finite element modeling of magnetostrictive smart structures

FINITE ELEMENT MODELING OF MAGNETOSTRICTIVE
SMART STRUCTURES
J.M. Bakhashwain
Electrical Engineering Department
and
M. Sunar* and S.J. Hyder
Mechanical Engineering Department
King Fahd University of Petroleum & Minerals
Dhahran, Saudi Arabia
ﺔــﺻﻼﺨﻟا
ﻰﻠﻋ يﻮﺘﺤﺗ ﻲﺘﻟا ﺔﻴآﺬﻟا ﺔﻳﺮﺼﺤﻟا ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا داﻮﻤﻠﻟ ﺔﻴﺴﻴﺳﺄﺘﻟا تﻻدﺎﻌﻤﻟا ﺔﻗرﻮﻟا ﻩﺬه مﺪﻘﺗ فﻮﺳ
ةدوﺪﺤﻤﻟا ﺮﺻﺎﻨﻌﻟا ﺔﻘﻳﺮﻃ ﺖﻣﺪﺨﺘﺳا ﺪﻗو . نﻮﺘﻠﻣﺎه أﺪﺒﻣ ماﺪﺨﺘﺳﺎﺑ ﻚﻟذو
، ﺔﻴﻜﻴﻧﺎﻜﻴﻣو ﺔﻴﺴﻴﻄﻨﻐﻣ تﻻﺎﺠﻣ
ﺪﻘﻓ ةرﻮآﺬﻤﻟا تﻻدﺎﻌﻤﻟا لﺎﻤﻌﺘﺳا ﺢﻴﺿﻮﺘﻟو . ﺔﻴآﺬﻟا داﻮﻤﻟا ﻩﺬﻬﻟ ﻲﻜﻴﻣﺎﻨﻳﺪﻟا وأ لﺎﻌﻔﻟا كﻮﻠﺴﻟا ﺔﺟﺬﻤﻨﻟ
ﻢﺗ ﺪﻗو . (CoFe2O4)
ﺔﻳﺮﺼﺣ ﺔﻴﺴﻴﻃﺎﻨﻐﻣ ﺔﻘﺒﻃ ﺎﻬﺤﻄﺳ ﻮﻠﻌﺗ ﺔﻴآذ
ﻪﺿرﺎﻋ ﺔﻟﺎﺣ ﺔﺳارﺪﻟ ﺖﻣﺪﺨﺘﺳا
ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا ﺔﻘﺒﻄﻟا نﺎﻜﻣ ﺮّﻴـﻐﺗ
ﻊﻣ ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا ﺔﺑﺎﺠﺘﺳﻻا ﻚﻟﺬآو
. ﺔﻣﺪﻘﻤﻟا ﺪﻨﻋ ﺮﻴﻴﻐﺘﻟا ﻲﺒﻴﺟ ﻞﻤﺣ وأ ﺊﺟﺎﻔﻣ
ﻞﻤﺤﻟ ضﺮﻌﺘﺗ ﺎﻣﺪﻨﻋ ﻚﻟذو
، ﻪﻴﺳأﺮﻟا
ﻪﺣازﻹا
ﺮّﻴـﻐﺗ
بﺎﺴﺣ
، ﺔﺿرﺎﻌﻟا لﻮﻃ ﻰﻠﻋ ﺔﻳﺮﺼﺤﻟا
ﺎﻣﺪﻨﻋ ﺔﻴﻟﺎﻌﻓ ﺮﺜآأ نﻮﻜﻳ ﺔﻳﺮﺼﺤﻟا ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا ةﺮهﺎﻇ ﻰﻠﻋ ﻲﻨﺒﻤﻟا ّﺲﺠﻤﻟا نأ ﺔﺳارﺪﻟا ﻦﻣ ﻦﻴﺒﺗو
. ﺔﺿرﺎﻌﻟا ﻦﻣ ﺖﺑﺎﺜﻟا فﺮﻄﻟا ﺪﻨﻋ ًﺎﺒآﺮﻣ نﻮﻜﻳ
*Address for Correspondence:
KFUPM Box 1205
King Fahd University of Petroleum & Minerals
Dhahran 31261
Saudi Arabia
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 125

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
ABSTRACT
Constitutive equations of magnetostrictive smart materials containing mechanical
and magnetic fields are presented via Hamilton’s principle. Finite element method is
used to model the dynamic behavior of these smart materials. A smart beam structure
consisting of a magnetostrictive (CoFe2O4) layer on its top surface is considered as a
case study to illustrate the use of magnetostrictive equations. The location of the
magnetostrictive layer is varied along the beam and the vertical displacement and
magnetic responses are computed when the system is subjected to step as well as
sinusoidal loads at the tip. It is found from the case study that the magnetostrictive
sensor is most effective when it is placed at the fixed end of the beam.
Keywords: magnetostrictive sensor, finite element method, vibrations, smart structure.
126 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
FINITE ELEMENT MODELING OF MAGNETOSTRICTIVE SMART STRUCTURES
INTRODUCTION
Due to advances in control related technologies there have been an enormous number of research activities in so-called
smart materials during the last decade. A smart material involves interactions between different fields, which allow the
material to be used as a sensor and/or actuator in some manner. Hence the idea of a smart or an intelligent structure has
been introduced to refer to structures that contain smart materials and have self-sensing and adaptive capabilities [1, 2].
The phenomenon of magnetostriction is defined as an interaction between mechanical and magnetic fields in a body.
This phenomenon was first discovered by Joule in 1842 [3]. Some of the magnetostrictive materials that can be
mentioned are CoFe2O4, Ni, Alfenol, and Terfenol-D. A numerical simulation scheme for magnetostrictive transducers
based on magnetic vector potential formulation was proposed by Kaltenbacher et al. [4]. It was observed that the
coupling of magnetic and mechanical systems induced mechanical strains and permeability changes. Experimental and
numerical results were compared and a reasonably good agreement was obtained between the results. The deformation of
the magnetic material caused by the magnetostriction was represented by an equivalent set of mechanical forces by
Delaere et al. [5]. The resulting magnetostriction force was superimposed on the other force distributions, which was the
factor for the coupling of mechanical and magnetic systems. A two-dimensional finite element model was used by
Mohammed [6] to evaluate the force components in magnetostriction phenomenon. The strains developed due to the
presence of the magnetic field generated electrical and mechanical forces that created undesirable acoustic noise at low
frequencies. A combined active and passive damping strategy was proposed by Bhattacharya et al. [7] for the control of
flexible structures using magnetostrictive and ferro-magnetic alloys. The performance of Terfenol-D was affected by
preloading applied to the material. Finite element simulations on a cantilever beam model was carried out to study the
dynamic characteristics resulting from the proposed damping approach. An approach was presented by Yamamoto et al.
[8] for monitoring and controlling the preloading in a compact giant magnetostrictive positioner. The proposed technique
provided a precise measurement of the preloading as well as a sensing capability proportional to the displacement of the
positioner.
A thermopiezomagnetic medium was formed by Sunar et al. [9, 10] via bonding together a piezoelectric and
magnetostrictive composites. Finite element equations for thermopiezomagnetic media were obtained using linear
constitutive equations in Hamilton’s principle together with the finite element approximations. It was shown that an
electrostatic field applied to the piezoelectric layer causes strain in the structure that in turn produces a magnetic field in
the magnetoceramic layer.
The objective of this paper is to develop the dynamic equations of a magnetostrictive medium via the Hamilton’s
principle and the finite element method, and to utilize these equations in obtaining the magnetic responses of a
magnetostrictive layer bonded to the top surface of a cantilever beam structure. The structure is acted upon by step and
sinusoidal forces at the tip. The location of the magnetostrictive sensor is varied along the beam in order to observe its
effect on the magnetic response of the sensor. It is well-known that the locations of sensors and actuators are very
important to their performance and effectiveness. Research studies were conducted for piezoelectric sensors and
actuators by Sunar and Rao [11], and Sunar [12] for the investigation on the impact of placement. The current work
extends these studies to the case of magnetostrictive sensors.
CONSTITUTIVE EQUATIONS OF MAGNETOSTRICTION
The following equations are written for a magnetostrictive medium [10]
T = G ∂
= cS − B
∂S
H = G
∂
∂B = −Τ S + µ −1 B (1)
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 127

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
where c and denote the matrices of elastic and magnetostrictive stress constants; µ represents the matrix of material
permeability; T, S, H, and B are the vectors of stress, strain, magnetic field intensity, and magnetic flux density,
respectively. It can be shown that [10] the thermodynamic potential G takes on the following quadratic form:
G = 1
2 S T cS + 1
2 BT µ −1 B − S T B
= 1
2 S T T + 1
2 BT µ −1 B, (2)
using the relation in Equation (1). The Hamilton’s principle is given as
t2
t1
δ∫ ( Ki−Π) dt=
0, (3)
where Ki is the kinetic energy and ∏ is an energy functional, which are defined by
and
1 T
Ki = ∫ ρuu
dt
(4)
V 2
Π =
∫ ∫ ∫ ∫ ∫
GdV u P dV u P dS A JdV A H′ n dS
(5)
T T T T
− b − s − +
V V S V S
where Pb and Ps are the vectors of body and surface forces, u, A, J, and n are the vectors of mechanical displacement,
magnetic potential, volume current density, and surface normal and H′ E is the matrix of external magnetic field intensity,
which is defined in cartesian coordinates as
The variation in G is found as
⎡ 0 Hz −H
⎤ y
⎢ ⎥
HE′ = ⎢−Hz 0 Hx
⎥
⎢ Hy −Hx
0 ⎥
⎣ ⎦
δG = δS T cS + δB T µ −1 B − δS T B− δB T Τ S
= δS T (cS − B) + δB T (− Τ S + µ −1 B)
E
128 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004
E
. (6)
= δS T T + δB T H, (7)
using Equation (1) and the variation in Ki is written as
t2 t2
T
∫ Ki dt
t ∫ dt dt
1 t ∫ uu . (8)
1 V
δ = − ρδ
Substitution of the above equations in Equation (3) yields the following result
t2 t2
⎡
T T T T T
∫ ( ) ∫ ⎢ ∫ ( uu ST BH uPb AJ)
δ Ki−Π dt= dt −ρδ −δ −δ + δ +δ dV
t1 t1 ⎣ V
T T
( s
E′
)
H dS
S
⎤
+ ∫ δu P −δA
n
⎥
= 0. (9)
⎦

Introducing the following relations
J.M. Bakhashwain, M. Sunar, and S.J. Hyder
S = Lu u , B = ∇ × A = LA A (10)
where Lu and LA are differential operators, and substituting Equation (10) into Equation (9) results in the following
equation
t2
t1V (
⎡
T T T T T
dt ⎢ −ρδuu − ( Luδu) T−( ∇ × δ A) H+ δ uP+ δAJ
dV
⎣
∫ ∫ b
T T
( E′
∫ s )
+ δ −δ
S
⎥
⎦
Note the following relations for some terms in Equation (11)
and
H dS ⎤
u P A n = 0. (11)
∫ ∫ ∫
T T T T
V
u
V
u
S
( L δ u) TdV =− δ u ( L T) dV + δu
( NT ) dS
∫ ∫ ∫
T T T
V V V
( ∇×δ A) HdV = ∇ ( δ A× H) dV + δA ( ∇× H ) dV
∫ ∫
T T
= − δ A H′ ndS + δA ( ∇× H ) dV.
(12)
S V
In Equation (12) N is defined in cartesian coordinates as
⎡nx 0 0 ny n 0 ⎤ z
⎢ ⎥
N = ⎢ 0 ny 0 nx 0 nz⎥
⎢ 0 0 nz0 nx n ⎥
⎣ y⎦
where nx, ny, and nz are the components of the surface normal n, and H′ is defined as in Equation (6). Equation (11) is
rewritten as
t2
⎡ T T T
∫ ⎢ ∫ δu ( −ρ u + uT
+ Pb) + ∫ δA ( −∇× H + J)
dt L dV dV
⎣
t1V V
T T
( N ) dS ( HEH ) dS
S S
⎤
+ δ − − δ ′ − ′
∫ u Ps T ∫ A n
⎥
= 0. (14)
⎦
Hence, the following equations must be satisfied with appropriate boundary conditions on the corresponding surfaces
−ρ u + L T + Pb=
0
T
u
−∇ × H + J = 0.
(15)
The above equations are the equations of motion for the mechanical field and the Maxwell’s equilibrium equation for the
magnetic field, respectively [13].
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 129
)
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J.M. Bakhashwain, M. Sunar, and S.J. Hyder
FINITE ELEMENT FORMULATION
For the finite element formulation, u and A are chosen variables for the mechanical and magnetic fields, respectively.
The finite element approximations are written as
ue = Nu ui
Ae = NA Ai
where the subscript e and i respectively stand for the element and nodes of the element and N’s are the appropriate shape
function matrices. Noting the following relations
and
Se = Lu ue = [Lu Nu] ui = Bu ui
Be = LA Ae = [LA NA] Ai = BA Ai
and substituting Equations (1), (16), and (17) in Equation (11) yields the finite element equations as
Muu u + Kuuu − KuA A = F
−KAuu+ KAA A = M (18)
where u, A, F, and M are the global vectors of mechanical displacement, magnetic potential, applied mechanical, and
magnetic excitations, respectively. The element matrices and applied excitations in Equation (18) are found as
[Muu]e =
[KuA]e =
Fe =
Me =
∫
Ve
e
T
u u
ρ N N dV , [Kuu]e =
T
∫ Bu eBAdV , [KAA]e =
Ve
∫ ∫
∫
∫
Ve
Ve
T
u e u
T T T
Nu Pbe dV + Nu PsedS + Nu
P ce
Ve Se
∫ ∫
T T
A e − A Ae
Ve Se
B c B dV
T −1
Aµ e A
B B dV
N J dV B H′ n dS
(19)
where KAu= KuA T and the concentrated force Pce is added to the definition of Fe.
CASE STUDY
A cantilever steel beam shown in Figure 1 is taken as an example to illustrate the use of the derived magnetostrictive
equations. A magnetostrictive (CoFe2O4) layer is bonded to the structure on its top surface and the distance of the layer
from the fixed end of the beam is denoted by d. The layer is acting as a sensor in response to excitations applied to the
beam. The material properties for the steel and CoFe2O4 are listed in Table 1. The length, width (out of plane) and
thickness of the beam are taken as 0.12 m, 0.02 m, and 0.001 m, respectively. For the magnetostrictive sensor, the
assumed dimensions are 0.004 m for the length, 0.02 m for the width, and 0.002 m for the thickness. Two-dimensional
rectangular elements are used for the finite element formulation of the magnetostrictive sensor and beam elements with
axial degrees of freedom for the beam structure. The sensor is divided into two finite elements and the division of the
beam into finite elements is tuned according to the location of the sensor from the fixed end of the beam. More
specifically, the number of beam elements varies from 8 elements to 13 elements. In all the simulations below, the
system is assumed to have a damping ratio of ζ = 0.01, which is typical for a physical system like this.
130 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004
(16)
(17)

Transient Response
d
Magnetostrictive Layer
Beam
Figure 1. Magnetostrictive system.
Table 1. Properties of Materials.
Magnetostrictive Ceramic (CoFe 2O 4)
c 11 (N/m 2 ) 1.54×10 11
31 (N/Wb or A/m) 2.86×10 8
µ 11, µ 33 (H/m) 20π×10 −7
ρ (kg/m 3 ) 7500
Steel
c 11 (N/m 2 ) 2.07×10 11
ρ (kg/m 3 ) 7800
J.M. Bakhashwain, M. Sunar, and S.J. Hyder
The beam is subjected to a unit step force acting upward at its tip. The variable d for the location of the sensor is
assumed to be taking on three values: 0 m, 0.03 m, and 0.06 m. The vertical tip deflection of the beam as well as the
vertical magnetic potential (Ay) of the sensor at its top-right corner are calculated. The time domain and frequency
spectrum plots of the system are shown in Figures 2 through 5. It is evident especially from the frequency spectrum plots
that the sensor output tends to fade away as the sensor is placed further away from the fixed end of the beam. This is due
to the fact that the first natural frequency is the dominant mode and the beam is strained highest at its fixed end. Hence it
seems reasonable to place the sensor close to or at the fixed end of the beam. This is advantageous for applications such
as rotating beams where it would be beneficial to place the sensor at the fixed end of the beam to protect it from damage
and at the same time to get the most reading from the sensor. As expected, the fundamental (1 st ) natural frequency of the
system at these sensor locations slightly differ, which are computed as 60.87 Hz at d = 0 m, 58.66 Hz at d = 0.03 m, and
57.07 Hz at d = 0.06 m. The variations of these natural frequencies are clearly seen in the zoomed magnetic plot of
Figure 5, which also shows the higher magnetic output at d = 0 m.
Steady-State Response
The steady-state response of the system is calculated for a unit sinusoidal force acting upward at the tip of the beam.
The forcing frequencies of the sinusoidal force are taken as f = 50 Hz, 100 Hz, and 200 Hz and the sensor locations are
assumed at the two limit positions of d = 0 m and 0.06 m. The tip deflection and vertical magnetic potential plots are
shown in Figures 6 to 11. It can be seen from these figures that the magnetostrictive sensor is able to track the response
of the beam at the two extreme locations of the sensor. However, higher magnetic peaks at the frequency spectrum are
visible for the sensor output at d = 0 m as compared with the sensor output at d = 0.06 m. Hence, it would again appear
to be reasonable to conclude that it is better to place the sensor at the fixed end than at a location farther away from the
fixed end for the steady-state response monitoring of the system. In order to observe the response of the magnetostrictive
sensor to a sinusoidal force acting at the system’s fundamental natural frequency, the forcing frequency of the sinusoidal
force is taken as f = 60.87 Hz for d = 0 m. The response is shown in Figure 12, where the forcing frequency is clearly
observed in the frequency spectrum.
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 131

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
Tip Deflection (m)
PSD of Tip Deflection
Tip Deflection (m)
PSD of Tip Deflection
3
2
1
x 10-3
0
0 0.02 0.04 0.06 0.08 0.1
Time (s)
5
4
3
2
1
x 10-5
0
0 100 200 300 400 500
Frequency (Hz)
-8
0 0.02 0.04 0.06 0.08 0.1
Time (s)
132 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004
Vertical MP (Wb/m)
PSD of Vertical MP
2
0
-2
-4
-6
2
1.5
1
0.5
x 10-5
x 10-8
0
0 100 200 300 400 500
Frequency (Hz)
Figure 2. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit step input.
4
3
2
1
x 10-3
0
0 0.02 0.04 0.06 0.08 0.1
Time (s)
6
4
2
x 10-5
0
0 100 200 300 400 500
Frequency (Hz)
Vertical MP (Wb/m)
PSD of Vertical MP
2
0
-2
-4
x 10-5
-6
0 0.02 0.04 0.06 0.08 0.1
Time (s)
1
0.8
0.6
0.4
0.2
x 10-8
0
0 100 200 300 400 500
Frequency (Hz)
Figure 3. Vertical tip deflection and magnetic potential (MP) at d = 0.03 m for unit step input.

PSD of Vertical MP
Tip Deflection (m)
PSD of Tip Deflection
4
3
2
1
x 10-3
0
0 0.02 0.04 0.06 0.08 0.1
Time (s)
6
4
2
x 10-5
0
0 100 200 300 400 500
Frequency (Hz)
Vertical MP (Wb/m)
PSD of Vertical MP
J.M. Bakhashwain, M. Sunar, and S.J. Hyder
-3
0 0.02 0.04 0.06 0.08 0.1
Time (s)
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 133
0
-1
-2
5
4
3
2
1
x 10-5
x 10-9
0
0 100 200 300 400 500
Frequency (Hz)
Figure 4. Vertical tip deflection and magnetic potential (MP) at d = 0.06 m for unit step input.
4.5
4
3.5
3
2.5
2
1.5
1
0.5
x 10-9
d = 0 m
d = 0.03 m
d = 0.06 m
0
0 20 40 60
Frequency (Hz)
80 100 120
Figure 5. Sensor output spectrum at various sensor locations.

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
Tip Deflection (m)
PSD of Tip Deflection
5
0
x 10-3
-5
0 0.02 0.04 0.06 0.08 0.1
Time (s)
8
6
4
2
x 10-3
0
0 100 200 300 400 500
Frequency (Hz)
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
134 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004
Vertical MP (Wb/m)
PSD of Vertical MP
1
0.5
0
-0.5
3
2
1
x 10-4
x 10-6
0
0 100 200 300 400 500
Frequency (Hz)
Figure 6. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 50 Hz.
Tip Deflection (m)
PSD of Tip Deflection
0.01
0.005
0
-0.005
-0.01
0 0.02 0.04 0.06 0.08 0.1
Time (s)
0.015
0.01
0.005
0
0 100 200 300 400 500
Frequency (Hz)
Vertical MP (Wb/m)
PSD of Vertical MP
5
0
x 10-5
-5
0 0.02 0.04 0.06 0.08 0.1
Time (s)
8
6
4
2
x 10-7
0
0 100 200 300 400 500
Frequency (Hz)
Figure 7. Vertical tip deflection and magnetic potential (MP) at d = 0.06 m for unit sinusoidal input with f = 50 Hz.

Tip Deflection (m)
PSD of Tip Deflection
1
0.5
0
-0.5
x 10-3
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
4
2
x 10-4
0
0 100 200 300 400 500
Frequency (Hz)
Vertical MP (Wb/m)
PSD of Vertical MP
J.M. Bakhashwain, M. Sunar, and S.J. Hyder
-4
0 0.02 0.04 0.06 0.08 0.1
Time (s)
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 135
4
2
0
-2
3
2
1
x 10-5
x 10-7
0
0 100 200 300 400 500
Frequency (Hz)
Figure 8. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 100 Hz.
Tip Deflection (m)
PSD of Tip Deflection
1
0.5
0
-0.5
x 10-3
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
3
2
1
x 10-4
0
0 100 200 300 400 500
Frequency (Hz)
Vertical MP (Wb/m)
PSD of Vertical MP
1
0.5
0
-0.5
x 10-6
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
3
2
1
x 10-10
0
0 100 200 300 400 500
Frequency (Hz)
Figure 9. Vertical tip deflection and magnetic potential (MP) at d = 0.06 m for unit sinusoidal input with f = 100 Hz.

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
Tip Deflection (m)
PSD of Tip Deflection
1
0.5
0
-0.5
x 10-4
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
2.5
2
1.5
1
0.5
x 10-6
0
0 100 200 300 400 500
Frequency (Hz)
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
136 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004
Vertical MP (Wb/m)
PSD of Vertical MP
1
0.5
0
-0.5
2.5
2
1.5
1
0.5
x 10-5
x 10-8
0
0 100 200 300 400 500
Frequency (Hz)
Figure 10. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 200 Hz.
Tip Deflection (m)
PSD of Tip Deflection
1
0.5
0
-0.5
x 10-4
-1
0 0.02 0.04 0.06 0.08 0.1
Time (s)
1.5
1
0.5
x 10-6
0
0 100 200 300 400 500
Frequency (Hz)
Vertical MP (Wb/m)
PSD of Vertical MP
5
0
x 10-6
-5
0 0.02 0.04 0.06 0.08 0.1
Time (s)
8
6
4
2
x 10-9
0
0 100 200 300 400 500
Frequency (Hz)
Figure 11. Vertical tip deflection and magnetic potential (MP) at d = 0.06 m for unit sinusoidal input with f = 200 Hz.

Vertical MP (Wb/m)
PSD of Vertical MP
CONCLUSIONS
2
1
0
-1
x 10-3
J.M. Bakhashwain, M. Sunar, and S.J. Hyder
-2
0 0.01 0.02 0.03 0.04 0.05
Time (s)
0.06 0.07 0.08 0.09 0.1
1.5
1
0.5
x 10-3
0
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
Figure 12. Vertical magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 60.87 Hz.
The linear constitutive equations of magnetostriction are presented. The differential equations governing dynamic
behavior of a magnetostrictive material are also given. The finite element method is applied to the material to obtain the
coupled finite element equations, which are used in modeling and analyzing the illustrative system consisting of a
cantilever beam and a magnetostrictive sensor. The location of the sensor is varied on the beam to observe its effect on
the sensor’s response.
The magnetic potential outputs of the sensor indicate that it can be utilized to monitor the dynamic response of the
system. It is noticed from the simulation results on this cantilever beam structure that the magnetostrictive sensor
produces magnetic outputs of higher magnitude when it is close to or at the fixed end of the beam. The location of the
sensor has also an effect on the fundamental natural frequency of the system. This effect can be minimized by keeping
the size of the sensor to be as small as possible. Hence, the conclusion is that it is advantageous to place the
magnetostrictive sensor at the fixed end of cantilever beam structures. Similar conclusions were reached in the literature
for piezoelectric sensors and actuators [11, 12]. It is hoped that this preliminary study will pave way for other research
works that will include both magnetostrictive sensors and actuators for dynamic system sensing and control.
ACKNOWLEDGMENT
The authors gratefully acknowledge the support provided by King Fahd University of Petroleum & Minerals through
Fast-Track Project # FT/2001-11.
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 137

J.M. Bakhashwain, M. Sunar, and S.J. Hyder
REFERENCES
[1] E.F. Crawley, “Intelligent Structures for Aerospace: A Technology Overview and Assessment,” Journal of Aircraft,
32(8) (1994), pp. 1689–1699.
[2] M. Sunar and S.S. Rao, “Recent Advances in Sensing and Control of Flexible Structures via Piezoelectric Materials
Technology”, ASME Applied Mechanics Reviews, 52 (1999), pp. 1–16.
[3] C. Body, G. Reyne, and G. Meunier, “Modeling of Magnetostrictive Thin Films, Application to a Micromembrane”, Journal of
Physics III France, 7 (1997), pp. 67–85.
[4] M. Kaltenbacher, S. Schneider, and H. Landes, “Nonlinear Finite Element Analysis of Magnetostrictive Transducers”, Smart
Structures and Materials 2001 – Modeling, Signal Processing and Control in Smart Structures, 4326 (2001), pp. 160–168.
[5] K. Delaere, W. Heylen, R. Belmans, and K. Hameyer, “Strong Magnetomechanical Coupling Using Local Magnetostriction
Forces”, EPJ Applied Physics, 13 (2001), pp. 115–119.
[6] O.A. Mohammed, “Coupled Magnetoelastic Finite Element Formulation of Anisotropic Magnetostatic Problems”, IEEE
Southeast Conference, 2001, pp. 183–187.
[7] B. Bhattacharya, B.R. Vidyashankar, S. Patsias, and G.R. Tomlinson, “Active and Passive Vibration Control of Flexible
Structures Using a Combination of Magnetostrictive and Ferro-magnetic Alloys”, SPIE – International Society for Optical
Engineering, 2000, pp. 204–214.
[8] Y. Yamamoto, H. Eda, and J. Shimizu, “Application of Giant Magnetostrictive Materials to Positioning Actuators”,
IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 1999, pp. 215–220.
[9] M. Sunar, “Modeling of Coupled Mechanical, Electrical, Thermal and Magnetic Fields”, International Thermal Energy
Congress, July 8–12, 2001, Cesme, Izmir, Turkey.
[10] M. Sunar, A.Z. Al-Garni, M.H. Ali, and R. Kahraman, “Finite Element Modeling of Thermopiezomagnetic Smart Structures”,
AIAA Journal, 40(9) (2002), pp. 1846–1851.
[11] M. Sunar and S.S. Rao, “Distributed Modeling and Actuator Location for Piezoelectric Control Systems”, AIAA Journal,
34(10) (1996), pp. 2209–2211.
[12] M. Sunar, “Thermopiezoelectric Sensor Design and Placement”, Recent Advances in Transport Phenomena, The 12 th
International Symposium on Transport Phenomena. ed. I. Dincer and M.F. Yardim. Paris, France: Elsevier, 2000, pp. 665–669.
[13] M.N.O. Sadiku, Elements of Electromagnetics, 2 nd edn. New York: Oxford University Press, 1995, ch. 9.
Paper Received 16 November 2003; Revised 23 May 2004; Accepted 13 October 2004.
138 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004