1 Introduction

Spherical shells or hollow spheres are frequently encountered in engineering industries,and the corresponding free vibration problem has become one of the basic problems in elastodynam-ics. Accidental failure of rotating cylinders due to flexural vibrations has frequently occurred inrotodynamic machinery such as gas turbines and steam turbines. For engineers and scientists,the analysis of the dynamic problems of elastic bodies is an interesting and important researchfield. This mater has attracted the attention of many researchers^{[ 1, 2, 3, 4 ]}. The hollow spheres arefrequently used as structural components,and their vibration characteristics are obviously im-portant for practical design. Mahmoud et al.^{[ 5, 6 ]} discussed the effect of the rotation on planevibrations in a transversely isotropic infinite hollow cylinder,and the effect of the rotation onwave motion through cylindrical bore in a micropolar porous cubic crystal. Mahmoud^{[ 7 ]} studiedthe wave propagation in the cylindrical poroelastic dry bones. Abd-Alla and Mahmoud^{[ 8 ]} solvedthe magneto-thermo elastic problem in rotating inhomogeneous orthotropic hollow cylindricalunder the hyperbolic heat conduction model. Wang^{[ 9 ]} studied the elastodynamic solution for ananisotropic hollow sphere. Ding et al.^{[ 10, 11 ]}discussed the elastodynamic solution of a inhomoge-neous orthotropic hollow cylinder and the solution of a inhomogeneous orthotropic cylindricalshell for axisymmetric plane strain dynamic thermo elastic problems. Hearmon^{[ 12 ]} studied anintroduction to applied anisotropic elasticity. Inclusions of arbitrary shape in magneto-electro-elastic composite materials have been investigated byWang and Shen^{[ 13 ]}. Mahmoud et al.^{[ 14, 15, 16, 17 ]}studied the analytical solution for electrostatic potential on wave propagation modeling in hu-man long wet bones and analytical solution for free vibrations of elastodynamic orthotropichollow sphere under the influence of rotation,the effect of rotation and magnetic field throughporous medium on peristaltic transport of a Jeffrey fluid in tube,free vibrations of elasto-dynamic problem in rotating non-homogeneous orthotropic hollow sphere. Marin et al.^{[ 18 ]} dis-cussed the non-simple material problems addressed by Lagrange’s identity. Abd-Alla et al.^{[ 19, 20, 21 ]}investigated radial vibrations in a inhomogeneous orthotropic elastic hollow sphere subjected torotation,the influence of the rotation and gravity field on Stonely waves in a inhomogeneous or-thotropic elastic medium,and the effect of the rotation on a inhomogeneous infinite cylinder oforthotropic material. Ding et al.^{[ 22 ]} obtained the analytical solution for the axisymmetric planestrain electro elastic dynamics of a non-homogeneous piezoelectric hollow cylinder. Hou and Leung^{[ 23 ]} further studied the corresponding problem of magneto-electro-elastic hollow cylinders.Buchanan and Liu^{[ 24 ]} discussed an analysis of the free vibration of thick-walled isotropic toroidalshells. Yu et al.^{[ 25 ]} investigated the wave propagation in inhomogeneous magneto-eletro-elatichollow cylinders. Recently,Mahmoud^{[ 26 ]} investigated the influence of rotation and generalizedmagneto-thermoelastic on Rayleigh waves in a granular medium under effect of initial stress and gravity field. Abd-Alla and Mahmoud^{[ 27 ]}discussed analytical solution of wave propagationin inhomogeneous orthotropic rotating elastic media. Abd-Alla et al.^{[ 28 ]} studied the effect ofthe rotation,magnetic field,and initial stress on peristaltic motion of micropolar fluid. Sharmaet al.^{[ 29 ]} studied free vibration analysis of a visco-thermo-elastic solid sphere. Abd-Alla^{[ 30, 31, 32 ]}investigated the problem of radial vibrations in inhomogeneous isotropic cylinder under influ-ence of initial stress and magnetic field. Ai et al.^{[ 33 ]} studied the state space solution to 3Dmultilayered elastic soils based on the order reduction method,Luo and Liu^{[ 34 ]} investigated theresearch on an orthogonal relationship for orthotropic elasticity. The present paper deals withthe problem of wave propagation in magneto-elastodynamic hollow sphere. The effect of mag-netic field in the equations of motion has been taken into account. Comparisons are made withthe result in the presence and absence of magnetic field in cases of orthotropic and isotropichollow sphere.2 Formulation of problem

Consider the elastodynamic hollow sphere with inner radius α and outer radius b. Take thespherical coordinates(γ,θ,φ),as shown in Fig. 1. From Maxwell’s equations,we use an initialmagnetic field vector H(0,0,H₀)in the spherical coordinates(γ,θ,φ).

The equations of the magneto-elasto-dynamic medium in the direction and θ are given by

where σ_rr,σ_rθ,σ_θθ,and σ_φφ are the mechanical stress components,f_r and f_θ are defined ascomponents of Lorentz’s force F,a ≤ r ≤ b,t ≥ 0. The stresses-strain relations for sphericallyorthotropic material in two dimensions are in the following form^{[ 35 ]}:where e_ij are the strain components,and cij are the orthotropic material constants. The straindisplacement relations for spherically orthotropic material are in the formwhere u_r(r,θ,t) and u_θ(r,θ,t)are,respectively,the components of displacement in the radial and tangential directions,u =(u_r(r,θ,t),u_θ(r,θ,t),0).where F is defined as Lorentz’s force,and the radial component of Lorentz’s force fr may bewritten aswhere τ_rr is the radial component of magnetic stress,h is the perturbed magnetic field overthe constant primary magnetic field H₀,μ is the magnetic permeability,and J is the electriccurrent density. Substituting(3)−(8)into(1)−(2),we obtainwhere the constants d01,d02,d03,and d04 are as follows:

3 Solution of problem

By Helmohltz’s theorem,the displacement vector u can be written as

where the two functions ξ1 and x are known in the theory of elasticity,by Lame’s potentialsirrotational and rotatinal parts of the displacement vector u,respectively. The displacementpotentials are introduced for facilitating the solution of the field equations(9)−(10). It ispossible to take only one component of the vector x to be non-zero asFrom(11) and (12),we obtainSubstituting(13)−(14)into(9)−(10) and after regrouping them leads to the equations for ξ1 and x1 as follows:where

To study the propagation of harmonic waves,we assume a solution of the form

where γ is the wave number,ω is the angular frequency,γ = 2π/λ,and λ is the wavelength.Substituting(17)−(18)into(15)−(16),we getwhere

One may obtain the solution of(19)−(20)numerically. From(15)−(18)by comparing the coef-ficients of those equations,applying the equation of harmonic waves(17)−(18),and regrouping and simplifing them,we get two independent equations for ξ2 and x2 as follows:

where

(21)−(22)are called spherical Bessel’s equation,whose general solutions are in the form

where c1,c2,c3,and c4 are arbitrary constants,and jn(Lr) and yn(Lr)denote spherical Bessel’sfunctions of the first and second kind of order n,respectively,which are defined in terms ofBessel’s function as follows:

From(17)−(18) and (23)−(24),we get

Substituting(25)−(26)into(13)−(14),we obtain the final solution of the displacementcomponent in the following forms:

Substituting(27)−(28)into(5)−(8),we obtain the final solution of the component of per-turbed magnetic field hz over the constant primary magnetic field H0 and the radial componentof the magnetic stress τrr in the following form:Substituting(27)−(28)into(3)−(4),we obtain the final solution of the components ofmechanical stress in the following forms:

Thus,we obtained the solution of the plane vibration by writing the solutions of elastic waveequations,and then we arrived to the systems of equations which involve sphere function whosearguments depend on the frequency parameter to be computed.4 Boundary conditions and frequency equationIn the following section,solutions of the hollow sphere with different boundary conditionsare sought. Here,we discuss the plane propagation of harmonic waves in hollow sphere withthe inner and the outer surfaces,i,e.,r = α and r = b,respectively,the condition that thesphere is free from traction means that

From(30)−(32) and (33)−(34),we get

The roots of these equations give the values of natural frequency for the free plane oscillationsof the sphere. According to(35)−(38),we can write the characteristic frequency equation asfollows:

where the coefficients aij are given in the following forms:

The roots of this equation give the values of the fundamental frequency for the free plane os-cillations of the sphere. Finally,to simplify the calculation of the eigenvalues of those equations,we confine our attention to make these quantities dimensionless.Also,we get the solutions of the hollow sphere with the second different boundary conditions(inner fixed surfaces and outer fixed surfaces)as follows:

According to(27)−(28) and (40)−(41),we write the characteristic frequency equation for thiscase in the following form:where the coefficients bij are given by similar manner to the previous equation(39)as follows:

5 Numerical results and discussion

The frequency equation of elastodynamic is solved in terms of displacement by using thetechnique of variables separation and Helmholtz’s theorem. Harmonic vibrations have beenstudied using a half-interval method. The frequency equations have been obtained under theeffect of magnetic field. The effects of magnetic field on variation of the fundamental frequencyw with the ratio h of homogeneous materials have been shown graphically. It is found that thefundamental frequency increases with the increase of h for all cases. As an illustrative example,the elastic constants for an orthotropic material are used here^{[ 35 ]},c₂₃ = 2.934,c₁₁ = 4.968,c₃₃ =4.981,c₁₃ = 1.109,c₁₂ = 0.93,ρ = 3.986,_c22 = 5.264. Figures 2-7 show the response histories ofthe non-dimensional frequencies of orthotropic material with the ratio h,in the case of the firstmode and free traction surfaces under the influence of magnetic field. Figures 3 and 6 showfrequency w versus the ratio h of orthotropic material under the influence of magnetic field H0in the case of inner fixed surfaces and outer fixed surfaces. Figures 4 and 7 show frequency wversus the ratio h in the case of isotropic under the influence of magnetic field H0 in the caseof free traction surfaces.

Figures 5,6,and 7 show the first mode,the second mode,and the third mode for thefrequency w versus the ratio h when magnetic field H₀ = 2.5×10². It is evident that orthotropyhas a significant influence on non-dimensional frequencies. Figure 2 shows frequency w versusthe ratio h of orthotropic material for the various values of magnetic field H₀=0.0,1.5×10²,2.5×10²,and 3.5×10² for free traction surfaces. Figure 3 shows frequency w versus the ratio hof orthotropic material for various values of magnetic field H₀=0.0,1.5×10²,2.5×10²,and 3.5×10² for fixed surfaces. Figure 4 shows that frequency w versus the ratio h of isotropic forthe various values of magnetic field H₀=0.0,1.5×10²,2.5×10²,and 3.5×10² for free tractionsurfaces. Figure 5 shows frequency w versus the ratio h of orthotropic material when magneticfield H₀=2.5×10²,free traction surfaces. Figure 6 shows frequency w versus the ratio h oforthotropic material when magnetic field H₀ = 2.5 × 10² for inner fixed surfaces and outerfixed surfaces. Figure 7 shows frequency w versus the ratio h of isotropic when magnetic fieldH₀ = 2.5 × 10²,free traction surfaces. It can be found that the distribution of the non-dimensional frequencies are increasing with the increase of the magnetic field H₀ and alsoincreasing with the increase of the ratio h. sWe can observe that the frequency is increasingwith the increase of the ratio h in the case of orthotropic hollow sphere more than the case ofisotropic infinite hollow sphere. Also,the influence of the magnetic field H₀ on non-dimensionalfrequencies is very pronounced.6 Conclusions

Equations of elastodynamic problem of orthotropic hollow sphere in terms of displacementare solved. The fundamental frequency for this problem is obtained from the frequency equationof the present problem. The effect of magnetic field on surface wave dispersion in elastodynamicorthotropic hollow sphere is discussed. A numerical method is presented for obtaining estimatesof the fundamental frequencies of vibration using half-interval method. The natural frequencies(eigenvalues)are calculated for different cases and compared with those reported in the absence and present of magnetic field in case of hollow sphere. The effects of magnetic field on thenatural frequencies are shown in figures. The results show that the effect of magnetic field isvery pronounced.