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Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field
ARTICLE IN PRESS
International Journal of Mechanical Sciences 50 (2008) 365–379 www.elsevier.com/locate/ijmecsci
Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field M. Higuchia,, R. Kawamurab, Y. Tanigawab a
Department of Mechanical Engineering, Oita National College of Technology Maki 1666, Oita City, 870-0152, Japan Department of Mechanical Systems Engineering, Osaka Prefecture University, Gakuencho 1-1, Nakaku, Sakai, Osaka 599-8531, Japan
b
Received 7 February 2007; accepted 8 November 2007 Available online 17 November 2007
Abstract In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field are investigated. It is assumed that a magnetic field defined by an arbitrary function of time acts on the outer surface in the direction parallel to its surface. Fundamental equations of plane axisymmetrical electromagnetic, temperature and elastic fields are formulated. Then, solutions of magnetic field, eddy current, temperature change and both dynamic solutions and quasi-static ones of stresses and deformations are analytically derived in the forms including the arbitrary function. The solutions of stresses are determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused by Lorentz force. For the case that the arbitrary function is given by the sine function, the dynamic and quasi-static behaviors of the stresses are examined by numerical calculations. r 2007 Elsevier Ltd. All rights reserved. Keywords: Magneto-thermo-elasticity; Eddy current loss; Lorentz force
1. Introduction Mechanical components or structural elements which are activated in magnetic field have been increasing with the rapid progress of electromechanics. Especially, induction heating has attracted attention in recent years because of its less electricity consumption, high efficiency, safety and cleanliness. Accordingly, this technology has widely been applied to several area such as industrial heating equipment, office automation equipment and cooking devices in home electric appliances. When a time-varying magnetic field acts on a conducting medium, the eddy currents are induced in the medium by electromagnetic induction, and the electric currents cause heat generation called the eddy current loss due to the Joule effect. Induction heating utilizes the heat generation induced by time variation of Corresponding author. Tel.: +81 97 552 6879; fax: +81 97 552 6975.
E-mail addresses: [email protected], [email protected] (M. Higuchi). 0020-7403/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2007.11.001
magnetic field in a conducting medium. The conducting medium is subjected to Lorentz force as well as the heat supply of eddy current loss. Thus, two kinds of stress should be generated by time variation of magnetic field: one is thermal stress caused by eddy current loss and the other is magnetic stress caused by Lorentz force. In the field of magneto-elasticity or magneto-thermoelasticity [1,2], many studies have been conducted on an analytical treatment of an interaction between elastic, electromagnetic and temperature fields [3–15]. However, few studies have been carried out on analytical development of thermal stresses induced by a time-varying magnetic field [16–18]. Moon and Chattopadhyay [16] have studied thermal stresses due to eddy current loss and magnetic stresses due to Lorentz force in a conducting halfspace caused by an applied jump in tangential magnetic field at the boundary. Chian and Moon [17] have extended the above work, investigating those stresses in a cylindrical conductor with a cavity caused by a pulsed magnetic field at the cavity. Wauer [18] has studied the dynamic behavior
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of magneto-thermo-elastic plate layer, of which the surfaces are subjected to a magnetic field which is composed of a constant and a harmonically oscillating part in the direction parallel to the surfaces. He has mentioned the stability of the plate due to the external magnetic field. Pantelyat and Fe´liachi [19] have studied the mechanical behavior of metals in induction heating devices by using of finite element method. They have calculated thermo-elastic-plastic stresses induced by an alternating magnetic field, taking into account temperature dependences of material properties. In the present paper, we have investigated dynamic and quasi-static behaviors of magneto-thermo-elastic stresses and deformations in a conducting hollow circular cylinder, which consists of non-ferromagnetic metals such as copper or aluminum, subjected to an arbitrary variation of magnetic field. Assuming that a time-varying magnetic field which is defined as an arbitrary function of time acts on the outer surface of the cylinder in the direction parallel to its surface, we have formulated fundamental equations of plane axisymmetrical electromagnetic field, temperature field and elastic field. Then, we have analytically derived solutions of electromagnetic field, temperature change and dynamic solutions and quasi-static solutions of stresses and displacements, respectively, in the form including the arbitrary function. The solutions of stresses have been determined to be sums of thermal stress and magnetic stress. Carrying out numerical calculations for the case that the arbitrary function of time is given by the sine function, we have examined the dynamic and quasi-static behaviors of stresses in the hollow circular cylinder. 2. Fundamental equation systems 2.1. Electromagnetic field Let us consider a conducting hollow circular cylinder of inner and outer radii a and b with cylindrical coordinate system, as shown in Fig. 1. We assume that a time-varying axial magnetic field H 0 fðtÞ, which is uniform along the y and z directions, acts on the outer surface in the direction parallel to its surface, from time t ¼ 0. H 0 is a reference magnetic field strength, and fðtÞ is an arbitrary function of time.
Assuming that the magnetic field vectors in the conducting cylinder and in the air region inside the cylinder have only the plane axisymmetric axial components, namely H ¼ ð0; 0; H z ðr; tÞÞ and H ðaÞ ¼ ð0; 0; H ðaÞ z ðr; tÞÞ, then the governing equations and the constitutive relations of electromagnetics in cylindrical coordinate are given by In the conducting hollow cylinder: 1q qBz ðrE y Þ þ ¼ 0, r qr qt
(1)
qH z ¼ J y, qr
(2)
J y ¼ sE y ,
(3)
Bz ¼ mH z .
(4)
In the air region: 1q qBðaÞ z ðrE ðaÞ ¼ 0, y Þþ r qr qt
(5)
ðaÞ qH ðaÞ qDy z ¼ , qr qt
(6)
ðaÞ DðaÞ y ¼ e0 E y ,
(7)
ðaÞ BðaÞ z ¼ m0 H z ,
(8)
where the displacement current is disregarded in Eq. (2). In Eqs. (1)–(4), E y , Bz and J y are the circumferential component of the electric field, the axial component of the magnetic flux, and the circumferential component of the electric current density in the conducting cylinder, respectively, and s and m are the electric conductivity and the magnetic permeability in the conducting cylinder, ðaÞ respectively. In Eqs. (5)–(8), E ðaÞ and DðaÞ y , Bz y are the circumferential component of the electric field, the axial component of the magnetic flux, and the circumferential component of the electric displacement in the air region, respectively, and e0 and m0 are the dielectric constant and the magnetic permeability in the air region, respectively. From Eqs. (1)–(8), the fundamental equations of magnetic fields are obtained as follows: 1 q qH z qH z r ðaorobÞ, (9a) ¼ ms r qr qr qt 1 q qH ðaÞ q2 H ðaÞ z r z ¼ m0 e 0 r qr qr qt
ð0oroaÞ.
(9b)
The boundary conditions and the initial conditions are written as
Fig. 1. Conditions and coordinate system of hollow cylinder.
r ¼ b;
H z ðb; tÞ ¼ H 0 fðtÞ,
(9c)
r ¼ a;
H z ða; tÞ ¼ H ðaÞ z ða; tÞ,
(9d)
1 q2 H z 1 qH ðaÞ z ¼ , s qtqr e0 qr
(9e)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
t ¼ 0;
H z ¼ 0,
(9f)
qH ðaÞ z ¼ 0, (9g) qt where the relation given by Eq. (9e) means the continuity of the electric field at the inner boundary. The electric current density J y ðr; tÞ induced by time variation of magnetic field is called the eddy current, which is obtained from Eq. (2). H ðaÞ z ¼
2.2. Temperature field The eddy current J y generates Joule heat called the eddy current loss. The eddy current loss wðr; tÞ per unit time per unit volume is given by ½J y ðr; tÞ2 . (10) s We assume that the hollow cylinder with zero initial temperature change is heated by the eddy current loss wðr; tÞ from time t ¼ 0, and that the outer and inner surfaces are subjected to surrounding media, of which temperature are zero, with relative heat transfer coefficients hb and ha . Then, the fundamental equation of heat conduction taking into account the eddy current loss, the boundary conditions and the initial condition are written as qT 1 q qT w ¼k r , (11a) þ qt r qr qr Cr
wðr; tÞ ¼
r ¼ b;
qT þ hb T ¼ 0, qr
(11b)
r ¼ a;
qT þ ha T ¼ 0, qr
(11c)
T ¼ 0,
(11d)
t ¼ 0;
where T ¼ Tðr; tÞ is temperature change, and k, C and r denote the thermal conductivity, the specific heat and the mass density, respectively. 2.3. Elastic field The conducting hollow cylinder is subjected to both temperature change and Lorentz force. Lorentz force vector f is defined by 0 1 0 1 0 0 B qH C B C zC B 0 C f ¼ J B ¼ B CB @ A @ qr A mH z 0 0 1 mq ½H ðr; tÞ2 B 2 qr z C B C ¼B ð12Þ C. 0 @ A 0
367
From Eq. (12), Lorentz force has only the radial component as follows: f r ðr; tÞ ¼
mq ½H z ðr; tÞ2 . 2 qr
(13)
Because temperature change and Lorentz force are plane axisymmetric, we analyze stresses and deformations under the axisymmetric plane strain state. The equation of motion in the radial direction taking into account Lorentz force is written as qsrr srr syy q2 u þ þ f r ¼ r 2, qt qr r
(14)
where srr and syy are the radial stress component and the circumferential stress component, respectively, and u is the radial displacement. Stress-displacement relations taking into account temperature change are given by 9 ð1 nÞE qu n u 1þn > srr ðr; tÞ ¼ þ aT ; > > = ð1 þ nÞð1 2nÞ qr 1 n r 1 n ð1 nÞE u n qu 1 þ n > > þ aT ; > syy ðr; tÞ ¼ ; ð1 þ nÞð1 2nÞ r 1 n qr 1 n (15) where E, n and a denote the Young’s modulus, the Poisson’s ratio and the coefficient of linear thermal expansion, respectively. The axial stress component is given by szz ðr; tÞ ¼ nðsrr þ syy Þ aET.
(16)
Substitution Eqs. (13) and (15) into Eq. (14) yields the displacement equation of motion: q 1 qðruÞ 1 q2 u 1 þ n qT þ a ¼ 2 qr r qr 1 n qr C L qt ð1 þ nÞð1 2nÞ m q ðH z Þ2 . þ ð1 nÞE 2 qr
ð17aÞ
The cylinder is at rest prior to time t ¼ 0 and we suppose that the inner and outer surfaces of the cylinder are traction free ðsrr ¼ 0Þ. Then, the mechanical boundary conditions and the initial conditions are given by r ¼ b; a;
t ¼ 0;
qu n u 1þn þ ¼ aT, qr 1 n r 1 n u¼
qu ¼ 0. qt
(17b)
(17c)
In Eq. (17a), C L denotes the velocity of longitudinal wave, which is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 nÞE . CL ¼ ð1 þ nÞð1 2nÞr
(18)
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2.4. Dimensionless quantities We define the following dimensionless quantities: 9 > ðr; aÞ ðH z ; H ðaÞ ðaÞ z Þ > ¯ ¯ ; ðH z ; H z Þ ¼ ;> ð¯r; a¯ Þ ¼ > > b H0 > > > > > t bJ y > ¯y ¼ > t¼ ; J ; > 2 > H0 > msb > > > > 2 > sb w CgT > > ¯ w¯ ¼ ; T ¼ ; > > 2 2 = H0 mH 0 bf r > > ; ðh¯ a ; h¯ b Þ ¼ bðha ; hb Þ; f¯ r ¼ > > > mH 20 > > > > ðsrr ; syy ; szz Þ > > > ðs¯ rr ; s¯ yy ; s¯ zz Þ ¼ ; > 2 > mH 0 > > > 2 > > > ð1 nÞE 2 > > > u¯ ¼ u > ; ð1 þ nÞð1 2nÞ bmH 2
Fundamental equation system: qT¯ 1 q qT¯ ¼ w1 r¯ þ w, ¯ qt r¯ q¯r q¯r
(19)
(24a)
r¯ ¼ 1;
qT¯ þ h¯ b T¯ ¼ 0, q¯r
(24b)
r¯ ¼ a¯ ;
qT¯ þ h¯ a T¯ ¼ 0, q¯r
(24c)
t ¼ 0;
T¯ ¼ 0.
(24d)
(3) Elastic field: Lorentz force: 1q ¯ ½H z ð¯r; tÞ2 . f¯ r ð¯r; tÞ ¼ 2 q¯r Stress–displacement relations:
0
9 q¯u n u¯ > ¯ w3 T; þ > > > q¯r 1 n r¯ = u¯ n q¯u ¯ w3 T; s¯ yy ð¯r; tÞ ¼ þ > > r¯ 1 n q¯r > > ¯ ; s¯ zz ð¯r; tÞ ¼ nðs¯ rr þ s¯ yy Þ ð1 2nÞw3 T:
and
(25)
s¯ rr ð¯r; tÞ ¼
msb x1 ¼ pffiffiffiffiffiffiffiffiffi; m 0 e0 w1 ¼ msk;
9 > > > =
ms2 b2 x2 ¼ ; e0 w2 ¼ msbC L ;
> 2aE > :> w3 ¼ ; ð1 2nÞCr
(20)
Fundamental equation system: q 1 qð¯ru¯ Þ 1 q2 u¯ qT¯ q ¯ 2 þ w3 þ ðH ¼ 2 zÞ , q¯r r¯ q¯r q¯r q¯r w2 qt
By use of these dimensionless quantities, Eqs. (2), (9)–(11), (13), (15)–(17) take the following form: (1) Electromagnetic field: Fundamental equation system: ¯z ¯z 1 q qH qH r¯ ð¯ao¯ro1Þ, ¼ r¯ q¯r q¯r qt ! ¯ ðaÞ ¯ ðaÞ 1 q qH 1 q2 H z z r¯ ð0o¯ro¯aÞ, ¼ 2 r¯ q¯r q¯r x1 qt
q¯u n u¯ ¯ ¼ w3 T, þ q¯r 1 n r¯
r¯ ¼ 1; a¯ ; (21a) t ¼ 0; (21b)
u¯ ¼
q¯u ¼ 0. qt
(26)
(27a)
(27b) (27c)
3. Solutions
r¯ ¼ 1;
¯ z ¼ fðtÞ, H
(21c)
3.1. Magnetic field
r¯ ¼ a¯ ;
¯z¼H ¯ ðaÞ H z ,
(21d)
In this section, the solutions of equation system (21) will be derived by using the Laplace transform: Z 1 f ðsÞ ¼ L½f ðtÞ ¼ f ðtÞest dt, (28)
ðaÞ
¯ ¯z q2 H qH ¼ x2 z , qtq¯r q¯r t ¼ 0;
¯ z ¼ 0, H
¯ ðaÞ qH z ¯ ðaÞ ¼ 0. ¼ H z qt
(21e)
0
(21f) (21g)
Eddy current: ¯ z ð¯r; tÞ qH . J¯ y ð¯r; tÞ ¼ q¯r
where the variable s is the Laplace transform parameter and asterisk denotes Laplace transformed variable. Applying the Laplace transform with respect to t to Eq. (21), we obtain the equation system in the transformed domain as follows:
(22)
¯z ¯ z 1 dH d2 H ¯ z ¼ 0 ð¯ao¯ro1Þ, þ k2 H þ 2 r d¯ r ¯ d¯r ðaÞ
¯z d2 H d¯r2
(2) Temperature field: Eddy current loss: w即 r; tÞ ¼ ½J¯ y ð¯r; tÞ2 .
(23)
r¯ ¼ 1;
(29a)
ðaÞ
þ
¯z 1 dH r¯ d¯r
¯ ðaÞ K 2H ¼0 z
¯ z ¼ f ðsÞ, H
ð0o¯ro¯aÞ,
(29b) (29c)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
r¯ ¼ a¯ ;
k2
¯ z ¯ ðaÞ dH dH z ¼ x2 , d¯r d¯r
(29d)
The residues Rn at s ¼ k2n are 2
Rn ¼ ekn t
where
2 D0 ðkn ; r¯ Þ, kn D0 ðkn Þ
k2 ¼ s,
ð30Þ
where D0 ðkn Þ are given by
k2 . x1
ð31Þ
D0 ðkn Þ ¼
K¼
Solving equation system (29), we obtain the solutions in the transformed domain as follows: ¯ z ð¯r; sÞ ¼ D0 ðk; r¯Þ sf ðsÞ, H sD0 ðk; 1Þ a3 ðkÞI0 ðK r¯Þ ¯ ðaÞ sf ðsÞ, r; sÞ ¼ H z ð¯ sD0 ðk; 1Þ
ð32Þ ð33Þ
where Dl ðk; r¯Þ ¼ a1 ðkÞJl ðk¯rÞ a2 ðkÞYl ðk¯rÞ ðl ¼ 0; 1Þ, 9 x2 > a1 ðkÞ ¼ kY1 ðka¯ ÞI0 ðK a¯ Þ Y0 ðka¯ ÞI1 ðK a¯ Þ; > > > x1 > = x2 a2 ðkÞ ¼ kJ1 ðka¯ ÞI0 ðK a¯ Þ J0 ðka¯ ÞI1 ðK a¯ Þ; > > x1 > > > a3 ðkÞ ¼ k½J0 ðka¯ ÞY1 ðka¯ Þ J1 ðka¯ ÞY0 ðka¯ Þ; ;
(34)
(35)
where Jl ðÞ is the Bessel function of the first kind of lth order, Yl ðÞ is the Bessel function of the second kind of lth order and Il ðÞ is the modified Bessel function of the first kind of lth order, respectively. The inverse Laplace transform of Eqs. (32) and (33) can be performed by the convolution theorem: Z t 1 L ½f 1 ðsÞ f 2 ðsÞ ¼ f 1 ðt t0 Þf 2 ðt0 Þdt0 . (36)
369
(41)
dD0 ðkn ; 1Þ dkn ¼ a¯ ½kn Y0 ðkn a¯ ÞI0 ðK n a¯ Þ þ 2K n Y1 ðkn a¯ ÞI1 ðK n a¯ Þ x2 þ a¯ Y1 ðka¯ ÞI1 ðK n a¯ Þ 2Y0 ðka¯ Þ x1 1 ðK n a¯ I0 ðK n a¯ Þ I1 ðK n a¯ ÞÞ J0 ðkn Þ kn a¯ ½kn J0 ðkn a¯ ÞI0 ðK n a¯ Þ þ 2K n J1 ðkn a¯ ÞI1 ðK n a¯ Þ x2 þ a¯ J1 ðka¯ ÞI1 ðK n a¯ Þ 2J0 ðka¯ Þ x1 1 ðK n a¯ I0 ðK n a¯ Þ I1 ðK n a¯ ÞÞ Y0 ðkn Þ kn a1 ðkn ÞJ1 ðkn Þ a2 ðkn ÞY1 ðkn Þ.
ð42Þ
Therefore, from Eqs. (40) and (41), Gð¯r; tÞ is determined to be 1 X
Gð¯r; tÞ ¼ 1 þ
2
ekn t
n¼1
2 D0 ðkn ; r¯Þ. kn D0 ðkn Þ
(43)
Substituting of Eq. (43) into Eq. (37), we obtain the magnetic field in the hollow cylinder as follows: ¯ z ð¯r; tÞ ¼ fðtÞ þ H
0
1 X
2 D0 ðkn ; r¯ Þa^ n ðtÞ, 0 k D ðkn Þ n n¼1
(44)
Using the convolution theorem (36) for Eq. (32), we obtain the following form: Z t dfðt0 Þ 0 ¯ z ð¯r; tÞ ¼ H Gð¯r; t t0 Þ dt , (37) dt0 0
where a^ n ðtÞ are determined by the function fðtÞ, which are given by Z t 0 2 0 dfðt Þ ekn ðtt Þ dt0 . (45) a^ n ðtÞ ¼ dt0 0
where
Substituting Eq. (44) into Eq. (22), we obtain the eddy current J¯ y as follows:
Gð¯r; tÞ ¼ L1
D0 ðk; r¯Þ . sD0 ðk; 1Þ
(38) J¯ y ð¯r; tÞ ¼
The inverse Laplace transform shown in (38) is performed by the use of the inversion theorem: Z zþi1 D0 ðk; r¯Þ 1 D0 ðk; r¯Þ est L1 ¼ ds, (39) sD0 ðk; 1Þ 2pi zi1 sD0 ðk; 1Þ pffiffiffiffiffiffiffi where i ¼ 1 and z is taken to be large enough so that all the singularities lie to the left of line of integration. The right-hand side of Eq. (39) is determined by the sum of the residues at poles of its integrand. The integrand est ½D0 ðk; r¯Þ=sD0 ðk; 1Þ has a pole at s ¼ 0 and poles at s ¼ k2n , and kn are the positive roots of transcendental equation D0 ðk; 1Þ ¼ 0. The residue R0 at s ¼ 0 is R0 ¼ 1.
(40)
1 X
2 D1 ðkn ; r¯ Þa^ n ðtÞ. 0 D ðk nÞ n¼1
(46)
In the same manner, the magnetic field in the air region is determined to be ¯ ðaÞ H r; tÞ ¼ fðtÞ þ z ð¯
1 X
2 a3 ðkn ÞI0 ðK n r¯Þa^ n ðtÞ. 0 k D ðkn Þ n n¼1
(47)
3.2. Temperature field By using the separation of variables technique, the solution of Eq. (24) will be assumed in the following form: ¯ r; tÞ ¼ Tð¯
1 X j¼1
bj ðtÞR0 ðpj ; r¯ Þ,
(48)
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where bj ðtÞ are unknown functions of t, and R0 ðpj ; r¯Þ are R0 ðpj ; r¯Þ ¼ b1 ðpj ÞJ0 ðpj r¯Þ b2 ðpj ÞY0 ðpj r¯Þ
(49)
in which b1 ðpj Þ ¼ h¯ a Y0 ðpj a¯ Þ þ pj Y1 ðpj a¯ Þ; b2 ðpj Þ ¼ h¯ a J0 ðpj a¯ Þ þ pj J1 ðpj a¯ Þ
)
a
(50)
and pj are the positive roots of the following eigenequation: ½h¯ a J0 ðp¯aÞ þ pJ1 ðp¯aÞ½h¯ b Y0 ðpÞ pY1 ðpÞ ½h¯ a Y0 ðp¯aÞ þ pY1 ðp¯aÞ½h¯ b J0 ðpÞ pJ1 ðpÞ ¼ 0.
j¼1
a¯
1
þ a¯
w即 r; tÞR0 ðpq ; r¯Þ¯r d¯r.
Substituting Eq. (58) into Eq. (48), we obtain temperature change as follows: ¯ r; tÞ ¼ Tð¯
ð51Þ
Substituting of Eq. (48) into Eq. (24a) with Eqs. (24b) and (24c), multiplying both side by r¯R0 ðpq ; r¯Þ, and then integrating with respect to r¯ from a¯ to 1, we obtain Z 1 X dbj ðtÞ 1 R0 ðpj ; r¯ÞR0 ðpq ; r¯Þ¯r d¯r dt a¯ j¼1 Z 1 1 X ¼ w1 p2j bj ðtÞ R0 ðpj ; r¯ÞR0 ðpq ; r¯Þ¯r d¯r Z
and I 1jmn are given by the following equation, and are calculated by numerical integration: Z 1 I 1jmn ¼ D1 ðkm ; r¯ÞD1 ðkn ; r¯ÞR0 ðpj ; r¯Þ¯r d¯r. (60)
ð52Þ
Using the orthogonality of Bessel functions, we can derive the following equation: ( Z 1 ðq ¼ jÞ; Mj R0 ðpj ; r¯ÞR0 ðpq ; r¯ Þ¯r d¯r ¼ (53) 0 ðqajÞ; a¯
i 1 h 2 ¯2 2 2 ¯ 2 Þ¯a2 R2 ðpj ; a¯ Þ . ðp þ h ÞR ðp ; 1Þ ðp þ h j j 0 j 0 b a 2p2j
Substitution of Eq. (53) into Eq. (52) gives Z 1 dbj ðtÞ 1 2 þ w1 pj bj ðtÞ ¼ w即 r; tÞR0 ðpj ; r¯Þ¯r d¯r. M j a¯ dt
(54)
(55)
The solutions of Eq. (55) under the initial condition (24d) are expressed as Z 1 Z t 1 w1 p2j ðtt0 Þ 0 0 wð¯ e bj ðtÞ ¼ ¯ r; t Þdt R0 ðpj ; r¯Þ¯r d¯r. (56) M j a¯ 0 Here, substitution of Eq. (46) into Eq. (23) yields w即 r; tÞ ¼ 4
1 D1 ðkm ; r¯ÞD1 ðkn ; r¯Þa^ m ðtÞa^ n ðtÞ. 0 D ðk ÞD0 ðkn Þ m m¼1 n¼1 (57)
1 X 1 I 1jmn 4 X b^jmn ðtÞ, M j m¼1 n¼1 D0 ðkm ÞD0 ðkn Þ
u¯ ð¯r; tÞ ¼ u1 ð¯r; tÞ þ u2 ð¯r; tÞ, where u1 ð¯r; tÞ satisfies q 1 qð¯ru1 Þ ¼ 0, q¯r r¯ q¯r qu1 n u1 ¯ ¼ w3 T. þ q¯r 1 n r¯
r¯ ¼ 1; a¯ ;
The solution of Eqs. (63) is expressed as 1n ¯ tÞ a¯ 2 T即 a; tÞ¯r w ½Tð1; u1 ¼ 1 a¯ 2 3 2 1 ¯ tÞ T即 a; tÞ a¯ . ½Tð1; þ 1 2n r¯
(62)
(63a)
(63b)
ð64Þ
Substitution of Eq. (62) with Eqs. (63a) into Eqs. (27a) yields q 1 qð¯ru2 Þ 1 q2 u2 1 q2 u1 qT¯ q ¯ 2 þ ¼ 2 2 þ 2 2 þ w3 H z , (65a) q¯r r¯ q¯r q¯r q¯r w2 qt w2 qt qu2 n u2 ¼ 0, þ q¯r 1 n r¯
t ¼ 0;
u2 ¼ u1 ;
(58)
where b^jmn ðtÞ are determined by the function fðtÞ, which are given by Z t 2 0 b^jmn ðtÞ ¼ ew1 pj ðtt Þ a^ m ðt0 Þa^ n ðt0 Þ dt0 (59) 0
ð61Þ
qu2 qu1 ¼ . qt qt
(65b)
(65c)
By using the separation of variables technique, the solution of Eqs. (65) will be assumed in the following form:
Substitution of Eq. (57) into Eq. (56) gives bj ðtÞ ¼
I 1jmn b^jmn ðtÞ. 0 0 D ðk m ÞD ðkn Þ m¼1 n¼1
3.3.1. Dynamic solutions In order to transform the inhomogeneous boundary conditions (27b) into the homogeneous one, we assume that u¯ ð¯r; tÞ is given by following expression:
r¯ ¼ 1; a¯ ;
1 X 1 X
1 X 1 X
3.3. Elastic field
where Mj ¼
1 X 4 R0 ðpj ; r¯Þ M j j¼1
u2 ð¯r; tÞ ¼
1 X
ci ðtÞV 1 ðZi ; r¯Þ,
(66)
i¼1
where ci ðtÞ are unknown functions of t, and V l ðZ; r¯Þ ðl ¼ 0; 1Þ are V l ðZi ; r¯Þ ¼ g1 ðZi ÞJl ðZi r¯Þ g2 ðZi ÞYl ðZi r¯ Þ ðl ¼ 0; 1Þ
(67)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
in which 9 1 2n Y1 ðZi a¯ Þ > ;> = a¯ 1n 1 2n J1 ðZi a¯ Þ > > g2 ðZi Þ ¼ Zi J0 ðZi a¯ Þ ; a¯ 1n
cTi ðtÞ ¼
g1 ðZi Þ ¼ Zi Y0 ðZi a¯ Þ
(68)
and Zi are the positive roots of the following eigenequation: 1 2n 1 2n Y1 ðZ¯aÞ J1 ðZÞ ZY0 ðZ¯aÞ ZJ0 ðZÞ 1n 1n a¯ 1 2n ZY0 ðZÞ Y1 ðZÞ 1n 1 2n J1 ðZ¯aÞ ZJ0 ðZ¯aÞ ¼ 0. ð69Þ 1 n a¯ In the same manner as Section 3.2, substituting Eq. (66) with Eq. (65b) into Eq. (65a), and utilizing the orthogonality of Bessel functions, we can derive the following equation: Z q2 ci ðtÞ 1 1 q2 u1 þ O2i ci ðtÞ ¼ V 1 ðZi ; r¯Þ¯r d¯r qt N i a¯ qt Z w22 1 qT¯ V 1 ðZi ; r¯Þ¯r d¯r w3 N i a¯ q¯r Z ¯ 2z w2 1 qH V 1 ðZi ; r¯Þ¯r d¯r, ð70Þ 2 N i a¯ q¯r
ð71Þ
and Oi are the natural angular frequencies of ith order mode in dimensionless form, which are given by O i ¼ w2 Zi .
9 > > > > > > > > > > > =
1 1 P 1 P 1 I 1jmn 4w3 P 2 Zi N i j¼1 M j m¼1 n¼1 D0 ðkm ÞD0 ðkn Þ
½I 2ij b^jmn ðtÞ þ I 3ij Oi c^Tijmn ðtÞ; 1 4w2 P I 4in Mð1Þ > cM ðtÞ ¼ ðtÞ c^ > i > Zi N i n¼1 D0 ðkn Þ in > > > > 1 1 > P P I 5imn > Mð2Þ > > ^ þ ðtÞ ; c ; imn 0 0 k k D ðk ÞD ðk Þ m n m n m¼1 n¼1
(75)
where cTi ðtÞ and cM i ðtÞ are the terms due to temperature change and due to Lorentz force, respectively, and c^Tijmn ðtÞ, Mð1Þ Mð2Þ c^in ðtÞ and c^imn ðtÞ are defined by the function fðtÞ, which are given by c^Tijmn ðtÞ ¼
Rt
c^Mð1Þ ðtÞ ¼ in c^Mð2Þ imn ðtÞ
¼
0
sin Oi ðt t0 Þb^jmn ðt0 Þ dt0 ;
Rt 0 Rt 0
sin Oi ðt t0 Þfðt0 Þa^ n ðt0 Þ dt0 ;
9 > > =
> > sin Oi ðt t0 Þa^ m ðt0 Þa^ n ðt0 Þ dt0 ;
(76)
and I 2ij ¼ R0 ðpj ; 1ÞV 1 ðZi ; 1Þ a¯ R0 ðpj ; a¯ ÞV 1 ðZi ; a¯ Þ,
I 3ij ¼
where 1 1 2n 2 N i ¼ 2 Zi V 21 ðZi ; 1Þ 2Zi ð1 nÞ2 1 2n 2 2 2 Zi a¯ V 1 ðZi ; a¯ Þ ð1 nÞ2
371
I 4in ¼
¯ Z2i hb 1 2n 1 R0 ðpj ; 1ÞV 1 ðZi ; 1Þ p2j Z2i Z2i 1 n ¯ ha 1 2n þ 2 þ 1 a¯ R0 ðpj ; a¯ ÞV 1 ðZi ; a¯ Þ , Zi a¯ 1 n
(77)
ð78Þ
1 1 2n D1 ðkn ; 1ÞV 1 ðZi ; 1Þ 2 2 k n Zi 1 n 1 2n D1 ðkn ; a¯ Þ kn D0 ðkn ; a¯ Þ a¯ V 1 ðZi ; a¯ Þ , ð79Þ 1n
(72)
The solutions of Eq. (70) under the initial condition (65c) are expressed as Z Z t 1 1 ci ðtÞ ¼ u1 ð¯r; tÞ þ Oi sin Oi ðt t0 Þu1 ð¯r; t0 Þdt0 N i a¯ 0 w2 V 1 ðZi ; r¯Þ¯r d¯r w3 Zi N i Z 1 Z t q 0 ¯ 0 0 sin Oi ðt t ÞTð¯r; t Þdt r 0 a¯ q¯ w V 1 ðZi ; r¯Þ¯r d¯r 2 Zi N i Z 1 Z t q ¯ z ð¯r; t0 Þg2 dt0 sin Oi ðt t0 ÞfH r 0 a¯ q¯
I 5imn ¼ a¯ D0 ðkm ; a¯ ÞD0 ðkn ; a¯ ÞV 1 ðZi ; a¯ Þ Z 1 þ Zi D0 ðkm ; r¯ ÞD0 ðkn ; r¯ ÞV 0 ðZi ; r¯Þ¯r d¯r.
ð80Þ
a¯
The second term of the right-hand side of Eq. (80) is calculated by numerical integration. From Eqs. (62), (66), (74), we have u¯ T ð¯r; tÞ ¼ u1 ð¯r; tÞ þ
1 P i¼1
u¯ M ð¯r; tÞ ¼
1 P i¼1
9 > cTi ðtÞV 1 ðZi ; r¯Þ; > > =
cM ¯ Þ; i ðtÞV 1 ðZi ; r
> > > ;
(81)
Substitution of Eqs. (44), (61) and (64) into Eq. (73) gives
where u¯ T ð¯r; tÞ and u¯ M ð¯r; tÞ are the radial displacements due to temperature change and due to Lorentz force, respectively, and satisfy the following relation:
ci ðtÞ ¼ cTi ðtÞ þ cM i ðtÞ,
u¯ ð¯r; tÞ ¼ u¯ T ð¯r; tÞ þ u¯ M ð¯r; tÞ.
V 1 ðZi ; r¯Þ¯r d¯r.
ð73Þ
(74)
(82)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
372
Substituting Eq. (82) into Eq. (26), we can obtain the dynamic solutions of the stress components as follows: 9 ¯ r; tÞ s¯ Trr ð¯r; tÞ ¼ s1 ð¯r; tÞ w3 Tð¯ > > > > 1 > P T 1 2n V 1 ðZi ; r¯Þ > > > ; þ ci ðtÞ Zi V 0 ðZi ; r¯ Þ > > 1 n r ¯ > i¼1 > > > T > ¯ s¯ yy ð¯r; tÞ ¼ s2 ð¯r; tÞ w3 Tð¯r; tÞ > = 1 P nZi 1 2n V 1 ðZi ; r¯Þ T V 0 ðZi ; r¯ Þ þ ;> þ ci ðtÞ > > 1n r¯ 1n i¼1 > > > > T ¯ > s¯ zz ð¯r; tÞ ¼ s3 ð¯r; tÞ w3 Tð¯r; tÞ > > > > 1 > P nZ > i T > V 0 ðZi ; r¯Þ; þ ci ðtÞ > ; 1n i¼1 (83) 9 1 2n V 1 ðZi ; r¯Þ > > ¼ Zi V 0 ðZi ; r¯Þ ; > > > 1n r¯ i¼1 > > > = 1 P nZ 1 2n V ðZ ; r Þ ¯ 1 i i M M s¯ yy ð¯r; tÞ ¼ V 0 ðZi ; r¯Þ þ ci ðtÞ ; , 1n r¯ 1n > i¼1 > > > 1 > P nZ > i M M > V 0 ðZi ; r¯ Þ; ci ðtÞ s¯ zz ð¯r; tÞ ¼ > ; 1n s¯ M r; tÞ rr ð¯
1 P
cM i ðtÞ
i¼1
(84) where s¯ Tii ði ¼ r; y; zÞ and s¯ M ii are thermal stresses and magnetic stresses, respectively, and satisfy the following relations: s¯ rr ¼ s¯ Trr þ s¯ M rr ;
s¯ yy ¼ s¯ Tyy þ s¯ M yy ;
s¯ yy ¼ s¯ Tzz þ s¯ M zz
and s1 ð¯r; tÞ; s2 ð¯r; tÞ; s3 ð¯r; tÞ in Eq. (83) are given by 9 w3 ¯ tÞ a¯ 2 T即 a; tÞ > > s1 ð¯r; tÞ ¼ ½ Tð1; > > 1 a¯ 2 > > > > 2 > > a ¯ > ¯ tÞ T即 a; tÞ > ½Tð1; ; > 2 > r¯ > > > = w3 ¯ tÞ a¯ 2 T即 a; tÞ s2 ð¯r; tÞ ¼ ½ Tð1; > 1 a¯ 2 > > > > 2 > a ¯ > > ¯ ¯ þ½Tð1; tÞ Tð¯a; tÞ 2 ; > > > r¯ > > > > 2nw3 ¯ > 2¯ > ; ½ Tð1; tÞ a s3 ð¯r; tÞ ¼ Tð¯ a ; tÞ: ¯ 2 1 a¯
(85)
(86)
3.3.2. Quasi-static solutions In this section, we derive the quasi-static solutions of the displacements and stresses. Disregarding the inertia term of the right hand side in Eq. (27a) gives the equilibrium equation: d 1 dð¯ru¯ Þ dT¯ d ¯ 2 þ ðH (87) ¼ w3 zÞ . d¯r r¯ d¯r d¯r d¯r Solving Eq. (87) with the boundary conditions (27b), we obtain the quasi-static solutions of the radial displacements due to temperature change and Lorentz force, respectively,
which are given as follows: 9 1 > ð2Þ ðtÞ¯ r þ H ðtÞ ;> u¯ T ð¯r; tÞ ¼ w3 F T ð¯r; tÞ þ ð1 2nÞH ð1Þ T T r¯ = > 1 > ; r þ H ð2Þ u¯ M ð¯r; tÞ ¼ F M ð¯r; tÞ þ ð1 2nÞH ð1Þ M ðtÞ¯ M ðtÞ ; r¯ (88) where 9 1R ¯ 1R ¯ 2 > r¯T d¯r; F M ð¯r; tÞ ¼ r¯ H z d¯r; > > > r¯ r¯ > > > > 1 > ð1Þ > H T ðtÞ ¼ ½F ð1; tÞ a F ð¯ a ; tÞ; ¯ > T T > 2 > 1 a¯ > > > a¯ > ð2Þ > > ½¯ a F ð1; tÞ F ð¯ a ; tÞ; H T ðtÞ ¼ T T > 2 > 1 a¯ > > > = 1 ð1Þ ½F ð1; tÞ a F ð¯ a ; tÞ H M ðtÞ ¼ ¯ M M 2 1 a¯ > > > 1n ¯2 > 2 ¯ 2 > ½H z ð1; tÞ a¯ H z ð¯a; tÞ ; > > > 1 2n > > > > a¯ > ð2Þ > > H M ðtÞ ¼ ½¯ a F ð1; tÞ F ð¯ a ; tÞ M M > 2 > 1 a¯ > > > > 1n ¯2 > 2 > ¯ > a¯ ½H z ð1; tÞ H z ð¯a; tÞ : ; 1 2n F T ð¯r; tÞ ¼
(89)
Substituting Eq. (88) with the relation of Eq. (82) into Eq. (26), we obtain the quasi-static solutions of the stress components as follows: " # 9 > 1 2n 1 H ð2Þ ð1Þ T ðtÞ > T s¯ rr ð¯r; tÞ ¼ w3 F T ð¯r; tÞ þ H T ðtÞ ;> > 2 > 1n r¯ > r¯ > > > > > 1 2n 1 > T > ¯ > Tð¯r; tÞ þ F T ð¯r; tÞ s¯ yy ð¯r; tÞ ¼ w3 = 1n r¯ # > > H ð2Þ > T ðtÞ > þH ð1Þ ; > T ðtÞ þ 2 > > r¯ > > > > > 1 2n > ð1Þ T > ¯ ½Tð¯r; tÞ þ 2nH T ðtÞ; s¯ zz ð¯r; tÞ ¼ w3 ; 1n (90) 9 1 2n 1 2 > ¯ > F M ð¯r; tÞ ¼ H z ð¯r; tÞ þ > > 1n r¯ > > # > > ð2Þ > > H M ðtÞ > ð1Þ > þH M ðtÞ ; > > > r¯2 > > > = n ¯2 1 2n 1 M F M ð¯r; tÞ s¯ yy ð¯r; tÞ ¼ H z ð¯r; tÞ þ > 1n 1 n r¯ > > # > > ð2Þ > > H ðtÞ > ð1Þ M > þH M ðtÞ þ ; > > 2 > r¯ > > > > > n 2 ð1Þ M ¯ ; ½H z ð¯r; tÞ þ 2ð1 2nÞH M ðtÞ: > s¯ zz ð¯r; tÞ ¼ 1n s¯ M r; tÞ rr ð¯
(91)
These quasi-static thermal stresses and magnetic stresses satisfy the relations of Eq. (85).
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
In this paper, we assume that an arbitrary function fðtÞ are defined as the sine function: fðtÞ ¼ sin ot,
(92)
where o is an angular frequency of magnetic field in the nondimensional form. For this case, the functions of t: Mð1Þ a^ n ðtÞ, b^jmn ðtÞ, c^Tijmn ðtÞ, c^in ðtÞ, c^Mð2Þ imn ðtÞ in Eqs. (45), (59) and (76) are determined by the function fðtÞ in Eq. (92). However, these functions are omitted here because they can be easily derived. Making use of the analytical results mentioned above, we carry out numerical calculations for aluminum, material properties of which are given by 9 m ¼ 4p 107 ðH=mÞ; s ¼ 3:42 107 ðS=mÞ; > > > > 3 3 3 C ¼ 2:7 10 ðJ=kgKÞ; r ¼ 0:9 10 ðkg=m Þ; = (93) > k ¼ 92:6 106 ðm2 =sÞ; > > > n ¼ 0:33; E ¼ 70ðGPaÞ; a ¼ 24 106 ð1=KÞ: ;
The natural angular frequencies given by Eq. (72) are depend on the radius ratio a¯ ¼ a=b as shown by the eigenequation (69). Here, we set a value of o as o ¼ e O1ð0:9Þ ,
25
The values of the nondimensional relative heat transfer coefficients h¯ a and h¯ b are given by h¯ a ¼ h¯ b ¼ 1:0.
(94)
In addition, since the nondimensional variables x1 , x2 and w2 in Eq. (20) include the outer radius b, this dimension needs to be fixed. We chose b ¼ 1:0 104 ðmÞ to ensure the convergence of the solutions.
(95)
where e is a parameter, and Oð0:9Þ is the natural angular 1 frequency of first order mode for the case of the radius ratio a¯ ¼ 0:9. Although angular frequencies of magnetic field and eddy current are o, Eqs. (10) and (13) indicate that those of temperature change caused by eddy current loss and Lorentz force become the twice of o.
Maximum absolute value of Jθ
4. Numerical results and discussion
= 0.4 × 1(0.9) = 2.0 × 1(0.9)
20
= 4.0 × 1(0.9)
15 10 5 0
0
0.2
r = 0.90 r = 0.95 r = 1.00
0.5
= 0.9 = 0.4
Hz
0 -0.5
0
0.5
1 τ
1.5
2
¯ z. Fig. 2. Time evolution of magnetic field H r = 0.90 r = 0.95 r = 1.00
4
Jθ
2
= 0.9 = 0.4
0 -2 -4 0
0.5
1 τ
0.4
0.6
0.8
1
Fig. 4. Maximum absolute value of eddy current J¯ y versus radius ratio a¯ .
1
-1
373
1.5
Fig. 3. Time evolution of eddy current J¯ y .
2
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
374
¯z Figs. 2 and 3 show the time evolutions of magnetic field H ¯ and eddy current J y for the radius ratio a¯ ¼ 0:9 and e ¼ 0:4, respectively. It can be seen from Figs. 2 and 3 that magnetic ¯ z and eddy current J¯ y vary sinusoidally. We here field H examine the maximum absolute values of eddy current J¯ y with radius ratio a¯ for some values of angular frequency o. Fig. 4 shows the maximum absolute values of eddy current J¯ y with the radius ratio a¯ . It can be seen from Fig. 4 that the maximum absolute values of eddy current J¯ y increases with the angular frequency o, and they also increase with the radius ratio a¯ because the area in which eddy current is
generated decreases with increasing radius ratio a¯ . However, for the case of o ¼ 4:0 Oð0:9Þ 1 , the maximum absolute value drops near a¯ ¼ 0:8. This is because, as shown in Fig. 5 which shows the distributions of the maximum absolute values of eddy current J¯ y for the case of o ¼ 4:0 Oð0:9Þ 1 , the maximum absolute value at the inner surface ð¯r ¼ 0:9Þ for the case of a¯ ¼ 0:8 is larger than for the case of a¯ ¼ 0:6. Fig. 6 shows the time evolution of temperature change T¯ for the radius ratio a¯ ¼ 0:9 and e ¼ 0:4. Temperature change T¯ increases with increasing time, and then it vibrates with the angular frequency 2o. In addition,
= 0.60 = 0.80 = 0.90 = 0.95
20 15
1200
10 5 0
= 0.4 × (0.9) 1 = 2.0 × (0.9) 1
1000
= 4.0 × 1(0.9)
Maximum value of T
Maximum absolute value of Jθ
25
= 4.0 × (0.9) 1
800 600 400 200
0.6
0.7
0.8
0.9
1
0
r Fig. 5. Distributions of maximum absolute values of eddy current J¯ y for o ¼ 4:0 O1ð0:9Þ .
0
0.2
r = 0.90 r = 0.95 r = 1.00
100
T
80
= 0.9 ε = 0.4
60
= 1.0 = 1.0
40 20
18 16 14 12 10 8 6 4 2 0
0
0
20
40
0.5
τ
1 τ
0.6
0.8
1
Fig. 7. Maximum values of temperature change T¯ versus radius ratio a¯ .
120
0
0.4
60
80
1.5
¯ Fig. 6. Time evolution of temperature change T.
100
2
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
temperature change T¯ takes long time to attain a steady state. This is because the value of w1 in Eq. (20), which is the ratio of the diffusion coefficient of temperature field k to that of magnetic field 1=ms, is very small as shown in the following equation: w1 ¼
k ffi 3:98 103 . 1=ms
(96)
= 0.9
0.06
ε = 0.4
375
Fig. 7 shows the maximum values of temperature change T¯ with the radius ratio a¯ . It can be seen from Fig. 7 that for the cases of e ¼ 0:4, the maximum value of temperature change T¯ decreases with the radius ratio a¯ , and that for the cases of e ¼ 2:0; 4:0, the maximum values decrease and then increase with the radius ratio a¯ . In all cases, the decreases of the maximum values with the increasing radius ratio are Dynamic solution Quasi-static solution
r = 0.95
0.04 0.02
rr
0 -0.02 -0.04 -0.06 -0.08 -0.1
0
1
= 0.9
0.06
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.95
0.04 0.02
M rr
0 -0.02 -0.04 -0.06 -0.08 -0.1
0
1
= 0.9
0.06
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.95
0.04 0.02
Trr
0 -0.02 -0.04 -0.06 -0.08 -0.1
0
1
2 τ
3
4
Fig. 8. Dynamic and quasi-static behaviors of the radial stress components, s¯ rr ; s¯ M ¯ Trr : (a) total stress, s¯ rr ¼ s¯ Trr þ s¯ M ¯ Trr , (c) magnetic rr ; s rr , (b) thermal stress, s stress, s¯ M . rr
ARTICLE IN PRESS 376
M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
due to the increase of the heat dissipation at the inner surface. In the cases of e ¼ 2:0; 4:0, because the maximum absolute values of eddy current J¯ y increase considerably beyond the value of about a¯ ¼ 0:8 as shown in Fig. 4, the maximum values of temperature change will increase. Next, we examine the dynamic and quasi-static behaviors of thermal stresses and magnetic stresses in the
= 0.9
25
ε = 0.4
cylinder. In order to examine inertia effects, numerical results for e ¼ 0:4 and a¯ ¼ 0:9 are presented in Figs. 8–10. Fig. 8 shows the dynamic and quasi-static behaviors of the radius stress components s¯ rr ð¼ s¯ Trr þ s¯ M ¯ Trr , s¯ M rr Þ, s rr at r¯ ¼ 0:95. Fig. 9 shows the dynamic and quasi-static behaviors of the circumferential stress components s¯ yy ð¼ s¯ Tyy þ s¯ M ¯ ¼ 0:9. Fig. 10 shows the ¯ Tyy , s¯ M yy Þ, s yy at r Dynamic solution Quasi-static solution
r = 0.95
20 15 10
5 0 -5 -10 -15 -20 -25
0
1
= 0.9
25
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.9
20 15 10 M
5 0 -5 -10 -15 -20 -25
0
1
= 0.9
25
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.9
20 15 10 T
5 0 -5 -10 -15 -20 -25
0
1
2 τ
3
4
Fig. 9. Dynamic and quasi-static behaviors of the circumferential stress components, s¯ yy ; s¯ M ¯ Tyy : (a) total stress, s¯ yy ¼ s¯ Tyy þ s¯ M ¯ Tyy , yy ; s yy , (b) thermal stress, s (c) magnetic stress, s¯ M . yy
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
dynamic and quasi-static behaviors of the radius stress components s¯ zz ð¼ s¯ Tzz þ s¯ M ¯ ¼ 1:0. It can be ¯ Tzz , s¯ M zz Þ, s zz at r seen from these figures that the amplifications of the amplitude involved in the dynamic behaviors of the total stresses s¯ rr ; s¯ yy ; s¯ zz in Figs. 8(a), 9(a), 10(a) are mainly caused by magnetic stress components in Figs. 8(c), 9(c),
= 0.5
10
ε = 0.4
377
10(c). Therefore, we should evaluate the magnetic stresses for the amplifications of the stress amplitudes, comparing dynamic solutions with quasi-static ones. Fig. 11 shows the variations of maximum absolute values of the circumferential magnetic stress s¯ M yy , which is the dominant component for the magnetic stresses as shown in Figs. 8–10, in
Dynamic solution Quasi-static solution
r = 1.0
0
zz
-10 -20 -30 -40 0
1
= 0.5
10
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 1.0
0
M zz
-10 -20 -30 -40 0
1
= 0.5
10
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 1.0
0
T zz
-10 -20 -30 -40 0
1
2 τ
3
4
Fig. 10. Dynamic and quasi-static behaviors of the axial stress components, s¯ zz ; s¯ M ¯ Tzz : (a) total stress, s¯ zz ¼ s¯ Tzz þ s¯ M ¯ Tzz , (c) zz ; s zz , (b) thermal stress, s magnetic stress, s¯ M . zz
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
378
Maximum abusolute value of
(0.9)
40
1
(0.9)
2
/2
/2
Dynamic solution Quasi-static solution
35 30
a = 0.9
25 20 15 10 5 0
0
2
4
6
8
10
12
14
16
18
(0.9)
20 [ ×1
]
Fig. 11. Variations of maximum absolute values of the circumferential magnetic stress in dynamic and quasi-static solutions with angular frequency.
Maximum absolute value of σ Tzz
1600
= 0.4 × (0.9) 1
1400
= 2.0 × = 4.0 ×
1200
(0.9) 1 (0.9) 1
1000 800 600 400 200 0
0
0.2
0.4
0.6
0.8
1
Fig. 12. Maximum absolute values of the quasi-static axial thermal stress s¯ Tzz with the radius ratio a¯ .
dynamic and quasi-static solutions with angular frequency. It can be seen from Fig. 11 that when angular frequency o tends to half of the natural frequency Oð0:9Þ or Oð0:9Þ 1 2 , the considerable increase due to the resonance in the dynamic stress is observed. However, the absolute values of the axial total stress s¯ zz , which is the dominant stress component as shown in Figs. 8–10, in both the dynamic and quasi-static solutions shown in Fig. 10(a) increase with increasing time due to the thermal stress shown in Fig. 10(b). In addition, there is no significant difference in magnitude between the dynamic and quasi-static behaviors of the axial total stress s¯ zz (and axial thermal stress s¯ Tzz ). Then, the maximum absolute values of the quasi-static axial thermal stresses s¯ Tzz with the radius ratio a¯ are shown in Fig. 12. Because maximum absolute value of s¯ Tzz as show in Fig. 12 are much larger than those of the magnetic stresses as shown in Fig. 11, the quasi-static thermal stresses play significant roles to estimate the magnitude of stresses in the cylinder. 5. Conclusion In the present study, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow
circular cylinder subjected to an arbitrary variation of magnetic field, which is defined by arbitrary function of time, have been investigated. Exact solutions of electromagnetic field, temperature change, stresses and deformations have been obtained. The stresses are determined to be sums of thermal stress and magnetic stress. Carrying out numerical calculations for the case that the arbitrary function of time is given by the sine function, we have examined the dynamic and quasi-static behaviors of the stresses. We conclude that quasi-static thermal stresses play significant roles to estimate the magnitude of stresses in the hollow cylinder because magnetic stresses are much smaller than thermal stresses and there is no significant deference in magnitude between dynamic and quasi-static behaviors of thermal stresses.
References [1] Paria G. Magneto-elasticity and magneto-thermo-elasticity. Advances in Applied Mechanics 1967;10:73–112. [2] Eringen AC, Maugin GA. Electrodynamics of continua, vol. I, Foundations and solid media. New York: Springer; 1990. [3] Kaliski S, Nowacki W. Combined elastic and electromagnetic waves produced by thermal shock. Bulletin de l Academie Polonaise des Sciences-Serie des Sciences Techniques 1962;10:159–68. [4] Kaliski S, Michalec J. The resonance amplification of a magnetoelastic wave radiated from a cylindrical cavity. Proceedings of Vibration Problems 1963;4:7–15. [5] Purushotham CM. Magneto-elastic coupling effects on the propagation of harmonic waves in an electrically conducting elastic plate. Proceedings of the Cambridge Philosophical Society 1967;63: 503–11. [6] Moon FC, Pao PH. Magnetoelastic buckling of a thin plate. Journal of Applied Mechanics, Transactions of the ASME 1968;35:53–8. [7] Moon FC, Pao PH. Vibration and dynamic instability of a beamplate in a transverse magnetic field. Journal of Applied Mechanics, Transactions of the ASME 1969;36:92–100. [8] Wauer J. Free and forced magneto-thermo-elastic vibrations in a conducting plate layer. Journal of Thermal Stresses 1996;19:671–91. [9] Banerjee S, Roychoudhuri SK. Magneto-thermo-elastic interactions in an infinite isotropic elastic cylinder subjected to a periodic loading. International Journal of Engineering Science 1997;35:437–44.
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379 [10] Wang X, Lu G, Guillow SR. Magnetothermodynamic stress and perturbation of magnetic field vector in a solid cylinder. Journal of Thermal Stresses 2002;25:909–26. [11] Wang X, Lee JS, Zheng X. Magneto-thermo-elastic instability of ferromagnetic plates in thermal and magnetic fields. International Journal of Solids and Structures 2003;40:6125–42. [12] Ezzat MA, Youssef HM. Generalized magneto-thermoelasticity in a perfectly conducting medium. International Journal of Solids and Structures 2005;42:6319–34. [13] Dai HL, Wang X. Magnetothermostress wave propagation and perturbation of magnetic field vector in an orthotropic laminated hollow cylinder. Journal of Thermal Stresses 2006;29:791–806. [14] Dai HL, Wang X. Magnetoelastodynamic stress and perturbation of magnetic field vector in an orthotropic laminated hollow cylinder. International Journal of Engineering Science 2006;44:365–78.
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[15] Hasanyana DJ, Librescua L, Amburb DR. Buckling and postbuckling of magnetoelastic flat plates carrying an electric current. International Journal of Solids and Structures 2006;43:4971–96. [16] Moon FC, Chattopadhyay S. Magnetically induced stress waves in a conducting solid-theory and experiment. Journal of Applied Mechanics, Transactions ASME 1974;41:641–6. [17] Chian CT, Moon FC. Magnetically induced cylindrical stress waves in a thermoelastic conductor. International Journal of Solids and Structures 1981;17:1021–35. [18] Wauer J. Parametric vibrations in a magneto-thermo–elastic layer of finite thickness. In Proceedings of the 1995 design engineering technical conferences, vol. 3A. New York; ASME: p. 407–14. [19] Pantelyat MG, Fe´liachi M. Magneto-thermo-elastic-plastic simulation of inductive heating of metals. EPJ Applied Physics 2002;17: 29–33.
International Journal of Mechanical Sciences 50 (2008) 365–379 www.elsevier.com/locate/ijmecsci
Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field M. Higuchia,, R. Kawamurab, Y. Tanigawab a
Department of Mechanical Engineering, Oita National College of Technology Maki 1666, Oita City, 870-0152, Japan Department of Mechanical Systems Engineering, Osaka Prefecture University, Gakuencho 1-1, Nakaku, Sakai, Osaka 599-8531, Japan
b
Received 7 February 2007; accepted 8 November 2007 Available online 17 November 2007
Abstract In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow circular cylinder subjected to an arbitrary variation of magnetic field are investigated. It is assumed that a magnetic field defined by an arbitrary function of time acts on the outer surface in the direction parallel to its surface. Fundamental equations of plane axisymmetrical electromagnetic, temperature and elastic fields are formulated. Then, solutions of magnetic field, eddy current, temperature change and both dynamic solutions and quasi-static ones of stresses and deformations are analytically derived in the forms including the arbitrary function. The solutions of stresses are determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused by Lorentz force. For the case that the arbitrary function is given by the sine function, the dynamic and quasi-static behaviors of the stresses are examined by numerical calculations. r 2007 Elsevier Ltd. All rights reserved. Keywords: Magneto-thermo-elasticity; Eddy current loss; Lorentz force
1. Introduction Mechanical components or structural elements which are activated in magnetic field have been increasing with the rapid progress of electromechanics. Especially, induction heating has attracted attention in recent years because of its less electricity consumption, high efficiency, safety and cleanliness. Accordingly, this technology has widely been applied to several area such as industrial heating equipment, office automation equipment and cooking devices in home electric appliances. When a time-varying magnetic field acts on a conducting medium, the eddy currents are induced in the medium by electromagnetic induction, and the electric currents cause heat generation called the eddy current loss due to the Joule effect. Induction heating utilizes the heat generation induced by time variation of Corresponding author. Tel.: +81 97 552 6879; fax: +81 97 552 6975.
E-mail addresses: [email protected], [email protected] (M. Higuchi). 0020-7403/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2007.11.001
magnetic field in a conducting medium. The conducting medium is subjected to Lorentz force as well as the heat supply of eddy current loss. Thus, two kinds of stress should be generated by time variation of magnetic field: one is thermal stress caused by eddy current loss and the other is magnetic stress caused by Lorentz force. In the field of magneto-elasticity or magneto-thermoelasticity [1,2], many studies have been conducted on an analytical treatment of an interaction between elastic, electromagnetic and temperature fields [3–15]. However, few studies have been carried out on analytical development of thermal stresses induced by a time-varying magnetic field [16–18]. Moon and Chattopadhyay [16] have studied thermal stresses due to eddy current loss and magnetic stresses due to Lorentz force in a conducting halfspace caused by an applied jump in tangential magnetic field at the boundary. Chian and Moon [17] have extended the above work, investigating those stresses in a cylindrical conductor with a cavity caused by a pulsed magnetic field at the cavity. Wauer [18] has studied the dynamic behavior
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of magneto-thermo-elastic plate layer, of which the surfaces are subjected to a magnetic field which is composed of a constant and a harmonically oscillating part in the direction parallel to the surfaces. He has mentioned the stability of the plate due to the external magnetic field. Pantelyat and Fe´liachi [19] have studied the mechanical behavior of metals in induction heating devices by using of finite element method. They have calculated thermo-elastic-plastic stresses induced by an alternating magnetic field, taking into account temperature dependences of material properties. In the present paper, we have investigated dynamic and quasi-static behaviors of magneto-thermo-elastic stresses and deformations in a conducting hollow circular cylinder, which consists of non-ferromagnetic metals such as copper or aluminum, subjected to an arbitrary variation of magnetic field. Assuming that a time-varying magnetic field which is defined as an arbitrary function of time acts on the outer surface of the cylinder in the direction parallel to its surface, we have formulated fundamental equations of plane axisymmetrical electromagnetic field, temperature field and elastic field. Then, we have analytically derived solutions of electromagnetic field, temperature change and dynamic solutions and quasi-static solutions of stresses and displacements, respectively, in the form including the arbitrary function. The solutions of stresses have been determined to be sums of thermal stress and magnetic stress. Carrying out numerical calculations for the case that the arbitrary function of time is given by the sine function, we have examined the dynamic and quasi-static behaviors of stresses in the hollow circular cylinder. 2. Fundamental equation systems 2.1. Electromagnetic field Let us consider a conducting hollow circular cylinder of inner and outer radii a and b with cylindrical coordinate system, as shown in Fig. 1. We assume that a time-varying axial magnetic field H 0 fðtÞ, which is uniform along the y and z directions, acts on the outer surface in the direction parallel to its surface, from time t ¼ 0. H 0 is a reference magnetic field strength, and fðtÞ is an arbitrary function of time.
Assuming that the magnetic field vectors in the conducting cylinder and in the air region inside the cylinder have only the plane axisymmetric axial components, namely H ¼ ð0; 0; H z ðr; tÞÞ and H ðaÞ ¼ ð0; 0; H ðaÞ z ðr; tÞÞ, then the governing equations and the constitutive relations of electromagnetics in cylindrical coordinate are given by In the conducting hollow cylinder: 1q qBz ðrE y Þ þ ¼ 0, r qr qt
(1)
qH z ¼ J y, qr
(2)
J y ¼ sE y ,
(3)
Bz ¼ mH z .
(4)
In the air region: 1q qBðaÞ z ðrE ðaÞ ¼ 0, y Þþ r qr qt
(5)
ðaÞ qH ðaÞ qDy z ¼ , qr qt
(6)
ðaÞ DðaÞ y ¼ e0 E y ,
(7)
ðaÞ BðaÞ z ¼ m0 H z ,
(8)
where the displacement current is disregarded in Eq. (2). In Eqs. (1)–(4), E y , Bz and J y are the circumferential component of the electric field, the axial component of the magnetic flux, and the circumferential component of the electric current density in the conducting cylinder, respectively, and s and m are the electric conductivity and the magnetic permeability in the conducting cylinder, ðaÞ respectively. In Eqs. (5)–(8), E ðaÞ and DðaÞ y , Bz y are the circumferential component of the electric field, the axial component of the magnetic flux, and the circumferential component of the electric displacement in the air region, respectively, and e0 and m0 are the dielectric constant and the magnetic permeability in the air region, respectively. From Eqs. (1)–(8), the fundamental equations of magnetic fields are obtained as follows: 1 q qH z qH z r ðaorobÞ, (9a) ¼ ms r qr qr qt 1 q qH ðaÞ q2 H ðaÞ z r z ¼ m0 e 0 r qr qr qt
ð0oroaÞ.
(9b)
The boundary conditions and the initial conditions are written as
Fig. 1. Conditions and coordinate system of hollow cylinder.
r ¼ b;
H z ðb; tÞ ¼ H 0 fðtÞ,
(9c)
r ¼ a;
H z ða; tÞ ¼ H ðaÞ z ða; tÞ,
(9d)
1 q2 H z 1 qH ðaÞ z ¼ , s qtqr e0 qr
(9e)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
t ¼ 0;
H z ¼ 0,
(9f)
qH ðaÞ z ¼ 0, (9g) qt where the relation given by Eq. (9e) means the continuity of the electric field at the inner boundary. The electric current density J y ðr; tÞ induced by time variation of magnetic field is called the eddy current, which is obtained from Eq. (2). H ðaÞ z ¼
2.2. Temperature field The eddy current J y generates Joule heat called the eddy current loss. The eddy current loss wðr; tÞ per unit time per unit volume is given by ½J y ðr; tÞ2 . (10) s We assume that the hollow cylinder with zero initial temperature change is heated by the eddy current loss wðr; tÞ from time t ¼ 0, and that the outer and inner surfaces are subjected to surrounding media, of which temperature are zero, with relative heat transfer coefficients hb and ha . Then, the fundamental equation of heat conduction taking into account the eddy current loss, the boundary conditions and the initial condition are written as qT 1 q qT w ¼k r , (11a) þ qt r qr qr Cr
wðr; tÞ ¼
r ¼ b;
qT þ hb T ¼ 0, qr
(11b)
r ¼ a;
qT þ ha T ¼ 0, qr
(11c)
T ¼ 0,
(11d)
t ¼ 0;
where T ¼ Tðr; tÞ is temperature change, and k, C and r denote the thermal conductivity, the specific heat and the mass density, respectively. 2.3. Elastic field The conducting hollow cylinder is subjected to both temperature change and Lorentz force. Lorentz force vector f is defined by 0 1 0 1 0 0 B qH C B C zC B 0 C f ¼ J B ¼ B CB @ A @ qr A mH z 0 0 1 mq ½H ðr; tÞ2 B 2 qr z C B C ¼B ð12Þ C. 0 @ A 0
367
From Eq. (12), Lorentz force has only the radial component as follows: f r ðr; tÞ ¼
mq ½H z ðr; tÞ2 . 2 qr
(13)
Because temperature change and Lorentz force are plane axisymmetric, we analyze stresses and deformations under the axisymmetric plane strain state. The equation of motion in the radial direction taking into account Lorentz force is written as qsrr srr syy q2 u þ þ f r ¼ r 2, qt qr r
(14)
where srr and syy are the radial stress component and the circumferential stress component, respectively, and u is the radial displacement. Stress-displacement relations taking into account temperature change are given by 9 ð1 nÞE qu n u 1þn > srr ðr; tÞ ¼ þ aT ; > > = ð1 þ nÞð1 2nÞ qr 1 n r 1 n ð1 nÞE u n qu 1 þ n > > þ aT ; > syy ðr; tÞ ¼ ; ð1 þ nÞð1 2nÞ r 1 n qr 1 n (15) where E, n and a denote the Young’s modulus, the Poisson’s ratio and the coefficient of linear thermal expansion, respectively. The axial stress component is given by szz ðr; tÞ ¼ nðsrr þ syy Þ aET.
(16)
Substitution Eqs. (13) and (15) into Eq. (14) yields the displacement equation of motion: q 1 qðruÞ 1 q2 u 1 þ n qT þ a ¼ 2 qr r qr 1 n qr C L qt ð1 þ nÞð1 2nÞ m q ðH z Þ2 . þ ð1 nÞE 2 qr
ð17aÞ
The cylinder is at rest prior to time t ¼ 0 and we suppose that the inner and outer surfaces of the cylinder are traction free ðsrr ¼ 0Þ. Then, the mechanical boundary conditions and the initial conditions are given by r ¼ b; a;
t ¼ 0;
qu n u 1þn þ ¼ aT, qr 1 n r 1 n u¼
qu ¼ 0. qt
(17b)
(17c)
In Eq. (17a), C L denotes the velocity of longitudinal wave, which is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 nÞE . CL ¼ ð1 þ nÞð1 2nÞr
(18)
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2.4. Dimensionless quantities We define the following dimensionless quantities: 9 > ðr; aÞ ðH z ; H ðaÞ ðaÞ z Þ > ¯ ¯ ; ðH z ; H z Þ ¼ ;> ð¯r; a¯ Þ ¼ > > b H0 > > > > > t bJ y > ¯y ¼ > t¼ ; J ; > 2 > H0 > msb > > > > 2 > sb w CgT > > ¯ w¯ ¼ ; T ¼ ; > > 2 2 = H0 mH 0 bf r > > ; ðh¯ a ; h¯ b Þ ¼ bðha ; hb Þ; f¯ r ¼ > > > mH 20 > > > > ðsrr ; syy ; szz Þ > > > ðs¯ rr ; s¯ yy ; s¯ zz Þ ¼ ; > 2 > mH 0 > > > 2 > > > ð1 nÞE 2 > > > u¯ ¼ u > ; ð1 þ nÞð1 2nÞ bmH 2
Fundamental equation system: qT¯ 1 q qT¯ ¼ w1 r¯ þ w, ¯ qt r¯ q¯r q¯r
(19)
(24a)
r¯ ¼ 1;
qT¯ þ h¯ b T¯ ¼ 0, q¯r
(24b)
r¯ ¼ a¯ ;
qT¯ þ h¯ a T¯ ¼ 0, q¯r
(24c)
t ¼ 0;
T¯ ¼ 0.
(24d)
(3) Elastic field: Lorentz force: 1q ¯ ½H z ð¯r; tÞ2 . f¯ r ð¯r; tÞ ¼ 2 q¯r Stress–displacement relations:
0
9 q¯u n u¯ > ¯ w3 T; þ > > > q¯r 1 n r¯ = u¯ n q¯u ¯ w3 T; s¯ yy ð¯r; tÞ ¼ þ > > r¯ 1 n q¯r > > ¯ ; s¯ zz ð¯r; tÞ ¼ nðs¯ rr þ s¯ yy Þ ð1 2nÞw3 T:
and
(25)
s¯ rr ð¯r; tÞ ¼
msb x1 ¼ pffiffiffiffiffiffiffiffiffi; m 0 e0 w1 ¼ msk;
9 > > > =
ms2 b2 x2 ¼ ; e0 w2 ¼ msbC L ;
> 2aE > :> w3 ¼ ; ð1 2nÞCr
(20)
Fundamental equation system: q 1 qð¯ru¯ Þ 1 q2 u¯ qT¯ q ¯ 2 þ w3 þ ðH ¼ 2 zÞ , q¯r r¯ q¯r q¯r q¯r w2 qt
By use of these dimensionless quantities, Eqs. (2), (9)–(11), (13), (15)–(17) take the following form: (1) Electromagnetic field: Fundamental equation system: ¯z ¯z 1 q qH qH r¯ ð¯ao¯ro1Þ, ¼ r¯ q¯r q¯r qt ! ¯ ðaÞ ¯ ðaÞ 1 q qH 1 q2 H z z r¯ ð0o¯ro¯aÞ, ¼ 2 r¯ q¯r q¯r x1 qt
q¯u n u¯ ¯ ¼ w3 T, þ q¯r 1 n r¯
r¯ ¼ 1; a¯ ; (21a) t ¼ 0; (21b)
u¯ ¼
q¯u ¼ 0. qt
(26)
(27a)
(27b) (27c)
3. Solutions
r¯ ¼ 1;
¯ z ¼ fðtÞ, H
(21c)
3.1. Magnetic field
r¯ ¼ a¯ ;
¯z¼H ¯ ðaÞ H z ,
(21d)
In this section, the solutions of equation system (21) will be derived by using the Laplace transform: Z 1 f ðsÞ ¼ L½f ðtÞ ¼ f ðtÞest dt, (28)
ðaÞ
¯ ¯z q2 H qH ¼ x2 z , qtq¯r q¯r t ¼ 0;
¯ z ¼ 0, H
¯ ðaÞ qH z ¯ ðaÞ ¼ 0. ¼ H z qt
(21e)
0
(21f) (21g)
Eddy current: ¯ z ð¯r; tÞ qH . J¯ y ð¯r; tÞ ¼ q¯r
where the variable s is the Laplace transform parameter and asterisk denotes Laplace transformed variable. Applying the Laplace transform with respect to t to Eq. (21), we obtain the equation system in the transformed domain as follows:
(22)
¯z ¯ z 1 dH d2 H ¯ z ¼ 0 ð¯ao¯ro1Þ, þ k2 H þ 2 r d¯ r ¯ d¯r ðaÞ
¯z d2 H d¯r2
(2) Temperature field: Eddy current loss: w即 r; tÞ ¼ ½J¯ y ð¯r; tÞ2 .
(23)
r¯ ¼ 1;
(29a)
ðaÞ
þ
¯z 1 dH r¯ d¯r
¯ ðaÞ K 2H ¼0 z
¯ z ¼ f ðsÞ, H
ð0o¯ro¯aÞ,
(29b) (29c)
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r¯ ¼ a¯ ;
k2
¯ z ¯ ðaÞ dH dH z ¼ x2 , d¯r d¯r
(29d)
The residues Rn at s ¼ k2n are 2
Rn ¼ ekn t
where
2 D0 ðkn ; r¯ Þ, kn D0 ðkn Þ
k2 ¼ s,
ð30Þ
where D0 ðkn Þ are given by
k2 . x1
ð31Þ
D0 ðkn Þ ¼
K¼
Solving equation system (29), we obtain the solutions in the transformed domain as follows: ¯ z ð¯r; sÞ ¼ D0 ðk; r¯Þ sf ðsÞ, H sD0 ðk; 1Þ a3 ðkÞI0 ðK r¯Þ ¯ ðaÞ sf ðsÞ, r; sÞ ¼ H z ð¯ sD0 ðk; 1Þ
ð32Þ ð33Þ
where Dl ðk; r¯Þ ¼ a1 ðkÞJl ðk¯rÞ a2 ðkÞYl ðk¯rÞ ðl ¼ 0; 1Þ, 9 x2 > a1 ðkÞ ¼ kY1 ðka¯ ÞI0 ðK a¯ Þ Y0 ðka¯ ÞI1 ðK a¯ Þ; > > > x1 > = x2 a2 ðkÞ ¼ kJ1 ðka¯ ÞI0 ðK a¯ Þ J0 ðka¯ ÞI1 ðK a¯ Þ; > > x1 > > > a3 ðkÞ ¼ k½J0 ðka¯ ÞY1 ðka¯ Þ J1 ðka¯ ÞY0 ðka¯ Þ; ;
(34)
(35)
where Jl ðÞ is the Bessel function of the first kind of lth order, Yl ðÞ is the Bessel function of the second kind of lth order and Il ðÞ is the modified Bessel function of the first kind of lth order, respectively. The inverse Laplace transform of Eqs. (32) and (33) can be performed by the convolution theorem: Z t 1 L ½f 1 ðsÞ f 2 ðsÞ ¼ f 1 ðt t0 Þf 2 ðt0 Þdt0 . (36)
369
(41)
dD0 ðkn ; 1Þ dkn ¼ a¯ ½kn Y0 ðkn a¯ ÞI0 ðK n a¯ Þ þ 2K n Y1 ðkn a¯ ÞI1 ðK n a¯ Þ x2 þ a¯ Y1 ðka¯ ÞI1 ðK n a¯ Þ 2Y0 ðka¯ Þ x1 1 ðK n a¯ I0 ðK n a¯ Þ I1 ðK n a¯ ÞÞ J0 ðkn Þ kn a¯ ½kn J0 ðkn a¯ ÞI0 ðK n a¯ Þ þ 2K n J1 ðkn a¯ ÞI1 ðK n a¯ Þ x2 þ a¯ J1 ðka¯ ÞI1 ðK n a¯ Þ 2J0 ðka¯ Þ x1 1 ðK n a¯ I0 ðK n a¯ Þ I1 ðK n a¯ ÞÞ Y0 ðkn Þ kn a1 ðkn ÞJ1 ðkn Þ a2 ðkn ÞY1 ðkn Þ.
ð42Þ
Therefore, from Eqs. (40) and (41), Gð¯r; tÞ is determined to be 1 X
Gð¯r; tÞ ¼ 1 þ
2
ekn t
n¼1
2 D0 ðkn ; r¯Þ. kn D0 ðkn Þ
(43)
Substituting of Eq. (43) into Eq. (37), we obtain the magnetic field in the hollow cylinder as follows: ¯ z ð¯r; tÞ ¼ fðtÞ þ H
0
1 X
2 D0 ðkn ; r¯ Þa^ n ðtÞ, 0 k D ðkn Þ n n¼1
(44)
Using the convolution theorem (36) for Eq. (32), we obtain the following form: Z t dfðt0 Þ 0 ¯ z ð¯r; tÞ ¼ H Gð¯r; t t0 Þ dt , (37) dt0 0
where a^ n ðtÞ are determined by the function fðtÞ, which are given by Z t 0 2 0 dfðt Þ ekn ðtt Þ dt0 . (45) a^ n ðtÞ ¼ dt0 0
where
Substituting Eq. (44) into Eq. (22), we obtain the eddy current J¯ y as follows:
Gð¯r; tÞ ¼ L1
D0 ðk; r¯Þ . sD0 ðk; 1Þ
(38) J¯ y ð¯r; tÞ ¼
The inverse Laplace transform shown in (38) is performed by the use of the inversion theorem: Z zþi1 D0 ðk; r¯Þ 1 D0 ðk; r¯Þ est L1 ¼ ds, (39) sD0 ðk; 1Þ 2pi zi1 sD0 ðk; 1Þ pffiffiffiffiffiffiffi where i ¼ 1 and z is taken to be large enough so that all the singularities lie to the left of line of integration. The right-hand side of Eq. (39) is determined by the sum of the residues at poles of its integrand. The integrand est ½D0 ðk; r¯Þ=sD0 ðk; 1Þ has a pole at s ¼ 0 and poles at s ¼ k2n , and kn are the positive roots of transcendental equation D0 ðk; 1Þ ¼ 0. The residue R0 at s ¼ 0 is R0 ¼ 1.
(40)
1 X
2 D1 ðkn ; r¯ Þa^ n ðtÞ. 0 D ðk nÞ n¼1
(46)
In the same manner, the magnetic field in the air region is determined to be ¯ ðaÞ H r; tÞ ¼ fðtÞ þ z ð¯
1 X
2 a3 ðkn ÞI0 ðK n r¯Þa^ n ðtÞ. 0 k D ðkn Þ n n¼1
(47)
3.2. Temperature field By using the separation of variables technique, the solution of Eq. (24) will be assumed in the following form: ¯ r; tÞ ¼ Tð¯
1 X j¼1
bj ðtÞR0 ðpj ; r¯ Þ,
(48)
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where bj ðtÞ are unknown functions of t, and R0 ðpj ; r¯Þ are R0 ðpj ; r¯Þ ¼ b1 ðpj ÞJ0 ðpj r¯Þ b2 ðpj ÞY0 ðpj r¯Þ
(49)
in which b1 ðpj Þ ¼ h¯ a Y0 ðpj a¯ Þ þ pj Y1 ðpj a¯ Þ; b2 ðpj Þ ¼ h¯ a J0 ðpj a¯ Þ þ pj J1 ðpj a¯ Þ
)
a
(50)
and pj are the positive roots of the following eigenequation: ½h¯ a J0 ðp¯aÞ þ pJ1 ðp¯aÞ½h¯ b Y0 ðpÞ pY1 ðpÞ ½h¯ a Y0 ðp¯aÞ þ pY1 ðp¯aÞ½h¯ b J0 ðpÞ pJ1 ðpÞ ¼ 0.
j¼1
a¯
1
þ a¯
w即 r; tÞR0 ðpq ; r¯Þ¯r d¯r.
Substituting Eq. (58) into Eq. (48), we obtain temperature change as follows: ¯ r; tÞ ¼ Tð¯
ð51Þ
Substituting of Eq. (48) into Eq. (24a) with Eqs. (24b) and (24c), multiplying both side by r¯R0 ðpq ; r¯Þ, and then integrating with respect to r¯ from a¯ to 1, we obtain Z 1 X dbj ðtÞ 1 R0 ðpj ; r¯ÞR0 ðpq ; r¯Þ¯r d¯r dt a¯ j¼1 Z 1 1 X ¼ w1 p2j bj ðtÞ R0 ðpj ; r¯ÞR0 ðpq ; r¯Þ¯r d¯r Z
and I 1jmn are given by the following equation, and are calculated by numerical integration: Z 1 I 1jmn ¼ D1 ðkm ; r¯ÞD1 ðkn ; r¯ÞR0 ðpj ; r¯Þ¯r d¯r. (60)
ð52Þ
Using the orthogonality of Bessel functions, we can derive the following equation: ( Z 1 ðq ¼ jÞ; Mj R0 ðpj ; r¯ÞR0 ðpq ; r¯ Þ¯r d¯r ¼ (53) 0 ðqajÞ; a¯
i 1 h 2 ¯2 2 2 ¯ 2 Þ¯a2 R2 ðpj ; a¯ Þ . ðp þ h ÞR ðp ; 1Þ ðp þ h j j 0 j 0 b a 2p2j
Substitution of Eq. (53) into Eq. (52) gives Z 1 dbj ðtÞ 1 2 þ w1 pj bj ðtÞ ¼ w即 r; tÞR0 ðpj ; r¯Þ¯r d¯r. M j a¯ dt
(54)
(55)
The solutions of Eq. (55) under the initial condition (24d) are expressed as Z 1 Z t 1 w1 p2j ðtt0 Þ 0 0 wð¯ e bj ðtÞ ¼ ¯ r; t Þdt R0 ðpj ; r¯Þ¯r d¯r. (56) M j a¯ 0 Here, substitution of Eq. (46) into Eq. (23) yields w即 r; tÞ ¼ 4
1 D1 ðkm ; r¯ÞD1 ðkn ; r¯Þa^ m ðtÞa^ n ðtÞ. 0 D ðk ÞD0 ðkn Þ m m¼1 n¼1 (57)
1 X 1 I 1jmn 4 X b^jmn ðtÞ, M j m¼1 n¼1 D0 ðkm ÞD0 ðkn Þ
u¯ ð¯r; tÞ ¼ u1 ð¯r; tÞ þ u2 ð¯r; tÞ, where u1 ð¯r; tÞ satisfies q 1 qð¯ru1 Þ ¼ 0, q¯r r¯ q¯r qu1 n u1 ¯ ¼ w3 T. þ q¯r 1 n r¯
r¯ ¼ 1; a¯ ;
The solution of Eqs. (63) is expressed as 1n ¯ tÞ a¯ 2 T即 a; tÞ¯r w ½Tð1; u1 ¼ 1 a¯ 2 3 2 1 ¯ tÞ T即 a; tÞ a¯ . ½Tð1; þ 1 2n r¯
(62)
(63a)
(63b)
ð64Þ
Substitution of Eq. (62) with Eqs. (63a) into Eqs. (27a) yields q 1 qð¯ru2 Þ 1 q2 u2 1 q2 u1 qT¯ q ¯ 2 þ ¼ 2 2 þ 2 2 þ w3 H z , (65a) q¯r r¯ q¯r q¯r q¯r w2 qt w2 qt qu2 n u2 ¼ 0, þ q¯r 1 n r¯
t ¼ 0;
u2 ¼ u1 ;
(58)
where b^jmn ðtÞ are determined by the function fðtÞ, which are given by Z t 2 0 b^jmn ðtÞ ¼ ew1 pj ðtt Þ a^ m ðt0 Þa^ n ðt0 Þ dt0 (59) 0
ð61Þ
qu2 qu1 ¼ . qt qt
(65b)
(65c)
By using the separation of variables technique, the solution of Eqs. (65) will be assumed in the following form:
Substitution of Eq. (57) into Eq. (56) gives bj ðtÞ ¼
I 1jmn b^jmn ðtÞ. 0 0 D ðk m ÞD ðkn Þ m¼1 n¼1
3.3.1. Dynamic solutions In order to transform the inhomogeneous boundary conditions (27b) into the homogeneous one, we assume that u¯ ð¯r; tÞ is given by following expression:
r¯ ¼ 1; a¯ ;
1 X 1 X
1 X 1 X
3.3. Elastic field
where Mj ¼
1 X 4 R0 ðpj ; r¯Þ M j j¼1
u2 ð¯r; tÞ ¼
1 X
ci ðtÞV 1 ðZi ; r¯Þ,
(66)
i¼1
where ci ðtÞ are unknown functions of t, and V l ðZ; r¯Þ ðl ¼ 0; 1Þ are V l ðZi ; r¯Þ ¼ g1 ðZi ÞJl ðZi r¯Þ g2 ðZi ÞYl ðZi r¯ Þ ðl ¼ 0; 1Þ
(67)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
in which 9 1 2n Y1 ðZi a¯ Þ > ;> = a¯ 1n 1 2n J1 ðZi a¯ Þ > > g2 ðZi Þ ¼ Zi J0 ðZi a¯ Þ ; a¯ 1n
cTi ðtÞ ¼
g1 ðZi Þ ¼ Zi Y0 ðZi a¯ Þ
(68)
and Zi are the positive roots of the following eigenequation: 1 2n 1 2n Y1 ðZ¯aÞ J1 ðZÞ ZY0 ðZ¯aÞ ZJ0 ðZÞ 1n 1n a¯ 1 2n ZY0 ðZÞ Y1 ðZÞ 1n 1 2n J1 ðZ¯aÞ ZJ0 ðZ¯aÞ ¼ 0. ð69Þ 1 n a¯ In the same manner as Section 3.2, substituting Eq. (66) with Eq. (65b) into Eq. (65a), and utilizing the orthogonality of Bessel functions, we can derive the following equation: Z q2 ci ðtÞ 1 1 q2 u1 þ O2i ci ðtÞ ¼ V 1 ðZi ; r¯Þ¯r d¯r qt N i a¯ qt Z w22 1 qT¯ V 1 ðZi ; r¯Þ¯r d¯r w3 N i a¯ q¯r Z ¯ 2z w2 1 qH V 1 ðZi ; r¯Þ¯r d¯r, ð70Þ 2 N i a¯ q¯r
ð71Þ
and Oi are the natural angular frequencies of ith order mode in dimensionless form, which are given by O i ¼ w2 Zi .
9 > > > > > > > > > > > =
1 1 P 1 P 1 I 1jmn 4w3 P 2 Zi N i j¼1 M j m¼1 n¼1 D0 ðkm ÞD0 ðkn Þ
½I 2ij b^jmn ðtÞ þ I 3ij Oi c^Tijmn ðtÞ; 1 4w2 P I 4in Mð1Þ > cM ðtÞ ¼ ðtÞ c^ > i > Zi N i n¼1 D0 ðkn Þ in > > > > 1 1 > P P I 5imn > Mð2Þ > > ^ þ ðtÞ ; c ; imn 0 0 k k D ðk ÞD ðk Þ m n m n m¼1 n¼1
(75)
where cTi ðtÞ and cM i ðtÞ are the terms due to temperature change and due to Lorentz force, respectively, and c^Tijmn ðtÞ, Mð1Þ Mð2Þ c^in ðtÞ and c^imn ðtÞ are defined by the function fðtÞ, which are given by c^Tijmn ðtÞ ¼
Rt
c^Mð1Þ ðtÞ ¼ in c^Mð2Þ imn ðtÞ
¼
0
sin Oi ðt t0 Þb^jmn ðt0 Þ dt0 ;
Rt 0 Rt 0
sin Oi ðt t0 Þfðt0 Þa^ n ðt0 Þ dt0 ;
9 > > =
> > sin Oi ðt t0 Þa^ m ðt0 Þa^ n ðt0 Þ dt0 ;
(76)
and I 2ij ¼ R0 ðpj ; 1ÞV 1 ðZi ; 1Þ a¯ R0 ðpj ; a¯ ÞV 1 ðZi ; a¯ Þ,
I 3ij ¼
where 1 1 2n 2 N i ¼ 2 Zi V 21 ðZi ; 1Þ 2Zi ð1 nÞ2 1 2n 2 2 2 Zi a¯ V 1 ðZi ; a¯ Þ ð1 nÞ2
371
I 4in ¼
¯ Z2i hb 1 2n 1 R0 ðpj ; 1ÞV 1 ðZi ; 1Þ p2j Z2i Z2i 1 n ¯ ha 1 2n þ 2 þ 1 a¯ R0 ðpj ; a¯ ÞV 1 ðZi ; a¯ Þ , Zi a¯ 1 n
(77)
ð78Þ
1 1 2n D1 ðkn ; 1ÞV 1 ðZi ; 1Þ 2 2 k n Zi 1 n 1 2n D1 ðkn ; a¯ Þ kn D0 ðkn ; a¯ Þ a¯ V 1 ðZi ; a¯ Þ , ð79Þ 1n
(72)
The solutions of Eq. (70) under the initial condition (65c) are expressed as Z Z t 1 1 ci ðtÞ ¼ u1 ð¯r; tÞ þ Oi sin Oi ðt t0 Þu1 ð¯r; t0 Þdt0 N i a¯ 0 w2 V 1 ðZi ; r¯Þ¯r d¯r w3 Zi N i Z 1 Z t q 0 ¯ 0 0 sin Oi ðt t ÞTð¯r; t Þdt r 0 a¯ q¯ w V 1 ðZi ; r¯Þ¯r d¯r 2 Zi N i Z 1 Z t q ¯ z ð¯r; t0 Þg2 dt0 sin Oi ðt t0 ÞfH r 0 a¯ q¯
I 5imn ¼ a¯ D0 ðkm ; a¯ ÞD0 ðkn ; a¯ ÞV 1 ðZi ; a¯ Þ Z 1 þ Zi D0 ðkm ; r¯ ÞD0 ðkn ; r¯ ÞV 0 ðZi ; r¯Þ¯r d¯r.
ð80Þ
a¯
The second term of the right-hand side of Eq. (80) is calculated by numerical integration. From Eqs. (62), (66), (74), we have u¯ T ð¯r; tÞ ¼ u1 ð¯r; tÞ þ
1 P i¼1
u¯ M ð¯r; tÞ ¼
1 P i¼1
9 > cTi ðtÞV 1 ðZi ; r¯Þ; > > =
cM ¯ Þ; i ðtÞV 1 ðZi ; r
> > > ;
(81)
Substitution of Eqs. (44), (61) and (64) into Eq. (73) gives
where u¯ T ð¯r; tÞ and u¯ M ð¯r; tÞ are the radial displacements due to temperature change and due to Lorentz force, respectively, and satisfy the following relation:
ci ðtÞ ¼ cTi ðtÞ þ cM i ðtÞ,
u¯ ð¯r; tÞ ¼ u¯ T ð¯r; tÞ þ u¯ M ð¯r; tÞ.
V 1 ðZi ; r¯Þ¯r d¯r.
ð73Þ
(74)
(82)
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
372
Substituting Eq. (82) into Eq. (26), we can obtain the dynamic solutions of the stress components as follows: 9 ¯ r; tÞ s¯ Trr ð¯r; tÞ ¼ s1 ð¯r; tÞ w3 Tð¯ > > > > 1 > P T 1 2n V 1 ðZi ; r¯Þ > > > ; þ ci ðtÞ Zi V 0 ðZi ; r¯ Þ > > 1 n r ¯ > i¼1 > > > T > ¯ s¯ yy ð¯r; tÞ ¼ s2 ð¯r; tÞ w3 Tð¯r; tÞ > = 1 P nZi 1 2n V 1 ðZi ; r¯Þ T V 0 ðZi ; r¯ Þ þ ;> þ ci ðtÞ > > 1n r¯ 1n i¼1 > > > > T ¯ > s¯ zz ð¯r; tÞ ¼ s3 ð¯r; tÞ w3 Tð¯r; tÞ > > > > 1 > P nZ > i T > V 0 ðZi ; r¯Þ; þ ci ðtÞ > ; 1n i¼1 (83) 9 1 2n V 1 ðZi ; r¯Þ > > ¼ Zi V 0 ðZi ; r¯Þ ; > > > 1n r¯ i¼1 > > > = 1 P nZ 1 2n V ðZ ; r Þ ¯ 1 i i M M s¯ yy ð¯r; tÞ ¼ V 0 ðZi ; r¯Þ þ ci ðtÞ ; , 1n r¯ 1n > i¼1 > > > 1 > P nZ > i M M > V 0 ðZi ; r¯ Þ; ci ðtÞ s¯ zz ð¯r; tÞ ¼ > ; 1n s¯ M r; tÞ rr ð¯
1 P
cM i ðtÞ
i¼1
(84) where s¯ Tii ði ¼ r; y; zÞ and s¯ M ii are thermal stresses and magnetic stresses, respectively, and satisfy the following relations: s¯ rr ¼ s¯ Trr þ s¯ M rr ;
s¯ yy ¼ s¯ Tyy þ s¯ M yy ;
s¯ yy ¼ s¯ Tzz þ s¯ M zz
and s1 ð¯r; tÞ; s2 ð¯r; tÞ; s3 ð¯r; tÞ in Eq. (83) are given by 9 w3 ¯ tÞ a¯ 2 T即 a; tÞ > > s1 ð¯r; tÞ ¼ ½ Tð1; > > 1 a¯ 2 > > > > 2 > > a ¯ > ¯ tÞ T即 a; tÞ > ½Tð1; ; > 2 > r¯ > > > = w3 ¯ tÞ a¯ 2 T即 a; tÞ s2 ð¯r; tÞ ¼ ½ Tð1; > 1 a¯ 2 > > > > 2 > a ¯ > > ¯ ¯ þ½Tð1; tÞ Tð¯a; tÞ 2 ; > > > r¯ > > > > 2nw3 ¯ > 2¯ > ; ½ Tð1; tÞ a s3 ð¯r; tÞ ¼ Tð¯ a ; tÞ: ¯ 2 1 a¯
(85)
(86)
3.3.2. Quasi-static solutions In this section, we derive the quasi-static solutions of the displacements and stresses. Disregarding the inertia term of the right hand side in Eq. (27a) gives the equilibrium equation: d 1 dð¯ru¯ Þ dT¯ d ¯ 2 þ ðH (87) ¼ w3 zÞ . d¯r r¯ d¯r d¯r d¯r Solving Eq. (87) with the boundary conditions (27b), we obtain the quasi-static solutions of the radial displacements due to temperature change and Lorentz force, respectively,
which are given as follows: 9 1 > ð2Þ ðtÞ¯ r þ H ðtÞ ;> u¯ T ð¯r; tÞ ¼ w3 F T ð¯r; tÞ þ ð1 2nÞH ð1Þ T T r¯ = > 1 > ; r þ H ð2Þ u¯ M ð¯r; tÞ ¼ F M ð¯r; tÞ þ ð1 2nÞH ð1Þ M ðtÞ¯ M ðtÞ ; r¯ (88) where 9 1R ¯ 1R ¯ 2 > r¯T d¯r; F M ð¯r; tÞ ¼ r¯ H z d¯r; > > > r¯ r¯ > > > > 1 > ð1Þ > H T ðtÞ ¼ ½F ð1; tÞ a F ð¯ a ; tÞ; ¯ > T T > 2 > 1 a¯ > > > a¯ > ð2Þ > > ½¯ a F ð1; tÞ F ð¯ a ; tÞ; H T ðtÞ ¼ T T > 2 > 1 a¯ > > > = 1 ð1Þ ½F ð1; tÞ a F ð¯ a ; tÞ H M ðtÞ ¼ ¯ M M 2 1 a¯ > > > 1n ¯2 > 2 ¯ 2 > ½H z ð1; tÞ a¯ H z ð¯a; tÞ ; > > > 1 2n > > > > a¯ > ð2Þ > > H M ðtÞ ¼ ½¯ a F ð1; tÞ F ð¯ a ; tÞ M M > 2 > 1 a¯ > > > > 1n ¯2 > 2 > ¯ > a¯ ½H z ð1; tÞ H z ð¯a; tÞ : ; 1 2n F T ð¯r; tÞ ¼
(89)
Substituting Eq. (88) with the relation of Eq. (82) into Eq. (26), we obtain the quasi-static solutions of the stress components as follows: " # 9 > 1 2n 1 H ð2Þ ð1Þ T ðtÞ > T s¯ rr ð¯r; tÞ ¼ w3 F T ð¯r; tÞ þ H T ðtÞ ;> > 2 > 1n r¯ > r¯ > > > > > 1 2n 1 > T > ¯ > Tð¯r; tÞ þ F T ð¯r; tÞ s¯ yy ð¯r; tÞ ¼ w3 = 1n r¯ # > > H ð2Þ > T ðtÞ > þH ð1Þ ; > T ðtÞ þ 2 > > r¯ > > > > > 1 2n > ð1Þ T > ¯ ½Tð¯r; tÞ þ 2nH T ðtÞ; s¯ zz ð¯r; tÞ ¼ w3 ; 1n (90) 9 1 2n 1 2 > ¯ > F M ð¯r; tÞ ¼ H z ð¯r; tÞ þ > > 1n r¯ > > # > > ð2Þ > > H M ðtÞ > ð1Þ > þH M ðtÞ ; > > > r¯2 > > > = n ¯2 1 2n 1 M F M ð¯r; tÞ s¯ yy ð¯r; tÞ ¼ H z ð¯r; tÞ þ > 1n 1 n r¯ > > # > > ð2Þ > > H ðtÞ > ð1Þ M > þH M ðtÞ þ ; > > 2 > r¯ > > > > > n 2 ð1Þ M ¯ ; ½H z ð¯r; tÞ þ 2ð1 2nÞH M ðtÞ: > s¯ zz ð¯r; tÞ ¼ 1n s¯ M r; tÞ rr ð¯
(91)
These quasi-static thermal stresses and magnetic stresses satisfy the relations of Eq. (85).
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
In this paper, we assume that an arbitrary function fðtÞ are defined as the sine function: fðtÞ ¼ sin ot,
(92)
where o is an angular frequency of magnetic field in the nondimensional form. For this case, the functions of t: Mð1Þ a^ n ðtÞ, b^jmn ðtÞ, c^Tijmn ðtÞ, c^in ðtÞ, c^Mð2Þ imn ðtÞ in Eqs. (45), (59) and (76) are determined by the function fðtÞ in Eq. (92). However, these functions are omitted here because they can be easily derived. Making use of the analytical results mentioned above, we carry out numerical calculations for aluminum, material properties of which are given by 9 m ¼ 4p 107 ðH=mÞ; s ¼ 3:42 107 ðS=mÞ; > > > > 3 3 3 C ¼ 2:7 10 ðJ=kgKÞ; r ¼ 0:9 10 ðkg=m Þ; = (93) > k ¼ 92:6 106 ðm2 =sÞ; > > > n ¼ 0:33; E ¼ 70ðGPaÞ; a ¼ 24 106 ð1=KÞ: ;
The natural angular frequencies given by Eq. (72) are depend on the radius ratio a¯ ¼ a=b as shown by the eigenequation (69). Here, we set a value of o as o ¼ e O1ð0:9Þ ,
25
The values of the nondimensional relative heat transfer coefficients h¯ a and h¯ b are given by h¯ a ¼ h¯ b ¼ 1:0.
(94)
In addition, since the nondimensional variables x1 , x2 and w2 in Eq. (20) include the outer radius b, this dimension needs to be fixed. We chose b ¼ 1:0 104 ðmÞ to ensure the convergence of the solutions.
(95)
where e is a parameter, and Oð0:9Þ is the natural angular 1 frequency of first order mode for the case of the radius ratio a¯ ¼ 0:9. Although angular frequencies of magnetic field and eddy current are o, Eqs. (10) and (13) indicate that those of temperature change caused by eddy current loss and Lorentz force become the twice of o.
Maximum absolute value of Jθ
4. Numerical results and discussion
= 0.4 × 1(0.9) = 2.0 × 1(0.9)
20
= 4.0 × 1(0.9)
15 10 5 0
0
0.2
r = 0.90 r = 0.95 r = 1.00
0.5
= 0.9 = 0.4
Hz
0 -0.5
0
0.5
1 τ
1.5
2
¯ z. Fig. 2. Time evolution of magnetic field H r = 0.90 r = 0.95 r = 1.00
4
Jθ
2
= 0.9 = 0.4
0 -2 -4 0
0.5
1 τ
0.4
0.6
0.8
1
Fig. 4. Maximum absolute value of eddy current J¯ y versus radius ratio a¯ .
1
-1
373
1.5
Fig. 3. Time evolution of eddy current J¯ y .
2
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
374
¯z Figs. 2 and 3 show the time evolutions of magnetic field H ¯ and eddy current J y for the radius ratio a¯ ¼ 0:9 and e ¼ 0:4, respectively. It can be seen from Figs. 2 and 3 that magnetic ¯ z and eddy current J¯ y vary sinusoidally. We here field H examine the maximum absolute values of eddy current J¯ y with radius ratio a¯ for some values of angular frequency o. Fig. 4 shows the maximum absolute values of eddy current J¯ y with the radius ratio a¯ . It can be seen from Fig. 4 that the maximum absolute values of eddy current J¯ y increases with the angular frequency o, and they also increase with the radius ratio a¯ because the area in which eddy current is
generated decreases with increasing radius ratio a¯ . However, for the case of o ¼ 4:0 Oð0:9Þ 1 , the maximum absolute value drops near a¯ ¼ 0:8. This is because, as shown in Fig. 5 which shows the distributions of the maximum absolute values of eddy current J¯ y for the case of o ¼ 4:0 Oð0:9Þ 1 , the maximum absolute value at the inner surface ð¯r ¼ 0:9Þ for the case of a¯ ¼ 0:8 is larger than for the case of a¯ ¼ 0:6. Fig. 6 shows the time evolution of temperature change T¯ for the radius ratio a¯ ¼ 0:9 and e ¼ 0:4. Temperature change T¯ increases with increasing time, and then it vibrates with the angular frequency 2o. In addition,
= 0.60 = 0.80 = 0.90 = 0.95
20 15
1200
10 5 0
= 0.4 × (0.9) 1 = 2.0 × (0.9) 1
1000
= 4.0 × 1(0.9)
Maximum value of T
Maximum absolute value of Jθ
25
= 4.0 × (0.9) 1
800 600 400 200
0.6
0.7
0.8
0.9
1
0
r Fig. 5. Distributions of maximum absolute values of eddy current J¯ y for o ¼ 4:0 O1ð0:9Þ .
0
0.2
r = 0.90 r = 0.95 r = 1.00
100
T
80
= 0.9 ε = 0.4
60
= 1.0 = 1.0
40 20
18 16 14 12 10 8 6 4 2 0
0
0
20
40
0.5
τ
1 τ
0.6
0.8
1
Fig. 7. Maximum values of temperature change T¯ versus radius ratio a¯ .
120
0
0.4
60
80
1.5
¯ Fig. 6. Time evolution of temperature change T.
100
2
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
temperature change T¯ takes long time to attain a steady state. This is because the value of w1 in Eq. (20), which is the ratio of the diffusion coefficient of temperature field k to that of magnetic field 1=ms, is very small as shown in the following equation: w1 ¼
k ffi 3:98 103 . 1=ms
(96)
= 0.9
0.06
ε = 0.4
375
Fig. 7 shows the maximum values of temperature change T¯ with the radius ratio a¯ . It can be seen from Fig. 7 that for the cases of e ¼ 0:4, the maximum value of temperature change T¯ decreases with the radius ratio a¯ , and that for the cases of e ¼ 2:0; 4:0, the maximum values decrease and then increase with the radius ratio a¯ . In all cases, the decreases of the maximum values with the increasing radius ratio are Dynamic solution Quasi-static solution
r = 0.95
0.04 0.02
rr
0 -0.02 -0.04 -0.06 -0.08 -0.1
0
1
= 0.9
0.06
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.95
0.04 0.02
M rr
0 -0.02 -0.04 -0.06 -0.08 -0.1
0
1
= 0.9
0.06
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.95
0.04 0.02
Trr
0 -0.02 -0.04 -0.06 -0.08 -0.1
0
1
2 τ
3
4
Fig. 8. Dynamic and quasi-static behaviors of the radial stress components, s¯ rr ; s¯ M ¯ Trr : (a) total stress, s¯ rr ¼ s¯ Trr þ s¯ M ¯ Trr , (c) magnetic rr ; s rr , (b) thermal stress, s stress, s¯ M . rr
ARTICLE IN PRESS 376
M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
due to the increase of the heat dissipation at the inner surface. In the cases of e ¼ 2:0; 4:0, because the maximum absolute values of eddy current J¯ y increase considerably beyond the value of about a¯ ¼ 0:8 as shown in Fig. 4, the maximum values of temperature change will increase. Next, we examine the dynamic and quasi-static behaviors of thermal stresses and magnetic stresses in the
= 0.9
25
ε = 0.4
cylinder. In order to examine inertia effects, numerical results for e ¼ 0:4 and a¯ ¼ 0:9 are presented in Figs. 8–10. Fig. 8 shows the dynamic and quasi-static behaviors of the radius stress components s¯ rr ð¼ s¯ Trr þ s¯ M ¯ Trr , s¯ M rr Þ, s rr at r¯ ¼ 0:95. Fig. 9 shows the dynamic and quasi-static behaviors of the circumferential stress components s¯ yy ð¼ s¯ Tyy þ s¯ M ¯ ¼ 0:9. Fig. 10 shows the ¯ Tyy , s¯ M yy Þ, s yy at r Dynamic solution Quasi-static solution
r = 0.95
20 15 10
5 0 -5 -10 -15 -20 -25
0
1
= 0.9
25
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.9
20 15 10 M
5 0 -5 -10 -15 -20 -25
0
1
= 0.9
25
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 0.9
20 15 10 T
5 0 -5 -10 -15 -20 -25
0
1
2 τ
3
4
Fig. 9. Dynamic and quasi-static behaviors of the circumferential stress components, s¯ yy ; s¯ M ¯ Tyy : (a) total stress, s¯ yy ¼ s¯ Tyy þ s¯ M ¯ Tyy , yy ; s yy , (b) thermal stress, s (c) magnetic stress, s¯ M . yy
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
dynamic and quasi-static behaviors of the radius stress components s¯ zz ð¼ s¯ Tzz þ s¯ M ¯ ¼ 1:0. It can be ¯ Tzz , s¯ M zz Þ, s zz at r seen from these figures that the amplifications of the amplitude involved in the dynamic behaviors of the total stresses s¯ rr ; s¯ yy ; s¯ zz in Figs. 8(a), 9(a), 10(a) are mainly caused by magnetic stress components in Figs. 8(c), 9(c),
= 0.5
10
ε = 0.4
377
10(c). Therefore, we should evaluate the magnetic stresses for the amplifications of the stress amplitudes, comparing dynamic solutions with quasi-static ones. Fig. 11 shows the variations of maximum absolute values of the circumferential magnetic stress s¯ M yy , which is the dominant component for the magnetic stresses as shown in Figs. 8–10, in
Dynamic solution Quasi-static solution
r = 1.0
0
zz
-10 -20 -30 -40 0
1
= 0.5
10
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 1.0
0
M zz
-10 -20 -30 -40 0
1
= 0.5
10
ε = 0.4
2 τ
3
4
Dynamic solution Quasi-static solution
r = 1.0
0
T zz
-10 -20 -30 -40 0
1
2 τ
3
4
Fig. 10. Dynamic and quasi-static behaviors of the axial stress components, s¯ zz ; s¯ M ¯ Tzz : (a) total stress, s¯ zz ¼ s¯ Tzz þ s¯ M ¯ Tzz , (c) zz ; s zz , (b) thermal stress, s magnetic stress, s¯ M . zz
ARTICLE IN PRESS M. Higuchi et al. / International Journal of Mechanical Sciences 50 (2008) 365–379
378
Maximum abusolute value of
(0.9)
40
1
(0.9)
2
/2
/2
Dynamic solution Quasi-static solution
35 30
a = 0.9
25 20 15 10 5 0
0
2
4
6
8
10
12
14
16
18
(0.9)
20 [ ×1
]
Fig. 11. Variations of maximum absolute values of the circumferential magnetic stress in dynamic and quasi-static solutions with angular frequency.
Maximum absolute value of σ Tzz
1600
= 0.4 × (0.9) 1
1400
= 2.0 × = 4.0 ×
1200
(0.9) 1 (0.9) 1
1000 800 600 400 200 0
0
0.2
0.4
0.6
0.8
1
Fig. 12. Maximum absolute values of the quasi-static axial thermal stress s¯ Tzz with the radius ratio a¯ .
dynamic and quasi-static solutions with angular frequency. It can be seen from Fig. 11 that when angular frequency o tends to half of the natural frequency Oð0:9Þ or Oð0:9Þ 1 2 , the considerable increase due to the resonance in the dynamic stress is observed. However, the absolute values of the axial total stress s¯ zz , which is the dominant stress component as shown in Figs. 8–10, in both the dynamic and quasi-static solutions shown in Fig. 10(a) increase with increasing time due to the thermal stress shown in Fig. 10(b). In addition, there is no significant difference in magnitude between the dynamic and quasi-static behaviors of the axial total stress s¯ zz (and axial thermal stress s¯ Tzz ). Then, the maximum absolute values of the quasi-static axial thermal stresses s¯ Tzz with the radius ratio a¯ are shown in Fig. 12. Because maximum absolute value of s¯ Tzz as show in Fig. 12 are much larger than those of the magnetic stresses as shown in Fig. 11, the quasi-static thermal stresses play significant roles to estimate the magnitude of stresses in the cylinder. 5. Conclusion In the present study, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting hollow
circular cylinder subjected to an arbitrary variation of magnetic field, which is defined by arbitrary function of time, have been investigated. Exact solutions of electromagnetic field, temperature change, stresses and deformations have been obtained. The stresses are determined to be sums of thermal stress and magnetic stress. Carrying out numerical calculations for the case that the arbitrary function of time is given by the sine function, we have examined the dynamic and quasi-static behaviors of the stresses. We conclude that quasi-static thermal stresses play significant roles to estimate the magnitude of stresses in the hollow cylinder because magnetic stresses are much smaller than thermal stresses and there is no significant deference in magnitude between dynamic and quasi-static behaviors of thermal stresses.
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