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Magnetoelastic modeling of circular cylindrical shells immersed in a magnetic field
International Journal of Engineering Science 41 (2003) 2023–2046 www.elsevier.com/locate/ijengsci
Magnetoelastic modeling of circular cylindrical shells immersed in a magnetic field Part II: Implications of finite dimensional effects on the free vibrations Zhanming Qin a, Davresh Hasanyan a, Liviu Librescu
a,*
, Damodar R. Ambur
b
a
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Mail Cose (0219), Blacksburg, VA 24061-0219, USA b Mechanics and Durability Branch, NASA Langley Research Center, Hampton, VA 23681-2199, USA Received 28 October 2002; accepted 27 January 2003
Abstract In the context of free vibration analysis of axi-symmetric perfectly electro-conductive circular cylindrical shells, four simplified magnetoelastic load models are investigated. Concerning the model of circular cylindrical shells, a linear theory based on Love–Kirchhoff hypothesis is adopted. Due to the high complexities involving singularity of integral equations, infinite integral domains and excessive time needed to evaluate some kernels, special treatments are designed toward achieving highly efficient and highly accurate numerical computation. The influence of applied magnetic field, thickness ratio and dimensionless radius on free vibrations of circular cylindrical shells are further investigated and pertinent conclusions are outlined. 2003 Elsevier Ltd. All rights reserved.
1. Introduction During the last several decades, the problem of the influence of applied magnetic field on the mechanical behavior of circular cylindrical elastic shells has been extensively studied (see e.g., [1,3,9,12,20], and the references therein). With the new emerging concept of multi-functional material/structure design paradigm, such interdisciplinary research is likely to be intensified in the years ahead. *
Corresponding author. Tel.: +1-540-231-5916; fax: +1-540-231-4574. E-mail address: [email protected] (L. Librescu).
0020-7225/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0020-7225(03)00135-6
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It is noted that in the previous work, the finite dimensional effects in the modeling of magnetoelastic loads, such as the influence of finite length and finite thickness, have not been considered in the analytical way. In Part I, the magnetoelastic loads incorporating these effects have been systematically derived and simplified magnetoelastic models were developed. However, due to the complexities involving singularity of integral equations, infinite integral domains and timeconsuming evaluation of some kernel functions, special treatments are essential toward achieving highly efficient and accurate numerical computation. It is noted that in the present paper, the evaluation of singular integrals is in the sense of CauchyÕs principal value [6]. The main objectives of the present paper are • investigation of the difference in the shell behavior when the formerly developed magnetoelastic load models in the context of axi-symmetric eigenfrequency solution of circular cylindrical thin shells are used. This will provide valuable information about the validity of these models when addressing specific problems, • investigation of the implications of various geometric parameters on the shellÕs magnetoelastic eigenfrequencies, • highly efficient and accurate implementation of the involved evaluation. 2. Governing system for axi-symmetric vibrations 2.1. Structural modeling Love–Kirchhoff (L–K) assumptions [8,13] are adopted in the modeling of isotropic closed circular cylindrical thin-shells featuring axi-symmetric vibrations. The geometric configuration of the shell and the coordinate system used are displayed in Fig. 1, Part I. The kinematic equations are: ur ¼ wðh; x; tÞ ¼ wðx; tÞ
ð1aÞ
uh ¼ uh ðh; x; tÞ ¼ 0
ð1bÞ
ux ¼ uðh; x; tÞ ðr RÞ
ow ow ¼ uðx; tÞ f ox ox
ð1cÞ
Based on the above adopted kinematics, the strain component err ¼ 0. Invoking, as usual that rrr ¼ 0 in the constitutive equations (see e.g., [8]), we get: E rxx 1 m exx ¼ ð2aÞ 2 rhh e m 1 1m hh err ¼
m ðexx þ ehh Þ 1m
ð2bÞ
For axi-symmetric vibration problem, the following shell stress resultants and stress couples are defined:
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
f E 2h3 w w rxx 1 þ 2hu þ mð2hÞ df ¼ ;x ;xx R 1 m2 R 3R h Z h i E 1 þ h=R E h w ð2hÞ rhh df ¼ mð2hÞu mð2hÞu þ w ln Nh þ ;x ;x 1 m2 1 h=R 1 m2 R h Z h i f 2Eh3 h u;x Mx w rxx f 1 þ df ¼ ;xx R 3ð1 m2 Þ R h
Nx
Z
2025
h
ð3aÞ
ð3bÞ
ð3cÞ
In Eq. (3b), the Taylor series expansion is used and truncated up to the second order: 3 5 1 þ h=R h 2 h h h þO 2 ¼2 þ ln 1 h=R R 3 R R R
ð4Þ
The governing equations of the magnetoelastic circular cylindrical shells and the consistent boundary conditions can be systematically derived by using the principle of virtual work in dynamic case [18]: Z
t2
ðdT dU þ dWe Þ dt ¼ 0
ð5aÞ
t1
with du ¼ dw ¼ 0
at t ¼ t1 and t2
ð5bÞ
where dT and dU denote the variation of the kinetic and strain energy, respectively, while dWe denotes the virtual work due to external forces. For the problem at hand, these terms are defined as: Virtual kinetic energy dT ¼ 2p
Z
Z
‘
h
dx ‘
h ih i q u_ fw_ ;x du_ f dw_ ;x ðR þ fÞ df
ð6Þ
h
Virtual strain energy Z ‘ 1 Nx du;x Mx dw;xx þ Nh dw dx dU ¼ 2pR R ‘
ð7Þ
Virtual work due to the magnetoelastic loads dWe ¼ 2pR
Z
‘
‘
h i r þ mx;x Þ dwj‘ x duj‘ M x dw;x j‘ þ ðQ ½qr dw þ qx du mx dw;x dx þ 2pR N ‘ ‘ ‘ ð8Þ
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In Eq. (8), the 2-D body forces qr , qx and shell moment mx are defined as: mx
Z
h
px fð1 þ f=RÞ df
ð9aÞ
pr ð1 þ f=RÞ df
ð9bÞ
px ð1 þ f=RÞ df
ð9cÞ
h
qr
Z
h h
qx
Z
h h
where, px and pr are the counterparts of 3-D body forces. In terms of the basic unknowns uðx; tÞ and wðx; tÞ, the governing equations are: E h2 m qx h2 þ u þ w w q€ u þ q€ w;x ¼ 0 du: ;xx ;xxx ;x 1 m2 R 3R 2h 3R dw:
ð10aÞ
u;xxx 3 m w 2h3 2h3 w u;x þ 2 D w;xxxx þ 2 q€ w;xx þ q€u;x ¼ 0 mx;x qr þ 2hq€ h R R R 3 3R ð10bÞ
It is noted that the terms underscored by double solid lines are associated with rotary inertia. The associated boundary conditions are: x Nx ¼ N x Mx ¼ M
du ¼ 0
ð11aÞ
or dw;x ¼ 0
ð11bÞ
or
Mx;x þ mx þ
2h3 2h3 r € ;x w q€ u¼Q 3 3R
or dw ¼ 0
ð11cÞ
r are the external loads at the shell ends (i.e., x ¼ ‘). Note that the positive x, Q x , M where N x defined in Eq. (9a) is along the negative direction of eh (see Fig. 1). From Eqs. (10) direction of M and (11), it is readily seen that the couplings between axial elongation u and flexural deformation w are entirely due to the presence of curvature 1=R. The boundary conditions corresponding to clamped, simply-supported and free ends are summarized in the following: 8 du ¼ 0; dw ¼ 0; dw;x ¼ 0 < clamped: simply-supported: du ¼ 0; dw ¼ 0; Mx ¼ 0 ð12Þ : free: Nx ¼ 0; Mx ¼ 0; Mx;x ¼ 0 When R ! 1, the preceding governing equations (10) and (11) reduce to the corresponding plate strip model. In such a case, the force–displacement relations reduce to:
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Nθ Nx
Mx θ
x
Fig. 1. Shell forces in the axi-symmetric problem of circular cylindrical shells (note that the direction of Mx is along the negative h direction).
Nx ¼
2hE u;x 1 m2
ð13aÞ
Nh
2hmE u;x 1 m2
ð13bÞ
Mx ¼
2Eh3 w;xx 3ð1 m2 Þ
ð13cÞ
Therefore, the governing equations corresponding to the plate strips are: du:
E qx u¼0 u;xx þ q€ 2 1m 2h
dw:
Dw;xxxx mx;x qr þ 2hq€ w
ð14aÞ 2h3 q€ w;xx ¼ 0 3
ð14bÞ
In such a case, the governing equations (14a) and (14b) and the associated boundary conditions become completely decoupled. In the following, for the sake of numerical illustration, only the clamped–clamped boundary conditions will be considered. 2.2. Magnetoelastic loads For thin-shells immersed in an applied uniform magnetic field along the axial direction (i.e., H0 ¼ H0 ex , see Fig. 1, Part I), the induced magnetic field within the perfectly electro-conductive medium is: h ¼ rot½u H0 ¼ H0
ow er ox
ð15Þ
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Therefore, the Lorentz force is: 2 f ðeÞ ¼ rot h B 0 ¼ l0 H0
o2 w er ox2
ð16Þ
In this article, only the induced magnetoelastic loads will be considered. In such a case, px ¼ 0 and pr ¼ ð1=l0 ÞH02 o2 w=ox2 . Therefore, the generalized shell forces defined in Eq. (9) now reduce to: mx ¼ 0;
qx ¼ 0;
qr ¼ l0 H02
ðþÞ o2 w ðÞ 2h þ r r rr rr ox2
ð17Þ
ðÞ in which, rðþÞ rr and rrr are the MaxwellÕs stress jumps on the upper and lower media surfaces, respectively. In the jump conditions,
f1 ðR þ h; xÞ ¼ H0
ow ; ox
f2 ðR h; xÞ ¼ H0
ow ox
ð18Þ
2.3. Solution methodology Due to the high complexity of magnetoelastic governing equations, the extended GalerkinÕs method (EGM) [11,14] is used to solve the intricate system of equations. The underlying idea of this method is to select weight functions that need only fulfill the geometric boundary conditions. As a result, the natural boundary conditions that may not be fulfilled appear as a residual in the functional which should be minimized in the GalerkinÕs sense [11]. We start from the governing equations (10a) and (10b). Construct the following variational functional dH: Z ‘ h i x Nx duj‘ ½l:h:s of ð10aÞ ð 2hÞ du þ l:h:s of ð10bÞ ðdwÞ dx þ N dH ¼ ‘ ‘ h i 2h3 2h3 x dw;x j‘ € ;x Mx;x þ mx þ ð19Þ w q€ u Qr dwj‘‘ þ Mx M ‘ 3 3R In Eq. (19), l.h.s of (10a) and (10b) denotes the left-hand side of Eqs. (10a) and (10b). It is noted that the coefficients 2h before du and the negative sign in expression dw are used to recover the functional ðdT dU þ dWe Þ=ð2pRÞ. Define the following non-dimensional parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E h ^ ðn; sÞ ¼ wðx; tÞ=h n ¼ x=‘; x0 ¼ ; s ¼ x0 t; u^ðn; sÞ ¼ uðx; tÞ=‘; w 2 3ð1 m Þq ‘2 ð20Þ Now, perform the spatial semi-discretization [2]: ^ T ðnÞ^ qu ðsÞ; u^ðn; sÞ ¼ U u
^ T ðnÞ^ ^ ðn; sÞ ¼ U w qw ðsÞ w
ð21Þ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
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^ u ðnÞ and U ^ w ðnÞ are where, ^ qu ðsÞ and ^ qw ðsÞ are Nm 1 generalized displacement vectors, whereas U the shape function vectors with dimensions Nm 1, which are required to fulfill only the geometric boundary conditions: ^ u ð1Þ ¼ 0 ^ u ð1Þ ¼ U U ^ w ð1Þ ¼ U ^ w ð1Þ ¼ 0; U
ð22aÞ ^ 0 ð1Þ ¼ U ^ 0 ð1Þ ¼ 0 U w w
ð22bÞ
In the present paper, the following shape functions are adopted: ^ T ðnÞ ¼ fsin pn; sin 2pn; . . . ; sin Nm png U u ^ T ðnÞ ¼ cos pn þ 1; cos 2pn 1; . . . ; cos Nm pn ð 1ÞNm U w The transformed governing equation is: # " € uw uu uu ^qu ^ K K 0 M qu þ ¼0 2 ww ^ € ^ q B ð K þ K Þ K K 0 M ^ qw wu ww H1 H2 w 0
ð23aÞ ð23bÞ
ð24Þ
where, B^0 is the non-dimensionalized applied magnetic field intensity (see Appendix A for its ww , K uu , K uw , M wu , M ww and K H 2 (related to the Lorentz uu , M definition). The details of matrices M force) are summarized in Appendix B, while KH 1 , which is related to the MaxwellÕs stress jump, are defined in the following special cases. 3. Calculation of the generalized magnetoelastic load in special cases with implementation details Due to the singular integrals, infinite integral domain and highly excessive time necessary for the evaluation of some regularized kernels, special treatment is needed for the efficient computation of the generalized magnetoelastic loads. 3.1. Generalized magnetoelastic load on the plate strip " Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # Z ‘ Z ‘ ‘ ðþÞ 1 ‘2 s2 w0 ðsÞ ðÞ 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rrr rrr dw dx ¼ l0 H0 ds dw dx p ‘ ‘2 x2 sx ‘ ‘ " Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ‘ ‘ 1 ‘2 s2 ½w0 ðsÞ w0 ðxÞ 22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ds ¼ l 0 H0 p ‘ ‘2 x2 sx ‘ # Z ‘ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘2 x2 dsw0 ðxÞ dw dx þ sx ‘
ð25Þ
In Eq. (25), the identity transformation is used to eliminate the singularity at point s ¼ x. Expressed in the modal space, the last equation is further transformed as follows:
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Z
" Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ‘ ‘ 0T ðþÞ 2 U ‘2 s2 ½U0T w 2 T w ðsÞ Uw ðxÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi dw dx ¼ l rrr rðÞ H dq ds 0 0 rr p w ‘ ‘2 x2 sx ‘ ‘ # Z ‘ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ‘2 x2 0T H 1 qw dsUw ðxÞ dxqw , B^20 dqTw K þ sx ‘ ‘
ð26Þ
in which, H1 K
qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 3 Z 1 Z 1 1 n2 ½U 0T 0T ^ ^ ðn Þ U ðnÞ ^ s Uw 1 ‘ w w s ^ 0T ðnÞ5 dn pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 dns pnU w 2 n p h n s 1 1 1n
ð27Þ
In Eq. (27), the following identity is used (see [7, p. 570]): Z
1 1
qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2s ns n
dns ¼ pn;
jxj < 1
ð28Þ
By using the standard Gauss–Chebyshev integration [5], we get:
H1 K
qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 3 X Z 1 1 n2 ½U Nd ^ 0T ðns Þ U ^ 0T ðnÞ < = 1 ‘ w w s ^ w ðnk Þ ^ 0T ðnk Þ U dns pnk U w : 1 ; Nd h ns nk k¼1
ð29Þ
where, nk are the locations of discrete points such that TNd ðnk Þ ¼ 0;
k ¼ 1; . . . ; Nd
ð30Þ
3.2. Generalized magnetoelastic load on the infinite cylindrical shells Z ‘ Z ‘( Z 1 h i ðþÞ ^ 0T G ^ ^ 0T U rrr rðÞ dw dx ¼ l0 H02 rr w ðs xÞ dsqw Uw ðxÞ dqw ‘
‘
2 þ p
Z
1 1
1
) h i ^ 0T U 0T w ^ ðxÞ dqw dx , l0 H 2 dqT ½KHH qw ð31Þ dsqw U w 0 w sx
where, 3
2 2
KHH h
Z 7 6 Z 1 ^ 0T ðns Þ 2 1 U 7 6 0T w ^ ^ ^ Uw ðnÞ6 Uw ðns ÞGðns nÞ dns þ dns 7 dn 5 4 p 1 ns n 1 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Z
1
^ H1 U
^ H2 U
ð32Þ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
^ðns nÞ is defined as: and the kernel G ( ) Z 1 ^ ^ 1 K ½j x jðR þ hÞ I ½j x jðR hÞ 0 0 ^ðns nÞ ^Þ ^ G | signðx þ 2 e|x^ ðns nÞ dx ^ jðR þ hÞ I00 ½jx ^ jðR hÞ 2p 1 K00 ½jx |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
2031
ð33Þ
^Þ F ðx
Next, noticing that U0 ðns Þ contains only sine-functions, then by exploiting the relationship between the Fourier integral and the inverse Fourier integral, we can dramatically simplify the first integral in the bracket of Eq. (32) as follows: Z 1 Z 1 Z 1 1 ^ ðns nÞ 0T 0T |x ^ ^ ^ ^ ^Þ e ^ dns Uw ðns ÞGðns nÞ dns ¼ Uw ðns Þ F ðx dx ð34Þ UH 1 2p 1 1 1 ^ 0T ðns Þ consists of sinðmi pns Þ functions (mi ¼ 1; . . . ; Nd ), we can convert Eq. (34) into the Since U w Fourier integral transform pairs. For example, taking an arbitrary element sinðmi pns Þ from ^ 0T ðns Þ, we get: U w Z 1 Z 1 1 ^ ðns nÞ |x ^Þ e ^ dns sin mi pns F ðx dx 2p 1 1 Z 1 Z 1 1 ^ ðns nÞ |x ^Þ e ^ dns ¼ sin mi pðns n þ nÞ F ðx dx 2p 1 1 Z 1 Z 1 1 ^ ðns nÞ |x ^Þ e ^ dðn ns Þ ¼ ð cos mi pnÞ sin mi pðns nÞ F ðx dx 2p 1 1 Z 1 Z 1 1 ^ ðns nÞ |x ^ ^ cos mi pðns nÞ F ðxÞ e dx dðn ns Þ þ sin mi pn 2p 1 1 ¼ ð cos mi pnÞIm ½F ðmi pÞ
ð35Þ
^ Þ at x ^ ¼ mi p. It is noted that in where, Im ½F ðmi pÞ denotes the imaginary part of the function F ðx Eq. (35), the real part of the function F ðmi pÞ ¼ 0. As to the second integral in the bracket of Eq. (32), using the following identity: Z 1 1 sin mi pns ð36Þ dns ¼ cos mi pn; due to mi > 0 p 1 ns n we get U2H ðnÞ ¼ fp cos pn; 2p cos 2pn; . . . ; Nd p cos Nd pngT
ð37Þ
Finally, we get H 1 l0 H 2 KHH =B^2 ¼ 1 K 0 0 2
3 Z 1 h i ‘ T T ^ ^ ^ Uw ðnÞ UH 1 ðnÞ þ UH 2 ðnÞ dn h 1
ð38Þ
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^ H 1 ðnÞ is defined as where, U 8 9 K0 ½pðR‘ þh‘ Þ I0 ½pðR‘ h‘ Þ > > > > p cosðpnÞ 2 þ K 0 ½pðRþhÞ I 0 ½pðRhÞ > > > > ‘ ‘ ‘ ‘ 0 0 > > > > > > > > RþhÞ RhÞ > > K ½2pð I ½2pð < 2p cosð2pnÞ 2 þ 0 R‘ h‘ 0 R‘ h‘ = 0 0 K0 ½2pð ‘ þ ‘Þ I0 ½2pð ‘ ‘ Þ ^ UH 1 ðnÞ > > > > .. > > > > . > > > > > > R h R h > K0 ½Nm pð ‘ þ ‘ Þ I0 ½Nm pð ‘ ‘Þ > > : Nm p cosðNm pnÞ 2 þ K 0 ½N pðRþhÞ I 0 ½N pðRhÞ > ; m m 0
‘
‘
0
‘
ð39Þ
‘
^ T ðnÞ ¼ 0. Such a case corresponds to When in addition, R ! 1, from Eq. (39), it follows that U H1 the infinite plate (both in length and width), and Eq. (38) reduces to H1 K
1 ¼ 2
3 Z 1 ‘ ^ w ðnÞU ^ T ðnÞ dn U H2 h 1
ð40Þ
3.3. Generalized magnetoelastic load of finite-length circular cylindrical shells with infinitely small thickness In non-dimensional form, wx ðnÞ should fulfill the following equation: Z 1 Z 1 1 wx ðns Þ h ~^ ^ 0 ðnÞ; jnj < 1 wx ðns Þk 11 ðns nÞ dns þ w dns ¼ H0 2p n ‘ n 1 1 s
ð41Þ
with the boundary condition wx ðnÞ Z
1
wx ðnÞ dn ¼ 0
ð42Þ
1
~ where, the regularized kernel k^11 ðns nÞ is defined as " # Z 1 | 1 1 ~^ ^Þ ^ jR=‘ÞK00 ðjx ^ jR=‘Þ ^ signðx I 0 ðjx e|x^ ðnns Þ dx k 11 ðns nÞ ¼ ^ jR=‘Þ 0 2p 1 Dðjx 2 # Z " h i 1 1 1 1 0 0 ^ ^ ^ ^ I ðjxjR=‘ÞK0 ðjxjR=‘Þ sin xðns nÞ dx ¼ ^ jR=‘Þ 0 p 0 Dðjx 2
ð43Þ
It is proved that the solution of wx ðns Þ fulfilling Eq. (41) and the boundary condition Eq. (42) has the following form (see e.g., Refs. [5,15]): 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi gðns Þ; 1 n2s
where gðns Þ is a bounded function
ð44Þ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2033
wx ðns Þ can be expressed by the following Chebyshev series: 1 X 1 Ak Tk ðns Þ wx ðns Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2s k¼0
ð45Þ
where, Tk ðns Þ is the first kind Chebyshev function of order k. Then the singular integral equation (41) is converted to the following: 1 X
Z
1
Ak 1
k¼0
Tk ðns Þ ~^ qffiffiffiffiffiffiffiffiffiffiffiffi ffi k 11 ðns nÞ dns þ 1 n2s
1 X k¼0
8 9 > > < 1 Z 1 = Tk ðns Þ h qffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 0 ðnÞ dns ¼ H0 Ak w > > 2p ‘ 2 1 : ; 1 ns ðns nÞ ð46Þ
where, n 2 ð1; 1Þ. Denote G11 ðk; nÞ
Z
1 1
1 G12 ðk; nÞ 2p
Tk ðns Þ ~^ ffi k 11 ðns nÞ dns qffiffiffiffiffiffiffiffiffiffiffiffi 1 n2s Z
1
1
Tk ðn Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi s dns ¼ 1 n2s ðns nÞ
ð47aÞ
0; 1 U ðnÞ; 2 k1
k¼0 k>0
ð47bÞ
In Eq. (47b), Uk1 ðnÞ is the second kind Chebyshev function of order k 1. Since Tk ðns Þ and ~^ k 11 ðns nÞ are functions with bounded amplitudes, Eq. (47a) can be efficiently evaluated by the Gauss–Chebyshev integration scheme as follows: Nd p X ~ G11 ðk; nÞ Tk ðni Þk^11 ðnis nÞ; Nd i¼1
with TNd ðnis Þ ¼ 0; i ¼ 1; 2; . . . ; Nd
ð48Þ
Therefore, from Eq. (46), we get 1 X k¼0
"
G11 ðk; ni Þ þ G12 ðk; ni Þ Ak |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Gðk;iÞ
#
h ^ 0 ðni Þ; ¼ H0 w ‘
i ¼ 1; . . . ; Nd
ð49Þ
where, ni fulfilling TNd ðnis Þ ¼ 0 are the points where the governing equation (41) is enforced to be strictly satisfied. From the boundary condition (42), we get A0 ¼ 0. After truncating the expansion order up to Nd , we get a set of algebraic equations expressed in terms of the unknown coefficients A1 ; . . . ; ANd :
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8 9 38 A 9 0 1 > > ^ ðn Þ w > > > 1 Gð1; 1Þ Gð2; 1Þ GðNd ; 1Þ > > > > > > > 0 2 > < A2 > .. > > ‘ > > > > > . > > > > : > ; > N Gð1; Nd Þ Gð2; Nd Þ GðNd ; Nd Þ : ; 0 d A ^ N ðn Þ w d |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2
ð50Þ
G
The solution of A1 ; . . . ; ANd is, therefore, 8 8 9 9 8 9 0 1 > 0 1 > > > A ^ ^ ðn Þ ðn Þ w w > > > > > > 1 > > > > > > > > > 0 2 > 0 2 > < A2 > < < = = = ^ ^ ðn Þ ðn Þ w w h h T G1 C1 CNd ¼ H0 , H0 .. . . .. .. > > > > > ‘ ‘ . > > > > > > > > > > > > > : > > ; > > :w : 0 Nd ; 0 Nd ; ANd ^ ðn Þ ^ ðn Þ w
ð51Þ
As a result, Z
‘ ‘
ðþÞ dw dx ¼ rrr rðÞ rr ¼
Z
1
1
ðþÞ T ^ d^qw ‘ dn hU rrr rðÞ rr w
qTw l0 H0 d^
Nd Z X k¼1
1
1
Tk ðnÞ ^ w ðnÞ dnAk pffiffiffiffiffiffiffiffiffiffiffiffiffi ðh‘ÞU 2 1n 2
¼ l0 H02 h2 d^ qTw
Nd X
(Z
k¼1
1
1
^ 0T ðn1 Þ U w
3
7 6 7 )6 ^ 0T ðn2 Þ 7 6U w 7 6 Tk ðnÞ ^ 7^qw pffiffiffiffiffiffiffiffiffiffiffiffiffi Uw ðnÞ dnCTk 6 7 6 2 . 7 6 1n .. 7 6 5 4 ^ 0T ðnNd Þ U w
ð52Þ
It is noted that the number of terms used in the Chebyshev series, is equal to that of the discrete points where the governing equation (41) is strictly enforced. By using the following two identities [4], the integrals in Eq. (52) can be precisely evaluated: Z
1 1
Z
1 1
Tk ðnÞ cos mi pn pffiffiffiffiffiffiffiffiffiffiffiffiffi dn ¼ 1 n2 Tk ðnÞ p; pffiffiffiffiffiffiffiffiffiffiffiffiffi dn ¼ 2 0; 1n
k=2
ð1Þ 0;
pJk ðmi pÞ;
when k ¼ 0 when k > 0
when k is even when k is odd
ð53aÞ
ð53bÞ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2035
Finally, we get
H1 K
8 9 Jk ðpÞ > > > > > > > > ð2pÞ J > > 3 k < = X 1 ‘ k=2 ð3pÞ J k ¼ ð1Þ Ak p > > 2 h k¼2;4;6;...;2bN =2c .. > > > > d > > . > > : ; Jk ðNm pÞ
ð54Þ
3.4. Efficient evaluation of the elements G(i,j) in Eq. (50) Direct numerical evaluation of the elements Gði; jÞ in Eq. (50) is heavily time-consuming. This is due to: ~ • infinite integral domain involved in the evaluation of the kernel k^11 (see Eq. (43)), which are also highly oscillatory when x becomes large, ~ • for each element Gði; jÞ, from Eq. (48), it is readily seen that the kernel k^11 needs be evaluated Nd times. The complexity of getting the matrix G in Eq. (50) involves totally ðNd Þ3 times of eval~ uation of k^11 . Since the discrete points on which to strictly enforce the governing equation (41) coincide with those used to implement the Gauss–Chebyshev integration, we need only evaluate the ~ kernel k^11 with all the different values ðni nj Þ once for all for each configuration of the shell (R=‘ and h=‘). Then store them to a suitable data structure (in the present paper, we use array) for look-up whenever they are needed later on. Due to the anti-symmetry of the kernel ~ ~ ~^ k 11 ðxÞ ¼ k^11 ðxÞ, totally only about 1=4Nd2 times of the evaluation of k^11 are needed to get the matrix G. Further steps toward significant improvement of the efficiency include: ~ • using binary search scheme [19] to find the value of k^11 ðxÞ in the array, ^ 1, • when x 1 1 ^ jR=‘ÞK00 ðjx ^ jR=‘Þ I00 ðjx ^ jR=‘Þ Dðjx 2
! !
‘2 ^ Þ2 cn ðRx
and replacing the infinite integral equation (43) by the following piecewise integral: # Z " h i 1 a 1 1 1 ~^ ^ ðns nÞ dx ^ jR=‘ÞK00 ðjx ^ jR=‘Þ ^ sin x k 11 ðns nÞ I00 ðjx ^ jR=‘Þ p 0 Dðjx 2 cn p ‘2 sin½aðns nÞ 2 ðns nÞCi ½aðns nÞ signðns nÞ þ a R
ð55Þ
ð56Þ
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Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
where, cn is a constant, a > 0 and Ci ðxÞ is the cosine integral function, which is defined as: Z 1 cos s Ci ðxÞ ¼ ds ð57Þ s x In the present article, a ¼ 100. In summary, the complexity of determining all the entries of matrix G reduces from ðNd Þ3 ~ integrations of k^11 to 1 2 N 4 d
~ ~ integrations of k^11 þ OðNd2 log2 Nd search Þ 14Nd2 integrations of k^11
ð58Þ
Therefore, the total speed-up is about 4cNd times faster, in which, c is the ratio of time used via ~ direct numerical evaluation of k^11 from Eq. (43) over the time used by the piecewise approximation of Eq. (56). The approximation in Eq. (58) is based on the fact that the searching time is much less than that used in the numerical integral evaluation. The numerical experience in the studied cases in the present article shows that on an average, c 10, therefore, about 1200 times of speedup has been achieved when Nd ¼ 30. 4. Test and validation In the preceding formulation of the FS model, approximations are made in three places: (i) EGM for the magnetoelastic system, (ii) evaluation of the integral equation (43) is approximated by piecewise functions (see Eq. (56)), (iii) Eq. (41) is enforced on discrete locations. The high accuracy of EGM has been tested in [16,17]. Accuracy test results of the approximation (ii) are listed in Table 1. It is readily seen that this approximation provides high accuracy with 0.2% error or less as compared to the predictions by method I. To investigate the accuracy of the approximation (iii), the following special case is used. Z 1 1 wx ðns Þ dn ¼ f ðnÞ; jnj < 1 ð59Þ 2p 1 ns n s Table 1 ~ Accuracy test of the piecewise approximation of the kernel k^11 ðnÞ ~ k^11 ðnÞ by method Ib na 0.1 0.2 0.5 1.0 1.5 1.8 1.9 2.0
)0.02115 )0.03391 )0.05622 )0.06817 )0.06642 )0.06292 )0.06159 )0.06023
~ Due to anti-symmetry of the k^11 , only n > 0 are considered. Direct numerical integration (see Eq. (43)), calculated by using MathematicaTM . c Piecewise approximation (see Eq. (56)). a
b
~^ k 11 ðnÞ by method IIc )0.02117 )0.03389 )0.05616 )0.06820 )0.06641 )0.06296 )0.06149 )0.06024
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2037
The solution of Eq. (59) in conjunction with the boundary condition (Eq. (42)) can be expanded in the following series: wx ðns Þ ¼
1 X k¼0
Nt X Tk ðns Þ Tk ðns Þ ffi ffi Ak qffiffiffiffiffiffiffiffiffiffiffiffi Ak qffiffiffiffiffiffiffiffiffiffiffiffi k¼0 1 n2s 1 n2s
ð60Þ
Substituting expression (60) into Eq. (59), invoking the identity (Eq. (47)) and the orthogonality property of the second kind Chebyshev function Uk , we get p Aiþ1 ¼ 4
Z
1
qffiffiffiffiffiffiffiffiffiffiffiffiffi f ðnÞUi ðnÞ 1 n2 dn;
i ¼ 0; 1; 2; . . . ; 1
ð61Þ
1
From the boundary condition (Eq. (42)), we get: A0 ¼ 0. For the purpose of accuracy test, f ðnÞ takes the form of sinðmpnÞ. Once Ak is solved, wx can be reconstructed from Eq. (60). It is noted that larger values of m put more strict test of the accuracy. In the present paper, f ðnÞ ¼ sinð7pnÞ is used in the test. Figs. 2–4 display the comparison of wx solved by discrete method and the synthesis with the truncated terms Nt ¼ 20, 25, 30, respectively. It is seen that when Nd ¼ Nt P 25, a very high accuracy is achieved by the discrete method. In order to validate the models developed so far and to get some insight on these models, the fundamental frequency of the magnetoelastic plate strips will be obtained. In such a case, R ! 1, which leads to 2 4 2 4 o2 w ^ ^2 ‘ o2 w ^ 3ð1 m2 Þ ‘ ðþÞ ^ 1 h ^ o4 w ow ðÞ B0 rrr rrr þ 2 ¼0 4 2 2 h 2E h 3 ‘ on os2 os on on
ð62Þ
3
2
1
0 x
-1
-2
-3 -1
-0.5
0
ξ
0.5
1
Fig. 2. Accuracy test of the discretization method (Nd ¼ Nt ¼ 20, f ðnÞ ¼ sin 7pn, solid curve––solution by expansion via Eqs. (60) and (61), dashed curve––solution by discretization via Eqs. (45) and (51)).
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Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046 2
1
0 x
-1
-2 -1
-0.5
0
ξ
0.5
1
Fig. 3. Accuracy test of the discretization method (Nd ¼ Nt ¼ 25, f ðnÞ ¼ sin 7pn, solid curve––solution by expansion via Eqs. (60) and (61), dotted curve––solution by discretization via Eqs. (45) and (51)).
2
1
x
0
-1
-2 -1
-0.5
0
ξ
0.5
1
Fig. 4. Accuracy test of the discretization method (Nd ¼ Nt ¼ 30, f ðnÞ ¼ sin 7pn, solid curve––solution by expansion via Eqs. (60) and (61), dotted curve––solution by discretization via Eqs. (45) and (51)).
The term underscored by the double solid line is associated with the rotary inertia. For thin plate strips (i.e., h=‘ 1) and mechanical vibrations (usually low-frequency), this term can be safely discarded. When using the magnetoelastic loading model of infinite plate (i.e., PL model), and considering the above simplification, we get
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2 3 Z 1 0 ^ ^2 ‘ o2 w ^ ^2 ‘ ^ ðn; sÞ ^ o4 w 1 o2 w w B B þ ¼0 dn s 0 0 4 2 h h p 1 ns n os2 on on
2039
ð63Þ
Similarly, when using the magnetoelastic loading model of plate strips (i.e., PS model), and considering the above simplification, we get qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 Z 1 1 n2 4 ^ ^2 ‘ o w ^ ^2 ‘ ^ ow 1 o2 w s 0 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ^ B B ðn; sÞ dn þ ¼0 ð64Þ w s 0 0 4 2 h h p 1 n2 1 ns n os2 on on In Eqs. (63) and (64), the terms underscored by single solid line are associated with the influence of Lorentz force, while the terms underscored by double solid lines are associated with MaxwellÕs stress jump at the media surfaces. In the case of clamped–clamped plate strips, after the use of the shape function UT ðnÞ ¼ fcos pn þ 1g (see Eq. (23)), we get an estimate of the fundamental frequency of the plate strip by using these two different magnetoelastic load models: ) 2 - 3 ( 2 x1 p4 ^2 ‘ h p p ð65aÞ þ - ¼ þ B0 h ‘ 3 3 x0 3 PL
2 -x1 x0 -
PS
) 3 ( 2 p4 ^2 ‘ h p þ 1:1818 ¼ þ B0 h ‘ 3 3
ð65bÞ
In Eq. (65), the subscripts PL and PS denote that the estimations are based on PL and PS magnetoelastic load models, respectively. The underscored terms by single solid line are associated with the influence of Lorentz body force. For thin plate strips (i.e., h=‘ 1), it is readily seen that they can be safely discarded. When using the shape function UT ðnÞ ¼ fcos 2:365n þ 0:133 cosh 2:365ng, which is the exact first-order eigenmode of the clamped–clamped beam, the following estimate is obtained: ) 2 - 3 ( x1 ‘ h 3:071 þ 1:1122 ð66aÞ - ¼ 31:284 þ B^20 h ‘ x0 PS
In the case of simply-supported plate strips after considering the shape function as UT ðnÞ ¼ fcosðpn=2Þg (see Eq. (23)), we get estimates of the fundamental frequency of the plate strip by using these two different magnetoelastic load models: ) 2 - 3 ( 2
p 4 x1 ‘ h p p 2 ð67aÞ þ þ B^0 - ¼ 2 h ‘ 4 2 x0 PL
2 -x1 x0 -
PS
¼
p 4 2
) 3 ( 2 ‘ h p þ B^20 þ 0:251p h ‘ 4
The results obtained from these special cases agree very well to those in literature [10].
ð67bÞ
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5. Numerical results and discussion Due to the complexities involving the exact solution of magnetoelastic loads on the circular cylindrical shells (see Part I), various approximations that attempt to simplify the calculations have been proposed. One main objective of the present study is to investigate the accuracy of various simplifications. Table 2 summarizes the geometric restrictions underlying the four magnetoelastic loads models. It is noted that the structure is still of the shell type, only the magnetoelastic loads acting on the shell are approximated by these simplified models. Characteristics of the influence of applied magnetic field on the free vibration of thin shells, and other important issues such as influence of thickness ratio h=‘, dimensionless radius R=‘ of the shell will also be investigated. Abbreviations of these models used in the following discussion are summarized in Table 1. Figs. 5 and 6 display the comparison of the first two eigenfrequencies of a circular cylindrical shell under the influence of applied magnetic fields predicted by different magnetoelastic load models. It is observed that: (i) in the total increase (3.7%) of x1 due to the presence of B^0 ¼ 0:03, the maximal difference of predictions among these four magnetoelastic load models can reach 2.5%; while for x2 , it can reach 2.5% out of the total increase of 5.8%. This implies that when the influence of applied magnetic field is strong, the difference of predictions among these four models Table 2 Summary of various simplified magnetoelastic load models Model
Abbreviation
Geometric restrictions
H 1 reference equation # K
Plate strip Infinite circular cylindrical shell Infinite plate (both length & width) Finite-length cylindrical shell
PS ISH PL FSH
R ! 1, h ! 0 ‘!1 R ! 1, ‘ ! 1, h ! 0 h!0
(29) (38) (40) (54) and (51)
PS 650
ISH PL FSH
645
1 0
640
635
630 0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 1 ( x1 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 0:5, h=‘ ¼ 0:005). Fig. 5. Comparison of x
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2041
PS ISH
680
PL FSH 670 2 0
660
650 0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 2 ( x2 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 0:5, h=‘ ¼ 0:005). Fig. 6. Comparison of x
will be very significant, (ii) among the four magnetoelastic load models, the ISH model gives the lowest prediction of the eigenfrequencies x1 and x2 , (iii) in terms of the closeness of the predictions, these four models are separated into two groups: ISH and PL models; PS and FSH models. It is scrutinized that the influence of dimensionless radius R=‘ is much less than the influence of finiteness of the length of shells. This inference can be confirmed by Figs. 7–9. With the increase of dimensionless radius (R=‘ ¼ 10 in this case), the difference of predictions between ISH and PL models or PS and FSH models almost vanishes. Figs. 7–9 also reconfirm the preceding first observation (i.e., i): the ISH or PL models predict 2.4 times of increase of x1 due to the presence of B^0 ¼ 0:03, while PS or FSH models predict 3.5 times of increase of x1 due to the 300
PS ISH 250
PL FSH 1 200 0
150
100
0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 1 ( x1 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 10, h=‘ ¼ 0:002). Fig. 7. Comparison of x
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PS ISH
600
PL 500
FSH
2 400 0
300
200
100 0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 2 ( x2 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 10, h=‘ ¼ 0:002). Fig. 8. Comparison of x
PS ISH
800
PL FSH 600 3 0
400
200
0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 3 ( x3 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 10, h=‘ ¼ 0:002). Fig. 9. Comparison of x
presence of B^0 ¼ 0:03. For x2 , the ISH or PL models predict 5.6 times of increase, while the PS or FSH models predict 7.8 times of increase. For x3 , the ISH or PL models predict 6.3 times of increase, while the PS or FSH models predict 9.0 times of increase. Having investigated the influence of these magnetoelastic load models on the prediction of shell eigenfrequencies, we now turn to investigate the influence of applied magnetic field itself on eigenfrequencies of shells. Fig. 10 displays the variation of the first six eigenfrequencies of a shell vs. the intensity of applied magnetic field B^0 . It is readily seen that the presence of applied magnetic field plays the role of increasing the structural stiffness and that the higher the modal
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2043
Fig. 10. Influence of the applied magnetic fields on the first six eigenfrequencies of a circular cylindrical shell predicted by the FS model (R=‘ ¼ 10, h=‘ ¼ 0:002).
eigenfrequency, the more the rate of increase, implying that the applied magnetic field cannot induce, by itself, static buckling of such structures. Put it another way, the presence of applied magnetic field tends to stabilize the structure. Fig. 11 shows the influence of thickness ratio h=‘ on the eigenfrequency of magnetoelastic shells. It is observed that the applied magnetic field B^0 has more prominent influence on thinner shells. With the increase of h=‘, the influence of the presence of B^0 on x1 will significantly decrease. Fig. 12 displays the influence of dimensionless radius R=‘ on the eigenfrequency of magnetoelastic shells. When the applied magnetic field B^0 increases from 0 to 0.03 in its intensity, in the
Fig. 11. The influence of the thickness ratio h=‘ on the variation of x1 vs. B^0 (FS model, R=‘ ¼ 0:5, xc1 x0 jh=‘¼0:002 ).
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Fig. 12. The influence of the dimensionless radius R=‘ on the variation of x1 vs. B^0 (FS model, h=‘ ¼ 0:01).
case of R=‘ ¼ 10, x1 increases to 3.7 times of its elastic value, while in the cases of R=‘ ¼ 1 and 0.5, x1 increases only to about 1.1 times of the corresponding elastic values. This implies that the applied magnetic field B^0 has more prominent influence on shells featuring larger dimensionless radius. 6. Conclusions Four different magnetoelastic load models have been developed and the accuracy of the model tested. Difference of these models on the prediction of eigenfrequencies of circular cylindrical shells has been investigated. The influence of applied magnetic field, thickness ratio and dimensionless radius on the free vibration of thin cylindrical shells have also been investigated. The major conclusions are: • when the influence of applied magnetic field is strong, the difference of predictions among these four models will be very significant, • in the modeling of magnetoelastic loads, the dimensionless radius R=‘ has much less influence than the finiteness of the length of shells, • the presence of applied magnetic field tends to stabilize the structure, and by itself, the applied magnetic field cannot induce static buckling of such structures, • in the case of simply-supported plate strips, the effect of finiteness of length or width must be considered when investigating their magnetoelastic behavior. Acknowledgement Partial support from NASA Langley Research Center through grant NAG-01101 is gratefully acknowledged.
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2045
Appendix A. Nomenclature pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B^0 non-dimensional applied magnetic field intensity, ð 3ð1 m2 Þl0 =EÞH0 D shell flexural stiffness, 2Eh3 =ð3ð1 m2 ÞÞ E, m, q YoungÕs modulus, PoissonÕs ratio and the mass density of the shell Lorentz force vector f ðeÞ f1 ðR þ h; xÞ lr h on the outer surface of the shell f2 ðR h; xÞ lr h on the inner surface of the shell h thickness of the shell induced magnetic field outside the electro-conductive medium h applied magnetic field H0 ‘ semi-length of the shell number of the discrete points used in the Gauss–Chebyshev integration (see Eq. (48)) Nd number of the truncated terms used in expansion (see Eq. (60)) Nt number of shape functions used in semi-discretization (see Eq. (21)) Nm r, M x , Q x shell stress resultants and tress couple, see Eqs. (3a)–(3c) N pr , px body forces (per volume) within the conductive media qr , qx , mx 2-D body forces, see Eqs. (9a)–(9c) R radius of the mid-surface of the shell Tk ðnÞ the first kind Chebyshev function of order k u, w axial and flexural displacements Uk ðnÞ the second kind Chebyshev function of order k u displacement vector within the conductive media f local thickness coordinate, r R magnetic permeability of the electro-conductive media le magnetic permeability of the free space l0 le =l0 , relative magnetic permeability lr s dimensionless time, defined by Eq. (20) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi reference frequency, ð E=ð3ð1 m2 ÞqÞÞh=ð‘Þ2 x0 ^i dimensionless frequency, xi =x0 x d variation operator pffiffiffiffiffiffiffi | 1 € ðð€ Þ; ð^ ÞÞ Þ ðo2 ð Þ=o2 t; o2 ð Þ=os2 Þ 0 00 ðð Þ ; ð Þ Þ ðoð Þ=ox; o2 ð Þ=ox2 Þ 0 ðð^ Þ ; ð^ Þ00 Þ ðoð^ Þ=on; o2 ð^ Þ=on2 Þ bxc operator, which gives the greatest integer no larger than x
Appendix B. Definition of matrices in Eq. (24) Z 1 Z 1 Z 1 1 h T T ^ ^ ^ ^ ^ wU ^ 00T dn; Muu Uu Uu dn; Mww Uw Uw dn U u 3 ‘ 1 1 1
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2 Z 1 ‘ ^ uU ^ 0T dn; U Kuu 3 u h 1
2 Z 1 ‘ ^ uU ^ 0T dn; U Kuw 3m w hR 1
3 Z 1 ‘ ‘ ^ wU ^ 0T dn; U Kwu 3m u h R 1 H2 K
ww K
Z
2 2 Z 1 ‘ ‘ 00 ^ 00T ^ ^ wU ^ T dn Uw Uw dn þ 3 U w h R 1 1 1
2 Z 1 ‘ ^ wU ^ 00T dn U w h 1
References [1] S.A. Ambartsumyan, G.Y. Bagadasyan, M.V. Belubekyan, Magnetoelasticity of Thin Shells and Plates, Nauka, Moscow, 1977 (in Russian). [2] T. Belytschko, An overview of semidiscretization and time integration procedures, in: T. Belytschko, T.J.R. Hughes (Eds.), Computational Methods for Transient Analysis, Elsevier Science, 1983, pp. 1–65. [3] A. Bogdanovich, Non-linear Dynamic Problems for Composite Cylindrical Shells, Elsevier Applied Science, London and New York, 1993. [4] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975, pp. 131–132. [5] F. Erdogan, Mixed boundary-value problems in mechanics, in: S. Nemat-Nasser (Ed.), Mechanics Today, vol. 4, 1978, pp. 1–86. [6] F.D. Gakhov, Boundary Value Problems (I.N. Sneddon (Ed.), Trans.), Addison-Wesley, 1966, pp. 7–10. [7] J. Katz, A. Plotkin, Low-Speed Aerodynamics: From Wing Theory to Panels Methods, McGraw-Hill, New York, 1991. [8] L. Librescu, Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures, Noordhoff International Publishing, Leyden, Netherlands, 1975, pp. 286–288. [9] L. Librescu, Recent contributions concerning the flutter problem of elastic thin bodies in an electrically conducting gas flow, a magnetic field being present, SM Archives 2 (1) (1977) 1–108. [10] L. Librescu, D. Hasanyan, D.R. Ambur, Electromagnetically conducting elastic plates in a magnetic field: modeling and dynamic implications, International Journal of Non-Linear Mechanics, in press. [11] L. Librescu, L. Meirovitch, S.S. Na, Control of cantilevers vibration via structural tailoring and adaptive materials, AIAA Journal 35 (8) (1997) 1309–1315. [12] F.C. Moon, Magneto-Solid Mechanics, John Wiley and Sons, Ithaca, New York, 1984, pp. 288–296. [13] V.V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, 1953, pp. 194–198. [14] A.N. Palazotto, P.E. Linnemann, Vibration and buckling characteristics of composite cylindrical panels incorporating the effects of a higher order shear theory, International Journal of Solids and Structures 28 (3) (1991) 341–361. [15] A.D. Polyanin, A.V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998, p. 629. [16] Z. Qin, L. Librescu, On a shear-deformable theory of anisotropic thin-walled beams: Further contribution and validation, Journal of Composite Structures 56 (4) (2002) 345–358. [17] Z. Qin, L. Librescu, Static/dynamic solutions and validation of a refined anisotropic thin-walled beam model, in: Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC structures, Structural Dynamics, and Materials Conference, Denver, Colorado, 22–25 April 2002, AIAA Paper 2002-1394. [18] J.N. Reddy, Mechanics of Laminated Composite Plates, CRC Press, Florida, 1997, p. 144. [19] C.A. Shaffer, A Practical Introduction to Data Structures and Algorithm Analysis, Prentice Hall, Upper Saddle River, New Jersey, 1997, pp. 60–62. [20] E.S. Suhubi, Small torsional oscillations of a circular cylinder with finite electric conductivity in a constant axial magnetic field, International Journal of Engineering Science 2 (1995) 441–459.
Magnetoelastic modeling of circular cylindrical shells immersed in a magnetic field Part II: Implications of finite dimensional effects on the free vibrations Zhanming Qin a, Davresh Hasanyan a, Liviu Librescu
a,*
, Damodar R. Ambur
b
a
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Mail Cose (0219), Blacksburg, VA 24061-0219, USA b Mechanics and Durability Branch, NASA Langley Research Center, Hampton, VA 23681-2199, USA Received 28 October 2002; accepted 27 January 2003
Abstract In the context of free vibration analysis of axi-symmetric perfectly electro-conductive circular cylindrical shells, four simplified magnetoelastic load models are investigated. Concerning the model of circular cylindrical shells, a linear theory based on Love–Kirchhoff hypothesis is adopted. Due to the high complexities involving singularity of integral equations, infinite integral domains and excessive time needed to evaluate some kernels, special treatments are designed toward achieving highly efficient and highly accurate numerical computation. The influence of applied magnetic field, thickness ratio and dimensionless radius on free vibrations of circular cylindrical shells are further investigated and pertinent conclusions are outlined. 2003 Elsevier Ltd. All rights reserved.
1. Introduction During the last several decades, the problem of the influence of applied magnetic field on the mechanical behavior of circular cylindrical elastic shells has been extensively studied (see e.g., [1,3,9,12,20], and the references therein). With the new emerging concept of multi-functional material/structure design paradigm, such interdisciplinary research is likely to be intensified in the years ahead. *
Corresponding author. Tel.: +1-540-231-5916; fax: +1-540-231-4574. E-mail address: [email protected] (L. Librescu).
0020-7225/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0020-7225(03)00135-6
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It is noted that in the previous work, the finite dimensional effects in the modeling of magnetoelastic loads, such as the influence of finite length and finite thickness, have not been considered in the analytical way. In Part I, the magnetoelastic loads incorporating these effects have been systematically derived and simplified magnetoelastic models were developed. However, due to the complexities involving singularity of integral equations, infinite integral domains and timeconsuming evaluation of some kernel functions, special treatments are essential toward achieving highly efficient and accurate numerical computation. It is noted that in the present paper, the evaluation of singular integrals is in the sense of CauchyÕs principal value [6]. The main objectives of the present paper are • investigation of the difference in the shell behavior when the formerly developed magnetoelastic load models in the context of axi-symmetric eigenfrequency solution of circular cylindrical thin shells are used. This will provide valuable information about the validity of these models when addressing specific problems, • investigation of the implications of various geometric parameters on the shellÕs magnetoelastic eigenfrequencies, • highly efficient and accurate implementation of the involved evaluation. 2. Governing system for axi-symmetric vibrations 2.1. Structural modeling Love–Kirchhoff (L–K) assumptions [8,13] are adopted in the modeling of isotropic closed circular cylindrical thin-shells featuring axi-symmetric vibrations. The geometric configuration of the shell and the coordinate system used are displayed in Fig. 1, Part I. The kinematic equations are: ur ¼ wðh; x; tÞ ¼ wðx; tÞ
ð1aÞ
uh ¼ uh ðh; x; tÞ ¼ 0
ð1bÞ
ux ¼ uðh; x; tÞ ðr RÞ
ow ow ¼ uðx; tÞ f ox ox
ð1cÞ
Based on the above adopted kinematics, the strain component err ¼ 0. Invoking, as usual that rrr ¼ 0 in the constitutive equations (see e.g., [8]), we get: E rxx 1 m exx ¼ ð2aÞ 2 rhh e m 1 1m hh err ¼
m ðexx þ ehh Þ 1m
ð2bÞ
For axi-symmetric vibration problem, the following shell stress resultants and stress couples are defined:
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
f E 2h3 w w rxx 1 þ 2hu þ mð2hÞ df ¼ ;x ;xx R 1 m2 R 3R h Z h i E 1 þ h=R E h w ð2hÞ rhh df ¼ mð2hÞu mð2hÞu þ w ln Nh þ ;x ;x 1 m2 1 h=R 1 m2 R h Z h i f 2Eh3 h u;x Mx w rxx f 1 þ df ¼ ;xx R 3ð1 m2 Þ R h
Nx
Z
2025
h
ð3aÞ
ð3bÞ
ð3cÞ
In Eq. (3b), the Taylor series expansion is used and truncated up to the second order: 3 5 1 þ h=R h 2 h h h þO 2 ¼2 þ ln 1 h=R R 3 R R R
ð4Þ
The governing equations of the magnetoelastic circular cylindrical shells and the consistent boundary conditions can be systematically derived by using the principle of virtual work in dynamic case [18]: Z
t2
ðdT dU þ dWe Þ dt ¼ 0
ð5aÞ
t1
with du ¼ dw ¼ 0
at t ¼ t1 and t2
ð5bÞ
where dT and dU denote the variation of the kinetic and strain energy, respectively, while dWe denotes the virtual work due to external forces. For the problem at hand, these terms are defined as: Virtual kinetic energy dT ¼ 2p
Z
Z
‘
h
dx ‘
h ih i q u_ fw_ ;x du_ f dw_ ;x ðR þ fÞ df
ð6Þ
h
Virtual strain energy Z ‘ 1 Nx du;x Mx dw;xx þ Nh dw dx dU ¼ 2pR R ‘
ð7Þ
Virtual work due to the magnetoelastic loads dWe ¼ 2pR
Z
‘
‘
h i r þ mx;x Þ dwj‘ x duj‘ M x dw;x j‘ þ ðQ ½qr dw þ qx du mx dw;x dx þ 2pR N ‘ ‘ ‘ ð8Þ
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In Eq. (8), the 2-D body forces qr , qx and shell moment mx are defined as: mx
Z
h
px fð1 þ f=RÞ df
ð9aÞ
pr ð1 þ f=RÞ df
ð9bÞ
px ð1 þ f=RÞ df
ð9cÞ
h
qr
Z
h h
qx
Z
h h
where, px and pr are the counterparts of 3-D body forces. In terms of the basic unknowns uðx; tÞ and wðx; tÞ, the governing equations are: E h2 m qx h2 þ u þ w w q€ u þ q€ w;x ¼ 0 du: ;xx ;xxx ;x 1 m2 R 3R 2h 3R dw:
ð10aÞ
u;xxx 3 m w 2h3 2h3 w u;x þ 2 D w;xxxx þ 2 q€ w;xx þ q€u;x ¼ 0 mx;x qr þ 2hq€ h R R R 3 3R ð10bÞ
It is noted that the terms underscored by double solid lines are associated with rotary inertia. The associated boundary conditions are: x Nx ¼ N x Mx ¼ M
du ¼ 0
ð11aÞ
or dw;x ¼ 0
ð11bÞ
or
Mx;x þ mx þ
2h3 2h3 r € ;x w q€ u¼Q 3 3R
or dw ¼ 0
ð11cÞ
r are the external loads at the shell ends (i.e., x ¼ ‘). Note that the positive x, Q x , M where N x defined in Eq. (9a) is along the negative direction of eh (see Fig. 1). From Eqs. (10) direction of M and (11), it is readily seen that the couplings between axial elongation u and flexural deformation w are entirely due to the presence of curvature 1=R. The boundary conditions corresponding to clamped, simply-supported and free ends are summarized in the following: 8 du ¼ 0; dw ¼ 0; dw;x ¼ 0 < clamped: simply-supported: du ¼ 0; dw ¼ 0; Mx ¼ 0 ð12Þ : free: Nx ¼ 0; Mx ¼ 0; Mx;x ¼ 0 When R ! 1, the preceding governing equations (10) and (11) reduce to the corresponding plate strip model. In such a case, the force–displacement relations reduce to:
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2027
Nθ Nx
Mx θ
x
Fig. 1. Shell forces in the axi-symmetric problem of circular cylindrical shells (note that the direction of Mx is along the negative h direction).
Nx ¼
2hE u;x 1 m2
ð13aÞ
Nh
2hmE u;x 1 m2
ð13bÞ
Mx ¼
2Eh3 w;xx 3ð1 m2 Þ
ð13cÞ
Therefore, the governing equations corresponding to the plate strips are: du:
E qx u¼0 u;xx þ q€ 2 1m 2h
dw:
Dw;xxxx mx;x qr þ 2hq€ w
ð14aÞ 2h3 q€ w;xx ¼ 0 3
ð14bÞ
In such a case, the governing equations (14a) and (14b) and the associated boundary conditions become completely decoupled. In the following, for the sake of numerical illustration, only the clamped–clamped boundary conditions will be considered. 2.2. Magnetoelastic loads For thin-shells immersed in an applied uniform magnetic field along the axial direction (i.e., H0 ¼ H0 ex , see Fig. 1, Part I), the induced magnetic field within the perfectly electro-conductive medium is: h ¼ rot½u H0 ¼ H0
ow er ox
ð15Þ
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Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
Therefore, the Lorentz force is: 2 f ðeÞ ¼ rot h B 0 ¼ l0 H0
o2 w er ox2
ð16Þ
In this article, only the induced magnetoelastic loads will be considered. In such a case, px ¼ 0 and pr ¼ ð1=l0 ÞH02 o2 w=ox2 . Therefore, the generalized shell forces defined in Eq. (9) now reduce to: mx ¼ 0;
qx ¼ 0;
qr ¼ l0 H02
ðþÞ o2 w ðÞ 2h þ r r rr rr ox2
ð17Þ
ðÞ in which, rðþÞ rr and rrr are the MaxwellÕs stress jumps on the upper and lower media surfaces, respectively. In the jump conditions,
f1 ðR þ h; xÞ ¼ H0
ow ; ox
f2 ðR h; xÞ ¼ H0
ow ox
ð18Þ
2.3. Solution methodology Due to the high complexity of magnetoelastic governing equations, the extended GalerkinÕs method (EGM) [11,14] is used to solve the intricate system of equations. The underlying idea of this method is to select weight functions that need only fulfill the geometric boundary conditions. As a result, the natural boundary conditions that may not be fulfilled appear as a residual in the functional which should be minimized in the GalerkinÕs sense [11]. We start from the governing equations (10a) and (10b). Construct the following variational functional dH: Z ‘ h i x Nx duj‘ ½l:h:s of ð10aÞ ð 2hÞ du þ l:h:s of ð10bÞ ðdwÞ dx þ N dH ¼ ‘ ‘ h i 2h3 2h3 x dw;x j‘ € ;x Mx;x þ mx þ ð19Þ w q€ u Qr dwj‘‘ þ Mx M ‘ 3 3R In Eq. (19), l.h.s of (10a) and (10b) denotes the left-hand side of Eqs. (10a) and (10b). It is noted that the coefficients 2h before du and the negative sign in expression dw are used to recover the functional ðdT dU þ dWe Þ=ð2pRÞ. Define the following non-dimensional parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E h ^ ðn; sÞ ¼ wðx; tÞ=h n ¼ x=‘; x0 ¼ ; s ¼ x0 t; u^ðn; sÞ ¼ uðx; tÞ=‘; w 2 3ð1 m Þq ‘2 ð20Þ Now, perform the spatial semi-discretization [2]: ^ T ðnÞ^ qu ðsÞ; u^ðn; sÞ ¼ U u
^ T ðnÞ^ ^ ðn; sÞ ¼ U w qw ðsÞ w
ð21Þ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2029
^ u ðnÞ and U ^ w ðnÞ are where, ^ qu ðsÞ and ^ qw ðsÞ are Nm 1 generalized displacement vectors, whereas U the shape function vectors with dimensions Nm 1, which are required to fulfill only the geometric boundary conditions: ^ u ð1Þ ¼ 0 ^ u ð1Þ ¼ U U ^ w ð1Þ ¼ U ^ w ð1Þ ¼ 0; U
ð22aÞ ^ 0 ð1Þ ¼ U ^ 0 ð1Þ ¼ 0 U w w
ð22bÞ
In the present paper, the following shape functions are adopted: ^ T ðnÞ ¼ fsin pn; sin 2pn; . . . ; sin Nm png U u ^ T ðnÞ ¼ cos pn þ 1; cos 2pn 1; . . . ; cos Nm pn ð 1ÞNm U w The transformed governing equation is: # " € uw uu uu ^qu ^ K K 0 M qu þ ¼0 2 ww ^ € ^ q B ð K þ K Þ K K 0 M ^ qw wu ww H1 H2 w 0
ð23aÞ ð23bÞ
ð24Þ
where, B^0 is the non-dimensionalized applied magnetic field intensity (see Appendix A for its ww , K uu , K uw , M wu , M ww and K H 2 (related to the Lorentz uu , M definition). The details of matrices M force) are summarized in Appendix B, while KH 1 , which is related to the MaxwellÕs stress jump, are defined in the following special cases. 3. Calculation of the generalized magnetoelastic load in special cases with implementation details Due to the singular integrals, infinite integral domain and highly excessive time necessary for the evaluation of some regularized kernels, special treatment is needed for the efficient computation of the generalized magnetoelastic loads. 3.1. Generalized magnetoelastic load on the plate strip " Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # Z ‘ Z ‘ ‘ ðþÞ 1 ‘2 s2 w0 ðsÞ ðÞ 22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rrr rrr dw dx ¼ l0 H0 ds dw dx p ‘ ‘2 x2 sx ‘ ‘ " Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ‘ ‘ 1 ‘2 s2 ½w0 ðsÞ w0 ðxÞ 22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ds ¼ l 0 H0 p ‘ ‘2 x2 sx ‘ # Z ‘ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘2 x2 dsw0 ðxÞ dw dx þ sx ‘
ð25Þ
In Eq. (25), the identity transformation is used to eliminate the singularity at point s ¼ x. Expressed in the modal space, the last equation is further transformed as follows:
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Z
" Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ‘ ‘ 0T ðþÞ 2 U ‘2 s2 ½U0T w 2 T w ðsÞ Uw ðxÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi dw dx ¼ l rrr rðÞ H dq ds 0 0 rr p w ‘ ‘2 x2 sx ‘ ‘ # Z ‘ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ‘2 x2 0T H 1 qw dsUw ðxÞ dxqw , B^20 dqTw K þ sx ‘ ‘
ð26Þ
in which, H1 K
qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 3 Z 1 Z 1 1 n2 ½U 0T 0T ^ ^ ðn Þ U ðnÞ ^ s Uw 1 ‘ w w s ^ 0T ðnÞ5 dn pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 dns pnU w 2 n p h n s 1 1 1n
ð27Þ
In Eq. (27), the following identity is used (see [7, p. 570]): Z
1 1
qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2s ns n
dns ¼ pn;
jxj < 1
ð28Þ
By using the standard Gauss–Chebyshev integration [5], we get:
H1 K
qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 9 3 X Z 1 1 n2 ½U Nd ^ 0T ðns Þ U ^ 0T ðnÞ < = 1 ‘ w w s ^ w ðnk Þ ^ 0T ðnk Þ U dns pnk U w : 1 ; Nd h ns nk k¼1
ð29Þ
where, nk are the locations of discrete points such that TNd ðnk Þ ¼ 0;
k ¼ 1; . . . ; Nd
ð30Þ
3.2. Generalized magnetoelastic load on the infinite cylindrical shells Z ‘ Z ‘( Z 1 h i ðþÞ ^ 0T G ^ ^ 0T U rrr rðÞ dw dx ¼ l0 H02 rr w ðs xÞ dsqw Uw ðxÞ dqw ‘
‘
2 þ p
Z
1 1
1
) h i ^ 0T U 0T w ^ ðxÞ dqw dx , l0 H 2 dqT ½KHH qw ð31Þ dsqw U w 0 w sx
where, 3
2 2
KHH h
Z 7 6 Z 1 ^ 0T ðns Þ 2 1 U 7 6 0T w ^ ^ ^ Uw ðnÞ6 Uw ðns ÞGðns nÞ dns þ dns 7 dn 5 4 p 1 ns n 1 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Z
1
^ H1 U
^ H2 U
ð32Þ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
^ðns nÞ is defined as: and the kernel G ( ) Z 1 ^ ^ 1 K ½j x jðR þ hÞ I ½j x jðR hÞ 0 0 ^ðns nÞ ^Þ ^ G | signðx þ 2 e|x^ ðns nÞ dx ^ jðR þ hÞ I00 ½jx ^ jðR hÞ 2p 1 K00 ½jx |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
2031
ð33Þ
^Þ F ðx
Next, noticing that U0 ðns Þ contains only sine-functions, then by exploiting the relationship between the Fourier integral and the inverse Fourier integral, we can dramatically simplify the first integral in the bracket of Eq. (32) as follows: Z 1 Z 1 Z 1 1 ^ ðns nÞ 0T 0T |x ^ ^ ^ ^ ^Þ e ^ dns Uw ðns ÞGðns nÞ dns ¼ Uw ðns Þ F ðx dx ð34Þ UH 1 2p 1 1 1 ^ 0T ðns Þ consists of sinðmi pns Þ functions (mi ¼ 1; . . . ; Nd ), we can convert Eq. (34) into the Since U w Fourier integral transform pairs. For example, taking an arbitrary element sinðmi pns Þ from ^ 0T ðns Þ, we get: U w Z 1 Z 1 1 ^ ðns nÞ |x ^Þ e ^ dns sin mi pns F ðx dx 2p 1 1 Z 1 Z 1 1 ^ ðns nÞ |x ^Þ e ^ dns ¼ sin mi pðns n þ nÞ F ðx dx 2p 1 1 Z 1 Z 1 1 ^ ðns nÞ |x ^Þ e ^ dðn ns Þ ¼ ð cos mi pnÞ sin mi pðns nÞ F ðx dx 2p 1 1 Z 1 Z 1 1 ^ ðns nÞ |x ^ ^ cos mi pðns nÞ F ðxÞ e dx dðn ns Þ þ sin mi pn 2p 1 1 ¼ ð cos mi pnÞIm ½F ðmi pÞ
ð35Þ
^ Þ at x ^ ¼ mi p. It is noted that in where, Im ½F ðmi pÞ denotes the imaginary part of the function F ðx Eq. (35), the real part of the function F ðmi pÞ ¼ 0. As to the second integral in the bracket of Eq. (32), using the following identity: Z 1 1 sin mi pns ð36Þ dns ¼ cos mi pn; due to mi > 0 p 1 ns n we get U2H ðnÞ ¼ fp cos pn; 2p cos 2pn; . . . ; Nd p cos Nd pngT
ð37Þ
Finally, we get H 1 l0 H 2 KHH =B^2 ¼ 1 K 0 0 2
3 Z 1 h i ‘ T T ^ ^ ^ Uw ðnÞ UH 1 ðnÞ þ UH 2 ðnÞ dn h 1
ð38Þ
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Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
^ H 1 ðnÞ is defined as where, U 8 9 K0 ½pðR‘ þh‘ Þ I0 ½pðR‘ h‘ Þ > > > > p cosðpnÞ 2 þ K 0 ½pðRþhÞ I 0 ½pðRhÞ > > > > ‘ ‘ ‘ ‘ 0 0 > > > > > > > > RþhÞ RhÞ > > K ½2pð I ½2pð < 2p cosð2pnÞ 2 þ 0 R‘ h‘ 0 R‘ h‘ = 0 0 K0 ½2pð ‘ þ ‘Þ I0 ½2pð ‘ ‘ Þ ^ UH 1 ðnÞ > > > > .. > > > > . > > > > > > R h R h > K0 ½Nm pð ‘ þ ‘ Þ I0 ½Nm pð ‘ ‘Þ > > : Nm p cosðNm pnÞ 2 þ K 0 ½N pðRþhÞ I 0 ½N pðRhÞ > ; m m 0
‘
‘
0
‘
ð39Þ
‘
^ T ðnÞ ¼ 0. Such a case corresponds to When in addition, R ! 1, from Eq. (39), it follows that U H1 the infinite plate (both in length and width), and Eq. (38) reduces to H1 K
1 ¼ 2
3 Z 1 ‘ ^ w ðnÞU ^ T ðnÞ dn U H2 h 1
ð40Þ
3.3. Generalized magnetoelastic load of finite-length circular cylindrical shells with infinitely small thickness In non-dimensional form, wx ðnÞ should fulfill the following equation: Z 1 Z 1 1 wx ðns Þ h ~^ ^ 0 ðnÞ; jnj < 1 wx ðns Þk 11 ðns nÞ dns þ w dns ¼ H0 2p n ‘ n 1 1 s
ð41Þ
with the boundary condition wx ðnÞ Z
1
wx ðnÞ dn ¼ 0
ð42Þ
1
~ where, the regularized kernel k^11 ðns nÞ is defined as " # Z 1 | 1 1 ~^ ^Þ ^ jR=‘ÞK00 ðjx ^ jR=‘Þ ^ signðx I 0 ðjx e|x^ ðnns Þ dx k 11 ðns nÞ ¼ ^ jR=‘Þ 0 2p 1 Dðjx 2 # Z " h i 1 1 1 1 0 0 ^ ^ ^ ^ I ðjxjR=‘ÞK0 ðjxjR=‘Þ sin xðns nÞ dx ¼ ^ jR=‘Þ 0 p 0 Dðjx 2
ð43Þ
It is proved that the solution of wx ðns Þ fulfilling Eq. (41) and the boundary condition Eq. (42) has the following form (see e.g., Refs. [5,15]): 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi gðns Þ; 1 n2s
where gðns Þ is a bounded function
ð44Þ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2033
wx ðns Þ can be expressed by the following Chebyshev series: 1 X 1 Ak Tk ðns Þ wx ðns Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2s k¼0
ð45Þ
where, Tk ðns Þ is the first kind Chebyshev function of order k. Then the singular integral equation (41) is converted to the following: 1 X
Z
1
Ak 1
k¼0
Tk ðns Þ ~^ qffiffiffiffiffiffiffiffiffiffiffiffi ffi k 11 ðns nÞ dns þ 1 n2s
1 X k¼0
8 9 > > < 1 Z 1 = Tk ðns Þ h qffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 0 ðnÞ dns ¼ H0 Ak w > > 2p ‘ 2 1 : ; 1 ns ðns nÞ ð46Þ
where, n 2 ð1; 1Þ. Denote G11 ðk; nÞ
Z
1 1
1 G12 ðk; nÞ 2p
Tk ðns Þ ~^ ffi k 11 ðns nÞ dns qffiffiffiffiffiffiffiffiffiffiffiffi 1 n2s Z
1
1
Tk ðn Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi s dns ¼ 1 n2s ðns nÞ
ð47aÞ
0; 1 U ðnÞ; 2 k1
k¼0 k>0
ð47bÞ
In Eq. (47b), Uk1 ðnÞ is the second kind Chebyshev function of order k 1. Since Tk ðns Þ and ~^ k 11 ðns nÞ are functions with bounded amplitudes, Eq. (47a) can be efficiently evaluated by the Gauss–Chebyshev integration scheme as follows: Nd p X ~ G11 ðk; nÞ Tk ðni Þk^11 ðnis nÞ; Nd i¼1
with TNd ðnis Þ ¼ 0; i ¼ 1; 2; . . . ; Nd
ð48Þ
Therefore, from Eq. (46), we get 1 X k¼0
"
G11 ðk; ni Þ þ G12 ðk; ni Þ Ak |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Gðk;iÞ
#
h ^ 0 ðni Þ; ¼ H0 w ‘
i ¼ 1; . . . ; Nd
ð49Þ
where, ni fulfilling TNd ðnis Þ ¼ 0 are the points where the governing equation (41) is enforced to be strictly satisfied. From the boundary condition (42), we get A0 ¼ 0. After truncating the expansion order up to Nd , we get a set of algebraic equations expressed in terms of the unknown coefficients A1 ; . . . ; ANd :
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8 9 38 A 9 0 1 > > ^ ðn Þ w > > > 1 Gð1; 1Þ Gð2; 1Þ GðNd ; 1Þ > > > > > > > 0 2 > < A2 >
ð50Þ
G
The solution of A1 ; . . . ; ANd is, therefore, 8 8 9 9 8 9 0 1 > 0 1 > > > A ^ ^ ðn Þ ðn Þ w w > > > > > > 1 > > > > > > > > > 0 2 > 0 2 > < A2 > < < = = = ^ ^ ðn Þ ðn Þ w w h h T G1 C1 CNd ¼ H0 , H0 .. . . .. .. > > > > > ‘ ‘ . > > > > > > > > > > > > > : > > ; > > :w : 0 Nd ; 0 Nd ; ANd ^ ðn Þ ^ ðn Þ w
ð51Þ
As a result, Z
‘ ‘
ðþÞ dw dx ¼ rrr rðÞ rr ¼
Z
1
1
ðþÞ T ^ d^qw ‘ dn hU rrr rðÞ rr w
qTw l0 H0 d^
Nd Z X k¼1
1
1
Tk ðnÞ ^ w ðnÞ dnAk pffiffiffiffiffiffiffiffiffiffiffiffiffi ðh‘ÞU 2 1n 2
¼ l0 H02 h2 d^ qTw
Nd X
(Z
k¼1
1
1
^ 0T ðn1 Þ U w
3
7 6 7 )6 ^ 0T ðn2 Þ 7 6U w 7 6 Tk ðnÞ ^ 7^qw pffiffiffiffiffiffiffiffiffiffiffiffiffi Uw ðnÞ dnCTk 6 7 6 2 . 7 6 1n .. 7 6 5 4 ^ 0T ðnNd Þ U w
ð52Þ
It is noted that the number of terms used in the Chebyshev series, is equal to that of the discrete points where the governing equation (41) is strictly enforced. By using the following two identities [4], the integrals in Eq. (52) can be precisely evaluated: Z
1 1
Z
1 1
Tk ðnÞ cos mi pn pffiffiffiffiffiffiffiffiffiffiffiffiffi dn ¼ 1 n2 Tk ðnÞ p; pffiffiffiffiffiffiffiffiffiffiffiffiffi dn ¼ 2 0; 1n
k=2
ð1Þ 0;
pJk ðmi pÞ;
when k ¼ 0 when k > 0
when k is even when k is odd
ð53aÞ
ð53bÞ
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2035
Finally, we get
H1 K
8 9 Jk ðpÞ > > > > > > > > ð2pÞ J > > 3 k < = X 1 ‘ k=2 ð3pÞ J k ¼ ð1Þ Ak p > > 2 h k¼2;4;6;...;2bN =2c .. > > > > d > > . > > : ; Jk ðNm pÞ
ð54Þ
3.4. Efficient evaluation of the elements G(i,j) in Eq. (50) Direct numerical evaluation of the elements Gði; jÞ in Eq. (50) is heavily time-consuming. This is due to: ~ • infinite integral domain involved in the evaluation of the kernel k^11 (see Eq. (43)), which are also highly oscillatory when x becomes large, ~ • for each element Gði; jÞ, from Eq. (48), it is readily seen that the kernel k^11 needs be evaluated Nd times. The complexity of getting the matrix G in Eq. (50) involves totally ðNd Þ3 times of eval~ uation of k^11 . Since the discrete points on which to strictly enforce the governing equation (41) coincide with those used to implement the Gauss–Chebyshev integration, we need only evaluate the ~ kernel k^11 with all the different values ðni nj Þ once for all for each configuration of the shell (R=‘ and h=‘). Then store them to a suitable data structure (in the present paper, we use array) for look-up whenever they are needed later on. Due to the anti-symmetry of the kernel ~ ~ ~^ k 11 ðxÞ ¼ k^11 ðxÞ, totally only about 1=4Nd2 times of the evaluation of k^11 are needed to get the matrix G. Further steps toward significant improvement of the efficiency include: ~ • using binary search scheme [19] to find the value of k^11 ðxÞ in the array, ^ 1, • when x 1 1 ^ jR=‘ÞK00 ðjx ^ jR=‘Þ I00 ðjx ^ jR=‘Þ Dðjx 2
! !
‘2 ^ Þ2 cn ðRx
and replacing the infinite integral equation (43) by the following piecewise integral: # Z " h i 1 a 1 1 1 ~^ ^ ðns nÞ dx ^ jR=‘ÞK00 ðjx ^ jR=‘Þ ^ sin x k 11 ðns nÞ I00 ðjx ^ jR=‘Þ p 0 Dðjx 2 cn p ‘2 sin½aðns nÞ 2 ðns nÞCi ½aðns nÞ signðns nÞ þ a R
ð55Þ
ð56Þ
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where, cn is a constant, a > 0 and Ci ðxÞ is the cosine integral function, which is defined as: Z 1 cos s Ci ðxÞ ¼ ds ð57Þ s x In the present article, a ¼ 100. In summary, the complexity of determining all the entries of matrix G reduces from ðNd Þ3 ~ integrations of k^11 to 1 2 N 4 d
~ ~ integrations of k^11 þ OðNd2 log2 Nd search Þ 14Nd2 integrations of k^11
ð58Þ
Therefore, the total speed-up is about 4cNd times faster, in which, c is the ratio of time used via ~ direct numerical evaluation of k^11 from Eq. (43) over the time used by the piecewise approximation of Eq. (56). The approximation in Eq. (58) is based on the fact that the searching time is much less than that used in the numerical integral evaluation. The numerical experience in the studied cases in the present article shows that on an average, c 10, therefore, about 1200 times of speedup has been achieved when Nd ¼ 30. 4. Test and validation In the preceding formulation of the FS model, approximations are made in three places: (i) EGM for the magnetoelastic system, (ii) evaluation of the integral equation (43) is approximated by piecewise functions (see Eq. (56)), (iii) Eq. (41) is enforced on discrete locations. The high accuracy of EGM has been tested in [16,17]. Accuracy test results of the approximation (ii) are listed in Table 1. It is readily seen that this approximation provides high accuracy with 0.2% error or less as compared to the predictions by method I. To investigate the accuracy of the approximation (iii), the following special case is used. Z 1 1 wx ðns Þ dn ¼ f ðnÞ; jnj < 1 ð59Þ 2p 1 ns n s Table 1 ~ Accuracy test of the piecewise approximation of the kernel k^11 ðnÞ ~ k^11 ðnÞ by method Ib na 0.1 0.2 0.5 1.0 1.5 1.8 1.9 2.0
)0.02115 )0.03391 )0.05622 )0.06817 )0.06642 )0.06292 )0.06159 )0.06023
~ Due to anti-symmetry of the k^11 , only n > 0 are considered. Direct numerical integration (see Eq. (43)), calculated by using MathematicaTM . c Piecewise approximation (see Eq. (56)). a
b
~^ k 11 ðnÞ by method IIc )0.02117 )0.03389 )0.05616 )0.06820 )0.06641 )0.06296 )0.06149 )0.06024
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2037
The solution of Eq. (59) in conjunction with the boundary condition (Eq. (42)) can be expanded in the following series: wx ðns Þ ¼
1 X k¼0
Nt X Tk ðns Þ Tk ðns Þ ffi ffi Ak qffiffiffiffiffiffiffiffiffiffiffiffi Ak qffiffiffiffiffiffiffiffiffiffiffiffi k¼0 1 n2s 1 n2s
ð60Þ
Substituting expression (60) into Eq. (59), invoking the identity (Eq. (47)) and the orthogonality property of the second kind Chebyshev function Uk , we get p Aiþ1 ¼ 4
Z
1
qffiffiffiffiffiffiffiffiffiffiffiffiffi f ðnÞUi ðnÞ 1 n2 dn;
i ¼ 0; 1; 2; . . . ; 1
ð61Þ
1
From the boundary condition (Eq. (42)), we get: A0 ¼ 0. For the purpose of accuracy test, f ðnÞ takes the form of sinðmpnÞ. Once Ak is solved, wx can be reconstructed from Eq. (60). It is noted that larger values of m put more strict test of the accuracy. In the present paper, f ðnÞ ¼ sinð7pnÞ is used in the test. Figs. 2–4 display the comparison of wx solved by discrete method and the synthesis with the truncated terms Nt ¼ 20, 25, 30, respectively. It is seen that when Nd ¼ Nt P 25, a very high accuracy is achieved by the discrete method. In order to validate the models developed so far and to get some insight on these models, the fundamental frequency of the magnetoelastic plate strips will be obtained. In such a case, R ! 1, which leads to 2 4 2 4 o2 w ^ ^2 ‘ o2 w ^ 3ð1 m2 Þ ‘ ðþÞ ^ 1 h ^ o4 w ow ðÞ B0 rrr rrr þ 2 ¼0 4 2 2 h 2E h 3 ‘ on os2 os on on
ð62Þ
3
2
1
0 x
-1
-2
-3 -1
-0.5
0
ξ
0.5
1
Fig. 2. Accuracy test of the discretization method (Nd ¼ Nt ¼ 20, f ðnÞ ¼ sin 7pn, solid curve––solution by expansion via Eqs. (60) and (61), dashed curve––solution by discretization via Eqs. (45) and (51)).
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1
0 x
-1
-2 -1
-0.5
0
ξ
0.5
1
Fig. 3. Accuracy test of the discretization method (Nd ¼ Nt ¼ 25, f ðnÞ ¼ sin 7pn, solid curve––solution by expansion via Eqs. (60) and (61), dotted curve––solution by discretization via Eqs. (45) and (51)).
2
1
x
0
-1
-2 -1
-0.5
0
ξ
0.5
1
Fig. 4. Accuracy test of the discretization method (Nd ¼ Nt ¼ 30, f ðnÞ ¼ sin 7pn, solid curve––solution by expansion via Eqs. (60) and (61), dotted curve––solution by discretization via Eqs. (45) and (51)).
The term underscored by the double solid line is associated with the rotary inertia. For thin plate strips (i.e., h=‘ 1) and mechanical vibrations (usually low-frequency), this term can be safely discarded. When using the magnetoelastic loading model of infinite plate (i.e., PL model), and considering the above simplification, we get
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2 3 Z 1 0 ^ ^2 ‘ o2 w ^ ^2 ‘ ^ ðn; sÞ ^ o4 w 1 o2 w w B B þ ¼0 dn s 0 0 4 2 h h p 1 ns n os2 on on
2039
ð63Þ
Similarly, when using the magnetoelastic loading model of plate strips (i.e., PS model), and considering the above simplification, we get qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 Z 1 1 n2 4 ^ ^2 ‘ o w ^ ^2 ‘ ^ ow 1 o2 w s 0 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ^ B B ðn; sÞ dn þ ¼0 ð64Þ w s 0 0 4 2 h h p 1 n2 1 ns n os2 on on In Eqs. (63) and (64), the terms underscored by single solid line are associated with the influence of Lorentz force, while the terms underscored by double solid lines are associated with MaxwellÕs stress jump at the media surfaces. In the case of clamped–clamped plate strips, after the use of the shape function UT ðnÞ ¼ fcos pn þ 1g (see Eq. (23)), we get an estimate of the fundamental frequency of the plate strip by using these two different magnetoelastic load models: ) 2 - 3 ( 2 x1 p4 ^2 ‘ h p p ð65aÞ þ - ¼ þ B0 h ‘ 3 3 x0 3 PL
2 -x1 x0 -
PS
) 3 ( 2 p4 ^2 ‘ h p þ 1:1818 ¼ þ B0 h ‘ 3 3
ð65bÞ
In Eq. (65), the subscripts PL and PS denote that the estimations are based on PL and PS magnetoelastic load models, respectively. The underscored terms by single solid line are associated with the influence of Lorentz body force. For thin plate strips (i.e., h=‘ 1), it is readily seen that they can be safely discarded. When using the shape function UT ðnÞ ¼ fcos 2:365n þ 0:133 cosh 2:365ng, which is the exact first-order eigenmode of the clamped–clamped beam, the following estimate is obtained: ) 2 - 3 ( x1 ‘ h 3:071 þ 1:1122 ð66aÞ - ¼ 31:284 þ B^20 h ‘ x0 PS
In the case of simply-supported plate strips after considering the shape function as UT ðnÞ ¼ fcosðpn=2Þg (see Eq. (23)), we get estimates of the fundamental frequency of the plate strip by using these two different magnetoelastic load models: ) 2 - 3 ( 2
p 4 x1 ‘ h p p 2 ð67aÞ þ þ B^0 - ¼ 2 h ‘ 4 2 x0 PL
2 -x1 x0 -
PS
¼
p 4 2
) 3 ( 2 ‘ h p þ B^20 þ 0:251p h ‘ 4
The results obtained from these special cases agree very well to those in literature [10].
ð67bÞ
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5. Numerical results and discussion Due to the complexities involving the exact solution of magnetoelastic loads on the circular cylindrical shells (see Part I), various approximations that attempt to simplify the calculations have been proposed. One main objective of the present study is to investigate the accuracy of various simplifications. Table 2 summarizes the geometric restrictions underlying the four magnetoelastic loads models. It is noted that the structure is still of the shell type, only the magnetoelastic loads acting on the shell are approximated by these simplified models. Characteristics of the influence of applied magnetic field on the free vibration of thin shells, and other important issues such as influence of thickness ratio h=‘, dimensionless radius R=‘ of the shell will also be investigated. Abbreviations of these models used in the following discussion are summarized in Table 1. Figs. 5 and 6 display the comparison of the first two eigenfrequencies of a circular cylindrical shell under the influence of applied magnetic fields predicted by different magnetoelastic load models. It is observed that: (i) in the total increase (3.7%) of x1 due to the presence of B^0 ¼ 0:03, the maximal difference of predictions among these four magnetoelastic load models can reach 2.5%; while for x2 , it can reach 2.5% out of the total increase of 5.8%. This implies that when the influence of applied magnetic field is strong, the difference of predictions among these four models Table 2 Summary of various simplified magnetoelastic load models Model
Abbreviation
Geometric restrictions
H 1 reference equation # K
Plate strip Infinite circular cylindrical shell Infinite plate (both length & width) Finite-length cylindrical shell
PS ISH PL FSH
R ! 1, h ! 0 ‘!1 R ! 1, ‘ ! 1, h ! 0 h!0
(29) (38) (40) (54) and (51)
PS 650
ISH PL FSH
645
1 0
640
635
630 0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 1 ( x1 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 0:5, h=‘ ¼ 0:005). Fig. 5. Comparison of x
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2041
PS ISH
680
PL FSH 670 2 0
660
650 0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 2 ( x2 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 0:5, h=‘ ¼ 0:005). Fig. 6. Comparison of x
will be very significant, (ii) among the four magnetoelastic load models, the ISH model gives the lowest prediction of the eigenfrequencies x1 and x2 , (iii) in terms of the closeness of the predictions, these four models are separated into two groups: ISH and PL models; PS and FSH models. It is scrutinized that the influence of dimensionless radius R=‘ is much less than the influence of finiteness of the length of shells. This inference can be confirmed by Figs. 7–9. With the increase of dimensionless radius (R=‘ ¼ 10 in this case), the difference of predictions between ISH and PL models or PS and FSH models almost vanishes. Figs. 7–9 also reconfirm the preceding first observation (i.e., i): the ISH or PL models predict 2.4 times of increase of x1 due to the presence of B^0 ¼ 0:03, while PS or FSH models predict 3.5 times of increase of x1 due to the 300
PS ISH 250
PL FSH 1 200 0
150
100
0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 1 ( x1 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 10, h=‘ ¼ 0:002). Fig. 7. Comparison of x
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PS ISH
600
PL 500
FSH
2 400 0
300
200
100 0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 2 ( x2 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 10, h=‘ ¼ 0:002). Fig. 8. Comparison of x
PS ISH
800
PL FSH 600 3 0
400
200
0
0.005
0.01
0.015
0.02
0.025
0.03
B0
^ 3 ( x3 =x0 ) predicted by different magnetoelastic models (R=‘ ¼ 10, h=‘ ¼ 0:002). Fig. 9. Comparison of x
presence of B^0 ¼ 0:03. For x2 , the ISH or PL models predict 5.6 times of increase, while the PS or FSH models predict 7.8 times of increase. For x3 , the ISH or PL models predict 6.3 times of increase, while the PS or FSH models predict 9.0 times of increase. Having investigated the influence of these magnetoelastic load models on the prediction of shell eigenfrequencies, we now turn to investigate the influence of applied magnetic field itself on eigenfrequencies of shells. Fig. 10 displays the variation of the first six eigenfrequencies of a shell vs. the intensity of applied magnetic field B^0 . It is readily seen that the presence of applied magnetic field plays the role of increasing the structural stiffness and that the higher the modal
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2043
Fig. 10. Influence of the applied magnetic fields on the first six eigenfrequencies of a circular cylindrical shell predicted by the FS model (R=‘ ¼ 10, h=‘ ¼ 0:002).
eigenfrequency, the more the rate of increase, implying that the applied magnetic field cannot induce, by itself, static buckling of such structures. Put it another way, the presence of applied magnetic field tends to stabilize the structure. Fig. 11 shows the influence of thickness ratio h=‘ on the eigenfrequency of magnetoelastic shells. It is observed that the applied magnetic field B^0 has more prominent influence on thinner shells. With the increase of h=‘, the influence of the presence of B^0 on x1 will significantly decrease. Fig. 12 displays the influence of dimensionless radius R=‘ on the eigenfrequency of magnetoelastic shells. When the applied magnetic field B^0 increases from 0 to 0.03 in its intensity, in the
Fig. 11. The influence of the thickness ratio h=‘ on the variation of x1 vs. B^0 (FS model, R=‘ ¼ 0:5, xc1 x0 jh=‘¼0:002 ).
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Fig. 12. The influence of the dimensionless radius R=‘ on the variation of x1 vs. B^0 (FS model, h=‘ ¼ 0:01).
case of R=‘ ¼ 10, x1 increases to 3.7 times of its elastic value, while in the cases of R=‘ ¼ 1 and 0.5, x1 increases only to about 1.1 times of the corresponding elastic values. This implies that the applied magnetic field B^0 has more prominent influence on shells featuring larger dimensionless radius. 6. Conclusions Four different magnetoelastic load models have been developed and the accuracy of the model tested. Difference of these models on the prediction of eigenfrequencies of circular cylindrical shells has been investigated. The influence of applied magnetic field, thickness ratio and dimensionless radius on the free vibration of thin cylindrical shells have also been investigated. The major conclusions are: • when the influence of applied magnetic field is strong, the difference of predictions among these four models will be very significant, • in the modeling of magnetoelastic loads, the dimensionless radius R=‘ has much less influence than the finiteness of the length of shells, • the presence of applied magnetic field tends to stabilize the structure, and by itself, the applied magnetic field cannot induce static buckling of such structures, • in the case of simply-supported plate strips, the effect of finiteness of length or width must be considered when investigating their magnetoelastic behavior. Acknowledgement Partial support from NASA Langley Research Center through grant NAG-01101 is gratefully acknowledged.
Z. Qin et al. / International Journal of Engineering Science 41 (2003) 2023–2046
2045
Appendix A. Nomenclature pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B^0 non-dimensional applied magnetic field intensity, ð 3ð1 m2 Þl0 =EÞH0 D shell flexural stiffness, 2Eh3 =ð3ð1 m2 ÞÞ E, m, q YoungÕs modulus, PoissonÕs ratio and the mass density of the shell Lorentz force vector f ðeÞ f1 ðR þ h; xÞ lr h on the outer surface of the shell f2 ðR h; xÞ lr h on the inner surface of the shell h thickness of the shell induced magnetic field outside the electro-conductive medium h applied magnetic field H0 ‘ semi-length of the shell number of the discrete points used in the Gauss–Chebyshev integration (see Eq. (48)) Nd number of the truncated terms used in expansion (see Eq. (60)) Nt number of shape functions used in semi-discretization (see Eq. (21)) Nm r, M x , Q x shell stress resultants and tress couple, see Eqs. (3a)–(3c) N pr , px body forces (per volume) within the conductive media qr , qx , mx 2-D body forces, see Eqs. (9a)–(9c) R radius of the mid-surface of the shell Tk ðnÞ the first kind Chebyshev function of order k u, w axial and flexural displacements Uk ðnÞ the second kind Chebyshev function of order k u displacement vector within the conductive media f local thickness coordinate, r R magnetic permeability of the electro-conductive media le magnetic permeability of the free space l0 le =l0 , relative magnetic permeability lr s dimensionless time, defined by Eq. (20) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi reference frequency, ð E=ð3ð1 m2 ÞqÞÞh=ð‘Þ2 x0 ^i dimensionless frequency, xi =x0 x d variation operator pffiffiffiffiffiffiffi | 1 € ðð€ Þ; ð^ ÞÞ Þ ðo2 ð Þ=o2 t; o2 ð Þ=os2 Þ 0 00 ðð Þ ; ð Þ Þ ðoð Þ=ox; o2 ð Þ=ox2 Þ 0 ðð^ Þ ; ð^ Þ00 Þ ðoð^ Þ=on; o2 ð^ Þ=on2 Þ bxc operator, which gives the greatest integer no larger than x
Appendix B. Definition of matrices in Eq. (24) Z 1 Z 1 Z 1 1 h T T ^ ^ ^ ^ ^ wU ^ 00T dn; Muu Uu Uu dn; Mww Uw Uw dn U u 3 ‘ 1 1 1
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2 Z 1 ‘ ^ uU ^ 0T dn; U Kuu 3 u h 1
2 Z 1 ‘ ^ uU ^ 0T dn; U Kuw 3m w hR 1
3 Z 1 ‘ ‘ ^ wU ^ 0T dn; U Kwu 3m u h R 1 H2 K
ww K
Z
2 2 Z 1 ‘ ‘ 00 ^ 00T ^ ^ wU ^ T dn Uw Uw dn þ 3 U w h R 1 1 1
2 Z 1 ‘ ^ wU ^ 00T dn U w h 1
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