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

Authors: S. R. Mahmoud; M. Marin; S. I. Ali; K. S. Al-Basyouni
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 form(1a)σrr=r2m(α11err+α12eθθ+α13eφφ),σθθ=r2m(α12err+α22eθθ+α23eφφ),σφφ=r2m(α13err+α23eθθ+α33eφφ),τrθ=r2mα44erθ,τrθ=0,τθφ=0.Figure 1
Problem geometry of non-homogeneous orthotropic material.The strain-displacements relations in two dimensions are in the form(1b)err=∂ur∂r,eθθ=1r(∂uθ∂θ+ur),eφφ=1rsinθ(ursinθ+uθcosθ),erθ=12(1r∂ur∂θ+∂uθ∂r-uθr),erφ=0,eθφ=0.Substituting from (1b) into (1a) we obtain the stresses-displacements relations in two dimensions in the form
(2)σrr=r1+2m(+α12∂uθ∂θ+rα11∂ur∂r)(α11+α13)ur+α13uθcotθ20000001+α12∂uθ∂θ+rα11∂ur∂r),σθθ=r1+2m(+α12∂uθ∂θ+rα11∂ur∂r)(α22+α22)ur+α23uθcotθ20000001+α22∂uθ∂θ+rα12∂ur∂r),σφφ=r1+2m(+α12∂uθ∂θ+rα11∂ur∂r)(α23+α33)ur+α33uθcotθ20000001+α23∂uθ∂θ+rα13∂ur∂θ),τrθ=12r1+2mα44(-uθ+∂ur∂θ+r∂uθ∂r),τrθ=0,τθφ=0,
where ur and uθ are, respectively, the components of displacement in the radial and tangential directions, eij are the strain components, and σij are the stress components. Where we have characterized the elastic constants cij and the density ρof non-homogeneous material in the form
(3)cij=αijr2m,ρ=ρ0r2m,i,j=1,2,3,
where αij and ρ0 are the values of cij 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 Ω⃑×(Ω⃑×u⃑)=(-Ω2ur,-Ω2uθ,0), due to time varying motion only, where Ω⃑=(0,0,Ω),  andu⃑=(ur,uθ,0).The elastodynamic equations of rotating non-homogeneous medium in two dimensions in the directionr,θ are given by:
(4)∂σrr∂r+1r∂τrθ∂θ+1r(2σrr-σθθ-σφφ+τrθcotθ)+ρ0r2mΩ2ur=ρ0r2m∂2ur∂t2,(5)∂τrθ∂θ+1r∂σθθ∂θ+1r((σθθ-σφφ)cotθ+3τrθ)+ρ0r2mΩ2uθ=ρ0r2m∂2uθ∂t2.Substituting from (1a), (1b), and (2) into (4) and (5), we obtain:
(6)r-1+m[+(2α12+α44)∂2uθ∂r∂θ+2α11r∂2ur∂r2]2(a0+r2ρ0Ω2)ur+a1uθcotθ+a2∂uθ∂θ+α44cotθ∂ur∂θ-∂uθ∂θ+∂2ur∂θ2+r-2rρ0∂2uθ∂t2+4(1+m)α11∂ur∂r+(2α13+α44)cotθ∂uθ∂r+(2α12+α44)∂2uθ∂r∂θ+2α11r∂2ur∂r2]=0,
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(7)r-1+m[+r(a5∂2ur∂r∂θ+rα44∂2uθ∂θ2)]2(α22-α33)urcotθ1-2(a3-r2ρ0Ω2+α33csc2θ)uθ1+a4∂ur∂θ+2α22cotθ∂uθ∂θ+∂2uθ∂θ21+r(-2rρ0∂2uθ∂t2+2(α12-α13)cotθ∂ur∂r+2(1+m)α44∂uθ∂r(-2rρ0∂2uθ∂t2+2(α12-α13)cotθ∂ur∂r)1+r(a5∂2ur∂r∂θ+rα44∂2uθ∂θ2)]=0,
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 vectoru← can be written as
(8)u←=∇_Φ1+∇_∧Ψ←,
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
(9)ur=cotθΨ1+(∂/∂θ)Ψ1r+∂Φ1∂r,uθ=-Ψ1-∂Φ1/∂θ+r(∂/∂r)Ψ1r.Substituting from (9) into (6) and (7) and regrouping them lead to the following equations for Φ1 and Ψ1:
(10)r-2+m[2(a1+r2ρ0Ω2)h1-a2cotθ(ψ1-∂ϕ1∂θ+r∂ψ1∂r)1-a3(∂ψ1∂θ-∂2ϕ1∂θ2+r∂2ψ1∂r∂θ)1+α44(cotθcsc2θψ1-csc2θ∂ψ1∂θ-∂2ϕ1∂θ21111111+2cotθ∂2ψ1∂θ2+∂3ψ1∂θ3+rh2)1-2ρ0r2(cotθ∂2ψ1∂t2+∂3ψ1∂θ∂t2+r∂3ϕ1∂r∂t2)1+4(1+m)α11r2h3+(2α13+α44)1×cotθh3+(2α12+α44)h41+2α11(2cotθψ1+2∂ψ1∂θ+rh5)]=0,
where
(11)h1=(cotθψ1+∂ψ1∂θ+r∂ϕ1∂r),h2=(cotθ∂2ϕ1∂r∂θ+∂2ψ1∂r∂θ+∂3ϕ1∂r∂θ2),h3=(∂2ϕ1∂r2-((cotθ((∂ψ_1)/∂r)+(∂^2ψ_1)/∂r∂θ)cotθψ1+(∂ψ1∂θ)1111-r(cotθ(∂ψ1∂r)+∂2ψ1∂r∂θ))111×r-2+∂2ϕ1∂r2),h3=(ψ1-∂ϕ1∂θ-r(∂ψ1∂r-∂2ϕ1∂r∂θ+r∂2ψ1∂r2)),h4=(∂ψ1∂θ-∂2ϕ1∂θ2-r(∂2ψ1∂r∂θ-∂3ϕ1∂r∂θ2+r∂3ψ1∂r2∂θ)),h5=(-2cotθ∂ψ1∂r-2∂2ψ1∂r∂θ200+r(cotθ∂2ψ1∂r2+∂3ψ1∂r2∂θ+r∂3ϕ1∂r3)).
One has
(12)r-2+m((2α12+α44)h6r2)2(α22-α33)cotθ(cotθψ1+∂ψ1∂θ+r∂ϕ1∂r)+2(a3-r2ρ0Ω2+α33csc2θ)(ψ1-∂ϕ1∂θ+r∂ψ1∂r)+2(a6(-csc2θψ1+cotθ∂ψ1∂θ+∂2ψ1∂θ2+r∂2ϕ1∂r∂θ)2000010-α22(∂2ψ1∂θ2-∂3ϕ1∂θ3111111111111+cotθ(∂ψ1∂θ-∂2ϕ1∂θ2+r∂2ψ1∂r∂θ)111111111111+r∂3ψ1∂r∂θ2))+r2(2ρ0(∂2ψ1∂t2-∂3ϕ1∂θ∂t2+r∂3ψ1∂r∂t2)2000010-α44(∂2ψ1∂θ2-∂3ϕ1∂θ3+r∂3ψ1∂r∂θ2)1111111+2(α12-α13)2000010×cotθ(-h8r2+∂2ϕ1∂r2)-2(1+m)α44h7r22000010+(2α12+α44)h6r2))=0,
where
(13)a6=(α22+α23+α44+mα44),h6=(-cotθ∂ψ1∂θ-∂2ψ1∂θ2+csc2θ(ψ1-r∂ψ1∂r)11+rcotθ∂2ψ1∂r∂θ+r∂3ψ1∂r∂θ2(r∂3ϕ1∂r2∂θ)),h7=(-ψ1+∂ϕ1∂θ+r(∂ψ1∂r-∂2ϕ1∂r∂θ+r∂2ψ1∂r2)),h8=cotθψ1+∂ψ1∂θ-r(cotθ∂ψ1∂r+∂2ψ1∂r∂θ).To study the propagation of harmonic waves, we assume a solution in the form(14)Φ1(r,θ,t)=Φ2(r)ei(γθ-ωt),Ψ1(r,θ,t)=Ψ2(r)ei(γθ-ωt).Substituting from (14) into (10) and (12) and regrouping them lead to the following equations for Φ2 and Ψ2:
(15)ⅇⅈ(γθ-tω)r-1+m[+r(4(1+m)a11d2Φ2dr2+(a14+a15cotθ)d2Ψ2dr2+2rα11d3Φ2dr3))]2γ(a7-a8cotθ)ϕ21+((2(a10+r2ρ0(ω2+Ω2))+α44csc2θ)-ⅈγ(a9-2r2ρ0(ω2+Ω2)111111111+α44csc2θ)2001+cotθ(2(a10+r2ρ0(ω2+Ω2))11111111111+α44csc2θ(2(a10+r2ρ0(ω2+Ω2))))Ψ21+r(+2rα11d3Φ2dr3))2(a11+r2ρ0(ω2+Ω2)11111111+i(a13+α44)γcotθ(a11+r2ρ0(ω2+Ω2))dΦ2dr200000+2(a12+a13cotθ)dΨ2dr200000+r(4(1+m)a11d2Φ2dr211111111111+(a14+a15cotθ)d2Ψ2dr211111111111+2rα11d3Φ2dr3))]=0,
where
(16)a7=(-2mα12+α22+α23+α44)γ,a8=ⅈ(-2mα13+α23+α33+α44),a9=-2α12+4m(α11-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+α44)γ2,a12=ⅈ(2mα11-2α12-2mα12+α22+α23)γ,a13=2(2mα11-2α13-2mα13+α23+α33),a14=ⅈ(2α11-2mα12-α44)γ+(2α11-2α13-α44),a15=(2mα11-2α13-α44).
One has
(17)ⅇⅈ(γθ-tω)r-1+m[-2(1+m)α44rd2Ψ2dr2)-ⅈγ(a16+α44(4+4m+r2γ2)111111-2r2ρ0(ω2+Ω2)-2ⅈα22γcotθ111111+2α33csc2θ)Φ2+2(α12-α13)Ψ2+(a17+(-1-2m+r2)α44γ21111-2α23(-1+γ2)-2r2ρ0(ω2+Ω2)1111-a18cotθ+a19csc2θ)Ψ2+r(d2Ψ2dr22ⅈ(a20-ⅈ(α22-α23)cotθ)dΦ2dr-(a21+2r2ρ0ω2+2r2ρ0Ω21111111,-a22cotθ+a23csc2θ)dΨ2dr+r(ⅈ(2α12+α44)γ11111111,+2(α12-α13)cotθ)d2Φ2dr2-2(1+m)α44rd2Ψ2dr2)]=0,
where
(18)a16=2α23-2α33+2α22γ2,a17=-2α22+4(1+m)α44+2α12γ2,a18=ⅈ(4α12-2(α13+α22+α23-α33)-(1+2m)α44)γ,a19=(2α13-2α23-α44-2mα44),a20=(α22+α23+2(1+m)α44)γ,a21=-2(α13+α23-α33)-2α22γ2+α44γ2-r2α44γ2+2α12(1+γ2),a22=ⅈ(4α12-2(α13+α22)+α44)γ,a23=(2α13-2α33+α44),
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
(19)Φ2(r)=A1jn(kr)+A2yn(kr),Ψ2(r)=A3jn(k1r)+A4yn(k1r),
where
(20)n(n+1)=(α22+α33+2α23)-(2m+1)(α12+α13)α11+m(m+1),k2=α44+ρ0ω2α11+L1-γ2(α11-2α43)+ρ0α11(Ω2+ω2),k12=ρ0α11(Ω2+ω2)+(L2+ρ0ω2)α11+2γ2(α12+2mα12-α22-α23),L1=((2+4m)α13-2(α23+α33)-mα44),L2=2α12+2mα13-α22-2αm23-α33,whereA1, A2, A3,  and A4are arbitrary constants  and jn(kr) and yn(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:  jn(kr)=π/2krJn+1/2(kr),  yn(kr)=π/2krYn+1/2(kr). From (19) and (14) we get the following solutions for Φ1 and Ψ1 as follows:
(21)Φ1(r,θ,t)=ei(γθ-wt)[A1jn(kr)+A2yn(kr)],Ψ1(r,θ,t)=ei(γθ-wt)[A3jn(k1r)+A4yn(k1r)].Substituting from (21) into (9), we obtain the final solution of the displacement components in the following form:
(22)ur=1rei(γθ-wt)[A1{njn(kr)-rkjn+1(kr)}1+A2{nyn+1(kr)+yn+1(kr)}1+A3(ⅈγ+cotθ)jn(k1r)+A4yn+1(k1r)],uθ=1rei(γθ-wt)[A1ⅈγjn(kr)+A2yn(kr)1-A3{(1+n)jn(k1r)+k1rjn+1(k1r)}1-A4{(1+n)yn(k1r)+rk1yn+1(k1r)}].Substituting from (22) into (2), we obtain the final solution of the stress  components in the following form:(23a)σrr=r-2+2mⅇⅈ(γθ-tω)×[A1{((-1+n)n-k2r2)((α12+α13)n1111111+α11((-1+n)n-k2r2)-α12γ21111111+ⅈα13γcotθ{((α12+α13)n)jn(kr)1111111+(2α11-α12-α13)krjn+1(kr)((-1+n)n-k2r2)}11+A3{(ⅈ(α13+α11(-1+n)-α12n)γ11110001+(α12+α11(-1+n)11111111111-α13n)cotθ)jn(k1r)11000111+k1r(-i(α11-α12)γ+(-α11+α13)11111111111111×cotθ(-i(α11-α12)γ+(-α11+α13))jn+1(k1r){(ⅈ(α13+α11(-1+n)-α12n)γ}11+A2{((α12+α13)n+α11((-1+n)n-k2r2)11000001-α12γ2+iα13γcotθ((α22+α23)n+α12((-1+n)n-k2r2)-α22γ2)yn(kr)1111111+(2α11-α12-α13)kryn+1(kr)((α22+α23)n+α12((-1+n)n-k2r2)-α22γ2}11+A4{(ⅈ(α13+α11(-1+n)-α12n)γ11100001+(α12+α11(-1+n)11111111111-α13n)cotθ(ⅈ(α13+α11(-1+n)-α12n)γ)yn(k1r)110000+k1r(-ⅈ(α11-α12)γ11111111111+(-α11+α13)cotθ)yn+1(k1r)}A1{((α12+α13)n],(23b)σθθ=r-2+2mⅇⅈ(γθ-tω)×[A1{((α22+α23)n1111111+α12((-1+n)n-k2r2)-α22γ2111100,+ⅈα23γcotθ((α22+α23)n)jn(kr)11111+(2α12-α22-α23)krjn+1(kr)}11+A3{(ⅈ(α23+α12(-1+n)-α22n)γ11101001+(α22+α12(-1+n)11111111111-α23n(α22+α12(-1+n))cotθ(ⅈ(α23+α12(-1+n)-α22n)γ)jn(k1r)10011101+k1r(-i(α12-α22)γ+(-α12+α23)11111111111111×cotθ)jn+1(k1r){(ⅈ(α23+α12(-1+n)-α22n)γ}11+A2{((α22+α23)n+α12((-1+n)n-k2r2)11000001-α22γ2+iα23γcotθ((α22+α23)n+α12((-1+n)n-k2r2)-α22γ2)yn(kr)111111+(2α12-α22-α23)kryn+1(kr)((α22+α23)n+α12((-1+n)n-k2r2)-α22γ2}11+A4{(ⅈ(α23+α12(-1+n)-α22n)γ11100001+(α22+α12(-1+n)11111111111-α23n)cotθ(α22+α12(-1+n))yn+1(k1r)1110000+k1r(-ⅈ(α12-α22)γ111111111111+(-α12+α23)cotθ)yn+1(k1r)}{((α23+α33)n],(23c)σφφ=r-2+2mⅇⅈ(γθ-tω)×[A1{((α23+α33)n1111111+α13((-1+n)n-k2r2)-α23γ21111111+ⅈα33γcotθ((α23+α33)n)jn(kr)1111111+(2α13-α23-α33)krjn+1(kr)}11+A3{(ⅈ(α33+α13(-1+n)-α23n)γ11110001+(α23+α13(-1+n)11111111111-α33n)cotθ)jn(k1r)11000111+k1r(-i(α13-α23)γ+(-α13+α33)11111111111111×cotθ)jn+1(k1r)}11+A2{((α23+α33)n+α13((-1+n)n-k2r2)11000001-α23γ2+ⅈα33γcotθ((α22+α23)n+α12((-1+n)n-k2r2)-α22γ2)yn(kr)111111+(2α13-α23-α33)kryn+1(kr)((α22+α23)n+α12((-1+n)n-k2r2)-α22γ2}11+A4{(ⅈ(α33+α13(-1+n)-α23n)γ111100001+(α23+α13(-1+n)111111111111-α33n)cotθ(ⅈ(α33+α13(-1+n)-α23n)γ)yn(k1r)11110000+k1r(-ⅈ(α13-α23)γ1111111111111+(-α13+α33)cotθ)yn+1(k1r)}{((α23+α33)n],(23d)τrθ=-r-2+2m2α44ⅇⅈ(γθ-tω)×[+Csα2θ)yn(k1r)+2k1ryn+1(k1r)}A1{-2ⅈ(-1+n)γjn(kr)+2ikrγjn+1(kr)}+A3{(-2+(-1+n)n-k12r2+γ2-ⅈγcotθ1100001+csc2θ)jn(k1r)+2k1rjn+1(k1r)}-A2{+Csα2θ)yn(k1r)+2k1ryn+1(k1r)}2ⅈ(-1+n)γyn(kr)+2ⅈkrγyn+1(kr)}+A4{+Csα2θ)yn(k1r)+2k1ryn+1(k1r)}(-2+(-1+n)n-k12r2+γ2-ⅈγcotθ1100001+csc2θ)yn(k1r)+2k1ryn+1(k1r)}].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,(24)ur=uθ=0,r=a,σrr=τrθ=0,r=b.In this case, from (22), (23a), (23b), (23c), (23d), and (24) we have(25a)A1{njn(ka)-akjn+1(ka)}+A2{nyn+1(ka)+yn+1(ka)}+A3(ⅈγ+cotθ)jn(k1a)+A4yn+1(k1a)=0,(25b)A1ⅈγjn(ka)+A2yn(ka)-A3{(1+n)jn(k1a)+k1ajn+1(k1a)}-A4{(1+n)yn(k1a)+ak1yn+1(k1a)}=0,(25c)A1{((α12+α13)n+α11((-1+n)n-k2b2)111-α12γ2+ⅈα13γcotθ((α12+α13)n+α11((-1+n)n-k2b2))jn(kb)111+(2α11-α12-α13)kbjn+1(kb)((α12+α13)n+α11((-1+n)n-k2b2)}+A2{((α12+α13)n+α11((-1+n)n-k2b2)11111111-α12γ2+ⅈα13γcotθ((α12+α13)n+α11((-1+n)n-k2b2))yn(kb)11111111+(2α11-α12-α13)kbyn+1(kb)((α12+α13)n+α11((-1+n)n-k2b2)}+A3{(ⅈ(α13+α11(-1+n)-α12n)γ1111111+(α12+α11(-1+n)-α13n)1111111×cotθ)jn(k1b)11111+k1b(-ⅈ(α11-α12)γ+(-α11+α13)11111111111×cotθ(-ⅈ(α11-α12)γ+(-α11+α13))jn+1(k1b){(ⅈ(α13+α11(-1+n)-α12n)γ}+A4{(ⅈ(α13+α11(-1+n)-α12n)γ11111111+(α12+α11(-1+n)-α13n)111111111×cotθ)yn(k1b)111111+k1b(-ⅈ(α11-α12)γ+(-α11+α13)111111111111×cotθ)yn+1(k1b)}=0,(25d)A1{-2i(-1+n)γjn(kb)+2ikbγjn+1(kb)}-A2{2ⅈ(-1+n)γyn(kb)+2ikbγyn+1(kb)}+A3{+Csα2θ)yn(k1b)+2k1byn+1(k1b)}(-2+(-1+n)n-k12b2+γ2-iγcotθ1111111+csc2θ)jn(k1b)+2k1bjn+1(k1b)}+A4{+Csα2θ)yn(k1b)+2k1byn+1(k1b)}(-2+(-1+n)n-k12b2+γ2-iγcotθ1111111+csc2θ)yn(k1b)+2k1byn+1(k1b)}.From (25a), (25b), (25c), and (25d) we get the following frequency equation:
(26)|aij|=0,i,j=1,2,3,4,
where the coefficients aij 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 aij are
(27)a11=njn(ka)-rkjn+1(ka),a12=nyn+1(ka)+yn+1(ka),a13=(ⅈγ+cotθ)jn(k1a),a14=yn+1(k1a),a21=iγjn(ka),a22=yn(ka),a23=-(1+n)jn(k1a)-k1ajn+1(k1a),a24=(1+n)yn(k1a)+ak1yn+1(k1a),a31=((α12+α13)n+α11((-1+n)n-k2b2)111-α12γ2+iα13γcotθ((-1+n)n-k2b2))jn(kb)+(2α11-α12-α13)kbjn+1(kb),a32=(111-α12γ2((α12+α13)n+α11((-1+n)n-k2b2)+iα13γαotθ)(α12+α13)n+α11((-1+n)n-k2b2)111-α12γ2((α12+α13)n+α11((-1+n)n-k2b2)+iα13γcotθ)yn(kb)+(2α11-α12-α13)kbyn+1(kb),a33={×cotθ(-i(α11-α12)γ+(-α11+α13))jn+1(k1b)}(αotθⅈi(α13+α11(-1+n)-α12n)γ1111+(α12+α11(-1+n)-α13n)1111×cotθ)jn(k1b)111+k1b(-i(α11-α12)γ+(-α11+α13)111111111×cotθ(-i(α11-α12)γ+(-α11+α13))jn+1(k1b){×cotθ(-i(α11-α12)γ+(-α11+α13))jn+1(k1b)}(αotθⅈi(α13+α11(-1+n)-α12n)γ},a34=(i(α13+α11(-1+n)-α12n)γ111+(α12+α11(-1+n)-α13n)111×cotθ)yn(k1b)+k1b(-i(α11-α12)γ1111111+(-α11+α13)cotθ)yn1+1(k1b),a41=-2i(-1+n)γjn(kb)+2ⅈkbγjn+1(kb),a42=-2i(-1+n)γyn(kb)-2ⅈkbγyn+1(kb),a43=(-2+(-1+n)n-k12b2+γ2111-iγcotθ+csc2θ(-2+(-1+n)n-k12b2+γ2)jn(k1b)+2k1bjn+1(k1b),a44=(csc2θ(-2+(-1+n)n-k12b2+γ2-2+(-1+n)n-k12b2+γ2111-iγcotθ+csc2θ(-2+(-1+n)n-k12b2+γ2)yn(k1b)+2k1byn+1(k1b).


## 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 radiusr 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 radiusr, 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 radiusr, 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 radiusr 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 radiusr 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 radiusr 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.



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*Source: 250567-2013-08-26.xml*