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Journal of Applied Mathematics and Physics, 2014, *, **
Published Online **** 2014 in SciRes.
http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.*****
Generalized Dynamic Modeling of Iron-Gallium Alloy (Galfenol)
for Transduc-ers Yimin Tan1, Zuguang Zhang2, Jean Zu1 1. Department
of Mechanical and Industrial Engineering, University of Toronto,
Toronto, Canada 2. Advanced Mechatronics of Toronto, Inc., Canada
Email: [email protected] Received: June 7th, 2015
Copyright © 2014 by author(s) and Scientific Research Publishing
Inc. This work is licensed under the Creative Commons Attribution
International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract Developments and applications of magnetostrictive
longitudinal transducers using iron-gallium alloy (Galfenol) have
been investigated in the last decade because of Galfenol's
outstanding mechanical properties. For the further development of
these transducers, a generalized multi-physics and
macroscopic-based model is significantly required although several
phenomenal and physical models were established. In this re-search,
using the energy approach, a generalized dynamic model is derived
for Galfenol based on the mechanical strain theory and the
Jiles-Atherton model. Experiments have been conducted to measure
the relationship between the strain and the magnetic field. Using
experimental data, unknown parameters in the model have been
identified by a developed optimization algorithm. Results show that
the novel dynamic model with iden-tified parameters is capable of
describing the performance of the Galfenol rod. Simula-tion and
experiment dynamic responses of Galfenol rods are derived. The
simulation and the experiment both agree that the magnitude of the
strain output decreases with the in-crease of the excitation
frequency.
Keywords Galfenol; Transducer; Dynamics; Magnetostrictive
Material; Magnetization Model
1. Introduction
http://www.scirp.org/journal/jamphttp://dx.doi.org/10.4236/jamp.2014.*****http://creativecommons.org/licenses/by/4.0/
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
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With the advancement of high precision engineering, smart
materials have attracted a lot of attention in decades. Currently,
literatures show that these materials have been applied in
precision machining [1, 2], active anti-vibration system [3],
micro-motor [4], sonar device and energy harvesting [5], etc. There
are several fa-vourite smart materials : shape memory alloy,
piezoelectric material and traditional magnetostrictive materials
(Metaglass 2605SC and Terfenol-D). Although the above listed
materials have been employed in versatile ap-plications, either the
geometry complexity or the brittle nature prevents these materials
from further applications. The research status starts to change
upon the arrival of a novel magnetostrictive material. Galfenol, is
an alloy of iron and gallium and was first invented by United
States Navy researchers in 1998. Galfenol has several advan-tages
over other similar types of materials. First, the high tensile
strength and the high Currie temperature allow this material to
work in harsh environment. Second, Galfenol has an excellent
machinability [6] that increases this material's popularity,
especially in miniature applications. Third, Galfenol has a high
relative permeability of 50-100, which means that the material can
be driven with a small power supply and a relatively small wind-ing
coil. Fourth, Galfenol can be used with a simple configuration and
assembly, because Galfenol does not need a pre-stress mechanism.
This benefits from the high tensile strength and the built-in
anisotropic behaviour through the stress annealing process of
Galfenol.
Because of the above listed properties, Galfenol has attracted
attentions in the development of miniature trans-ducers. A
micro-magnetostrictive vibration speaker [7] using Galfenol as the
vibration source was built and this vibrator achieved a 1.2μm
displacement output with a wide bandwidth of 30 kHz . Thereafter,
instead of using a continuous Galfenol rod or bar as the driving
element, Ueno, Higuchi, Saito, Imaizumi and Wun-Fogle [8] precisely
machined the Galfenol into the U-shape to suppress the eddy-current
generation and form an efficient magnetic flux path. Later, this
design was used to build a linear motor [9] based on the smooth
impact drive mechanism. Similar to the work [8], Zhang, Ueno,
Yamazaki and Higuchi [10] built a closed magnetic loop Galfenol
transducer that ran at its resonant frequency of 4 kHz . Then, an
E-core Galfenol miniature vibrator [11] is developed which works
under tensile stress.
Since magnetostrictive transducers of Galfenol have attracted
tremendous attentions, a generalized dynamic model for the design
and control of Galfenol transducers is significantly required. For
the motion control pur-pose, Braghin, Cinquemani and Resta [12]
introduces a linear model for magnetostrictive transducers which is
valid in the frequency range within 2 kHz . However, this model is
built using the linear magnetomechanical equations which cannot
predict the nonlinear hysteresis behaviour of Galfenol
transducers.
To date, efforts have been made to model the hysteresis
behaviour of magnetostrictive materials. One of the most prevalent
and earliest models is the Preisach model [13, 14]. The Preisach
model is a pure mathematical model that employs hysteron that takes
the value of 1 or 1− depending upon current and previous states.
The model has been applied to address the hysteresis of
piezoelectric materials [15, 16], shape memory alloys [15, 17] and
Terfenol-D [18] so far. The major disadvantage of the Preisach
model is the lack of physical information. Another novel model is
the Armstrong model [19, 20] which is prevalently used to model
Galfenol. The total energy in this model consists of three parts:
(1) magnetocrystalline, (2) magnetoelastic and (3) magnetic field
energy terms. The magnetization of Galfenol under the applied
magnetic field and stress at a given direction de-pends on the
total energy corresponding to the prescribed orientation. The lower
the energy, the higher the probability that the magnetization would
occupy at the given direction. The general magnetization in
macro-scope is derived by performing the summation with probability
weighted over all directions. However, on the one hand, the
Armstrong model acquires the final response by the microscopic
analysis that requires a large amount of computation even for
simple geometrical structures. On the other hand, it is hard to
incorporate extra physical energy terms in the model, especially
when the material is employed in a complicated system. A new free
energy model [21, 22] has been established to account for the
hysteresis behaviour of magnetostrictive transducers from the point
of energy conservation. This free energy model considers that the
Gibbs energy for a given magnetic field has a certain relationship
with the Helmholtz free energy. More recently, a framework [23, 24]
based on the first and second laws of thermodynamics was developed
to describe the hysteresis behaviour of the magnetostrictive
material. It assumes that hysteresis results from domain changes
that were modelled by the elementary hysteron. The difference
between this framework and the Preisach model is that the hysteron
status in the new framework depends on the coercive energy that has
a comprehensive physical meaning. Thus, it is
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
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different from the Preisach model that this framework does not
rely on complex density functions to describe the hysteresis.
Another model for the magnetostrictive material is the
Jiles-Atherton magnetization model [25, 26]. In the ear-lier
research, the domain rotation and the domain wall pinning were put
in place to account for the energy loss caused by the hysteresis
while the modelling of anhysteretic magnetization was based on the
Langevin equation. Later, this model was redeveloped to consider
the temperature effect, the magnetocrystalline anisotropy and the
spatial effect [27-29]. Since the Jiles-Atherton magnetization
model is based on physics, not only this model can interpret the
response of Galfenol under the excitation, but also it can be
easily modified by adding additional energy terms to describe
additional physical effects. This feature leaves room for further
modifications based on specific applications. Recently, the
Jiles-Atherton model [30-32] has been applied to the ferromagnetic
and the magnetostrictive materials. However, this model has not
been applied in the modelling of Galfenol yet.
Mostly, the equation of motion for smart materials was derived
based on the Newtonian mechanics [30]. In the work of Sarawate
[33], a magnetostrictive transducer was modelled as a
single-degree-of-freedom resonator while the magnetostrictive rod
was simplified as a spring with the stiffness related to the
Young's modulus. Similarly, Braghin, Chinquemani and Resta [12]
modelled the magnetostrictive transducer as a lumped system with an
inertial mass and a spring-damper element. In his work, a linear
model was made to describe the me-chanical behaviour of the
magnetostrictive transducer. Systems, consisted of smart materials,
are usually multi-physical where the complexity of modelling is
significantly increased by electric field, magnetic field or
thermal field, etc. It's an universal way to analyse multi-physics
problems with energy approach which renders researchers the ability
to model transducers from an overall perspective.
The objective of this paper is to present a generalized dynamic
model of Galfenol for longitudinal magnetostric-tive transducers
using the energy approach. The Jiles-Atherton model is applied to
describe the hysteresis be-haviour of Galfenol. The newly developed
model is capable of describing the inherent nonlinear behaviour.
Ex-periments have been conducted for the characterization and the
validation of the new model. The unknown pa-rameters from the
Jiles-Atherton model have been identified using the experimental
data.
2. Modeling
2.1. Dynamic Modelling Using Energy Approach The Hamilton's
principle is applied in classical fields, e.g. the magnetic field
and the electrical field, and the even quantum field. So far, the
Hamilton's principle has already been employed in, for instance,
the modelling of energy harvesters using smart materials [5, 34].
Unlike previous researches who make the assumption of linear
material properties, this study considers the inherent nonlinear
behaviour of Galfenol.
The sketch, Figure 1, simplified axial-deformation applications
of Galfenol. The parameter LK denotes the stiffness of a spring,
the parameter ( )Lf t is used to model a time-variant external
forces applied on the system, and the accessories' equivalent mass
and the system's structural damping are modelled as LM and LC ,
respec-tively. Under the assumption that this system is
scleronomous, Galfenol is modelled as a continuous rod in the
scheme. For Galfenol materials, an appropriate pres-tress can
enlarge the final strain output significantly [35, 36]. However,
Galfenol can be developed with a built-in stress using the stress
annealing process to avoid the redundant pre-stress mechanism. In
this modelling, the possible pre-stress is considered for the
general applica-tion purpose. Pre-stress and amplification
mechanisms in the system are modelled as an equivalent linear
spring.
The equations of motion are derived from the Hamilton's
principle,
2
1
[ ( ) ] 0t
MtT U W W dtδ δ− + + =∫ (1)
where T is the kinetic energy, U is the strain energy, mW is the
magnetic energy, and W is the work done by the external force. For
the Galfenol rod, the relationships between strain, stress and
magnetostriction are formulated as E Eσ ε λ= − . The parameter E is
the Young's modulus of Galfenol.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
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Figure 1. Mechanical model for a Galfenol rod.
According to Figure 1, energy terms in the Hamilton's principle
are expressed as,
( )2 21 12 2m L x LVm
T u dV M uρ=
= +∫
2 200
1 1+ ( ) +2 2LVm VmL x L
AU d dV K u dV
K Eσ
σ ε σ=
= −∫ ∫ ∫
22 20
01 1 1+2 2 2LVm Vm VmL
x L
AE dV E dV K u dV
K Eσ
ε ελ σ=
= − − +
∫ ∫ ∫
0 0( ) ( )
t
m VW M H dH t dVµ= +∫ ∫
212( )
L x L
L x L x Lx L
C uW f t u u
uδ δ δ
=
= ==
∂ = − −
∂
(2)
In the above equations, u is the mechanical deformation of the
Galfenol rod, A is the cross-sectional area of the Galfenol rod, mρ
is the material's density, 0σ is the pre-stress, and 0µ is the
permeability of vacuum. The parameter M and H represent the
magnetization and the magnetic field strength. Applying expressions
of the above energy terms into Equation (1), the following
expression is derived,
2
10[ ( ) | ( ) ( ) ( ) | ]
t
m L x L L x Lt Vm Vm Vmu udV M u u E dV E dV K u A u dtρ δ δ ε
δε δε λ σ δ= =+ − + − −∫ ∫ ∫ ∫
( ) ( )21
00[ ( ) ( ) ( ) | ] 0
t t
L L x Lt Vm VE dV M H dH t dV f t C u u dtε δλ δ µ δ =+ + + − +
=∫ ∫ ∫ ∫ (3)
The term represents the magnetic energy is derived from the
magnetization model in the later section. Solving Equation (3), the
equations of motion are written as,
0mu Eu Eρ λ′′ ′− + = (4) with boundary conditions of
0| 0xu = =
( )0| | | ( ) | 0L x L x L L x L L L x LM u EAu K u A EA t f t C
uσ λ= = = =′+ + − − + + = (5) and initial conditions of
(0, ) 0u x = ( )0, 0u x = (6)
2.2. Model of Magnetization Hysteresis
The Jiles-Atherton model, a physical-based model for the
magnetization, models the magnetization hysteresis as a function of
domain rotation and wall pinning. Because the Jiles-Atherton model
is a physical-based model with limited parameters, redundant
experiments are not needed for the identification. This model is
expanded into the following equations [37],
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
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( ) cothan sH M aM H M
a H Mα
α + = − +
(7)
( )**1
1irr anM M c M
c= −
− (8)
eH H Mα= + (9)
* *(1 ) irrane
dMM M k c
dHδ= − − (10)
where anM is the local anhysteretic magnetization, irrM
represents the irreversible magnetization, α is the domain
interactions coefficient, *c is the reversibility coefficient, and
the constant k is the hysteresis loss coefficient. The parameter *δ
is defined as 1 when 0dH dt ≥ and as -1 when 0dH dt ≤ . The
parameter
sM is the saturation magnetization, a characterizes the shape
parameter, and eH denotes the effective field.
The energy input is represented by 0 an eM dHµ ∫ . Integrating M
over eH , the magnetization energy 0 eMdHµ ∫ is derived. In a
similar way, the hysteresis loss is obtained by integrating the
irreversible magnetiza-
tion,
( )* *0 1hysloss irrE k c dMµ δ= − ∫
* * *0 0
ane e
e e
dMdMk dH k c dHdH dH
µ δ µ δ= −∫ ∫ (11)
The energy method is used to derive the relationship between the
magnetization and the applied magnetic field. For Galfenol
materials, the energy loss due to the eddy-current cannot be
omitted. Based on the law of electro-magnetic induction, the eddy
current power loss per volume is derived as ( ) ( )221 2 ed dB dtρ
β ⋅ [38]. The term assumes an uniform distribution of the magnetic
flux density for materials with low permeability. For materials
with high permeability, it is inaccurate to directly assume that
the magnetic flux density distributes uniformly within materials.
Thus, in order to describe the behaviour of Galfenol that has a
relatively high permeability, a correction coefficient η is
introduced to the classic eddy-current loss term to compensate the
inaccuracy of the
assumption. Then, the eddy-current loss can be written as ( ) (
) ( ) ( )2 22 2 21 0 12 d 2 de ed dB t d dM tρ βη µ ρ βη⋅ ≈ ⋅ ,
where 1d is the diameter of rod, eρ is the resistivity, β is the
geometry factor. For the system, the following energy equation [37]
is satisfied,
22 2* * * 0 1
0 0 0 0 + 2 dan
an e e e ee e e
dM ddM dMM dH MdH k dH k c dH dtdH dH t
µµ µ µ δ µ δ
ρ βη = + − ∫ ∫ ∫ ∫ ∫ (12)
Through mathematical manipulations, the above equation is
rewritten as,
22* * * * *0 1 +
2an an
an ane e e
d dM dMdH dM dMM k k c M M M k cdt dH dH dH dH
µα δ α δ α δ
ρ βη + − − = − +
(13)
Combining equations (7), (9) and (13), the magnetization M is
numerically obtained.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
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According to the quadratic moment domain rotation model [39],
the relationship between the magnetization and the magnetostriction
is modeled as follows,
( ) ( )223, ,2
s
s
t x M t xMλ
λ = (14)
where sλ
is the saturation magnetostriction.
2.3. Solution
To solve equations of motion shown in Equation (4), the model
analysis is conducted. The mode shape of the Galfenol rod is shown
below,
( ) sin nn x xcω
φ =
(15)
Using the discrete Galerkin method, differential equations can
be written in the following matrix form as,
( ) ( ) ( ) ,0t t t t T+ + = ≤ ≤Mc Cc Kc f (16)
(0) =c 0 (17)
(0) =c 0 (18)
where the N N×
mass matrix M , the damping matrix C , and the stiffness matrix
K are defined as,
( )0
( ) ( ) ( ) ( )L
ij m i j L i jM x x dx M A L Lρ φ φ φ φ= +∫
( ) ( ) ( )ij L i jC C A L Lφ φ=
( ) ( ) ( )0
( ) ( )L
ij i j L i jK E x x dx K A L Lφ φ φ φ′ ′= +∫ , respectively.
The source term f is rewritten as ( )( ) ( )( ) ( )j L j jf f t
A L E t Lφ λ φ= − + . So far, the magnetization model, the
magnetostriction model and the equation of motion have been
combined by a set of ordinary differential equations. The equations
are solved for the relationship between Galfenol's dis-placement
and the applied magnetic field. The Jiles-Atherton model is used to
describe the magnetization proc-ess within the Galfenol rod. As
this model is first employed on Galfenol, five typical parameters
of the Jiles-Atherton model are unidentified at this stage. Also,
the energy loss related to the eddy-current also brings a
correction parameter. In order to employ the built model in
practical applications, these unknown parameters are to be
identified. For this highly nonlinear system, the simulated
annealing algorithm [40, 41] has been developed to find the
solution. In the developed algorithm, the strain output of Galfenol
has a certain function with respect
to parameters, which is represented by ( ), , , , ,su f a c k Mα
η= . The error is defined as 2
1
n
i ii
y y∧
=
−
∑ , where iy
represents the experimental data and iy∧
is the simulation result based on the current parameter set. In
the fol-lowing process, the nonlinear least-square algorithm is
used to acquire the optimal solution.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
7
3. Experimental Design
3.1. Structure Design
In the previous section, a generalized model for Galfenol has
been built and the solution is provided. It is neces-sary to
develop and conduct experiments for the characterization and the
validation of this model. The stress annealed Galfenol sample (
81.6 18.4Fe Ga ) for this research is provided by ETREMA
PRODUCTS.
Figure 2 shows the compact design for experiments. A
reciprocating output is generated by providing an alter-nating
current to the excitation coil surrounding the Galfenol rod.
According to the Ampere's circuital law, the relationship between
the current and the magnetic field can be acquired. To achieve the
largest energy efficiency, two end caps are used to ensure that the
magnetic field distributes axially within the Galfenol rod. The
assem-bled belleville spring and the set screw allow for the fine
adjustment of the pre-stress. However, because the sample is stress
annealed, there is no pre-stress applied to Galfenol in this
experiment. An internal halting ring is employed to locate entire
components. A housing is fabricated to offer protection for the
Galfenol rod from the ambient disturbance. To prevent the
generation of the eddy-current, the housing is grooved along the
axial direc-tion. Furthermore, to make the magnetic reluctance as
small as possible, all parts of the assembly are made of stainless
steel. According to the manufacturer, Galfenol is capable of
generating a strain output up to 250 ppm . In this design, the mass
of the output rod is negligible compared to the stiffness of the
system and there is no external force applied. Thus, the end mass
LM , the structural damping LC , and the external force ( )Lf t in
Equation (2) are assumed to be zeros.
Figure 2. Experiment structure design.
3.2. Test Platform Harmonic current without bias is applied to
the driving coil using a current amplifier (LA220) whose power is
supplied by ST-1 Regulated Power Supply (Power Output 60W). The
following instruments are employed to evaluate the response of the
Galfenol rod. First, the output displacement of the Galfenol rod is
measured using the laser unit with the modulus of Polytec OFV534F.
The modulus of the controller is OFV5000 that measures the
displacement of the output rod's top surface in micro-scale. The
resolution chosen for this measurement is 2μm/V . The bandwidth of
the laser unit for the displacement mode is from DC to 10 kHz which
covers the frequency range of the measurement. Second, a current
probe (Tektronix TCP0030) is used to measure the exact current that
is supplied by the amplifier. Finally, signal outputs from these
sensors are acquired and processed by the Tecktronix DPO 3014
oscilloscope. Figure 3 shows the described experimental scheme.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
8
Figure 3. Experiment scheme.
4. Results and Discussion
4.1. Experimental Results
The Galfenol rod chosen for this test has a diameter of 5mm and
a length of 20 mm . The experiment was conducted at frequencies
ranging from 0.1Hz to 100 Hz . Figure 4 shows experimental results
of the strain output versus the magnetic field under the
quasi-static condition. The general trend of the Galfenol sample is
that this material begins to reach saturation when the magnetic
field is larger than 150Oe . The nonlinear be-haviours, both
hysteresis and saturation, are exhibited by the experimental
result. It is noticed that the hysteresis of the response increases
with the rising of the excitation frequency. Physically, there is
an explanation for the increased severity of hysteresis. The
increasing frequency of the applied magnetic field leads to the
generation of a larger eddy-current which results in a much severer
hysteresis.
Figure 4. Experimental strain vs. magnetic field at different
excitation frequencies.
Because Galfenol is a ferromagnetic material, the effect of eddy
current on Galfenol has to be verified for the further application
purpose. A contrast test was conducted to prove the eddy-current's
influence on the system's nonlinear behaviour. The comparison
between the original Galfenol rod 5mmd = and the Galfenol rod
3mmd = is shown in Figure 5. In order to compare the hysteresis
in a uniform scale, the ratio of the strain
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
9
output and the saturation output is plotted. The response of the
Galfenol rod 3mmd = is displayed with the cyan line while the
Galfenol rod 5mmd = is displayed with the pink line. At the
quasi-static frequency of 0.1Hz , the responses of two rods do not
have remarkable differences. While the excitation frequency
increases to 1Hz , the discrepancy between the responses of two
rods is gradually marked. When the frequency reaches 10 Hz , not
only the curves of the strain output versus the magnetic field for
both rods are significantly 'fattened', but also the hysteresis of
the Galfenol rod 5mmd = is remarkably larger than the Galfenol rod
3mmd = . The results indicate that the increased hysteresis is
mainly caused by the eddy-current loss.
Figure 5. Experimental results of galfenol rods with different
diameters.
4.2. Model Results
The parameter identification process were conducted based on the
previously described algorithm. The experi-mental response under
the excitation frequency of 0.1Hz was used to derive parameters by
fitting the model to the experimental data. The experimental data
whose frequency equals to 100 Hz was selected to identify the
eddy-current loss correction coefficient. The fitting result and
the material property in the fitting are listed in Table 1.
Figure 6 shows the comparison between the simulated result and
the experimental result at various excitation frequencies. The
horizontal axis indicates the alternating of the applied magnetic
field strength while the vertical axis shows the strain output of
the rod. Nonlinear behaviours, e.g. the quadratic behaviour, the
hysteresis, and the saturation, are characterized by this model.
The quadratic behaviour reveals that the deformation of the
Galfenol rod only relates to the strength of the applied magnetic
field. The hysteresis behaviour and the energy dissipation by the
eddy-current are explicitly observed. When the excitation frequency
is 0.1Hz , the response rarely exhibits hysteresis. With the
increase of the excitation frequency, the hysteresis becomes
severer. While the frequency is 100 Hz , the hysteresis becomes
distinct for the system. The built model accurately describes the
influence of the eddy-current on the hysteresis of Galfenol.
Experimental and model responses in the time domain are shown in
Figure 7. In addition, the frequency analysis is conducted on the
model response and the experimental response of the Galfenol rod
5mmd = , as shown in Figure 8. It is noted that the model
accu-rately describes frequency components of experimental results.
It is also disclosed that discrepancies between model results and
experimental results gradually become slightly larger with the
rising of the excitation fre-quency. This could be because the
model is built based on the amendatory classic eddy-current loss.
Although a correction coefficient is introduced to account for the
non-uniform distribution of the magnetic flux density, it is not
precise enough for the high-precision application of Galfenol.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
10
Table 1. Material properties and fitted parameters.
Parameter Physical Meaning Value
E Young's Modulus 40Gpa
Material Properties
ρ Density 37800 kg/m
sλ Magnetostriction 250 ppm
eρ Resistivity 78.5 10 Ω×m−×
a Shape Parameter 32.2622 10 A/m×
α Domain Interactions Coef-
ficient 0.0044−
Fitted
Parameters c Reversibility Coefficient 0.7476
k Hysteresis Loss Coefficient 31.4842 10 A/m×
sM Saturation Magnetization 61.3447 10 A/m×
η Eddy-Current Correction
Coefficient 0.026
In order to evaluate the coincidence of the model result and the
experimental data, an error analysis has been conducted. The
simulation error of the model compared to the experimental result
is defined as an acceptance factor given by the following
equation,
( )
2
1
2
1
n
i ii
n
ii
y yError
y
∧
=
=
− =
∑
∑ (19)
The parameter iy represents the measured experimental data while
iy∧
is the estimated data using the model. Figure 9 shows the
fitting error against the excitation frequency. Although the error
gets larger when the frequency is above 10 Hz , the result reveals
that the error of the fitting is generally below 10% . This model
can still describe the mechanical response under given experimental
conditions.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
11
Figure 6. Galfenol rod strain responses at different excitation
frequencies.
Figure 7. Galfenol rod time domain responses at different
excitation frequencies.
Figure 8. Strain orders at different excitation frequencies.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
12
Figure 9. Error (acceptance factor) of fittings vs. excitation
frequencies.
Simulation and model results of the dynamic response are plotted
in Figure 10. The horizontal axis represents the excitation
frequency of the Galfenol. In Figure 10, the solid blue curve and
the blue dashed curve reveal the model frequency response and the
experimental frequency response of the Galfenol rod ( 5mmd = ),
respec-tively. The red curves exhibit frequency responses of the
Galfenol rod ( 3mmd = ). The build model can de-scribe the dynamic
response of the material. Both strain outputs of Galfenol rods show
a decline trend along with the rising of the frequency due to the
eddy-current. It should be noticed that when the diameter of
Galfenol is reduced, the strain output ability of Galfenol is
weakened because of the enlarged equivalent boundary spring
stiffness ( /LK A ). This phenomenon also shows that the built
model is able to describe the mechanical behav-iour of
Galfenol.
4.3. Discussion
The purpose of this work is to provide a generalized model that
is used for the subsequent design and control of Galfenol
transducers. The correction coefficient in the model can be
conveniently modified based on the ex-perimental data for specific
applications. For the precision application, the accuracy of the
model can be im-proved by deriving the distribution of the magnetic
flux density based on the Maxwell equations. Instead of us-ing the
eddy-current power loss term, a more precise model result can be
derived employing the specific distri-bution of the magnetic flux
density.
Figure 10. Dynamic reponses of Galfenol rods with different
diameters.
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YIMIN TAN, ZUGUANG ZHANG, JEAN ZU
13
The experiment was conducted at frequencies ranging from 0.1Hz
100 Hz− . When the excitation frequency is larger than 100 Hz , the
response of the experiment is too small due to the increased energy
consumption by the eddy-current. Not only the reduced response will
lead to the decreased signal to noise ratio in measurement, but
also the response is meaningless to transducers. Theoretically, the
verification of the model at higher fre-quencies can be conducted
using smaller samples.
Due to the limitation of authors' manufacturing facilities,
Galfenol rods with smaller diameters were not fab-ricated. However,
in authors' following work, the fork-shape Galfenol (rectangular
cross-section dimension
1mmd = ) was made using the wire EDM method. In the follow-up
experiment, it is found that Galfenol with small geometries suffers
less from the eddy-current effect, which leads to a broader
bandwidth. More or less, this phenomenon can be noticed in Figure
10. When the frequency increases, strain outputs of Galfenol rod (
5mmd = ) decrease significantly. Meanwhile, the Galfenol rod ( 3mmd
= ) does not show a strong decline trend. It indicates that
Galfenol has a strong potential to be utilized in the development
of vibrators with minia-ture sizes.
5. Conclusion
In this paper, a generalized dynamic model of Galfenol for
longitudinal magneostrictive transducers is devel-oped. The
equations of motion are derived for Galfenol rods using the energy
approach. This approach enables transducers to be analyzed from the
macro perspective with energy terms correlated. The Jiles-Atherton
model, a nonlinear magnetization model, is employed to quantify the
magnetization energy of the Galfenol rod. In order to identify
unknown parameters in the Jiles-Atherton model for Galfenol, an
optimization program has been de-veloped based on the simulated
annealing and the nonlinear least square algorithms. A compact
testing structure and a test rig have been developed to
characterize and validate the built model. Experimental results
show that the time response and the frequency response of the
material are accurately predicted by the model. In the fre-quency
range of 0.1Hz 100 Hz− , the discrepancy between the simulative
strain output and the experimental strain output is generally
within 10% . The accuracy of the model can also be improved by
employing the spe-cific magnetic flux density distribution instead
of using the eddy-current loss term. In addition, it is observed
that the hysteresis of the response increases with the rising of
the excitation frequency. Meanwhile, the dynamic simulation result
and the experiment result both agree that the magnitude of the
strain output decreases with the increase of the excitation
frequency. Not only the work provides a generalized model for
longitudinal transduc-ers of Galfenol, but also the energy-based
model can be easily modified for complex applications of the
Galfenol material. The future work will experimentally focus on the
dynamics of miniature Galfenol structures at higher frequencies.
Also, the evaluation of the model will be conducted at higher
frequencies based on minia-ture Galfenol structures.
Acknowledgements
This work was supported by the Natural Science and Engineering
Research Council of Canada and the State Key Program of National
Natural Science of China (No. 11232009).
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