Natural Frequencies of Vibration of a Magnetoelastic Hollow Cylinder in a Magnetic Field Under Large Deformation

In this work, the frequency equation of vibration of elastic hollow cylinder in a magnetic field under large deformation is obtained for a semi-linear material. Also, the natural frequencies are numerically calculated and the effect of the magnetic field on the frequency modes are considered. We describe the problem using the equations of elasticity and the Maxwell equations of electromagnetism taking into consideration the effect of the magnetic field on the frequency of vibration of the cylinder. We invoke the appropriate boundary conditions on the Maxwell stress tensor within and on the surface of the cylinder. In the result, the obtained frequency equation showed that it is a generalization of the frequency equation obtained for small deformation theory. The natural frequency of the body increases as the magnetic intensity increases. Keyword-Natural Frequency, Vibration, Semi-linear material, Magnetoelastic cylinder, Large Deformation


INTRODUCTION
The phenomenon of vibration involves an alternating interchange of potential energy to kinetic energy and vice-versa. Any body having mass and elasticity is capable of oscillatory motion. In engineering, an understanding of the vibratory behavior of mechanical and structural systems is important for the safe design, construction and operation of variety of machines and structures. The failure of most mechanical and structural elements and systems can be associated with vibration. Rumerman and Raynor (1971) considered the natural frequencies of axially symmetric longitudinal vibration of circular cylinders. Laura et al (1974) derived the frequency equation of a cantilever beam that has additional mass attach to it, which is considered as shear force that acted on the free end of the beam. Hutchinson and El-Azhari (1986) developed a series solution of the general threedimensional equations of linear elasticity which was used to find the natural frequencies of the vibration of hollow elastic cylinders with traction free surfaces. Oz and Ozkaya (2005) investigated the natural frequencies of transverse vibration of beam-mass systems for different boundary conditions. Abbas (2006) examined the natural frequencies of vibration of a poroelastic hollow cylinder. Yazdanparast (2011) investigated the vibrations of hollow cylinder in rotation. Abd-Alla (2012) examined the effect of magnetic field and non-homogeneity on the radial vibrations in hollow elastic cylinder under rotation. Yahya and Abd-Alla (2014) considered the radial vibrations of an isotropic elastic rotating hollow cylinder. Using Biot's extension theory, Perati and Gurijala (2015) investigated the torsional vibrations in thick walled hollow poroelastic cylinder. Ebenezer and Ravichandran (2015) considered the free and forced vibrations of hollow elastic cylinders of finite length. Wang et al (2017) examined the frequency equation of flexural vibration of a cantilever beam considering the rotary inertial moment of an attached mass. The objective of this work is to derive the frequency equation of vibration of a magnetoelastic hollow cylinder in a magnetic field under large deformation for a semi-linear material in form of a determinant. Also, the natural frequencies for the modes of a magnetoelastic hollow cylinder were numerically calculated and the effect of the magnetic field on the frequency modes are considered.

Geometry of deformation
Let Ω be the subset of a three-dimensional Euclidean space 3 (i.e Ω ∁ 3 ) occupied an isotropic semi-linear elastic body with 1 and 2 as the inner and outer radii respectively of the hollow cylinder. We seek for the plane finite deformation of Ω from an initial configuration of Ω into a current configuration of Ω by the action, say, of externally applied magnetic field. The transformation from the initial configuration Ω into the current configuration Ω is the form = ( , ), Ф = , = , (1) Where ( , , ) are the material coordinates in the initial configuration Ω and ( , , ) are the material coordinate in the current configuration Ω . The position vectors of every particle in the initial configuration Ω and the current configuration Ω are respectively given as ⃗ = ⃗⃗⃗⃗ + , ⃗⃗⃗⃗⃗ ⃗⃗ = ( , )⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗ , (2) where ⃗⃗⃗⃗, ⃗⃗⃗⃗⃗, ⃗⃗⃗⃗ are the orthogonal local basis vectors associated with the cylindrical coordinates ( , , ) in Ω and ⃗⃗⃗⃗⃗, ⃗⃗⃗⃗⃗, ⃗⃗⃗⃗ are the corresponding local basis vector associated with the cylindrical coordinates ( , , ) in Ω .
Let the geometry of deformation of Ω from initial configuration Ω to current configuration Ω be the deformation gradient ∇ ⃗⃗ , where ∇ is the gradient operator in the initial configuration Ω . The governing equations of the magneto-elasticity problem we are considering are given as where ⃗⃗ is the electric field intensity vector, ⃗ is the Lorentz force, ⃗ ⃗⃗ is the magnetic intensity, ⃗⃗ is the magnetic induction vector, ⃗ is the speed of light, ⃗ is the current density vector, is the mass density, * is the charge density, and ̃ is the Piola-Kirchhoff stress tensor.
where , and are components of Lorentz force ⃗ acting on the body. (John, 1960) constructed the energy function for isotropic semi-linear material under large deformation which is given as

Energy function and Piolar-kirchhoff stress tensor
where 1 (̃−̃) is the first invariant of the tensor (̃−̃) and ̃ is the right stretch tensor. We invoke the hypothesis of hyperelasticity and take the Frechet derivative of the energy function (5) with respect to the deformation gradient ∇ ⃗⃗ to obtain the first Piola-Kirchhoff stress tensor ̃ which is given as where ̃ is the second rank rotation tensor and is the magnetic permeability of the body. For a perfect conductor ( → ∾), Ohm's law written as equation (8)

Radial vibration in perfect conductor
For radial vibration, the radial component of the displacement does not vanish identically, i.e = , = = 0 (10) Let the magnetic field ⃗ ⃗⃗ be such that is the externally applied magnetic field acting parallel to the axis of the cylinder and ℎ ⃗⃗ = ℎ ⃗⃗ ( , ) is the perturbation in the magnetic field due to deformation in the electrically conducting cylinder. Substituting equation (9) into equation (3) 4 of Faraday's law and making use of equation (11) we have, In component form, equation (12) can be expressed as ℎ ⃗⃗ = (0,0, ℎ) , (13) where ℎ satisfies the equation The Lorentz force ⃗ acting on the body is

Zeroth-level stress field in the cylinder
In order to obtain the stress fields associated with solution (23), we employ the relations and respectively.

BOUNDARY CONDITIONS AND FREQUENCY EQUATION
In this section, we are going to obtain the frequency equation for the boundary conditions of an electrically conducting hollow cylinder. We recall that the magnetic wave ℎ * ⃗⃗⃗⃗ ( , ) and * ⃗⃗⃗⃗⃗ ( , ) in vacuum satisfy the electromagnetic field equations

Zeroth-level frequency equation
where ℎ( , ), ℎ * ( , ) are the magnetic waves in the cylindrical body and vacuum respectively.

First-level frequency equation
In order to obtain the frequency equation at first-level of approximation, we employ the boundary conditions: Equation (67) is the first level frequency equation for a perfectly conducting semi-linear elastic hollow cylinder under consideration.

NUMERICAL RESULTS
In this section, we evaluated the natural frequencies for the first four modes of an elastic hollow cylinder of various thickness for two different values of externally applied magnetic field. The results are shown in the tables below    6. CONCLUSIONS The frequency equation of vibration of a magnetoelastic hollow cylinder in a magnetic field under large deformation for a semilinear material was obtained. Also, the natural frequencies for the first four modes of a magnetoelastic hollow cylinder of various thickness for two different values of externally applied magnetic field were numerically calculated. It is shown in the tables above that the natural frequencies of the magnetoelastic hollow cylinder increases as the thickness of the hollow cylinder increases. Furthermore, the effect of the magnetic field on the frequency modes were considered. It is clearly shown from the tables above that the natural frequencies of the magnetoelastic hollow cylinder increases as the externally applied magnetic field intensity increases. The results shows that the natural frequencies obtained are greater than the corresponding small deformation case.