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Original Article

Journal of Intelligent Material Systemsand Structures2015, Vol. 26(16) 2242–2255� The Author(s) 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X14551434jim.sagepub.com

A hysteresis nonlinear model of giantmagnetostrictive transducer

Yuchuan Zhu and Yuesong Li

AbstractIn order to describe the hysteresis nonlinear performance of giant magnetostrictive transducer, based on the relation-ship between complex permeability and magnetization, eddy current effects of Terfenol-D, and kinetic characteristic ofgiant magnetostrictive transducer, a new hysteresis nonlinear model is developed, which consists of quasi-static modeland dynamic model considering eddy current effect of Terfenol-D, dynamic of power amplifier, and magnetic aftereffectin the magnetization process. After that, to identify model parameters and improve the accuracy of the proposed model,an experiment platform is established and a modified hysteresis nonlinear model is put forward. Finally, experiments areconducted to demonstrate the effectiveness of the modified model and the corresponding identification method. Themodified model prediction results of giant magnetostrictive transducer agree well with experiment results not onlyunder quasi-static operating conditions but also under dynamic operating conditions. Additionally, the proposed model’sparameters have definite physical meaning, but the parameters to be identified are less and model expression is moresimple than that of the other hysteresis nonlinear models such as the Jiles–Atherton model. The present model can beused in the simulation and performance prediction of giant magnetostrictive transducer, which is especially suitable forthe real-time control system of giant magnetostrictive transducer.

KeywordsEddy current effect, magnetic hysteresis, giant magnetostrictive transducer, dynamic model, complex permeability

Introduction

Giant magnetostrictive transducer opened the way todevelop totally new electromechanical devices withhigher energy density, faster response, and better pre-cision than previously possible, such as sound andvibration (Wakiwaka et al., 1997), sonar system(Jenner et al., 2000), active vibration control (Braghinet al., 2012; Hiller et al., 1989), magnetostrictivemotors (Kiesewetter, 1988), hydraulics (Karunanidhiand Singaperumal, 2010; Li et al., 2011a; Yoo andWereley, 2004), and sensors (Yamamoto et al., 1997).However, like other smart material actuators (e.g.piezoelectric actuator and shape memory alloy actua-tor), giant magnetostrictive transducers display stronghysteresis, which makes their effective use quite chal-lenging, and hysteresis nonlinearity makes them diffi-cult to control giant magnetostrictive transducer,which has serious influence on the positioning preci-sion of giant magnetostrictive transducer.

Hysteresis nonlinear model plays an important roleto predict and optimize the performance of giant mag-netostrictive transducer, and at present, the hysteresisnonlinear modeling for giant magnetostrictive

transducer can roughly be classified into two kinds:physics-based models and phenomenological models.An example of physics-based models is the Jiles–Atherton model of ferromagnetic hysteresis (Jiles andAtherton, 1986), which is built on first principles ofphysics. Phenomenological models only reflect thecharacteristics between input and output, which is ablack box model based on experimental data; a popularphenomenological hysteresis model adopted for giantmagnetostrictive transducer is the Preisach model(Adly et al., 1991; Cruz-Hernandez and Hayward,2001; Gorbet et al., 1998; Hughes and Wen, 1994).Smith (1997) employed a Preisach model to describesaturation and hysteresis nonlinearities observed in amagnetostrictive actuator mounted on a cantilever

College of Mechanical and Electrical Engineering, Nanjing University of

Aeronautics and Astronautics, Nanjing, China

Corresponding author:

Yuchuan Zhu, College of Mechanical and Electrical Engineering, Nanjing

University of Aeronautics and Astronautics, 29 Yudao St., Nanjing

210016, China.

Email: [email protected]

http://crossmark.crossref.org/dialog/?doi=10.1177%2F1045389X14551434&domain=pdf&date_stamp=2014-09-19

beam. Because the classical Preisach operator is rateindependent, which is different from the actual hyster-esis nonlinearity in giant magnetostrictive transducer,‘‘dynamic’’ generalizations of the Preisach operatorwere proposed by assuming output-rate-dependentPreisach density functions (Mayergoyz, 1991) or input-rate-dependent behavior of delayed relays (Bertotti,1992). Then, the frequency-dependent hysteresis (Tanand Baras, 2004) is conveniently described using thePreisach integral method. Similar to the Preisachmodel, another popular phenomenological hysteresismodel is the Prandtl–Ishlinskii model; unlike the dis-continuous relay operators in the Preisach model, thePrandtl–Ishlinskii (Al Janaideh, 2009) model is con-structed using continuous play or stop hysteresis opera-tors which are characterized by the input and thethreshold. The Prandtl–Ishlinskii model has been pro-posed as an operator-based model to characterize thehysteresis properties of giant magnetostrictive transdu-cer, which offers greater flexibility and unique propertyso that its inverse can be attained analytically.However, the Prandtl–Ishlinskii model can neitherexhibit asymmetric hysteresis loops nor saturated hys-teresis output; in response to this issue, a rate-dependent Prandtl–Ishlinskii model (Aljanaideh et al.,2014) integrating a memoryless function of deadbandoperator was subsequently formulated to describe boththe rate dependence and the asymmetric hysteresisloops of the magnetostrictive actuator in addition tothe output saturation.

Physics-based models, such as the Jiles–Athertonmodel (Jiles and Atherton, 1984, 1986), have theadvantage of relying on intuitive development, whichhelps with design and facilitates extension of the modelto new types of actuators, and then, the Jiles–Athertonmodel is extended to include the effects of stress(Dapino et al., 2000; Sablik and Jiles, 1993).Furthermore, the Jiles–Atherton model is developed toquantify the strains and output displacements by thegiant magnetostrictive transducers (Dapino et al.,2000a,b). Another physics-based model is the Stoner–Wohlfarth model; the original Stoner–Wohlfarth modelis modified for magnetostrictive materials (Jiles andThoelke, 1994; Park et al., 2002). In this model, therotation of noninteracting magnetic domains is themain mechanism, and magnetostriction is introducedby adding an energy term to the domain energy, butthe model is not useful in some applications, such ascontroller design, where accuracy is very important.

However, physics-based models usually includemany complicated physical parameters that are difficultto be identified (Almeida et al., 2001; Calkins et al.,2000; Cao et al., 2004), such as saturation magnetiza-tion intensity, saturation magnetostriction, averagedenergy of the pinning sites, and. Phenomenologicalmodels, such as Preisach model, provide an empiricalestimation of the relationship between input and

output of experimental data, which need more experi-mental data and involve complicated calculation.

Besides aiming to describe the complex nonlinearconstitutive behavior of Terfenol-D measured in theexperiments, numerous nonlinear models have beenestablished, including the dynamic hysteresis constitu-tive model (Zheng et al., 2009), energy-averaged model(Chakrabarti and Dapino, 2012), dynamic loss hyster-esis model (Xu et al., 2013), nonlinear transient consti-tutive model (Wang and Zhou, 2010), and so on.However, among these models, it is a purely materialconstitutive model and does not incorporate the struc-tural dynamic behavior arising from the transduceroperation; nonetheless, magnetostriction is the phe-nomenon of strong coupling between magnetic stateand mechanical state of magnetostrictive materials. Ageneral hysteresis nonlinear model considering materialconstitutive model and the structural dynamic behavioris extremely significant from a practical point of view.

Additionally, generally speaking, a good hysteresisnonlinear model should have the following properties:agree well with experiments, computationally simpleand efficient, and reasonably easy to determine transdu-cer parameters. In this article, we mainly focus on thecomplex hysteresis behavior of the giant magnetostric-tive actuator system under magnetically biased condi-tions and mechanically biased conditions. Based on therelationship between complex permeability and magne-tization intensity, a new hysteresis nonlinear model isdeveloped, which consists of quasi-static hysteresis non-linear model and dynamic hysteresis nonlinear modelconsidering eddy current effect and dynamic of poweramplifier. In order to improve the accuracy of themodel, additionally, a modified hysteresis nonlinearmodel is presented as well. The present model’s para-meters have definite physical meaning just like theJiles–Atherton model, but the parameters to be identi-fied are lesser than those of the Jiles–Atherton modeland model expression is more simple than that ofPreisach model, which is more suitable for the real-timecontrol system of giant magnetostrictive transducer.The present model can be used in the simulation andperformance prediction of giant magnetostrictivetransducer.

Configuration of giant magnetostrictivetransducer

Figure 1 illustrates the configuration of giant magnetos-trictive transducer, mainly consisting of permanentmagnet, giant magnetostrictive material (GMM) rod(Terfenol-D rod is used in this research), output rod,coil, and spring. The spring applies a prestress onGMM rod because a larger magnetostrictive strain canbe obtained with same magnetic field when GMM rodis compressed (Evans and Dapino, 2010). Permanent

Zhu and Li 2243

magnet is used for providing a longitudinal magneticbias, which can improve the sensitivity of GMM rodand eliminate frequency doubling. When the sign ofmagnetic field produced by the coil is the same as thesign of magnetic bias, the length of GMM rod elon-gates. If the sign of magnetic field produced by the coiland the sign of magnetic bias are opposite, the length ofGMM rod is shorter than that in the initial condition.

Quasi-static hysteresis nonlinear model

At quasi-static condition, the energy conversion pro-cess in giant magnetostrictive transducer involves threestages: the stage from the electrical energy to the mag-netic energy, the stage from the magnetic energy to theelastic potential energy, and the stage from the elasticpotential energy to the mechanical energy. So, themodel of giant magnetostrictive transducer involvesthree submodels from the viewpoint of energy conver-sion. The first submodel is the magnetization modelwhich describes the relationship between applied excit-ing current and magnetization of GMM rod. The sec-ond submodel is the magnetoelastic model describingthe relationship between magnetostrictive strain andmagnetization. The third submodel is output displace-ment model of giant magnetostrictive transducerdescribing the relationship between the output displace-ment and magnetostrictive strain of giant magnetostric-tive transducer.

Magnetization model of GMM rod

In a closed magnetic circuit, the applied magnetic fieldH generated by an alternating current i is given by

H =Ni

kfLG=

NIm

kfLGcos vt =Hm cos vt ð1Þ

where N is the number of excitation coil turns, Im is theamplitude of the alternating current, kf is the leakagecoefficient of the magnetic flux, LG is the length ofGMM rod, Hm is the amplitude of applied magneticfield, and v is the angular frequency.

The complex number form of the applied magneticfield is written as (Engdahl, 1999)

~H =Hmejvt ð2Þ

In a sinusoidal magnetic field, the relative permeabil-ity of GMM rod is a complex number, and the complexnumber form of magnetic flux density is given by

~B=m0(m0 � jm00) ~H =m0Hm m0e jvt +m00e j(vt�p=2)

h ið3Þ

So, the magnetic flux density B in GMM rod can bewritten as

B=m0m0Hm cos vt+m0m

00Hm cos vt �p

2

� �=m0H m

0+m00 tan vtð Þð4Þ

where m0 is the permeability of free space, m0 is the

real part of the complex relative permeability, and m00

is the imaginary part of the complex relativepermeability.

The magnetization M can be deduced from magneticflux density B and the applied magnetic field H

M =B

m0� H =H m0 � 1+m00 tan vtð Þ ð5Þ

Consider relative permeability mr =m0 � jm00, so

m0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimrj j

2 � (m00)2q

ð6Þ

Substituting equations (1) and (6) into equation (5)yields

M =Hm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimrj j

2 � (m00)2q

� 1� �

cos vt+m00 sin vt

� �ð7Þ

Rewriting equation (7) leads to

M =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimrj j2 � m00hð Þ

2

q� 1

� �2+ m00hð Þ

2

sHm cos (vt � uh)

ð8Þ

where uh is the lag angle caused by hysteresis

uh = arctanm00hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mrj j2 � m00hð Þ

2q

� 1

0B@

1CA ð9Þ

Equations (8) and (9) are valid for low-frequencysinusoidal magnetic field.

Magnetoelastic model of GMM rod

As discussed in Li and Zhu (2012), the magnetostrictivel is given by

Figure 1. Configuration of giant magnetostrictive transducer.

2244 Journal of Intelligent Material Systems and Structures 26(16)

l=3

2

lS

M2SM2 ð10Þ

where lS and MS are the saturation magnetostrictiveand the saturation magnetization, respectively.

GMM rod practical magnetic field distribution con-sists of bias magnetic field Hb and control magneticfield Hc, and accordingly, GMM rod practical magneti-zation consists of bias magnetization Mb and controlmagnetization Mc; hence, equation (10) can be writtenas follows

l=3

2

lS

M2SM2 Hc +Hbð Þ �

3

2

lS

M2SM2b

= kl M2 Hc +Hbð Þ �M2b

� � ð11Þwhere kl is the magnetostrictive coefficient, which canbe obtained from saturation magnetostrictive and thesaturation magnetization or from experimental data.M( � ) is the function of magnetic field.

If an experimental point (l0, M0) has been known,kl can be calculated from the following equation

kl =l0

M2 H0 +Hbð Þ �M2bð12Þ

Output displacement model of giant magnetostrictivetransducer

Output displacement y of giant magnetostrictive trans-ducer can be written as follows according to Hooke’slaw

Ky

AG=EHl ð13Þ

where K is the equivalent stiffness of GMM rod.Substituting equation (11) into equation (13) yields

y=EHAGl

K= kl

EHAG

KM2 Hc +Hbð Þ �M2b� �

ð14Þ

If an experimental point (H0, y0) has been known

y0 = klEHAG

KM20 �M2b

=klE

HAG mrj j � 1ð Þ2

K

H0 +Hbð Þ2 � H2bh i ð15Þ

Substituting equation (15) into equation (14) yields

y=M2 Hc +Hbð Þ � mrj j � 1ð Þ

2H2b

mrj j � 1ð Þ2

H0 +Hbð Þ2 � H2bh i y0 ð16Þ

Because the applied magnetic field Hc is small, theresultant magnetic field varied around the bias mag-netic field Hb; based on equation (7) and the principleof linear superposition, the resultant magnetization inlow excitation frequency yields

M Hc +Hbð Þ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimrj j�1ð Þ

2� m00hð Þ2

q�1

� �2þ m00hð Þ

2

s

Hm cos (vt � uh)+ mrj j � 1ð ÞHb ð17Þ

Dynamic hysteresis nonlinear model

At dynamic condition, two dynamic processes are con-sidered: dynamic of power amplifier and dynamic ofgiant magnetostrictive transducer. Dynamic of poweramplifier is the dynamic process from input voltage ofpower supply to input current of drive coil. Dynamicof giant magnetostrictive transducer is the dynamicprocess from the point of view of the energy; the energyconversion process in giant magnetostrictive transducerinvolves three stages: the stage from the electricalenergy to the magnetic energy, where eddy currenteffect and hysteresis effect will be included; the stagefrom the magnetic energy to the elastic potentialenergy; and the stage from the elastic potential energyto the mechanical energy. So, the model of giant mag-netostrictive transducer involves three submodels. Thefirst submodel is the dynamic magnetization model

Figure 2. Circuit diagram of power amplifier.

Zhu and Li 2245

which describes the relationship between applied excit-ing current and magnetization of GMM rod. The mag-netoelastic model describing the relationship betweenmagnetostrictive strain and magnetization is the secondsubmodel, which is the same as the above-mentionedquasi-static hysteresis model of giant magnetostrictivetransducer. The final submodel is the kinetic model ofgiant magnetostrictive transducer describing the rela-tionship between the output displacement and magne-tostrictive strain.

Dynamic model of power amplifier

Drive coil is an inductance element which makes inputcurrent lag behind input voltage. So, in order to obtainfast current response, we designed a power amplifier withintegrated power amplifiers by depth electric current neg-ative feedback technology, which can eliminate input cur-rent lag, and its circuit diagram and transfer functionblock diagram are shown in Figures 2 and 3, respectively.

The transfer function of power amplifier can be writ-ten as follows according to Figure 3

i ¼ R2R1 þ R2

Ku

Lsþ Rþ R5uc

1þ KuLsþ Rþ R5

R3R0

R3þR4

ð18Þ

Assuming R1 = R2 and R3 = R4, equation (18) canbe rewritten as follows

i ¼ KuLsþ Rþ R5

uc

2 1þ KuLsþ Rþ R5

R02

!

¼ Kuuc2Lsþ ð2Rþ 2R5 þ KuR0Þ

ð19Þ

where Ku is the open-loop gain of the integrated opera-tional amplifier, uc is the input voltage, L is the induc-tance of drive coil, and R and R0 are resistance of thedrive coil and sampling resistance, respectively.

Dynamic magnetization model of GMM rod

A time-varying magnetic field Maxwell equation can bewritten as follows

r3H= J+ ∂D∂t

ð20Þ

r3E= � ∂B∂t

ð21Þ

r � B= 0 ð22Þr �D= r ð23ÞD= eE ð24ÞB=mH ð25ÞJ= gE ð26Þ

Consider the rotation on both sides of equation (20)

r3r3H=r3 J+r3 ∂D∂t

ð27Þ

Substituting equations (21), (24), and (26) into equa-tion (27) yields

r3r3H= � g ∂B∂t� e ∂

2B

∂t2ð28Þ

Assuming the change in magnetic flux density B tobe independent of the change in stress on the GMMand substituting equation (25) into equation (28) yield

r3r3H= � gm ∂H∂t� em ∂

2H

∂t2ð29Þ

As is well known, the vector identical relation

r3r3H=r(r �H)�r2H ð30Þ

Substituting equations (22) and (25) into equation(30) yields

r3r3H= �r2H ð31Þ

Substituting equation (31) into equation (29) yields

r2H= gm ∂H∂t

+ em∂2H

∂t2ð32Þ

The second term on the right-hand side of equation(32) can be neglected for the frequency range in whichmagnetostrictive transducers are operated (g � ve).So, equation (32) can be rewritten as follows

r2H= gm ∂H∂t

ð33Þ

In a high-frequency sinusoidal magnetic field, we canobtain GMM rod practical magnetic field distributionin a frequency characteristic form from equation (33)

r2H= jvgmH ð34Þ

We can simplify GMM rod practical magnetic fielddistribution as one-dimensional axial direction (z direc-tion) as shown in Figure 4.

Figure 3. Transfer function block diagram of power amplifier.


According to the cylindrical coordinate system ofGMM rod, in the z direction, equation (34) can berewritten as follows

r2H = jvgmH ð35Þ

Equation (35) can be rearranged, and the appliedmagnetic field H yields (Braghin et al., 2011; Yang etal., 2008)

d2 _H

dr2+

1

r

d _H

dr= jvgm _H ð36Þ

Consider boundary conditions

_H rG, tð Þ=Hmejvt ð37Þ

where rG is the radius of GMM rod.Thus, the solution of equation (36) is

_H r, tð Þ= J0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jvgmpð Þ

J0 rGffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jvgmpð ÞHmejvt ð38Þ

where J0( � ) is the zero-order Bessel function.In GMM rod axes

_H 0, tð Þ= J0 0ð ÞJ0 rG

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jvgmpð ÞHmejvt = 1J0 rG ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jvgmpð ÞHmejvtð39Þ

By the definition of imaginary variation Bessel func-tion (Li et al., 2011a)

J0 rGffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jvgm

p� �=ber rG

ffiffiffiffiffiffiffiffiffiffivgmp

+ jbei rGffiffiffiffiffiffiffiffiffiffivgmp

ð40Þ

where ber( � ) and bei( � ) are the real part and the ima-ginary part of Bessel function, respectively.

Substituting equation (40) into equation (39), GMMrod practical magnetic field distribution

_H =ber rG

ffiffiffiffiffiffiffiffiffiffivgmp � jbei rG ffiffiffiffiffiffiffiffiffiffivgmp

ber2 rGffiffiffiffiffiffiffiffiffiffivgmp

+bei2 rGffiffiffiffiffiffiffiffiffiffivgmp Hmejvt ð41Þ

Rewriting equation (40) in real form yields

H =ber rG


Hm cos vt


+bei2 rGffiffiffiffiffiffiffiffiffiffivgmp

+ber rG


Hm cos vt � p2



ð42Þ

Rearranging equation (42) leads to

H =1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


+bei2 rGffiffiffiffiffiffiffiffiffiffivgmp q Hm cos vt � ueð Þ

ð43Þ

where ue is the lag angle caused by eddy

ue = arctanbei rG


ber rGffiffiffiffiffiffiffiffiffiffivgmp

!ð44Þ

From simultaneous equations (8), (9), (43), and (44),the dynamic magnetization model of GMM rod can bewritten as follows considering the eddy current effect

M =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimrj j

2 � m00hð Þ2

q� 1

� �2+ m00hð Þ

2



vuuuutHm cos (vt � uh � ue)

ð45Þ

Kinetic model of giant magnetostrictive transducer

The lumped parameter model of giant magnetostrictivetransducer is a mass–spring–damping system (Tan andBaras, 2004), as shown in Figure 5. Therefore, themodel of giant magnetostrictive transducer can bedescribed as

F =md2y

dt2+ c

dy

dt+Ky ð46Þ

where y is the output displacement of giant magnetos-trictive transducer; and m, c, and K are the equivalentmass, equivalent damping, and equivalent stiffness ofgiant magnetostrictive transducer, respectively.

The magnetostrictive force F is defined as

F =AGEHl ð47Þ

Z

Y

X

Figure 4. Cylindrical coordinate system of GMM rod.GMM: giant magnetostrictive material.

c

Km

F

i

Figure 5. Lumped parameter model of giant magnetostrictivetransducer.

Zhu and Li 2247

where AG and EH are the cross-sectional area and the

elastic modulus of GMM rod, respectively.Inserting equation (46) into equation (47) and using

Laplace transforms yield

y=AGE

Hl

ms2 + cs+Kð48Þ

From simultaneous equations (11), (45), and (48), adynamic hysteresis nonlinear displacement model ofgiant magnetostrictive transducer can be obtained.

Experiment platform and parameteridentification

Experiment platform

The precision of the present giant magnetostrictive trans-ducer hysteresis nonlinear model needs to be verified byexperimental data; therefore, the experiment platformused for measuring the performance of giant magnetos-trictive transducer is built and is shown in Figure 6, whichconsists of signal generator, adjustable power supply,constant current power amplifier, giant magnetostrictivetransducer, eddy current sensor, and oscilloscope.

The giant magnetostrictive transducer is driven byalternating current from power amplifier which is con-trolled by the sinusoidal signal generated by signal gen-erator, and the sinusoidal signal is displayed in theoscilloscope with the output signal of displacement sen-sor. Due to current negative feedback, the phase of theoutput current of the power amplifier is basically syn-chronous with the input voltage signal. The gain ofconstant current power amplifier is 2 A/V.

Parameter identification

Coil inductance L with GMM rod and coil inductanceL0 without GMM rod in giant magnetostrictive trans-ducer meet

L= mrj jL0 ð49Þ

We can obtain mrj j= 5 by the measured value of Land L0 around the bias magnetic field.

In the present model and at quasi-static operationconditions, the output displacement of giant magnetos-trictive transducer meets equation (16), and only twoparameters need to be identified: the magnetostrictivecoefficient kl and the imaginary part of relative com-plex permeability m00h; the other parameters are knownand are shown in Table 1.

Because the lag angle caused by hysteresis increaseswith the amplitude value of the applied magnetic field,the value of imaginary part of relative complex perme-ability has a positive correlation with the amplitudevalue of the input current, so we assume

m00h = kuIm ð50Þ

where ku is the parameter to be identified and Im is theamplitude value of the input current.

Assuming the amplitude value of the input currentI0 = 0.5 A and frequency of the input currentf = 1 Hz, we make the minimum square sum of differ-ence value between simulation curves with test curves,which is shown in Figure 7. Thus, we obtain the ima-ginary part of the relative complex permeabilitym00h = 1:1.

If an experimental point (H0, y0) has been knownbased on equation (15), the magnetostrictive coefficientkl can be written as follows

kl =y0K

EHAG mrj j � 1ð Þ2

H0 +Hbð Þ2 � H2bh i ð51Þ

Table 1. Model parameters for giant magnetostrictivetransducer.

Parameters Value

Radius of GMM rod (rG) (mm) 5Length of GMM rod (LG) (mm) 80Equivalent mass of giantmagnetostrictive transducer (m) (kg)

0.1

Equivalent damping of giantmagnetostrictive transducer (c) (N s/m)

3000

Equivalent stiffness of giantmagnetostrictive transducer (K) (N/m)

9.9 3 106

Bias magnetization (Mb) (kA/m) 250Leakage coefficient of the magnetic flux (Kf) 1.2Number of the excitation coil (N) (turns) 1300Young’s modulus of GMM rod (EH) (GPa) 20Equivalent time constant of poweramplifier t(ms)

0.2

Electric conductivity of GMM rod (g) (S) 1.67 3 106

Inductance of drive coil (L) (mH) 6Resistance of drive coil (R) (O) 6Sampling resistance (R0) (O) 0.5

4

2

3

15

6

Figure 6. Photograph of the experiment platform: (1) giantmagnetostrictive transducer, (2) oscilloscope, (3) signalgenerator, (4) power amplifier, (5) displacement sensor, and (6)sensing coil.


Therefore, we can obtain kl = 1:4694 3 10�14

m2=A2.

Hysteresis nonlinear model validation

Quasi-static hysteresis nonlinear model validation

To verify the validity and applicable scope of theabove-mentioned model equations (16) and (17), we

obtained quasi-static output displacement under inputcurrent amplitude Im = 0.25, 0.5, 0.7, and 1 A bysimulation and experiment, respectively, which areshown in Figure 8.

As shown in Figure 8, under quasi-static operatingconditions, the model prediction results overestimatethe amplitude and phase of hysteresis loop under inputcurrent amplitude of 0.25 A and, on the other hand,underestimate the amplitude and phase of hysteresisloop under input current amplitudes of 0.7 and 1 A buthave a good agreement with the experimental dataunder input current amplitude of 0.5 A.

Dynamic hysteresis nonlinear model validation

Consider the same input current amplitude value of0.8 A and the different input current exciting frequen-cies of 10, 50, 80, 100, 150, and 200 Hz. Thus, weobtained dynamic output displacement simulationcurves and experiment curves as shown in Figure 9.

As shown in Figure 9, under dynamic operating con-ditions, the model prediction results underestimate theamplitude and phase of hysteresis loop under input cur-rent exciting frequencies of 10, 50, and 80 Hz and, on

-5

-2.5

0

2.5

5

y /µm

Model resultsExperimental data

-10

-5

0

5

10

y /µm


-0.8 -0.4 0 0.4 0.8-20

-10

0

10

20

i/A

y /µm


-1 -0.5 0 0.5 1-30

-15

0

15

30

i/A

y /µm


(d)(c)

-0.6 -0.3 0 0.3 0.6i/A

(b)

-0.3 -0.15 0 0.15 0.3i/A

(a)

Figure 8. Verification for the quasi-static hysteresis nonlinear model of giant magnetostrictive transducer: (a) Im = 0.25 A, (b)Im = 0.5 A, (c) Im = 0.7 A, and (d) Im = 1 A.

-0.6 -0.3 0 0.3 0.6-10

-5

0

5

10

i/A

y/µm

Figure 7. Identification of the imaginary part for relativecomplex permeability.

Zhu and Li 2249

-1 -0.5 0 0.5 1-30

-20

-10

0

10

20

30

Experimental dataModel results

-1 -0.5 0 0.5 1-20

-10

0

10

20


(a) (b)

-1 -0.5 0 0.5 1-20

-10

0

10

20

-1 -0.5 0 0.5 1-20

-10

0

10

20

(c) (d)

i/A-1 -0.5 0 0.5 1

-15

-10

-5

0

5

10

15

y/µm


i/A

i/A i/A

i/A i/A

-1 -0.5 0 0.5 1-15

-10

-5

0

5

10

15

y/µm

y/µm

y/µm

y/µm

y/µm


(e) (f)

Model results

Experimental data

Model results

Experimental data

Figure 9. Verification for the dynamic hysteresis nonlinear model of giant magnetostrictive transducer: (a) 10 Hz, (b) 50 Hz, (c)80 Hz, (d) 100 Hz, (e) 150 Hz, and (e) 200 Hz.


the other hand, have a good agreement with the experi-mental data under input current exciting frequencies of100, 150, and 200 Hz.

Modified hysteresis nonlinear model andmodel validation

Modified quasi-static hysteresis nonlinear model andmodel validation

Figure 8 shows that simulation results have a betteragreement with experiment result under input currentamplitude I0 = 0.5 A than that under other input cur-rent amplitudes. The reason is that the imaginary partof relative complex permeability is identified underinput current amplitude Im = 0.5 A. So, when inputcurrent amplitude is not equal to 0.5 A, the above-mentioned quasi-static hysteresis nonlinear modelneeds to be modified. We assume that the correctionterm of quasi-static hysteresis nonlinear model is pro-portional to the difference between the input currentamplitude Im and the reference input current amplitudeI0 = 0.5 A; thus, we can obtain the modified quasi-static hysteresis nonlinear model of giant magnetostric-tive transducer as follows

y=M2 Hc +Hbð Þ � mrj j � 1ð Þ

2H2b

H0 +Hbð Þ2 � H2b

y0 + Im � I0ð Þŷ0mrj j � 1ð Þ

2

ð52Þ

where ŷ0 is the unknown parameter and has to beidentified.

Assuming the input current amplitude value of0.25 A and the input current frequency f = 1 Hz, wemake the minimum square sum of difference valuebetween simulation curves based on equation (52) withtest curves, which is shown in Figure 10. Thus, weobtained the unknown parameter to be identifiedŷ0’13.

In order to verify the validity and reliability of themodified hysteresis quasi-static nonlinear model, quan-titative comparisons between its predictions and experi-mental data are given. We obtained output

displacement under input current amplitude Im = 0.25,0.5, 0.7, and 1 A by the modified model formula (52)and experimental data, respectively, which is shown inFigure 11.

Comparing with Figures 8 and 11, it is observed thatthe modified quasi-static hysteresis nonlinear modelyields output displacement precision higher than that ofunmodified one. The reason for the difference betweenthe mentioned hysteresis nonlinear model and the mod-ified one is the nonlinear constitutive behavior ofTerfenol-D and the structural behavior of transducermechanical system.

As shown in Figure 11, although the modified modelprecision decreases slightly with the amplitude value ofthe input current, however, when the amplitude valueof the input current is within 1 A, the modified modelresults of giant magnetostrictive transducer have a per-fect agreement with the majority experiment results.

Additionally, as shown in Figure 11, the differencebetween the modified hysteresis nonlinear model andexperimental data is mainly symmetry of hysteresis loopof giant magnetostrictive transducer, that is, the hyster-esis loop controlled by the modified hysteresis nonlinearmodel is symmetrical; however, on this actual transdu-cer system, the hysteresis loop drawn by experimentaldata is asymmetrical for discontinuous damping reduc-tion in the magnetization process and structure defectof GMM.

Modified dynamic hysteresis nonlinear model andmodel validation

Similar to the method used by modified quasi-statichysteresis nonlinear model, based on the kinetic modelof giant magnetostrictive transducer, we can obtain themodified dynamic hysteresis nonlinear model of giantmagnetostrictive transducer as follows

y sð Þ= M2 Hc +Hbð Þ � mrj j � 1ð Þ

2H2b

mrj j � 1ð Þ2

H0 +Hbð Þ2 � H2bh i y0 + Im � I0ð Þŷ0

ms2 +Cs+KK

ð53Þ

where M (Hc + Hb) is the synthesis of magnetizationconsidering eddy effect and bias magnetization effect.Considering equations (17) and (43), M (Hc + Hb)can be written as follows

M Hc +Hbð Þ= Lf Hm cos (vt � uh � ue)+ mrj j � 1ð ÞHbð54Þ

Lf =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimrj j

2 � m00hð Þ2

q� 1

� �2+ m00hð Þ

2



vuuuut ð55Þ

-0.3 -0.15 0 0.15 0.3-4

-2

0

2

4

i/A

y /µm

Figure 10. Identification of the correction term.

Zhu and Li 2251

where Lf represents the impact on output displacementconsidering eddy effect and hysteresis effect.

Equation (53) is the output displacement theoreticalmodel of giant magnetostrictive transducer; in theactual application system, the dynamic process of giantmagnetostrictive transducer also includes power ampli-fier dynamic, magnetic aftereffect dynamic, and so on;thus, equation (53) can be rewritten as follows

y sð Þ= M2 Hc +Hbð Þ � mrj j � 1ð Þ

2H2b

mrj j � 1ð Þ2

H0 +Hbð Þ2 � H2bh i

y0 + Im � I0ð Þŷ0ms2 +Cs+K

K

ts+ 1

ð56Þ

where t is the time constant of power amplifier dynamicand magnetic aftereffect dynamic for giant magnetos-trictive transducer.

Consider the same input current amplitude value of0.8 A and the different input current exciting frequen-cies of 10, 50, 80, 100, 150, and 200 Hz. The compari-sons between the simulation results predicted by themodified dynamic hysteresis nonlinear model and

experimental data under dynamic operating conditionsare shown in Figure 12. As is evident in these six fig-ures, the above modified model simulation results areperfectly coincident with the experimental data. It con-firms that the modified dynamic nonlinear model estab-lished in this article can quantitatively describe thecomplex hysteresis behavior of the giant magnetostric-tive transducer system under dynamic operating condi-tions within exciting frequencies of 100 Hz.

Conclusion

Based on the relationship between complex perme-ability and magnetization, eddy current effects ofTerfenol-D, and kinetic model of giant magnetostric-tive transducer, a novel general hysteresis nonlinearmodel is established in this article for the giant mag-netostrictive transducer system, in which strong cou-pling interaction between the nonlinear constitutivebehavior of Terfenol-D and the structural dynamicbehavior of transducer system itself is modeled, whichconsists of quasi-static hysteresis nonlinear model and

-0.3 -0.15 0 0.15 0.3-4

-2

0

2

4

i/A

y /µm

Modified model resultsExperimental data

-0.6 -0.3 0 0.3 0.6-12

-6

0

6

12

i/A

y /µm


(a) (b)

(c) (d)

-0.8 -0.4 0 0.4 0.8-20

-10

0

10

20

i/A

y /µm


-1 -0.5 0 0.5 1-40

-20

0

20

40

i/A

y/µ

m

Experimental dataModified model results

Figure 11. Verification for the quasi-static modified hysteresis nonlinear model of giant magnetostrictive transducer: (a)Im = 0.25 A, (b) Im = 0.5 A, (c) Im = 0.7 A, and (d) Im = 1 A.


dynamic hysteresis nonlinear model considering eddycurrent effect, power amplifier dynamic process, andmagnetic aftereffect dynamic process. After that, on thebasis of experimental data, a modified quasi-static anddynamic hysteresis nonlinear model is proposed, andsubsequently, the validity and reliability of the proposedmodified nonlinear model are verified by quantitatively

comparing its predicted results with the existing experi-mental data. The excellent agreements between the pre-dicted results and experimental data indicate thatnonlinear dynamic model established in this article canaccurately describe the complex hysteresis behavior ofthe giant magnetostrictive transducer system not onlyunder quasi-static operating conditions but also under

(a) (b)

(c) (d)

(e) (e)

-1 -0.5 0 0.5 1-30

-20

-10

0

10

20

30

i/A

y/µm


-1 -0.5 0 0.5 1-20

-10

0

10

20

i/A

y/µ

m


-1 -0.5 0 0.5 1-20

-10

0

10

20

y/µm

i/A

Experimental dataModified model results

-1 -0.5 0 0.5 1-20

-10

0

10

20

y/µ

m

i/A

Experimental data Modified model results

-1 -0.5 0 0.5 1-20

-10

0

10

20

i/A

y/µ

m


-1 -0.5 0 0.5 1-20

-10

0

10

20

i/A

y/µm

Experimetal dataModel results

Figure 12. Verification for the dynamic modified hysteresis nonlinear model of giant magnetostrictive transducer: (a) 10 Hz, (b)50 Hz, (c) 80 Hz, (d) 100 Hz, (e) 150 Hz, and (e) 200 Hz.

Zhu and Li 2253

dynamic operating conditions of less than 100 Hz.Furthermore, the present modified hysteresis nonlinearmodel of giant magnetostrictive transducer in this articlehas the following characteristics: agree well with experi-mental data, computationally simple and efficient, andreasonably easy to determine transducer parameters. Thepresent model’s parameters have definite physical mean-ing just like the Jiles–Atherton model, but the parametersto be identified are less than those of the other hysteresisnonlinear models such as the Jiles–Atherton model. Thepresent model’s expression is more simple than that ofthe other hysteresis nonlinear models such as Preisachmodel, which is more suitable for the real-time controlsystem, the simulation, and performance prediction ofgiant magnetostrictive transducer. Therefore, theresearch in this article provides a basic theoretical modelfor accurate characterization of the giant magnetostric-tive transducers, which can be used in model-based activevibration control design as well.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This work was supported by the National NaturalScience Foundation of China (grant number 51175243), theNatural Science Foundation of Jiangsu Province(BK20131359), the Aeronautical Science Foundation ofChina (grant number 20130652011), and FundamentalResearch Funds for the Central Universities (NS2013046).

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Zhu and Li 2255

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