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

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

Olokuntoye B. A. , Akinola A. P. , Layeni O. P. , Fadodun O. O.

Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria.

Abstract-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

  1. 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 three- dimensional 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 Biots 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.

  2. PROBLEM SETTING

      1. Geometry of deformation

        Let be the subset of a three-dimensional Euclidean space 3 (i.e 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 from an initial configuration of into a current configuration of by the action, say, of externally applied magnetic field.

        The transformation from the initial configuration into the current configuration is the form

        = (, ), = , = , (1)

        Where (, , ) are the material coordinates in the initial configuration and (, , ) are the material coordinate in the current configuration .

        The position vectors of every particle in the initial configuration and the current configuration are respectively given as

        = + ,

        = (, ) + , (2)

        where , , are the orthogonal local basis vectors associated with the cylindrical coordinates (, , ) in and , , are

        the corresponding local basis vector associated with the cylindrical coordinates (, , ) in .

        Let the geometry of deformation of from initial configuration to current configuration be the deformation gradient , where is the gradient operator in the initial configuration .

        The governing equations of the magneto-elasticity problem we are considering are given as

        . + = ,

        . = 4 ,

        . = 0 ,

        (3)

        × = 1 ,

        { × = 4 ,

        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.

        2

        2

        The component form of the equation of motion in equation (3)1 are

        + 1

        + 1 (

        ) +

        +

        = (, ) ,

        2

        + 1

        + 1 (

        ) +

        +

        = 0 ,

        (4)

        + 1

        + 1

        +

        +

        = 0 ,

        {

        where , and are components of Lorentz force acting on the body.

      2. Energy function and Piolar-kirchhoff stress tensor

        (John, 1960) constructed the energy function for isotropic semi-linear material under large deformation which is given as

        2

        2

        = ( + 1 2( ), (5)

        1 ) 2 1

        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

        (, ) = [ (

        2

        2

        + 1 2( )] , (6)

        () 1

        ) 2 1

        (, ) = 2 + (1( ) 2) . (7) The set of equations in (3) above are complemented with constitutive relations given as

        = ( + 1 × ) ,

        (, ) = 2 + (1( ) 2),

        (8)

        { = ,

        where is the second rank rotation tensor and is the magnetic permeability of the body. For a perfect conductor ( ), Ohms law written as equation (8)1 becomes

        = 1 × (9)

      3. Radial vibration in perfect conductor

    For radial vibration, the radial component of the displacement does not vanish identically, i.e

    = , = = 0 (10)

    = 0 + ,

    (11)

    where 0 (constant) is the externally applied magnetic field acting parallel to the axis of the cylinder and = (, ) 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,

    = + × ( × ) + × ( × ) . (12)

    0 0

    In component form, equation (12) can be expressed as

    = (0,0, ) , (13)

    where satisfies the equation

    = 0 ((, )) 1 ( (, )) . (14)

    0

    The Lorentz force acting on the body is

    = × 0 , (15)

    substituting the Ampere-Maxwell equation in (3)5 into equation (15) and using equation (11) yields

    = ( × ) × . (16)

    4 0

    Introducing = (, ) in × gives

    × = (0, (, ), 0,0) = (, ) . (17)

    Using equation (17) in equation (16) gives

    = ( (, ), 0,0) , (18)

    so that

    4 0

    = (, ), = 0, = 0 . (19)

    4

    0

    Substituting equation (19) into equation (4), we have

    + 1

    + 1 (

    ) +

    (, ) =

    ,

    4

    0

    + 1

    + 1 (

    ) +

    = 0 ,

    (20)

    + 1

    + 1

    +

    = 0 .

    {

    Substituting the components of Piola stress tensor in equation (20), (Olokuntoye et al, 2020) obtained the wave equation in the magnetoelastic semi-linear cylinder as

    1

    1

    (+2) ( ((, )))

    (, ) =

    (21)

    4

    0

    (, ) = 0 ((, )) 1 ((, ) (, )) (22)

    0

  3. STRESS FIELDS IN THE CYLINDER

    (Olokuntoye et al., 2020) obtained the solution of equation (21) by taking the first approximation of equation (22) i.e by approximating (, ) as 0 . The solution was given as

    (, ) = (11() + 21()) (23)

      1. Zeroth-level stress field in the cylinder

        In order to obtain the stress fields associated with solution (23), we employ the relations

        () = () + 1 () , (24)

        1

        2 1

        () = () + 1 () . (25)

        1

        2 1

        The non zero components , and of Piola-Kirchhoffs stress tensor are

        = (2 + ) (, ) + (,) (2 + ) , (26)

        = (2 + ) (,) + (,) (2 + ) , (27)

        = ( (, ) + (,) 2) . (28)

        Substituting the wave solution (23) in equations (26)-(28) and using the relations (24) and (25), we obtain

        = (2 + ) [ ( () + ()) 1] + 1 ( () + ()) ,

        1 1 2 1

        1 1 2 1

        = (2 + )[( () + ()) 1] + 2( + ) 1 ( () + ()), (29)

        2 2 2 2

        1 1 2 1

        = (2 + ) 1 ( () + ()) + ( () + ()) (2 + ) ,

        1 1 2 1

        1 1 2 1

        = ( () + ()) + 2( + ) 1 ( () + ()) (2 + ), (30)

        1 2 2 2

        1 1 2 1

        = ( ( () + ()) + 1 ( () + ()) 2) ,

        1 1 2 1

        1 1 2 1

        = (( () + ()) + 2 ( () + ()) 2) . (31)

        1 2 2 2

        1 1 2 1

        The current density electric field intensity , and Maxwells stress generated in the body as a result of (, )

        0(, ) = 0 are

        = 1 × (, ) = 1 × ( + ) = 0 , (32)

        4

        4 0

        = 1 × = × ( + ) =0 , (33)

        and

        0

        = = 2 , (34)

        respectively.

        4 0

        4 0

      2. First-level stress field in the cylinder

    (Olokuntoye et al., 2020) obtained the first-level solution of equation (21) by taking the second approximation of equation (22)

    i.e by approximating (, ) 0

    0 ((, )). The solution obtained was given as

    (, ) = ( () + ()) , =

    (35)

    1 1 2 1

    2+

    2

    where =

    and 2 = (

    + 0 ) ,

    4

    1, 2 are constants and 1(), 1() are Bessel functions of first and second kinds of order one respectively. Using the relation in equation (24) and (25), the non-zero Piola-Kirchhoff stresses , and are =

    (2 + )[( () + ()) 1] + 2( + ) 1 ( () + ()) , (36)

    2 2 2

    1 1 2 1

    = ( () + ()) + 2( + ) 1 ( () + ()) (2 + ) , (37)

    1 2 2 2

    1 1 2 1

    = (( () + ()) + 2 ( () + ()) 2) . (38)

    1 2 2 2

    1 1 2 1

    The current density = (0, , 0) and the electric field = (0, , 0) generated in the cylinder under consideration are

    = 1 = 1

    (

    0 ((, ))) ,

    4

    4

    0

    and

    = 0

    4

    1 ((, ))) , (39)

    (

    (

    = 0 (, ) ((, )) , (40)

    where (, ) is the solution of wave equation (21).

  4. 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.

      1. Zeroth-level frequency equation

        We proceed to construct the zeroth-level frequency equation which describes various modes of vibration of an electrically conducting semi-linear elastic cylinder. For this, we employ this set of boundary conditions compatible with the problem under consideration:

        + + 0 = at = and = , (41)

        = at = and = , (42)

        where 0 is a given prescribed stress at the boundary of the cylinder, (, ), is the Maxwells stress inside the cylinder,

        (, ) is the Maxwells stress in the vacuum, = (, ) is the electric wave generated in the cylinder, (, ) is the electric

        wave in the vacuum, and (, ) is the radial component of the Piola-Kirchhoff stress .

        2

        2

        We recall that the magnetic wave (, ) and (, ) in vacuum satisfy the electromagnetic field equations

        2 = 1 , (43)

        2

        2

        2

        2 = 1 , (44)

        2

        × = 1 , (45)

        × = 1 . (46)

        The solution of equation (43) is

        (, ) = () () = ( ( ) + ( )) , (47)

        1 2 1 0 2 0

        where , are constants and ( ) , ( ) are the Bessel function of the first and second kinds of order zero respectively.

        1 2 0 0

        Using equation (47) in equation (46) we obtain

        (, ) = ( ( ) + ( )) . (48)

        1 0

        2 0

        For the purpose of the nature of the problem under consideration we set

        (, ) = {

        10

        ( ) ,

        (49)

        ( ) , ,

        2 0

        and

        (, ) = {

        11

        ( ) ,

        (50)

        ( ) , .

        2 1

        The Maxwells stresses (, ) and (, ) in equation (41) are defined as

        (, ) = (, ) , (51)

        4 0

        (, ) = (, ) , (52)

        4 0

        where (, ), (, ) are the magnetic waves in the cylindrical body and vacuum respectively. Introducing equation (49) in equation (52) gives

        ( ) ,

        (, ) = {

        4

        0 1 0

        (53)

        ( ) , .

        Setting 0 as

        4

        0 2 0

        = (2 + ) 2 , (54)

        0 4 0

        we obtain the following from equations (41) and (42)

        111 + 122 = 0, 211 + 222 = 0, (55)

        1 = 2 = 0 , (56)

        where the coefficient 11, 12, 21 and 22 in equation (55) are defined as

        11 = (2 + )2() +

        2( + )

        1(),

        12 = (2 + )2() +

        2( + )

        1(),

        and

        21 = (2 + )2() +

        2( + )

        1(),

        respectively.

        22 = (2 + )2() +

        2( + )

        1(),

        For non-zero solution 1, 2 in equation (55), we set

        = |11 12| = 0 (57)

        21 22

        Equation (57) is the zeroth-level frequency equation of an electrically conducting semi-linear elastic cylinder.

      2. First-level frequency equation

        In order to obtain the frequency equation at first-level of approximation, we employ the boundary conditions:

        + + 0 = , = = , (58)

        and

        (, ) + 0 = (, ) = = . (59)

        The solution of the electromagnetic field equations are

        1 0

        1 0

        ( ) ,

        (, ) = {

        (60)

        ( )

        2 0

        and

        ( ) ,

        (, ) = {

        1 1

        (61)

        ( ) .

        2 1

        The associated Maxwells stress in vacuum is

        ( ) ,

        (, ) = {

        4

        0 1 0

        (62)

        ( )

        4

        0 2 0

        Equations (58) and (59) gives the system of linear equations in variables 1, 2, 1 and 2

        111 + 122 + 131 + 142 = 0 , (63)

        211 + 222 + 231 + 242 = 0 , (64)

        311 + 322 + 331 + 342 = 0 , (65)

        411 + 422 + 431 + 442 = 0 . (66)

        The coefficient of 11, 12, , 44, in the above equation (63)-(66) are written as

        2( + ) 2

        = (2 + ) () + () + 2 ( () ())

        11 2

        1

        4 0

        1 2

        2( + ) 2

        = (2 + ) () + () + 2 ( () ())

        12 2

        1

        4 0

        1 2

        = ( ),

        = 0, = 0,

        = ( )

        13 4 0 0 14

        34 44

        0

        2( + ) 2

        = (2 + ) () + () + 2 ( () ())

        21 2

        1

        4 0

        1 2

        2( + ) 2

        = (2 + ) () + () + 2 ( () ())

        22 2

        1

        4 0

        1 2

        = ( ),

        = 0, = 0,

        = ( )

        24 4 0 0 23

        43 33

        0

        =

        (2 () ()),

        =

        (2 () ())

        31 0

        1 2

        32 0

        1 2

        =

        (2 () ()),

        =

        (2 () ())

        41 0

        1 2

        42 0

        1 2

        For non-zero solution 1, 2, 1, 2,

        11

        = |21

        31

        41

        12

        22

        32

        42

        13 0

        33 0

        0

        0

        0

        24| = 0 . (67)

        44

        Equation (67) is the first level frequency equation for a perfectly conducting semi-linear elastic hollow cylinder under consideration.

  5. 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

    Table 1: First mode of Natural Frequencies of an magnetoelastic hollow cylinder of various thickness () for two different values of externally applied magnetic field

    =

    =

    0.1

    4.600489912

    4.741371586

    0.2

    5.911873584

    6.092304748

    0.3

    7.597071168

    7.828151931

    0.4

    9.762639461

    10.05858459

    0.5

    12.5455096

    12.92452227

    0.6

    16.12164534

    16.60703595

    0.7

    20.71716949

    21.33878818

    0.8

    26.6226618

    27.41873278

    0.9

    34.21153269

    44.77870011

    Table 2: Second mode of Natural Frequencies of magnetoelastic hollow cylinder of various thickness () for two different values of externally applied magnetic field

    =

    =

    0.1

    12.20800396

    12.7371271

    0.2

    15.68793269

    16.52081535

    0.3

    20.15982573

    21.42848523

    0.4

    25.9064455

    27.79402647

    0.5

    33.29115675

    36.05051403

    0.6

    42.78090245

    46.75967202

    0.7

    54.97572909

    60.65009019

    0.8

    70.64672824

    78.66679301

    0.9

    119.4536756

    128.6219678

    Table 3: Third mode of Natural Frequencies of magnetoelastic hollow cylinder of various thickness () for two different values of externally applied magnetic field

    =

    =

    0.1

    20.43234346

    20.80118

    0.2

    25.43096458

    27.3854

    0.3

    32.68013854

    36.05372

    0.4

    41.99571165

    47.46582

    0.5

    53.96671726

    62.49021

    0.6

    69.3500945

    82.27028

    0.7

    89.11855032

    108.3113

    0.8

    114.5220647

    142.5952

    0.9

    185.3921045

    199.6127

    Table 4: Fourth mode of Natural Frequencies of magnetoelastic hollow cylinder of various thickness () for two different values of externally applied magnetic field

    =

    =

    0.1

    27.3691174

    27.930159

    0.2

    35.28349447

    36.62412121

    0.3

    45.48648622

    48.02429711

    0.4

    58.63989551

    62.97306358

    0.5

    75.5968999

    82.57500838

    0.6

    97.45739184

    108.2785499

    0.7

    125.6393217

    141.9829632

    0.8

    161.970671

    186.17872

    0.9

    208.8080222

    244.1315142

  6. CONCLUSIONS

The frequency equation of vibration of a magnetoelastic hollow cylinder in a magnetic field under large deformation for a semi- linear 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.

REFERENCES

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      3. Ebenezer, D.D., Ravichandran, K. and Padmanabhan, C. Free and forced vibrations of hollow elastic cylinders of finite length. The Journal of the Acoustical Society of America, 2015; 137(5): 2927-2938.

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