−2𝛼₁₂+4𝑚𝛼₁₁−𝛼₁₃−2𝛼₁₃+2𝛼₂₃+2𝛼₃₃+𝛼₄₄−1+𝛾²,𝑎₁₀=𝛼₁₂+2𝑚−𝛼₁₁+𝛼₁₂+𝛼₁₃−𝛼₂₂−𝛼₂₃+𝛼₄₄−𝛼₄₄𝛾²,𝑎₁₁=𝛼₁₂+2𝑚𝛼₁₂+𝛼₁₃+2𝑚𝛼13−𝛼₂₂−2𝛼₂₃−𝛼₃₃−𝛼₁₂+𝛼₄₄𝛾²,𝑎₁₂=𝑖2𝑚𝛼₁₁−2𝛼₁₂−2𝑚𝛼₁₂+𝛼₂₂+𝛼₂₃𝑎𝛾,₁₃=22𝑚𝛼₁₁−2𝛼₁₃−2𝑚𝛼₁₃+𝛼₂₃+𝛼₃₃,𝑎₁₄=𝑖2𝛼₁₁−2𝑚𝛼₁₂−𝛼₄₄𝛾+2𝛼₁₁−2𝛼₁₃−𝛼₄₄,𝑎₁₅=2𝑚𝛼₁₁−2𝛼₁₃−𝛼₄₄. (16) One has 𝑒^{𝑖(𝛾𝜃−𝑡𝜔)}𝑟^−1+𝑚−𝑖𝛾𝑎₁₆+𝛼₄₄4+4𝑚+𝑟²𝛾²−2𝑟²𝜌₀𝜔²+Ω²−2𝑖𝛼₂₂𝛾cot𝜃+2𝛼33csc²𝜃Φ₂+2𝛼₁₂−𝛼₁₃Ψ₂+𝑎₁₇+−1−2𝑚+𝑟²𝛼₄₄𝛾²−2𝛼₂₃−1+𝛾²−2𝑟²𝜌₀𝜔²+Ω²−𝑎₁₈cot𝜃+𝑎₁₉csc²𝜃Ψ₂+𝑟2𝑖𝑎₂₀−𝑖𝛼₂₂−𝛼₂₃cot𝜃𝑑Φ₂−𝑎𝑑𝑟₂₁+2𝑟²𝜌₀𝜔²+2𝑟²𝜌₀Ω²−𝑎₂₂cot𝜃+𝑎₂₃csc²𝜃𝑑Ψ₂𝑖𝑑𝑟+𝑟2𝛼₁₂+𝛼₄₄𝛾+2𝛼₁₂−𝛼₁₃𝑑cot𝜃²Φ₂𝑑𝑟²−2(1+𝑚)𝛼₄₄𝑟𝑑²Ψ₂𝑑𝑟²=0, (17) where 𝑎₁₆=2𝛼₂₃−2𝛼₃₃+2𝛼₂₂𝛾²,𝑎₁₇=−2𝛼₂₂+4(1+𝑚)𝛼₄₄+2𝛼₁₂𝛾²,𝑎₁₈=𝑖4𝛼₁₂−2𝛼₁₃+𝛼₂₂+𝛼₂₃−𝛼₃₃−(1+2𝑚)𝛼₄₄𝑎𝛾,₁₉=2𝛼₁₃−2𝛼₂₃−𝛼₄₄−2𝑚𝛼₄₄,𝑎₂₀=𝛼₂₂+𝛼₂₃+2(1+𝑚)𝛼₄₄𝑎𝛾,₂₁=−2𝛼₁₃+𝛼₂₃−𝛼₃₃−2𝛼₂₂𝛾²+𝛼₄₄𝛾²−𝑟²𝛼₄₄𝛾²+2𝛼₁₂1+𝛾²,𝑎₂₂=𝑖4𝛼₁₂−2𝛼₁₃+𝛼₂₂+𝛼₄₄𝑎𝛾,₂₃=2𝛼₁₃−2𝛼₃₃+𝛼₄₄, (18) where 𝛾 is the wave number, 𝜔 is the angular frequency, 𝛾=2𝜋/𝜆, and 𝜆 is the wavelength. Substituting from (14) into (15) and (17) and after regrouping them leads to two independent equations for Φ₂ and Ψ₂; these equations are called spherical Bessel’s equations whose general solution is in the form Φ₂(𝑟)=𝐴₁𝑗_𝑛(𝑘𝑟)+𝐴₂𝑦_𝑛(),Ψ𝑘𝑟₂(𝑟)=𝐴₃𝑗_𝑛𝑘₁𝑟+𝐴₄𝑦_𝑛𝑘₁𝑟, (19) where 𝑛()=𝛼𝑛+1²²+𝛼₃₃+2𝛼₂₃−(2𝑚+1)𝛼₁₂+𝛼₁₃𝛼₁₁(),𝑘+𝑚𝑚+1₂=𝛼₄₄+𝜌₀𝜔²𝛼₁₁+𝐿₁−𝛾²𝛼₁₁−2𝛼₄₃+𝜌₀𝛼₁₁Ω²+𝜔²,𝑘²₁=𝜌₀𝛼₁₁Ω²+𝜔²+𝐿₂+𝜌₀𝜔²𝛼₁₁+2𝛾²𝛼₁₂+2𝑚𝛼₁₂−𝛼₂₂−𝛼₂₃,𝐿₁=(2+4𝑚)𝛼₁₃−2𝛼₂₃+𝛼₃₃−𝑚𝛼₄₄,𝐿₂=2𝛼₁₂+2𝑚𝛼₁₃−𝛼₂₂−2𝛼𝑚₂₃−𝛼₃₃, (20)

where 𝐴₁, 𝐴₂, 𝐴₃, and 𝐴₄are arbitrary constants and 𝑗_𝑛(𝑘𝑟) and 𝑦_𝑛(𝑘𝑟) denote spherical Bessel’s functions of the first and second kind of order 𝑛, respectively, which are defined in terms of Bessel’s function as follows: 𝑗_𝑛(𝑘𝑟)=𝜋/2𝑘𝑟𝐽_𝑛+1/2(𝑘𝑟), 𝑦_𝑛(𝑘𝑟)=𝜋/2𝑘𝑟𝑌_𝑛+1/2(𝑘𝑟). From (19) and (14) we get the following solutions for Φ₁ and Ψ₁ as follows: Φ₁(𝑟,𝜃,𝑡)=𝑒^{𝑖(𝛾𝜃−𝑤𝑡)}𝐴₁𝑗_𝑛(𝑘𝑟)+𝐴₂𝑦_𝑛(,Ψ𝑘𝑟)₁(𝑟,𝜃,𝑡)=𝑒^{𝑖(𝛾𝜃−𝑤𝑡)}𝐴₃𝑗_𝑛𝑘₁𝑟+𝐴₄𝑦_𝑛𝑘₁𝑟. (21)

Substituting from (21) into (9), we obtain the final solution of the displacement components in the following form: 𝑢_𝑟=1𝑟𝑒^{𝑖(𝛾𝜃−𝑤𝑡)}𝐴₁𝑛𝑗_𝑛(𝑘𝑟)−𝑟𝑘𝑗_𝑛+1(𝑘𝑟)+𝐴₂𝑛𝑦_𝑛+1(𝑘𝑟)+𝑦_𝑛+1(𝑘𝑟)+𝐴₃𝑖𝛾+cot𝜃𝑗_𝑛𝑘₁𝑟+𝐴₄𝑦_𝑛+1𝑘₁𝑟,𝑢_𝜃=1𝑟𝑒^{𝑖(𝛾𝜃−𝑤𝑡)}𝐴₁𝑖𝛾𝑗_𝑛(𝑘𝑟)+𝐴₂𝑦_𝑛(𝑘𝑟)−𝐴₃(1+𝑛)𝑗_𝑛𝑘₁𝑟+𝑘₁𝑟𝑗_𝑛+1𝑘₁𝑟−𝐴₄(1+𝑛)𝑦_𝑛𝑘₁𝑟+𝑟𝑘₁𝑦_𝑛+1𝑘₁𝑟. (22)

Substituting from (22) into (2), we obtain the final solution of the stress components in the following form:𝜎_𝑟𝑟=𝑟^−2+2𝑚𝑒^{𝑖(𝛾𝜃−𝑡𝜔)}×𝐴₁𝛼₁₂+𝛼₁₃𝑛+𝛼₁₁(−1+𝑛)𝑛−𝑘²𝑟²−𝛼₁₂𝛾²+𝑖𝛼₁₃𝛾cot𝜃𝑗_𝑛()+𝑘𝑟2𝛼₁₁−𝛼₁₂−𝛼₁₃𝑘𝑟𝑗_𝑛+1(𝑘𝑟)+𝐴₃𝑖𝛼₁₃+𝛼₁₁(−1+𝑛)−𝛼₁₂𝑛𝛾+𝛼₁₂+𝛼₁₁(−1+𝑛)−𝛼₁₃𝑛cot𝜃)𝑗_𝑛𝑘₁𝑟+𝑘₁𝑟−𝑖𝛼₁₁−𝛼₁₂𝛾+−𝛼₁₁+𝛼₁₃×cot𝜃𝑗_𝑛+1𝑘₁𝑟+𝐴₂𝛼₁₂+𝛼₁₃𝑛+𝛼₁₁(−1+𝑛)𝑛−𝑘²𝑟²−𝛼₁₂𝛾²+𝑖𝛼₁₃𝛾cot𝜃𝑦_𝑛()+𝑘𝑟2𝛼₁₁−𝛼₁₂−𝛼₁₃𝑘𝑟𝑦_𝑛+1(𝑘𝑟)+𝐴₄𝑖𝛼₁₃+𝛼₁₁(−1+𝑛)−𝛼₁₂𝑛𝛾+𝛼₁₂+𝛼₁₁(−1+𝑛)−𝛼₁₃𝑛cot𝜃𝑦_𝑛𝑘₁𝑟+𝑘₁𝑟−𝑖𝛼₁₁−𝛼₁₂𝛾+−𝛼₁₁+𝛼₁₃cot𝜃𝑦_𝑛+1𝑘₁𝑟,𝜎_𝜃𝜃=𝑟^−2+2𝑚𝑒^{𝑖(𝛾𝜃−𝑡𝜔)}×𝐴₁𝛼₂₂+𝛼₂₃𝑛+𝛼₁₂(−1+𝑛)𝑛−𝑘²𝑟²−𝛼₂₂𝛾²+𝑖𝛼₂₃𝛾cot𝜃𝑗_𝑛()+𝑘𝑟2𝛼₁₂−𝛼₂₂−𝛼₂₃𝑘𝑟𝑗_𝑛+1(𝑘𝑟)+𝐴₃𝑖𝛼₂₃+𝛼₁₂(−1+𝑛)−𝛼₂₂𝑛𝛾+𝛼₂₂+𝛼₁₂(−1+𝑛)−𝛼₂₃𝑛cot𝜃𝑗_𝑛𝑘₁𝑟+𝑘₁𝑟−𝑖𝛼₁₂−𝛼₂₂𝛾+−𝛼₁₂+𝛼₂₃×cot𝜃)𝑗_𝑛+1𝑘₁𝑟+𝐴₂𝛼₂₂+𝛼₂₃𝑛+𝛼₁₂(−1+𝑛)𝑛−𝑘²𝑟²−𝛼₂₂𝛾²+𝑖𝛼₂₃𝛾cot𝜃𝑦_𝑛()+𝑘𝑟2𝛼₁₂−𝛼₂₂−𝛼₂₃𝑘𝑟𝑦_𝑛+1(𝑘𝑟)+𝐴₄𝑖𝛼₂₃+𝛼₁₂(−1+𝑛)−𝛼₂₂𝑛𝛾+𝛼₂₂+𝛼₁₂(−1+𝑛)−𝛼₂₃𝑛cot𝜃𝑦_𝑛+1𝑘₁𝑟+𝑘₁𝑟−𝑖𝛼₁₂−𝛼₂₂𝛾+−𝛼₁₂+𝛼₂₃cot𝜃𝑦_𝑛+1𝑘₁𝑟,𝜎_𝜑𝜑=𝑟^−2+2𝑚𝑒^{𝑖(𝛾𝜃−𝑡𝜔)}×𝐴₁𝛼₂₃+𝛼₃₃𝑛+𝛼₁₃(−1+𝑛)𝑛−𝑘²𝑟²−𝛼₂₃𝛾²+𝑖𝛼₃₃𝛾cot𝜃𝑗_𝑛()+𝑘𝑟2𝛼₁₃−𝛼₂₃−𝛼₃₃𝑘𝑟𝑗_𝑛+1(𝑘𝑟)+𝐴₃𝑖𝛼₃₃+𝛼₁₃(−1+𝑛)−𝛼₂₃𝑛𝛾+𝛼₂₃+𝛼₁₃(−1+𝑛)−𝛼₃₃𝑛cot𝜃)𝑗_𝑛𝑘₁𝑟+𝑘₁𝑟−𝑖𝛼₁₃−𝛼₂₃𝛾+−𝛼₁₃+𝛼₃₃×cot𝜃)𝑗_𝑛+1𝑘₁𝑟+𝐴₂𝛼₂₃+𝛼₃₃𝑛+𝛼₁₃(−1+𝑛)𝑛−𝑘²𝑟²−𝛼₂₃𝛾²+𝑖𝛼₃₃𝛾cot𝜃𝑦_𝑛()+𝑘𝑟2𝛼₁₃−𝛼₂₃−𝛼₃₃𝑘𝑟𝑦_𝑛+1(𝑘𝑟)+𝐴₄𝑖𝛼₃₃+𝛼₁₃(−1+𝑛)−𝛼₂₃𝑛𝛾+𝛼₂₃+𝛼₁₃(−1+𝑛)−𝛼₃₃𝑛cot𝜃𝑦_𝑛𝑘₁𝑟+𝑘₁𝑟−𝑖𝛼₁₃−𝛼₂₃𝛾+−𝛼₁₃+𝛼₃₃cot𝜃𝑦_𝑛+1𝑘₁𝑟,𝜏_𝑟𝜃𝑟=−^−2+2𝑚2𝛼₄₄𝑒^{𝑖(𝛾𝜃−𝑡𝜔)}×𝐴₁−2𝑖(−1+𝑛)𝛾𝑗_𝑛(𝑘𝑟)+2𝑖𝑘𝑟𝛾𝑗_𝑛+1(𝑘𝑟)+𝐴₃−2+(−1+𝑛)𝑛−𝑘²₁𝑟²+𝛾²−𝑖𝛾cot𝜃+csc²𝜃𝑗_𝑛𝑘₁𝑟+2𝑘₁𝑟𝑗_𝑛+1𝑘₁𝑟−𝐴₂2𝑖(−1+𝑛)𝛾𝑦_𝑛(𝑘𝑟)+2𝑖𝑘𝑟𝛾𝑦_𝑛+1(𝑘𝑟)+𝐴₄−2+(−1+𝑛)𝑛−𝑘²₁𝑟²+𝛾²−𝑖𝛾cot𝜃+csc²𝜃𝑦_𝑛𝑘₁𝑟+2𝑘₁𝑟𝑦_𝑛+1𝑘₁𝑟. (23a)(23b)(23c)(23d)

From the solutions of elastic wave equations, the systems of equations depend on non-homogenity, rotation and the frequency.

4. Boundary Conditions and Frequency Equation

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, that is, 𝑢_𝑟=𝑢_𝜃=0,𝑟=𝑎,𝜎_𝑟𝑟=𝜏_𝑟𝜃=0,𝑟=𝑏. (24)

In this case, from (22), (23a), (23b), (23c), (23d), and (24) we have𝐴₁𝑛𝑗_𝑛(𝑘𝑎)−𝑎𝑘𝑗_𝑛+1(𝑘𝑎)+𝐴₂𝑛𝑦_𝑛+1(𝑘𝑎)+𝑦_𝑛+1(𝑘𝑎)+𝐴₃𝑖𝛾+cot𝜃𝑗_𝑛𝑘₁𝑎+𝐴₄𝑦_𝑛+1𝑘₁𝑎𝐴=0,₁𝑖𝛾𝑗_𝑛(𝑘𝑎)+𝐴₂𝑦_𝑛(𝑘𝑎)−𝐴₃(1+𝑛)𝑗_𝑛𝑘₁𝑎+𝑘₁𝑎𝑗_𝑛+1𝑘₁𝑎−𝐴₄(1+𝑛)𝑦_𝑛𝑘₁𝑎+𝑎𝑘₁𝑦_𝑛+1𝑘₁𝑎𝐴=0,₁𝛼₁₂+𝛼₁₃𝑛+𝛼₁₁(−1+𝑛)𝑛−𝑘²𝑏²−𝛼₁₂𝛾²+𝑖𝛼₁₃𝛾cot𝜃𝑗_𝑛()+𝑘𝑏2𝛼₁₁−𝛼₁₂−𝛼₁₃𝑘𝑏𝑗_𝑛+1(𝑘𝑏)+𝐴₂𝛼₁₂+𝛼₁₃𝑛+𝛼₁₁(−1+𝑛)𝑛−𝑘²𝑏²−𝛼₁₂𝛾²+𝑖𝛼₁₃𝛾cot𝜃𝑦_𝑛()+𝑘𝑏2𝛼₁₁−𝛼₁₂−𝛼₁₃𝑘𝑏𝑦_𝑛+1(𝑘𝑏)+𝐴₃𝑖𝛼₁₃+𝛼₁₁(−1+𝑛)−𝛼₁₂𝑛𝛾+𝛼₁₂+𝛼₁₁(−1+𝑛)−𝛼₁₃𝑛×cot𝜃)𝑗_𝑛𝑘₁𝑏+𝑘₁𝑏−𝑖𝛼₁₁−𝛼₁₂𝛾+−𝛼₁₁+𝛼₁₃×cot𝜃𝑗_𝑛+1𝑘₁𝑏+𝐴₄𝑖𝛼₁₃+𝛼₁₁(−1+𝑛)−𝛼₁₂𝑛𝛾+𝛼₁₂+𝛼₁₁(−1+𝑛)−𝛼₁₃𝑛×cot𝜃)𝑦_𝑛𝑘₁𝑏+𝑘₁𝑏−𝑖𝛼₁₁−𝛼₁₂𝛾+−𝛼₁₁+𝛼₁₃×cot𝜃)𝑦_𝑛+1𝑘₁𝑏𝐴=0,₁−2𝑖(−1+𝑛)𝛾𝑗_𝑛(𝑘𝑏)+2𝑖𝑘𝑏𝛾𝑗_𝑛+1(𝑘𝑏)−𝐴₂2𝑖(−1+𝑛)𝛾𝑦_𝑛(𝑘𝑏)+2𝑖𝑘𝑏𝛾𝑦_𝑛+1(𝑘𝑏)+𝐴₃−2+(−1+𝑛)𝑛−𝑘²₁𝑏²+𝛾²−𝑖𝛾cot𝜃+csc²𝜃𝑗_𝑛𝑘₁𝑏+2𝑘₁𝑏𝑗_𝑛+1𝑘₁𝑏+𝐴₄−2+(−1+𝑛)𝑛−𝑘²₁𝑏²+𝛾²−𝑖𝛾cot𝜃+csc²𝜃𝑦_𝑛𝑘₁𝑏+2𝑘₁𝑏𝑦_𝑛+1𝑘₁𝑏. (25a)(25b)(25c)(25d)

From (25a), (25b), (25c), and (25d) we get the following frequency equation: 𝑎_𝑖𝑗=0,𝑖,𝑗=1,2,3,4, (26) where the coefficients 𝑎_𝑖𝑗 are functions of rotation, non-homogenity, frequency, the radius 𝑟. Finally, we confined our attention to make these quantities dimensionless to simplify the calculation of the eigenvalues of equations. The coefficients 𝑎_𝑖𝑗 are 𝑎₁₁=𝑛𝑗_𝑛(𝑘𝑎)−𝑟𝑘𝑗_𝑛+1(),𝑎𝑘𝑎₁₂=𝑛𝑦_𝑛+1(𝑘𝑎)+𝑦_𝑛+1(),𝑎𝑘𝑎₁₃=𝑖𝛾+cot𝜃𝑗_𝑛𝑘₁𝑎,𝑎₁₄=𝑦_𝑛+1𝑘₁𝑎,𝑎₂₁=𝑖𝛾𝑗_𝑛(),𝑎𝑘𝑎₂₂=𝑦_𝑛(),𝑎𝑘𝑎₂₃=−(1+𝑛)𝑗_𝑛𝑘₁𝑎−𝑘₁𝑎𝑗_𝑛+1𝑘₁𝑎,𝑎₂₄=(1+𝑛)𝑦_𝑛𝑘₁𝑎+𝑎𝑘₁𝑦_𝑛+1𝑘₁𝑎,𝑎₃₁=𝛼₁₂+𝛼₁₃𝑛+𝛼₁₁(−1+𝑛)𝑛−𝑘²𝑏²−𝛼₁₂𝛾²+𝑖𝛼₁₃𝛾cot𝜃𝑗_𝑛()+𝑘𝑏2𝛼₁₁−𝛼₁₂−𝛼₁₃𝑘𝑏𝑗_𝑛+1(),𝑎𝑘𝑏₃₂=𝛼₁₂+𝛼₁₃𝑛+𝛼₁₁(−1+𝑛)𝑛−𝑘²𝑏²−𝛼₁₂𝛾²+𝑖𝛼₁₃𝛾cot𝜃𝑦_𝑛()+𝑘𝑏2𝛼₁₁−𝛼₁₂−𝛼₁₃𝑘𝑏𝑦_𝑛+1(),𝑎𝑘𝑏₃₃=𝑖𝛼₁₃+𝛼₁₁(−1+𝑛)−𝛼₁₂𝑛𝛾+𝛼₁₂+𝛼₁₁(−1+𝑛)−𝛼₁₃𝑛×cot𝜃)𝑗_𝑛𝑘₁𝑏+𝑘₁𝑏−𝑖𝛼₁₁−𝛼₁₂𝛾+−𝛼₁₁+𝛼₁₃×cot𝜃𝑗_𝑛+1𝑘₁𝑏,𝑎₃₄=𝑖𝛼₁₃+𝛼₁₁(−1+𝑛)−𝛼₁₂𝑛𝛾+𝛼₁₂+𝛼₁₁(−1+𝑛)−𝛼₁₃𝑛×cot𝜃)𝑦_𝑛𝑘₁𝑏+𝑘₁𝑏−𝑖𝛼₁₁−𝛼₁₂𝛾+−𝛼₁₁+𝛼₁₃cot𝜃𝑦_𝑛1+1𝑘₁𝑏,𝑎₄₁=−2𝑖(−1+𝑛)𝛾𝑗_𝑛(𝑘𝑏)+2𝑖𝑘𝑏𝛾𝑗_𝑛+1(),𝑎𝑘𝑏₄₂=−2𝑖(−1+𝑛)𝛾𝑦_𝑛(𝑘𝑏)−2𝑖𝑘𝑏𝛾𝑦_𝑛+1(),𝑎𝑘𝑏₄₃=−2+(−1+𝑛)𝑛−𝑘²₁𝑏²+𝛾²−𝑖𝛾cot𝜃+csc²𝜃𝑗_𝑛𝑘₁𝑏+2𝑘₁𝑏𝑗_𝑛+1𝑘₁𝑏,𝑎₄₄=−2+(−1+𝑛)𝑛−𝑘²₁𝑏²+𝛾²−𝑖𝛾cot𝜃+csc²𝜃𝑦_𝑛𝑘₁𝑏+2𝑘₁𝑏𝑦_𝑛+1𝑘₁𝑏. (27)

5. Numerical Results and Discussion

To examine the influence of non-homogenity, rotation and variation of the non-dimensional frequency in hollow sphere with the radius 𝑟 have been shown graphically. Free vibrations have been studied using a half-interval method. The frequency equations have been obtained under the effects of non-homogenity and rotation. It is found that the non-dimensional frequency increases with the increases of radius 𝑟 for all cases. As an illustrative example, the elastic constants for an orthotropic non-homogeneous material from Hearmon [36, 37] are 𝛼₂₃=3.945,   𝛼₁₁=3.198,   𝛼₃₃=2.7951, 𝛼₁₃=2.310,   𝛼₁₂=0.713, 𝛼₂₂=4.560,   and 𝜌=2.680. Numerical calculations are carried out for the displacement and the stress components along the 𝑟-direction at different values of the rotation Ω=0.0,1.3,2.6 in the case of non-homogeneous material and orthotropic material. Figure 2 shows the response histories of the non-dimensional frequencies with the radius 𝑟 for rotating hollow sphere Ω=2.5. With the effect of various values of non-homogeneous 𝑚=0.0,0.5,0.9 in the case of orthotropic material, it can be found that the distribution of the non-dimensional frequencies is increasing as the increase in the radius 𝑟 and the non-dimensional frequencies are increasing with the increase in the non-homogenity. Figure 3 shows the variation of the non-dimensional frequencies with the radius 𝑟 for hollow sphere with the effect of various values of rotation Ω=0.0,1.3,2.6 in the case of non-homogeneous 𝑚=0.65 orthotropic material. It can be found that the distribution of the non-dimensional frequencies is increasing with the increase in the radius 𝑟, but the non-dimensional frequencies are increasing with the decrease in the rotation. Figure 4 shows the response histories of the non-dimensional frequencies (the first mode, the second mode, and the third mode) with the radius 𝑟 at value of non-homogeneous 𝑚=0.65 and the rotation Ω=2.5. It can be found that the distribution of the non-dimensional frequencies is increasing with the increase in the radius 𝑟, for various boundary conditions, inner fixed surface, and outer free surface. Figure 5 shows the variation of the non-dimensional frequencies (three modes) with the radius 𝑟 for orthotropic sphere in the absence of rotation Ω=0.0 in the case of non-homogeneous material 𝑚=0.65. We observed that the frequency is increasing with the increase of the radius 𝑟 in the case of free traction surfaces, 𝑛=0. Figure 6 shows the first three modes of non-dimensional frequency for homogeneous 𝑚=0.0 orthotropic material in presence of the rotation Ω=2.5. We observed that the frequency is increasing with the increase in the radius 𝑟 in the case of orthotropic homogeneous hollow sphere more than in the case non-homogeneous hollow sphere. It is evident; non-homogenity, rotation, and orthotropic have a significant influence on non-dimensional frequencies.

Figure 2 

Variation of non-dimensional frequency versus the radius 𝑟, for the various values of rotation Ω and non-homogeneous 𝑚=0.65  (inner fixed surfaces and outer free surfaces) and  𝑛=0.

Figure 3 

Variation of non-dimensional frequency versus the radius 𝑟, for the various values of non-homogeneous 𝑚 and rotation Ω=2.5 (inner fixed surfaces and outer free surfaces) and  𝑛=0.

Figure 4 

Variation of non-dimensional frequency (three modes) versus the radius 𝑟 of the hollow sphere at rotation Ω=2.5 and non-homogeneous 𝑚=0.65 (inner fixed surfaces and outer free surfaces) and  𝑛=0.

Figure 5 

Variation of non-dimensional frequency (three modes) versus the radius 𝑟 of the hollow sphere at absent of rotation Ω=0.0 and non-homogeneous 𝑚=0.5 (inner fixed surfaces and outer free surfaces) and  𝑛=0.

Figure 6 

Variation of non-dimensional frequency (three modes) versus the radius 𝑟 of the hollow sphere at rotation Ω=2.5 and non-homogeneous 𝑚=0.0 (inner fixed surfaces and outer free surfaces) and  𝑛=0.

6. Conclusion

The effect of non-homogenity and rotation on surface wave dispersion in elastodynamic problem in orthotropic hollow sphere is studied. The vibration of sphere with the mixed surfaces boundary conditions is evaluated. The natural frequencies (eigenvalues) are calculated and compared with those reported in the absence and presence of non-homogenity and rotation. The effects of non-homogenity and rotation on the natural frequencies are shown graphically.

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant. The authors, therefore, acknowledge with thanks the DSR technical and financial support.