Nonlinear phenomena in magnetostrictive elastic resonators

Int. J. EngngSci. Vol. 27, No. 12, pp. 1613-1619, 1989 Printed in Great Britain. All rights reserved

NONLINEAR

PHENOMENA IN MAGNETOSTRICTIVE ELASTIC RESONATORS

ABO-EL-NOUR UniversitC

Abstract-The purpose the problems in obtains thus in amplitude anisochronism the basis of

Pierre

0020-7225/89 $3.00 + 0.00 Copyright @ 1989 Pergamon Press plc

and GERARD

ABD-ALLA?

A. MAUGIN

et Marie Curie, Laboratoire de ModClisation en MCcanique, Jussieu, 75252 Paris Cedex 05, France

Tour 66, 4 place

work considers the nonlinear vibrations in magnetostrictive resonators. To that Galerkin method, which is frequently employed to solve nonlinear wave propagation finite regions (e.g. resonators), is used jointly with a multiple-time scale technique. One the displacement at the first partial mode, placed in a bias magnetic field, as altered both and velocity. Moreover, the change of velocity in the first partial mode provides the due to magnetostrictive couplings at the second order. The problem is investigated on previously given, rotationally invariant nonlinear equations.

1. INTRODUCTION

The present work offers a study of the nonlinear vibrations in magnetostrictive resonators. To that purpose the Galerkin method is used jointly with a multiple-time scale technique in the manner of Googerdy and Peddleson [l-3]. There is exhibited a time-dependent defect called anisochrunism (relative change in velocity of the fundamental frequency component) due to magnetostrictive couplings at the second order and it involves a time effect which was overlooked in a previous work [4]. Magnetostriction is the production of a stress eve12 in the applied magnetic field. It is, per se, a nonlinear effect in contradistinction, say, with piezoelectricity [2,4]. It is the only one type of electromagnetomechanical couplings and nonlinear effects which can be either useful or undesirable in resonators, transducers and S.A.W. systems using magnetostrictive couplings [4-71. The effect of magnetostriction has been considered from the point of view of second order hyperbolic wave equations with accompanying initial and boundary conditions, and this mathematical problem is interesting as a model problem, motivated by natural phenomena, in applied mathematics, applied physics and electrical engineering. Before focussing on these nonlinear effects, we need recall the basic nonlinear, rotationally invariant continuum equations of magnetostrictive insulators that we have developed elsewhere [8]. This work presents fundamental differences with other studies concerned with electromechanical devices [2,4,6] and this materializes in two facts: (i) the Galerkin method accompanied by a multiple-scale technique gives the displacement at the first partial mode as altered both in amplitude and velocity; (ii) the change of velocity, in the first partial mode, differs from its analogue in [4, paragraph 91, by a time-varying quantity of small order and this is due to the use the slow-time variable while in the above mentioned study the treatment was performed by using a straightforward-expansion technique.

2.

EQUATIONS

OF

NONLINEAR

MAGNETOELASTICITY

We consider elastic insulators in the framework of quasi-magnetostatics. Nonlinearities may have elastic, magnetic and mixed origins. Of necessity all equations are written down in the material description so that all boundaries are fixed even for nonlinear strains. The equations consist of the equations of motion, Maxwell’s equations and the associated boundary conditions together with the nonlinear, coupled, stress and magnetic-induction constitutive equations. We have thus (for a domain Q, of regular boundary a& equipped with outward unit normal N) [8]: t On leave from Sohag

University,

Egypt. 1613

1614

A. N. ABD-ALLA

and G. A. MAUGIN

Field equations in DO:

(2.1) Boundary conditions at aDO: &UTIKiD= 0,

~&%ll=

0,

Ml=0

(2.2)

Constitutive equations

+6

KRMNPQAB"M,NuP,QuA,B

+ B'LRMN~M~N

(2.3)

+B'YCRMNPQ~M~NUP,Q)

(2.4)

where upper and lower case Latin indices refer to material and present Cartesian coordinates, respectively. The symbols po, Ui = %i -XK6,, Zi> 2~; %K, $, and T’,i denote, respectively, the matter density at KR (the reference configuration), the elastic displacement, the nonlinear motion, the “Lagrangian” magnetic induction and field, the magnetostatic scalar potential and the total Piola-Kirchhoff nonsymmetric stress tensor, 8Ki is a shifter. CKLMN and pKL are tensors of elasticity coefficients of second order and magnetic permeabilities, xKLMNis a tensor of magetic susceptibilities of the fourth order, while Y&,MNPQ and G;LMNpQAB are effective tensors of elasticity coefficients of the third and fourth orders, B’lfCRMN and BgLMN are effective tensors of magnetostriction coefficients of the first order and B’);(RMNpQand AOKRMNpQ are effective tensors of magnetostriction of the second order. The terms of third order jointly in displacement and magnetic field are neglected in the following analysis for the sake of simplicity. The material has been assumed centrosymmetric and, obviously, does not exhibit any natural piezomagnetism (a seldom encountered coupling in any case).

3. GOVERNING

EQUATIONS

We shall invoke the monomode hypothesis [2] and consider a medium of finite extent [length normalized to the interval (0, n)] in one of its dimensions, plane waves travelling back and forth between these two limiting surfaces (Fig. 1). This is called a resonator if the vibration is adapted to the thickness it. This allows us to extract from the system (2.1)-(2.2) a nonlinear model problem (for the one-dimensional displacement u-not necessarily longitudinal-and the magnetic scalar potential @) in adimensional form as u, -

UXX(l+ 2Y%)

@XX - Bz(buA

(3.1)

= m3x

o
= 0,

(3.2)

u(0, t) = u(n, t) = 0, U(X, 0) = sin x, II& - Bz(4VG)ll= 0 &=O uH(oqj=0

for at

(3.3)

u&X, 0) = 0

(3.4)

at

(3.5)

x=0,

x=n

xn

(3.6)

x = 0, x = Ed

(3.7)

in spacetime (x, t), where y accounts for elastic nonlinearities and fir and fi2 for magnetostriction. We have chosen (3.4) as initial conditions. Small parameters c1 and cZ can be introduced

Nonlinear phenomena

in magnetostrictive

elastic resonators

1615

X

Fig. 1

in such a way that (see [4]) El =

yu, =

A= E&4,

u, = 0(10-a) (3.8)

&2=

P 24

=x

=

O(h),

it is clear that .sl and .s2 are of the order O(10-4-10-5).

4. TREATMENT

BY

MEANS

OF THE

GALERKIN

METHOD

The Galerkin method is frequently employed to solve nonlinear wave propagation problems in finite regions (in our case, resonators). This essentially consists in assuming the spatial variation (such as the linear mode shapes), using the orthogonality properties of the linear mode shapes and obtaining the temporal behavior. Then the latter equations are solved by using a perturbation technique such as the one of multiple scales. We assume Galerkin representations for u and $I in the form u(x, t) = jJ fn(f)sin u n=l

(4.1) &(x, t) = - H(O) + 5 Qn(t)cos U n=l

Substituting from (4.1) into (3.1) and (3.2) leads to + m’f,)sin

2 kak sin kx

mx = - 2fi,H”)

k=l m -

Y m$, kzl km2fmf,[sin(k

m cc + 8, c c kGmak[sin(k m=l k=l

+ m)x - Wk

-

mbl

- m)x - sin(k + m)x]

(4.2)

A. N. ABD-ALLA

1616

7 k% sinkx =

and G. A. MAUGIN

P2( H(O) s,m*f, sin mx

-

m

+i ? m

m

z

m2@kfm[sin(k - m)x - sin(k + m)x]

lk-1

*cc

m

- i z,

,

zI mkakf,[sin(k

- m)x + sin(k + m)x])

(4.3)

Multiplying (4.2) and (4.3) by sin nx and integrating over (0, n) on account of the orthogonality properties of the basis {sin nx}, leads to &+

n2fn =-

2fi,H’“‘n@,,

m2fmKm + ~lfm+n+ (m - n)(fn-m-fn-,)I

+ Yi m=l

+

815

%J(m

+

n)@

m+n

+

(m

-

~)(~n-m

-

m=l

@)m-dl

(4.4)

and n@,, + /32H(o)n2f, =; /32 2

m=l

rnf,[(m

+ n)@m+n + (n - m)+,+,]

- g B* mT, ~*fmPm+,

- RI-,

-

%-ml

(4.5)

Introducing now the slow time as q = st while the fast one is 5 = t, then we can write

d=d+,d dt

ag

a*

*

a$

az

a2

$=-•+2Eac

ag aq

+s*aq2

(4.6)

and seek a solution for fn and Q,, in the form of asymptotic expansions f”(E9 77) = ElfnO(& 7) + Gfnl(E, 7) + * * *

(4.7) %(E> rl) =

&*@no(S,

rl)

+

dfnl(E9

rl)

+

* * *

where we recall that s1 and .s2 are of the order of (10-4-10-5) and s1 is related to the nonlinear elastic coefficients while s2 is related to the magnetostriction coefficients. Substituting (4.6) and and (4.5) and equating coefficients of like powers of E = s1 = .s2, we obtain

f”. = n2fno = 0 @,,o+ /?znH(o)f,o = 0

(4.8)

At order two in E = q = Ed: x1+

n2fnl = - 2j31H(“)n@~o - 2a&f,.

m=l

n@,,+ B2H(“)n2f,l

= ; h(

-

Zl

i

m=l

mfmo[(m + n)@,(m+njo - (m - n)%~~%-n~ol

~*fmoP’(m+n)o - %a-“)O - %-NO)}

(4.9)

The general solution of equation (4.8), is fno(5, rl) = A,(rl)cos(nS)

+ &(rl)sin(nE)

(4.10)

Nonlinear phenomena in magnetostrictive elastic resonators

Introducing,

instead of

A,(q)and B,(q), the amplitudes C, and phase angles a;, by

A,(V) = C,(rl)cos cu,(tl), then,

1617

B,(V) = C&)sin

(4.11)

Qrl),

on account of (4.11) we can write (4.10) in the following form: -

4

(Pno= - B*H(0)nCn(rl)cos[a;l(rl)

-

ME,

rl) = C,(rl)cos]cu,(rl)

(4.12)

Substituting from (4.12) into (4.8)2 yields

4

(4.13)

Substituting from (4.12) and (4.13) into (4.9), and noting that the coefficients of cos(nQ and sin(n5) on the right-hand side of the resulting equation should vanish in order to eliminate secular terms (secularity condition) yields the governing equations for the An'sand B,'s or, equivalently, for Cn’s and cu,‘s. We obtain thus: Ch sin(a;, - n5) + C,(cuL + /Q32H2(0)n2)COS(% - nf) -&k

m”C,{(n

+ m)G+,[cos(G

+ % - cu,+,)

+ cos(a;, + cu,,, - (2m + n)zj)] + (m - n)[C,_,{cos(n5 + cos(cu, - an_, - (2m - n)Q} - C,_,{cos(nS + cos(cu, + mm-, - (2m -

- Lu, - Ly,-,)

- cu, + %-,) (4.14)

~N3>1~ =0

For nc - q, = lt/2, we obtain the following equations for the C,,‘s:

CA -

*z, km2Cm{(m +n)C,+,[sin(cu,+ + (m - n)C,_,[sin(cu,

a;, - ~+,)l

- cu, - (u,-,)I - (m - n)C,_,[sin(cu,

- a;, + an_,)]}

=0

(4.15)

and for a;, = n5 C,(a:,+ /r3J&H(")2n2) - m&$n2cm{(n

+ m)C,+,[cos(~"

+ (Ym - a;,-m)]

Gt-,)I - (??I- n)C,_,[cos(a;,- am+ am_,)]}= 0 - (m - n)C,+m[cos(a;, - %I -

5. TWO-MODE

GALERKIN

(4.16)

SOLUTION

Here we retain only Cr, C2, (Y~and m2 in equations (4.15) and (4.16) corresponding and n = 2. Then equations (4.15), on account of cu, = (nZj - ~r/2), read c;+$c,c,=o c;+:=o

to n = 1

(5.1) (5.2)

and from equations (4.16) on account of cu, = ng we obtain a; -;

c2 + bJ32H(0)2 = 0

C,((Y; + 4fi$2H(0)2) + & CT = 0 where (’ = dldv).

(5.3) (5.4)

A. N. ABD-ALLA

1618

and G. A. MAUGIN

Finally, and on account of (3.4), the solution of equations (5.1) and (5.2) are (cf. [2], pp. 88) Cl = sech i 0 C,=itanh

i

(5.6)

0

Substituting (5.6) into (5.3) yields o1 = In cash i 0 where the constant of integration into (5.4) we then obtain

- 6,~~~2(o)~ + Ci

(5.7)

is equal to zero when rl= 0 and crl = 0. From (5.5) and (5.6)

oZ=ln[coth(f)]

- 4fir&H2(0)?7 + cz

(5.8)

where C2 is a constant of integration which must be determined when t > 0. Substituting from (5.5), (5.7) and (5.12) into (4.1), we can get, for the partial mode of the first order: rl fir, = sech - cos $ + ~~~~~‘(‘)~ - In cash 12 sinx 08 [ ( 8 !I

(5.9)

and this can be rewritten in the form a,,=

V sech - cos[t(l+ 08

dAc) - lncosh

)I

21 sinx (8

(5.10)

where we have set s, = ?l&H2(0)

(5.11)

which is the so-called anisochronism of the resnoator. Considering the expansion In cosh(r]/8) and neglecting the very small terms of the fourth order in the small parameter we rewrite (5.10) as n ulo = t sech a sin(x - 7) + sin@ - i), 0

of E,

(5.12)

where (5.13)

6. CONCLUSION It is clear that the displacement at the first partial mode is changed in both its amplitude and velocity, The change of velocity occurs by the small quantity aA = BAc = (c&3&(0)2) which is the alteration in the fundamental mode of vibrations of the resonator resulting from the nonlinear magneJoe1astic properties of the body. But in this case we find the change of velocity by the quantity 6, and an additional term which is a function of time and of order higher than c2. We note here a change of velocity which differs from the one obtained in a previous study ([4] Paragraph 9) by a factor E. However, this is normal because of the use in the present study of the slow time variable Q = Et while in the above-mentioned study we used a straightforwardexpansion technique. The time-dependent contribution did not appear in that simple approach. An anisochronism due solely to nonlinear elastic properties can be exhibited if elasticity of the fourth order is retained (see [9] for the approach using a straightforward expansion, also [5]).

Nonlinear phenomena

in magnetostrictive

elastic resonators

1619

REFERENCES [1] A. GOOGERDY and J. PEDDLESON JR, An evaluation of the Galerkin method for the solution of nonlinear wave-propagation problems. Commun. to the 21s~ Society of Engineering Science Meeting, Blacksburg, Va (October 1984). [2] G. A. MAUGIN, Nonlinear Electromechanical Effects and Applications-A Series of Lectures. World Scientific, Singapore (1985). [3] A. H. NAYPEH and D. T. MOOK, Nonlinear Oscillations. Wiley, New York (1979). [4] A. N. ABD-ALLA and G. A. MAUGIN, Harmonic generation and anisochronism in magnetostrictive materials. ht. J. Solids Struct., 25, 683-705 (1989). [5] G. A. MAUGIN, In Electromagnetic Znferacfions in Elastic solids (Edited by H. PARKUS), pp. 243-324. Springer, Vienna (1979). [6] V. M. RISTIC, Principles of Acoustic Devices. Wiley-Interscience, New York (1983). [7] A. N. ABD-ALLA and G. A. MAUGIN, Linear and nonlinear surface waves on magnetostrictive substrates. Eur. J. Mech. A9, in press. (81 A. N. ABD-ALLA and G. A. MAUGIN, Nonlinear magnetoacoustic equations. J. Acoust. Sot. Am. 82, 1746-1752 (1987). [9] M. PLANAT, G. THEOBALD and J. J. GAGNEPAIN, Propagation nonlineaire d’ondes elastiques dans un milieu anisotrope-II. Ondes de Surface. L’onde Electrique 60, 61-67 (1980). (Received

12 June 1989)