A Structural-Magnetic Strain Model for Magnetostrictive Transducers

A STRUCTURAL-MAGNETIC STRAIN MODEL FOR

MAGNETOSTRICTIVE TRANSDUCERS

Marcelo J. Dapino

Department of Aerospace Engineering

and Engineering Mechanics

Iowa State University

Ames, IA 50011

marcelo [email protected]

Ralph C. Smith

Center for Research in Scienti c Computation

Department of Mathematics

North Carolina State University

Raleigh, NC 27695-8205

[email protected]

Alison B. Flatau

Department of Aerospace Engineering

and Engineering Mechanics

Iowa State University

Ames, IA 50011

[email protected]

Abstract

This pap er addresses the mo deling of strains generated by magnetostrictive transducers in re-

sp onse to applied magnetic elds. The measured strains are dep endent up on b oth the rotation of

moments within the material in resp onse to the eld and the elastic prop erties of the material.

The magnetic b ehavior is characterized through the consideration of the Jiles-Atherton mean eld

theory for ferromagnetic hysteresis in combination with a quadratic moment rotation mo del for

magnetostriction. The incorp oration of elastic prop erties is necessary to account for the dynamics

of the material as it vibrates. This is mo deled through force balancing which yields a wave equation

with magnetostrictive inputs. The validityof the resulting transducer mo del is illustrated through

comparison with exp erimental data. i

1 Intro duction

The phenomenon of magnetostriction is characterized by the changes in shap e which o ccur in cer-

tain materials when the materials are sub jected to magnetic elds. For rare-earth alloys such as

Terfenol-D Tb Dy Fe , the generated strains and forces are suciently large to prove advanta-

0:3 0:7 1:9

geous in transducer design. Initial investigations have demonstrated the utilityof such transducers

in applications ranging from ultrasonic transduction to vibration control in heavy structures.

This pap er addresses the mo deling of strains generated by magnetostrictive materials when em-

ployed in transducer design. To illustrate, we consider the prototypical broadband transducer de-

picted in Figure 1 and detailed in [1 ]. While transducer design will vary according to sp eci c require-

ments, this design is typical for control applications and illustrates the various physical comp onents

whichmust b e mo deled to fully utilize the magnetostrictive actuator capabilities. The primary com-

p onents of the actuator consist of a cylindrical Terfenol-D ro d, a wound wire solenoid, an enclosing

p ermanent magnet and a prestress mechanism. The ro d is manufactured so that magnetic moments

are primarily oriented p erp endicular to the longitudinal axis. The prestress mechanism increases

the distribution of moments p erp endicular to the ro d axis and allows the transducer to b e op erated

in compression. Application of currentto the solenoid then pro duces a magnetic eld which causes

the moments to rotate so as to align with the eld. The resulting strains and forces provide the

actuator capabilities for the transducer. The capability for attaining bidirectional strains and forces

is provided by a magnetic bias generated by either the surrounding p ermanent magnet or an applied

DC current to the solenoid.

For control applications, it is necessary to accurately quantify the relationship b etween the current

I t applied to the solenoid and the strains et generated by the transducer. This necessitates

mo deling the electric, magnetic, mechanical and thermal regimes within the system. While all four

regimes are fully coupled, we fo cus here on the magnetic and mechanical asp ects of the system with

nearly constant temp eratures maintained to reduce thermal e ects.

To illustrate the nature of the magnetic and mechanical phenomena, exp erimental data collected

from the transducer depicted in Figure 1 is plotted in Figure 2. In Figure 2a, it is observed that the

relationship b etween the input magnetic eld H and magnetization M is nonlinear with signi cant

saturation and is irreversible due to hysteresis. These e ects must be incorp orated when mo deling

the magnetic regime. The relationship b etween the magnetization M and strain e, plotted in Fig-

ure 2b, also exhibits hysteresis and nonlinear features whichmust b e mo deled when characterizing

the mechanical prop erties of the system. An imp ortant feature of the magneto elastic mo del consid-

ered here is that it incorp orates the observed hysteresis in the strain whereas previously considered

mo dels yielded single-valued strain outputs.

Wound Wire Solenoid

Spring Washer Compression Bolt Terfenol-D Rod End Mass

Direction of Rod Motion

Permanent Magnet

Figure 1. Cross section of a prototypical Terfenol-D magnetostrictive transducer. 1

Initial mo dels quantifying the magnetomechanical coupling were based on the linear constitutive

piezomagnetic equations

e = s + d H 1a

B = d + H 1b

which are derived from thermo dynamic principles in combination with empirical laws. In these re-

lations, e and denote the longitudinal strain and axial stress in the material while s denotes

the mechanical compliance at a xed eld strength H . Moreover, B and resp ectively denote the

@B @e

j and d j are magneto e- magnetic ux and p ermeabilityat constant stress while d

H 33

@H @

lastic coupling co ecients. It is noted in 1a that the generated strains are dep endent up on b oth

the elastic prop erties of the material mo deled by the term s and magnetic inputs mo deled by

d H . Equation 1b mo dels the converse magnetostrictive e ect in which magnetic ux is gener-

ated by stresses in the material. This latter prop erty provides the magnetostrictive materials with

their sensor capabilities.

While the linear mo del 1 is commonly employed in magnetostrictive transducer applications, the

relations are accurate only at low op erating levels. They do not provide mechanisms for incorp orating

the hysteresis and nonlinearities observed in the data in Figure 2 at higher drive levels and will b e

highly de cient in such regimes. For example, the p ermeability is not only nonconstant for the

data, but is in fact a multivalued map dep ending on b oth H and . As detailed in [2 , 3], the

H H

assumption of a constant Young's mo dulus E and corresp onding compliance s is also invalid for

large eld uctuations, and a variable Young's mo dulus E H; and compliance sH; must be

employed to attain accurate mo dels.

There are numerous approaches for extending the magnetomechanical term d H in 1a to in-

clude the nonlinear dynamics and hysteresis observed at mo derate to high drive levels. However,

most previous investigations have fo cussed on sp eci c magnetic or magnetostrictive comp onents of

the system and few results are currently available which address the coupled magneto elastic prop er-

ties of magnetostrictive materials. For the magnetic regime, mo deling techniques include microme-

chanical characterizations [4 ], phenomenological and Preisach approaches [5, 6, 7], the inclusion of

sp eci c nonlinear e ects [8 , 9], and domain theory based up on mean eld equilibrium thermo dy-

namics [10 , 11, 12 ]. The mo deling of strain e ects due to the magnetostriction has received less

attention and is less develop ed than the theory for magnetization. Current magnetostriction mo dels

are typically based up on either energy-based theories which quantify the interaction b etween atomic

moments in a crystal lattice [13 , 14, 15 , 16 ] or p olynomial expansions constructed to quantify the

phenomenological b ehavior of the magnetostriction [17 , 18 ]. With suitable assumptions, b oth ap-

proaches yield mo dels in which the magnetostriction is characterized in terms of even p owers of the

magnetization such a relation can b e observed in the exp erimental data of Figure 2. To extend 1a

to a nonlinear mo del whichcharacterizes strains in terms of input elds, it is necessary to quantify

the coupled magnetic, magnetostrictive, and elastic prop erties of the material. Certain asp ects of

this problem are considered in [16 , 19 , 20] for magnetostrictive materials and [21 ] for electrostric-

tives. Mo dels and corresp onding numerical metho ds appropriate for quantifying strains generated

by magnetostrictive transducers in general control applications are still lacking, however, and it is

this problem whichwe address here.

To mo del the relationship between the input current I t and output strains et, we consider

the magnetic, magnetostrictive and elastic comp onents in the system. To mo del the rst, the Jiles-

Atherton mean eld theory for ferromagnetic hysteresis is mo di ed to provide an energy-based re-

lationship between the current I t applied to the solenoid and the resulting magnetization M t. 2

A quadratic moment rotation mo del then yields the output magnetostriction t. This provides

a nonlinear and hysteretic analogue to the linear term d H in 1a. As demonstrated in [22 ], the

magnetostriction provides adequate ts to exp erimental strain data at low to mo derate drive levels.

At high input levels, however, it is inadequate since it incorp orates only active contributions to the

strain and neglects material or passive strain e ects. To mo del these e ects, force balancing is used

to derive a dynamic PDE mo del quantifying the ro d dynamics. This PDE mo del has the form of

a wave equation with magnetostrictive inputs and b oundary conditions which mo del the prestress

mechanism and mechanical return. The solution to this system provides the ro d displacements and

corresp onding total strains. A comparison b etween the strain relations employed in this mo del and

the linear relation 1a is provided in Section 4.

Due to the generality of the PDE mo del, it is not p ossible to obtain an analytic solution sp ecifying

the ro d displacements. To address this issue, we present in Section 5 appropriate numerical metho ds

for approximating the spatial and temp oral comp onents of the PDE mo del. We consider a Galerkin

nite element discretization in space which reduces the PDE to a matrix ODE system whichevolves in

time. The dynamics of the ODE system are approximated through a nite di erence discretization to

obtain a discrete time system having the measured current to the solenoid as input. The validity of the

mo del and approximation metho d are illustrated in Section 6 through comparison with exp erimental

data. It is demonstrated that the mo del characterizes the inherent magnetic hysteresis and accurately

quanti es the strains and displacements output by the transducer.

5 −3 x 10 x 10 1.2

1 4

0.8 2

0 0.6 Strain (e)

Magnetization (M) −2 0.4

−4 0.2

−6

0 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 8 4 5

Magnetic Field (H) x 10 Magnetization (M) x 10

a b

Figure 2. Relationship in exp erimental data b etween a the magnetic eld H and the magnetization

M , and b the magnetization M and the generated strains e.

2 Magnetization and Magnetostriction Mo dels

The magnetization and magnetostriction mo dels whichwe employ are based up on domain and do-

main wall theory for ferromagnetic materials. In ferromagnetic materials such as Terfenol-D, mo-

ments are highly aligned in regions termed domains at temp eratures b elow the Curie p oint. The

transition regions between domains are termed domain walls. Magnetization in such materials can

then be describ ed through quanti cation of domain con gurations while magnetostriction can be

characterized through the determination of the deformations which o ccur when moment con gura-

tions change. 3

2.1 Magnetization Mo del

The mo del which is employed for the magnetic comp onent of the system is based up on the ther-

mo dynamic mean eld theory of Jiles and Atherton [10 , 11 , 12 ]. In this approach, hysteresis-free

anhysteretic, irreversible and reversible comp onents of the magnetization are quanti ed and used

to characterize the total magnetization generated by an input magnetic eld. The anhysteretic mag-

netization M is attributed to moment rotation within domains and is completely reversible. Such

magnetization curves are rarely observed in lab oratory materials, however, due to the presence of

defects or second-phase materials e.g., Dysprosium in Terfenol-D which provide minimum energy

states that imp ede domain wall movement and subsequent bulk moment reorientation. These in-

clusions or defects are often referred to as pinning sites. The e ects of pinning on domain wall

movement is quanti ed through the theory of Jiles and Atherton through consideration of reversible

M and irreversible M comp onents of the magnetization. For low eld variations ab out an equi-

rev ir r

librium level, the magnetization is reversible since the domain walls bulge but remain pinned at

the inclusions. At higher input levels, the walls attain sucient energy to break the pinning sites

move out of the minimum energy state and intersect remote pinning sites. This leads to an irre-

versible change in magnetization and provides a signi cant mechanism for hysteresis. The reader is

referred to [10 , 13 , 23 ] for additional details and discussion of other exp erimental phenomena, suchas

Barkhausen discontinuities, which are attributed to domain wall e ects. This approachwas initially

employed in [22 , 24 , 25 ] to mo del magnetostrictive transducers. We summarize here p ertinent details

and indicate extensions from the original mo del.

To quantify M ;M and M , it is necessary to rst determine the e ective eld H which

an rev ir r

ef f

acts up on magnetic moments in the Terfenol ro d. As detailed in [10 , 11 ], H is dep endent up on

ef f

the magnetic eld generated by the solenoid, magnetic moment interactions, crystal and stress

anisotropies, temp erature and the transducer architecture e.g., end e ects. In [10 , 22], it is il-

lustrated that for large prestresses, stress anisotropies dominate crystalline anisotropies; hence for

this mo del, crystalline anisotropies are neglected. Under the assumption of xed temp erature and

quasi-static op erating conditions, the e ective eld is then mo deled by

H t; x=Ht; x+ M t; x+H t; x

ef f

where x denotes the longitudinal co ordinate. Here H is the eld generated by a solenoid with n

turns p er unit length, M quanti es the eld due to magnetic interactions between moments, and

H is the eld due to magneto elastic domain interactions. The parameter quanti es the amountof

domain interaction. For the prestress mechanism under consideration, it is demonstrated in [22 ] that

the approximation H = M provides an adequate average of the stress contributions to the

e ective eld. Here and M resp ectively denote the saturation magnetostriction and magnetization

s s

while is the free space p ermeability. The magnetic interactions and stress co ecient can then b e

whichmust b e exp erimentally determined for a combined into the single co ecient = +

given system.

Empirical studies have indicated that under a variety of op erating conditions, a reasonable ap-

proximation to the e ective eld is provided by

H t; x=nI t'x+ M t; x 2

ef f

where I t is the current to the solenoid and 'x is an empirically-determined function which

incorp orates transducer anomalies, such as end e ects, that pro duce nonuniform eld characteristics

along the length of the ro d. It should b e noted that while the expression 2 is time-dep endent, it

must be restricted to low frequencies since the present mo del do es not incorp orate AC losses. The

extension of this mo del to incorp orate eddy current losses is under investigation. 4

For a computed e ective eld H , Boltzmann statistics are used to quantify the anhysteretic

ef f

magnetization in terms of the Langevin function

H t; x

ef f

: 3 coth M t; x=M

an s

a H t; x

ef f

N k T

The constant a = , where k is Boltzmann's constant, N denotes the domain density and

k T represents the Boltzmann thermal energy, is treated as a parameter to b e identi ed since N is

unknown.

As detailed in [10 , 11 ], quanti cation of the energy required to break pinning sites yields the

expression

@M dI M t; x M t; x

ir r an ir r

t; x= 'x 4

ir r

@t dt

k [M t; x M t; x]

an ir r

for the time rate of change of the irreversible magnetization curve. The constant k has the form

hpih" i

k = , where hpi is the average densityof pinning sites, h" i is the average energy for 180

2m 1c

walls, c is a reversibility co ecient, and m is the magnetic moment per unit volume of a typical

domain. The parameter k provides a measure of the average energy required to break pinning sites

and is also treated as a parameter to b e estimated since hpi;c and h" i are unknown. The parameter

dH dH

is de ned to have the value +1 when > 0 and 1 when < 0 to guarantee that pinning

dt dt

always opp oses changes in magnetization.

The reversible magnetization quanti es the degree to which domain walls bulge b efore attaining

the energy necessary to break the pinning sites. To a rst approximation, the reversible magnetization

is given by

M t; x=c[M t; x M t; x] 5

rev an ir r

see [11 ]. The reversibility co ecient c can b e estimated from the ratio of the initial and anhysteretic

di erential susceptibilities [12 ] or through a least squares t to data.

The total magnetization is then given by

M t; x=M t; x+M t; x 6

rev ir r

where M and M are de ned in 4 and 5 and the anhysteretic magnetization is given by 3.

ir r rev

For implementation purp oses, it is necessary to numerically integrate the expression 4 to obtain

M . For the results in Section 4, this was accomplished via Euler's metho d. If higher accuracy is

ir r

required, metho ds such as a trap ezoid rule or Runge-Kutta metho d can b e employed.

2.2 Magnetostriction Mo del

The second magnetomechanical comp onent to be mo deled is the deformations which o ccur when

moment con gurations are altered by an applied eld H . These deformations are typically quanti ed

through either an energy formulation [13 , 14 , 15 , 16 ] or a phenomenological series expansion involving

even p owers of the magnetization [17 , 18 ].

In the rst case, general relations quantifying the material deformations are obtained through the

minimization of various energy functionals. For example, one choice is the total energy expression

E = E + E + E 7

mag anis

where the magneto elastic energy E quanti es the interactions b etween atomic magnetic moments

mag

in a crystal lattice, E denotes the elastic energy, and E is the crystal anisotropy energy. As de-

anis

tailed in [13 , 15 ], minimization of 7 yields a general expression for the anisotropic magnetostriction. 5

The situation is simpli ed in the regime considered here since the magnetic moments are essentially

p erp endicular to the applied eld due to the manner of ro d solidi cation and the compression pro-

vided by the prestress mechanism. In this case, energy minimization yields the isotropic single-valued

relation

M t; x 8 t; x=

between the magnetization M and magnetostriction .

A second approach for mo deling the magnetostriction is to employ the symmetry ab out M =0

to formulate a series expansion

t; x= M t; x 9

i=0

which empirically relates the magnetization and magnetostriction [17 , 18 ]. The series is typically

truncated after i =1 or i = 2 to obtain a mo del which can b e eciently implemented. Note that the

constant term yields elastic strains while i = 1 yields the quadratic term obtained in 8 through an

energy formulation.

The use of the quadratic expression 8 or a truncation of 9 in a transducer mo del requires

the identi cation of the physical constants M ; or the empirical constants ; ; ; . As will

s s 0 1 N

be observed in the exp erimental results of Section 5, the saturation magnetization M varies little

between samples of Terfenol-D, and values identi ed for the transducer are very close to published

material sp eci cations. The saturation magnetostriction exhibits more dep endence on op erating

conditions due to its dep endence on the initial orientation of moments; hence it must b e estimated for

each transducer con guration. For the remainder of this discussion, the quadratic magnetostriction

mo del 8 is employed.

3 Strain Mo del for the Terfenol Ro d

The expression 8 quanti es the magnetostriction which o ccurs when moments within the material

reorient in resp onse to an applied eld. This provides a generalization of the term d H in 1a to

accommo date the nonlinear dynamics inherentto the material at mo derate to high drive levels. It

ignores, however, the elastic prop erties of the material which are quanti ed in 1a by the term s .

In this section, we build on prior work to address this issue through consideration of a PDE mo del

for the Terfenol ro d which employs the eld-induced magnetostriction t; x as input. This provides

a partially coupled mo del which incorp orates b oth structural dynamics and magnetic hysteresis.

For mo deling purp oses, we consider the Terfenol ro d, prestress mechanism and end mass from

the transducer depicted in Figure 1. The ro d is assumed to have length L, cross-sectional area A

and longitudinal co ordinate x. The density, Young's mo dulus and internal damping co ecient are

denoted by ; E and c , resp ectively. The left end of the ro d x = 0 is assumed xed while the right

end is constrained by the spring washer which is mo deled by a linear translational spring having

sti ness k and damping co ecient c . It is noted that due to the compression b olt and spring

L L

washers, the ro d is sub jected to a prestress . Finally, the attached end mass is mo deled by a p oint

mass M .

The displacements of the ro d at a p oint x and time t can be sp eci ed relative to either the

unstressed state or the equilibrium state attained by the material after a prestress is applied. The

longitudinal displacements relative to the unstressed and prestressed equilibrium states are denoted

by ut; x and ut; x, resp ectively. To relate the two, it is noted that when the prestress is applied,

0 6

the free end of the ro d displaces by an amount

u =

0 0

as illustrated in Figure 3. Under the assumption of homogeneous material prop erties and a uniform

cross-sectional area, this pro duces a displacement

u x=u

0 0

for 0 x L. The time-dep endent displacements relative to the unstressed and prestressed equi-

librium states are then related by the equation

ut; x=ut; x u x : 10

Note that for a compressive prestress , u and hence u x will b e negative.

0 0 0

We consider rst a dynamic mo del in terms of the displacementat ut; x. Under the assumptions

of linear elasticity, small displacements, and Kelvin-Voigt damping, the stress at a p oint x; 0

is given by

@u @ u

t; x=E t; x+c t; x Et; x 11

@x @x@t

where is given by 8. When integrated across the ro d, this yields the inplane resultant

@u @ u

N t; x=EA t; x+c A t; x EAt; x : 12

Tot D

@x @x@t

Force balancing then yields the wave equation

@ u @N

Tot

A =

@t @x

as a mo del for the internal ro d dynamics.

To obtain appropriate b oundary conditions, it is rst noted that the xed end of the ro d satis es

ut; x = 0. At the end x = L, we consider an in nitesimal section having the the condition

orientation depicted in Figure 4. Force balancing then yields the b oundary condition

@u @ u

N t; L+k [ut; L u ]+c t; L+ A = M t; L 13

tot L 0 L 0 L

@t @t

a more general discussion regarding the derivation of general elastic b oundary conditions can be

@ @ u u

found in [26 ]. Note that when = ut; L=u and = 0, the b oundary condition 13 reduces

0 2

@t @t

(a)

u0 < 0 (b) σ 0 < 0

u(t,L) (c)

u(t,L)

Figure 3. a Unstressed ro d, b End displacement u = L =E due to a prestress , and

0 0 0

c Time-dep endent end displacements ut; L and ut; L=ut; L u .

0 7

to the equilibrium condition

N t; L= A:

tot 0

When combined with initial conditions, the strong form of the mo del is then

@ @ u N

tot

A =

@t @x

ut; 0 = 0

@ @ u u

N t; L=k [ut; L u ] c t; L M t; L A

tot L 0 L L 0

@t @t

u0;x=u x

0;x=0:

It is noted that the initial conditions incorp orate the equilibrium displacement u x due to the

prestress .

To de ne a weak or variational form of the mo del, the state u is considered in the state space

2 1 1

X = L 0;L and the space of test functions is taken to b e V = H 0;L f 2H 0;Lj0 = 0g.

Multiplication by test functions followed byintegration then yields the weak form

Z Z

2 2

L L

@ @ @ u u @ u

EA A dx = + c A EA + A dx

D 0

@t @x @x@t @x

0 0

@u @ u

k [ L ut; L u ]+c t; L+M t; L

L 0 L L

@t @t

for all 2 V . The consideration of the prestress as a distributed input rather than a b oundary

condition follows from integration by parts. This formulation illustrates that if the prestress is

negated by a constant input magnetostriction = =E , the system has the equilibrium solution

ut; x = 0. Conversely, the retention of the prestress as a b oundary condition illustrates that in

the absence of an applied current and resulting magnetostriction, the system will have the solution

ut; x=u x and hence retain the initial o set for all time. 0

+ u kLu

N ML tot Rod

σ0 < 0

ρ u

x = L c L ρ t

Figure 4. Orientation of spring forces, edge reactions and resultants for the Terfenol ro d. 8

For many applications, the o set u x due to the prestress is not consistent with strain or

0 0

displacement data measured from the equilibrium prestressed state. For such cases, it is advantageous

to consider the system satis ed by the p erturb ed displacement ut; x. Substitution of the expression

10 into 11 yields the stress relation

@ u @u

t; x+c Et; x+ 16 t; x=E

D 0

@x @x@t

while substitution into 12 and 13 yields the strong form of the mo del

@ u @N

tot

A =

@t @x

ut; 0 = 0

@ u @u

t; L M t; L N t; L=k ut; L c

L tot L L

@t @t

u0;x=0

0;x=0:

The initial displacement in this case is ux; 0 = 0 since u denotes displacements from the prestressed

equilibrium state.

The corresp onding weak form of the mo del is

Z Z

2 2

L L

@u @ @ u @ u

EA A dx = + c A EA dx

@t @x @x@t @x

0 0

@u @ u

k ut; L+c t; L+M t; L L

L L L

@t @t

for all 2 V . The solution ut; x to 17 or 18 provides the longitudinal displacements of the

ro d from the p erturb ed state u x pro duced by the prestress . In the absence of an applied

0 0

input t; x, the system 18 will have the equilibrium solution ut; x 0 as compared with the

o set solution ut; x = u x which satis es 15. Hence the mo del 18 and corresp onding stress

relation 16 are preferable when data is measured relative to the prestressed state. This structural

mo del is summarized along with the previously-de ned magnetization and magnetostriction mo dels

in Table 1. Note that in the weak form 18, displacements and test functions are di erentiated only

once compared with the second derivatives required in the strong form. This reduces the smo othness

requirements on the nite element basis when constructing an approximation metho d. 9

H t; x=nI t'x

H t; x=Ht; x+ M t; x

ef f

! "

H t; x

ef f

coth M t; x=M

an s

a H t; x

ef f

Magnetization

Mo del

@M dI M t; x M t; x

ir r an ir r

t; x=n 'x

ir r

@t dt

k [M t; x M t; x]

an ir r

M t; x=c[M t; x M t; x]

rev an ir r

M t; x=M t; x+M t; x

rev ir r

Magnetostric-

t; x= M t; x

tion Mo del

Z Z

2 2

L L

@u @ @ u @ u

EA A dx = + c A EA dx

Structural

@t @x @x@t @x

0 0

Dynamics of

@u @ u

Terfenol ro d

k ut; L+c L t; L+M t; L

L L L

@t @t

Table 1. Time-dep endent mo del quantifying the magnetization M t; x, the output magnetostriction

t; x and ro d displacements ut; x.

4 Comparison with the Linear Mo del

The stress relation 16 generalizes the linear constitutive law 1a to include b oth Kelvin-Voigt

damping and hysteretic and nonlinear magnetomechanical inputs. Toverify this, it is noted that in

so that 16 can b e reformulated as this regime, s =1=E and e =

e = s + e:_

To illustrate that provides the active strain contributions mo deled by d H in the linear mo del

1a, we take c = = 0 and employ the relation 8 to obtain

D 0

H 2

e = s + M :

2 M

Linearization ab out a biasing magnetization level M then yields the strain expression

H 2

e = s + 2MM M : 19

2 M

s 10

To express the magnetization and resulting strain in terms of the applied magnetic eld, it is

noted that in the absence of hysteresis, the magnetization satis es the anhysteretic relation

a H

ef f

coth M = M

a H

ef f

ef f ef f

= M + O` :

3a a

To linearize, the high order terms in the Taylor expansion are neglected to obtain

M = H +~ M

which implies that

H: M =

3a M ~

Hence 19 can b e written as

3 M

s s

H 2

e = s + 20 2HH H

2 M 3a M ~

where H is the magnetic eld required to pro duce the bias magnetization M .

0 0

As depicted in Figure 5, the total strain e is comp osed of a bias e due to the biasing magnetization

M and bidirectional strains e ab out e . To sp ecify e and e,we reformulate 20 as

0 0 0

2 2

3 M 3 M

s s s s

e = s + + H H H H

0 0 0

2 2

M 3a M ~ 2 M 3a M ~

s s

= e + e

M 3

s s

H and where e =

0 0

2 3aM ~

M 3

s s

e = s + H H H

0 0

M 3a M ~

Note that e = s when H = H . With the de nitions

3 M

s s

H d =

0 33

M 3a M ~

H = H H ;

the bidirectional strains are then given by

e = s + d H

which is equivalent to the original linear expression 1a when it is used to mo del bidirectional strains

ab out a preset magnetization level M .

We conclude this section by noting that the reader can nd further details regarding the decom-

p osition of strains into passive comp onents due to elastic prop erties of the material and nonlinear

active comp onents due to magnetostriction in [19 , p. 180]. 11 Strain

Linearization e ∆ e e0

M Magnetization

Figure 5. Linearization ab out the biasing magnetization M and the resulting bias strain e ,

0 0

bidirectional strain e; and total strain e = e +e.

5 Approximation Metho d

To approximate the solution of 18, we consider a Galerkin discretization in space followed by a

nite-di erence approximation of the resulting temp oral system. To this end, we consider a uniform

partition of the interval [0;L] with p oints x = ih; i = 0; 1; ;N and stepsize h = L=N where N

denotes the numb er of subintervals. The spatial basis f g is comprised of linear splines, or `hat

i=1

functions,' of the form

xx ; x x

i1 i1 i

x x ; x x x

x= ; i =1; ;N 1

i+1 i i+1

0 ; otherwise

xx ; x xx

N1 N1 N

0 ; otherwise

see [27 ] for details. A general basis function and nal basis function are plotted in Figure 6.

i N

The solution ut; x to 18 is then approximated by the expansion

u t; x= u t x ;

j j

j=1

N N

in the subspace H = spanf g . It should b e noted that through the construction of the basis

i=1

functions, the approximate solution satis es u t; 0 = 0 and allows arbitrary displacements at

x = L.

t; x in A semi-discrete matrix system is obtained by considering the approximate solution u

18 with the basis functions employed as test functions this is equivalent to pro jecting the system

18 onto the nite dimensional subspace H . This yields the second-order time-dep endentvector

system

Q~ut+C~ut+K~ut=ft 21 12

where ~ut=[u t; ;u t]. The mass, sti ness and damping matrices have the comp onents

1 N

A dx ; i 6= n and j 6= n

i j

[Q] =

A dx + M ; i = n and j = n

i j L

0 0

EA dx ; i 6= n and j 6= n

i j

[K ] =

0 0

EA dx + k ; i = n and j = n

i j

0 0

c A dx ; i 6= n and j 6= n

i j

[C ] =

0 0

c A dx + c ; i = n and j = n

D L

i j

while the force vector is de ned by

[f t] = EAt; x x dx

0 T

here denotes a spatial derivative. With the de nitions ~y t=[~ut;~ut] ,

" "

0 I

; F t= ; W =

1 1

Q K Q C

Q ft

the second-order system 21 can b e p osed as the rst-order system

~y t=W~yt+Ft

~y0 = ~y ;

where the 2N 1 vector ~y denotes the pro jection of the initial conditions into the approximating

space.

The system 22 must b e discretized in time to p ermit numerical or exp erimental implementation.

The choice of approximation metho d is dictated by accuracy and stability requirements, storage

capabilities, sample rates, et cetera. A trap ezoidal metho d can be advantageous for exp erimental

implementation since it is mo derately accurate, is A-stable, and requires minimal storage when

implemented as a single step metho d. For temp oral stepsizes t, a standard trap ezoidal discretization

yields the iteration

h i

~ ~

F F t +Ft ~y = W ~y +

j j+1 j +1 j

~y = ~y 0 ;

where t = j t and ~y app oximates ~y t . The matrices

j j j

1 1

t t t

W = I W I + W ; F =t I W

2 2 2 13

need only b e created once when numerically or exp erimentally implementing the metho d. This yields

2 2

approximate solutions having O`h ;t accuracy. For applications in which data at future times

t is unavailable, the algorithm 23 can b e replaced by the mo di ed trap ezoidal iteration

j +1

~y = W ~y + F F t

j +1 j j

~y = ~y 0 :

While this decreases slightly the temp oral accuracy, for large sample rates with corresp ondingly small

stepsizes t, the accuracy is still commensurate with that of the data.

φ φ i (x) N (x)

xxxi-1 i i+1 xN-1 xN

(a) (b)

Figure 6. Linear basis functions a x and b x.

i N

6 Exp erimental Validation

As summarized in Table 1, the magnetization mo del 6, magnetostriction mo del 8 and PDE 18

can b e combined to yield time-dep endent displacementvalues of the Terfenol ro d for all p oints along

its length in resp onse to an input current I t to the solenoid. This mo del incorp orates magnetic

hysteresis, nonlinear strain prop erties and the coupling between the external strains generated by

the material and the dynamics of the ro d. In its present form, however, the mo del is not fully

coupled since it do es not yet incorp orate the dynamic stress e ects on the e ective eld and ensuing

magnetization. This topic is under currentinvestigation.

For the results which follow, exp erimental data was collected from a broadband transducer devel-

op ed at Iowa State University the general con guration of the transducer can b e noted in Figure 1.

A solid Terfenol-D Tb Dy Fe rodhaving a length of 115 mm 4:53 in and diameter of 12:7mm

0:3 0:7 1:9

0:5inwas employed in the transducer. Mechanical prestresses to the ro d were generated byavari-

able prestress b olt at one end of the transducer and Belleville washers tted at the opp osite end of

the ro d. The results rep orted here were obtained with a prestress of =1:0 ksi. Surrounding the

ro d were two coils consisting of an inner single layer 150-turn pickup coil and a multi-layer 900-turn

drive coil. A current control ampli er Techron 7780 provided the input to the drive coil to pro duce

an applied AC magnetic eld and DC bias as necessary. The reference signal to this ampli er was

provided bya Tektronix sp ectrum analyzer and the applied magnetic eld H generated by the drive

coil had a frequency of 0:7 Hz and magnitude up to 70 kA=m.

A cylindrical p ermanent magnet surrounding the coils provided the capability for generating

additional DC bias if necessary. This p ermanent magnet was constructed of Alnico V and was slit

to reduce eddy current losses. Note that for the exp eriments rep orted here, biases generated in this

manner were unnecessary and the p ermanent magnet was demagnetized to obtain unbiased data. 14

To determine a function 'x see 2 which characterizes transducer anomalies such as end

e ects, an axial Hall e ect prob e connected to a F.W. Bell Mo del 9500 gaussmeter was used to map

the ux density and corresp onding eld along the length of the ro d. The resulting function 'x is

plotted in Figure 7 with the predominance of end e ects readily noted.

The measured output from the transducer during op eration included the current and voltage in

the drive coil, the voltage induced in the pickup coil, and the ro d displacement. The current I t

was used to compute the eld H t; x=nI t'x applied to the ro d. From the induced voltage in

the pickup coil, the Faraday-Lenz lawwas used to compute the temp oral derivative of the magnetic

induction B . Integration then yielded the induction and magnetization M = B H . Both the eld

and magnetization were numerically ltered to remove the small biases due to instrumentation. A

Lucas LVM-110 linear variable di erential transformer, based up on changing reluctance, was used to

measure the displacement of the ro d tip. Corresp onding strains at this p ointwere then computed by

dividing by the ro d's length. The exp erimental data collected in this manner is plotted in Figures 8-

10. Throughout the exp eriments, temp erature was monitored using two thermo couples attached to

o o

the Terfenol-D sample and maintained within 5 C of the ambient temp erature 23 C.

To employ the magnetization and structural mo del summarized in Table 1, appropriate parame-

ters must b e ascertained. These include the magnetization parameters ; c; k ; a; ;M , the structural

s s

parameters ; E ; c , the spring constants k ;c and the end mass M . The values used here are

D L L L

summarized in Table 2. The magnetization parameters were estimated through a least squares t

to data as detailed in [22 ]. The values of and E are published sp eci cations for Terfenol-D while

the damping parameters c and c were chosen within a range typical for the material. The spring

D L

sti ness co ecient k was measured through a compression test while the end mass M was mea-

L L

sured directly. We note that while the sp eci cation of parameters in this manner provided adequate

mo del ts, they are not optimal. To obtain optimal parameters and corresp onding mo del ts, it is necessary to estimate all parameters through a least squares t to data.

1.2

1.1

Filter Value 0.9

0.8

0.7

0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Longitudinal Rod Axis

Figure 7. Empirical function 'x used to qualify transducer e ects. 15

Magnetization Parameters Structural Parameters Spring and End Mass Dimension

M =7:65 10 A=m = 9250 kg =m M =0:5kg L= 115 mm

s L

6 10 6

= 1005 10 E =310 N=m k =210 N=m d =12:7mm

s L

6 3

a= 7012 A=m c =310 Ns=m c =110 Ns=m

D L

k = 4000 A=m =1:0 ksi

= 0:01

c =0:18

Table 2. Physical parameters and dimensions employed in the magnetization and structural mo dels.

Magnetization Mo del

The domain wall mo del discussed in Section 3.1 and summarized in Table 1 provides a charac-

terization of the magnetization M generated by an applied magnetic eld H . The p erformance of

the mo del under quasi-static 0:7 Hz op erating conditions is illustrated in Figure 8. It is observed

that while the mo del accurately characterizes the measured magnetization over most of the range,

certain asp ects of the transducer b ehavior are not completely quanti ed at low eld levels. The

constricted b ehavior in the magnetization at low eld levels has b een observed by other researchers

[23 , 28 ] and is hyp othesized to b e due to 180 domain rotations. While quanti cation of this e ect

is ultimately desired, the accuracy and exibilityof the current magnetization mo del are sucient

for control applications in this op erating regime.

5 x 10 8

0 M (A/m)

−2

−4

−6

−8 −6 −4 −2 0 2 4 6 4

H (A/m) x 10

Figure 8. Exp erimental M -H data { {{ and mo del dynamics || at x=L with the magneti-

zation M computed using ltered magnetic eld data H t; x=nI t'x; 16

Strain Mo del

The mo del summarized in Table 1 characterizes two asp ects of the strain in the Terfenol ro d. The

magnetostriction quanti es the external or active comp onent of the strain while e = provides

the total strain in the ro d. The relationship b etween the two can b e noted in the stress expressions

11 or 16. The total strain incorp orates b oth the elastic prop erties of the material and the

magnetomechanical e ects due to domain rotation; hence it is the quantity which mo dels the strains

measured in the transducer ro d during exp eriments.

ut;L

The mo deled strain et; L= at the ro d tip is plotted with exp erimental data in Figures 9a

and 10a. For comparison, the magnetostriction given by 8 is plotted with exp erimental data in Fig-

ures 9b and 10b. Recall that while the magnetic eld, magnetization and strains are time-dep endent,

data was collected at a suciently low frequency 0:7Hzto avoid AC losses and harmonic e ects.

It is observed in Figure 9 that some discrepancy o ccurs in b oth the strain and magnetostriction due

to limitations in the quadratic mo del 8. The total strain provided by the dynamic mo del do es,

however, include the hysteresis observed in the exp erimental data. This is a signi cant advantage

over the mo deled magnetostriction whichis single-valued. This leads to the highly accurate mo del

t observed in Figure 10 where the relation b etween the input eld H and the output strain et; L

at the ro d tip is plotted.

For comparison purp oses, we include in Figure 11 the corresp onding mo del ts obtained when the

ltering function 'xwas omitted and an averaged eld measurementwas used to compute M and

. This was the regime considered in [22 ] where it was demonstrated that use of magnetostriction

rather than total strain led to an adequate mo del at low to mo derate drive levels but provided

inadequate ts at high input levels. These conclusions are reinforced by the mo del ts observed in

Figure 11. It is noted that even when the saturation magnetostriction was scaled to the maximum

exp erimentally observed value, the lack of hysteresis in the relation between M and led to an

inadequate characterization of in terms of H . A comparison with Figures 9a and 10a again

indicates the necessity of considering the total strains.

−3 −3 x 10 x 10

1 1

0.8 0.8

0.6 0.6 Strain Magnetostriction

0.4 0.4

0.2 0.2

0 0 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 0 2 4 6 8 5 5

M (A/m) x 10 M (A/m) x 10

a b

Figure 9. Exp erimental M -strain data { { { and mo del dynamics || with the magnetiza-

tion M computed using ltered magnetic eld data H t; x = nI t'x; a Total strain et; L,

b Magnetostriction t; L. 17 −3 −3 x 10 x 10

1 1

0.8 0.8

0.6 0.6 Strain Magnetostriction

0.4 0.4

0.2 0.2

0 0 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 4 4

H (A/m) x 10 H (A/m) x 10

a b

Figure 10. Exp erimental H -strain data { { { and mo del dynamics || computed using ltered

magnetic eld data H t; x=nI t'x; a Total strain et; L, b Magnetostriction t; L.

−3 −3 x 10 x 10

1 1

0.8 0.8

0.6 0.6 Magnetostriction Magnetostriction

0.4 0.4

0.2 0.2

0 0 −8 −6 −4 −2 0 2 4 6 8 −6 −4 −2 0 2 4 6 5 4

M (A/m) x 10 H (A/m) x 10

a b

Figure 11. Exp erimental data strain { { { and mo del dynamics || computed using un ltered

magnetic eld data H t = nI t and a scaled saturation magnetostriction ; a M - relation,

b H - relation.

7 Concluding Remarks

This pap er addresses the mo deling of strains generated by magnetostrictive materials when employed

in high p erformance transducers. The mo del extends the classical relation 1a, which includes elastic

e ects and a linear magnetomechanical comp onent, to the regime in which magneto elastic inputs

are nonlinear and exhibit signi cant hysteresis. This is necessary to accommo date the dynamics 18

observed in current transducers at high drive levels and to provide a mo del for design and control

applications in such regimes.

The mo del was constructed in three steps. In the rst, the mean eld theory of Jiles and Atherton

was used to quantify the relation between the current input to the solenoid and the magnetization

pro duced in the ro d. This comp onent of the mo del incorp orates b oth inherent magnetic hysteresis

and saturation e ects at high eld levels. In the second step, the magnetostriction due to the

rotation of moments was quanti ed through consideration of a quadratic mo del p osed in terms of

the magnetization. When combined with the mean eld magnetization mo del, this provided a means

for extending the linear magneto elastic relation 1a to include the nonlinear dynamics and hysteresis

observed at high eld levels. Finally, force balancing provided a PDE mo del which quanti ed material

displacements due to the magnetostriction. For a given input current, the solution to the PDE yields

the displacements and strains pro duced by the ro d.

While the PDE has the form of a wave equation, the nature of the b oundary conditions mo deling

the prestress mechanism and end mass precluded analytic solution. Hence we approximated the

solution through a nite element discretization in the spatial variable followed by a nite di erence

discretization in time. This yielded a vector system which could b e iterated in time with the measured

electric current or magnetic eld data employed as input.

The examples illustrated that the resulting mo del yields the hysteresis observed in exp erimental

strain data when plotted as function of the magnetization. This is a signi cant improvement over

the magnetostriction which is mo deled as a single-valued function of the magnetization. Finally,

the accuracy of the mo del is re ected in the full relation between the input eld H and strains e

pro duced by the transducer.

Acknowledgements

The authors express sincere appreciation to Tad Calkins and David Jiles for input regarding the

mo deling techniques employed here and to Brian Lund for his assistance in collecting the exp erimental

data. The research of R.C.S. was supp orted in part by the Air Force Oce of Scienti c Research

under the grantAFOSR F49620-95-1-023 6. Financial supp ort for M.J.D. and A.B.F. was provided by

the NSF Young Investigator Award CMS9457288 of the Division of Civil and Mechanical Systems.

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