finite element modeling of magnetostrictive smart structures
finite element modeling of magnetostrictive smart structures
finite element modeling of magnetostrictive smart structures
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
FINITE ELEMENT MODELING OF MAGNETOSTRICTIVE<br />
SMART STRUCTURES<br />
J.M. Bakhashwain<br />
Electrical Engineering Department<br />
and<br />
M. Sunar* and S.J. Hyder<br />
Mechanical Engineering Department<br />
King Fahd University <strong>of</strong> Petroleum & Minerals<br />
Dhahran, Saudi Arabia<br />
ﺔــﺻﻼﺨﻟا<br />
ﻰﻠﻋ يﻮﺘﺤﺗ ﻲﺘﻟا ﺔﻴآﺬﻟا ﺔﻳﺮﺼﺤﻟا ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا داﻮﻤﻠﻟ ﺔﻴﺴﻴﺳﺄﺘﻟا تﻻدﺎﻌﻤﻟا ﺔﻗرﻮﻟا ﻩﺬه مﺪﻘﺗ فﻮﺳ<br />
ةدوﺪﺤﻤﻟا ﺮﺻﺎﻨﻌﻟا ﺔﻘﻳﺮﻃ ﺖﻣﺪﺨﺘﺳا ﺪﻗو . نﻮﺘﻠﻣﺎه أﺪﺒﻣ ماﺪﺨﺘﺳﺎﺑ ﻚﻟذو<br />
، ﺔﻴﻜﻴﻧﺎﻜﻴﻣو ﺔﻴﺴﻴﻄﻨﻐﻣ تﻻﺎﺠﻣ<br />
ﺪﻘﻓ ةرﻮآﺬﻤﻟا تﻻدﺎﻌﻤﻟا لﺎﻤﻌﺘﺳا ﺢﻴﺿﻮﺘﻟو . ﺔﻴآﺬﻟا داﻮﻤﻟا ﻩﺬﻬﻟ ﻲﻜﻴﻣﺎﻨﻳﺪﻟا وأ لﺎﻌﻔﻟا كﻮﻠﺴﻟا ﺔﺟﺬﻤﻨﻟ<br />
ﻢﺗ ﺪﻗو . (CoFe2O4)<br />
ﺔﻳﺮﺼﺣ ﺔﻴﺴﻴﻃﺎﻨﻐﻣ ﺔﻘﺒﻃ ﺎﻬﺤﻄﺳ ﻮﻠﻌﺗ ﺔﻴآذ<br />
ﻪﺿرﺎﻋ ﺔﻟﺎﺣ ﺔﺳارﺪﻟ ﺖﻣﺪﺨﺘﺳا<br />
ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا ﺔﻘﺒﻄﻟا نﺎﻜﻣ ﺮّﻴـﻐﺗ<br />
ﻊﻣ ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا ﺔﺑﺎﺠﺘﺳﻻا ﻚﻟﺬآو<br />
. ﺔﻣﺪﻘﻤﻟا ﺪﻨﻋ ﺮﻴﻴﻐﺘﻟا ﻲﺒﻴﺟ ﻞﻤﺣ وأ ﺊﺟﺎﻔﻣ<br />
ﻞﻤﺤﻟ ضﺮﻌﺘﺗ ﺎﻣﺪﻨﻋ ﻚﻟذو<br />
، ﻪﻴﺳأﺮﻟا<br />
ﻪﺣازﻹا<br />
ﺮّﻴـﻐﺗ<br />
بﺎﺴﺣ<br />
، ﺔﺿرﺎﻌﻟا لﻮﻃ ﻰﻠﻋ ﺔﻳﺮﺼﺤﻟا<br />
ﺎﻣﺪﻨﻋ ﺔﻴﻟﺎﻌﻓ ﺮﺜآأ نﻮﻜﻳ ﺔﻳﺮﺼﺤﻟا ﺔﻴﺴﻴﻃﺎﻨﻐﻤﻟا ةﺮهﺎﻇ ﻰﻠﻋ ﻲﻨﺒﻤﻟا ّﺲﺠﻤﻟا نأ ﺔﺳارﺪﻟا ﻦﻣ ﻦﻴﺒﺗو<br />
. ﺔﺿرﺎﻌﻟا ﻦﻣ ﺖﺑﺎﺜﻟا فﺮﻄﻟا ﺪﻨﻋ ًﺎﺒآﺮﻣ نﻮﻜﻳ<br />
*Address for Correspondence:<br />
KFUPM Box 1205<br />
King Fahd University <strong>of</strong> Petroleum & Minerals<br />
Dhahran 31261<br />
Saudi Arabia<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 125
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
ABSTRACT<br />
Constitutive equations <strong>of</strong> <strong>magnetostrictive</strong> <strong>smart</strong> materials containing mechanical<br />
and magnetic fields are presented via Hamilton’s principle. Finite <strong>element</strong> method is<br />
used to model the dynamic behavior <strong>of</strong> these <strong>smart</strong> materials. A <strong>smart</strong> beam structure<br />
consisting <strong>of</strong> a <strong>magnetostrictive</strong> (CoFe2O4) layer on its top surface is considered as a<br />
case study to illustrate the use <strong>of</strong> <strong>magnetostrictive</strong> equations. The location <strong>of</strong> the<br />
<strong>magnetostrictive</strong> layer is varied along the beam and the vertical displacement and<br />
magnetic responses are computed when the system is subjected to step as well as<br />
sinusoidal loads at the tip. It is found from the case study that the <strong>magnetostrictive</strong><br />
sensor is most effective when it is placed at the fixed end <strong>of</strong> the beam.<br />
Keywords: <strong>magnetostrictive</strong> sensor, <strong>finite</strong> <strong>element</strong> method, vibrations, <strong>smart</strong> structure.<br />
126 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
FINITE ELEMENT MODELING OF MAGNETOSTRICTIVE SMART STRUCTURES<br />
INTRODUCTION<br />
Due to advances in control related technologies there have been an enormous number <strong>of</strong> research activities in so-called<br />
<strong>smart</strong> materials during the last decade. A <strong>smart</strong> material involves interactions between different fields, which allow the<br />
material to be used as a sensor and/or actuator in some manner. Hence the idea <strong>of</strong> a <strong>smart</strong> or an intelligent structure has<br />
been introduced to refer to <strong>structures</strong> that contain <strong>smart</strong> materials and have self-sensing and adaptive capabilities [1, 2].<br />
The phenomenon <strong>of</strong> magnetostriction is defined as an interaction between mechanical and magnetic fields in a body.<br />
This phenomenon was first discovered by Joule in 1842 [3]. Some <strong>of</strong> the <strong>magnetostrictive</strong> materials that can be<br />
mentioned are CoFe2O4, Ni, Alfenol, and Terfenol-D. A numerical simulation scheme for <strong>magnetostrictive</strong> transducers<br />
based on magnetic vector potential formulation was proposed by Kaltenbacher et al. [4]. It was observed that the<br />
coupling <strong>of</strong> magnetic and mechanical systems induced mechanical strains and permeability changes. Experimental and<br />
numerical results were compared and a reasonably good agreement was obtained between the results. The deformation <strong>of</strong><br />
the magnetic material caused by the magnetostriction was represented by an equivalent set <strong>of</strong> mechanical forces by<br />
Delaere et al. [5]. The resulting magnetostriction force was superimposed on the other force distributions, which was the<br />
factor for the coupling <strong>of</strong> mechanical and magnetic systems. A two-dimensional <strong>finite</strong> <strong>element</strong> model was used by<br />
Mohammed [6] to evaluate the force components in magnetostriction phenomenon. The strains developed due to the<br />
presence <strong>of</strong> the magnetic field generated electrical and mechanical forces that created undesirable acoustic noise at low<br />
frequencies. A combined active and passive damping strategy was proposed by Bhattacharya et al. [7] for the control <strong>of</strong><br />
flexible <strong>structures</strong> using <strong>magnetostrictive</strong> and ferro-magnetic alloys. The performance <strong>of</strong> Terfenol-D was affected by<br />
preloading applied to the material. Finite <strong>element</strong> simulations on a cantilever beam model was carried out to study the<br />
dynamic characteristics resulting from the proposed damping approach. An approach was presented by Yamamoto et al.<br />
[8] for monitoring and controlling the preloading in a compact giant <strong>magnetostrictive</strong> positioner. The proposed technique<br />
provided a precise measurement <strong>of</strong> the preloading as well as a sensing capability proportional to the displacement <strong>of</strong> the<br />
positioner.<br />
A thermopiezomagnetic medium was formed by Sunar et al. [9, 10] via bonding together a piezoelectric and<br />
<strong>magnetostrictive</strong> composites. Finite <strong>element</strong> equations for thermopiezomagnetic media were obtained using linear<br />
constitutive equations in Hamilton’s principle together with the <strong>finite</strong> <strong>element</strong> approximations. It was shown that an<br />
electrostatic field applied to the piezoelectric layer causes strain in the structure that in turn produces a magnetic field in<br />
the magnetoceramic layer.<br />
The objective <strong>of</strong> this paper is to develop the dynamic equations <strong>of</strong> a <strong>magnetostrictive</strong> medium via the Hamilton’s<br />
principle and the <strong>finite</strong> <strong>element</strong> method, and to utilize these equations in obtaining the magnetic responses <strong>of</strong> a<br />
<strong>magnetostrictive</strong> layer bonded to the top surface <strong>of</strong> a cantilever beam structure. The structure is acted upon by step and<br />
sinusoidal forces at the tip. The location <strong>of</strong> the <strong>magnetostrictive</strong> sensor is varied along the beam in order to observe its<br />
effect on the magnetic response <strong>of</strong> the sensor. It is well-known that the locations <strong>of</strong> sensors and actuators are very<br />
important to their performance and effectiveness. Research studies were conducted for piezoelectric sensors and<br />
actuators by Sunar and Rao [11], and Sunar [12] for the investigation on the impact <strong>of</strong> placement. The current work<br />
extends these studies to the case <strong>of</strong> <strong>magnetostrictive</strong> sensors.<br />
CONSTITUTIVE EQUATIONS OF MAGNETOSTRICTION<br />
The following equations are written for a <strong>magnetostrictive</strong> medium [10]<br />
T = G ∂<br />
= cS − B<br />
∂S<br />
H = G<br />
∂<br />
∂B = −Τ S + µ −1 B (1)<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 127
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
where c and denote the matrices <strong>of</strong> elastic and <strong>magnetostrictive</strong> stress constants; µ represents the matrix <strong>of</strong> material<br />
permeability; T, S, H, and B are the vectors <strong>of</strong> stress, strain, magnetic field intensity, and magnetic flux density,<br />
respectively. It can be shown that [10] the thermodynamic potential G takes on the following quadratic form:<br />
G = 1<br />
2 S T cS + 1<br />
2 BT µ −1 B − S T B<br />
= 1<br />
2 S T T + 1<br />
2 BT µ −1 B, (2)<br />
using the relation in Equation (1). The Hamilton’s principle is given as<br />
t2<br />
t1<br />
δ∫ ( Ki−Π) dt=<br />
0, (3)<br />
where Ki is the kinetic energy and ∏ is an energy functional, which are defined by<br />
and<br />
1 T<br />
Ki = ∫ ρuu<br />
dt<br />
(4)<br />
V 2<br />
Π =<br />
∫ ∫ ∫ ∫ ∫<br />
GdV u P dV u P dS A JdV A H′ n dS<br />
(5)<br />
T T T T<br />
− b − s − +<br />
V V S V S<br />
where Pb and Ps are the vectors <strong>of</strong> body and surface forces, u, A, J, and n are the vectors <strong>of</strong> mechanical displacement,<br />
magnetic potential, volume current density, and surface normal and H′ E is the matrix <strong>of</strong> external magnetic field intensity,<br />
which is defined in cartesian coordinates as<br />
The variation in G is found as<br />
⎡ 0 Hz −H<br />
⎤ y<br />
⎢ ⎥<br />
HE′ = ⎢−Hz 0 Hx<br />
⎥<br />
⎢ Hy −Hx<br />
0 ⎥<br />
⎣ ⎦<br />
δG = δS T cS + δB T µ −1 B − δS T B− δB T Τ S<br />
= δS T (cS − B) + δB T (− Τ S + µ −1 B)<br />
E<br />
128 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004<br />
E<br />
. (6)<br />
= δS T T + δB T H, (7)<br />
using Equation (1) and the variation in Ki is written as<br />
t2 t2<br />
T<br />
∫ Ki dt<br />
t ∫ dt dt<br />
1 t ∫ uu . (8)<br />
1 V<br />
δ = − ρδ<br />
Substitution <strong>of</strong> the above equations in Equation (3) yields the following result<br />
t2 t2<br />
⎡<br />
T T T T T<br />
∫ ( ) ∫ ⎢ ∫ ( uu ST BH uPb AJ)<br />
δ Ki−Π dt= dt −ρδ −δ −δ + δ +δ dV<br />
t1 t1 ⎣ V<br />
T T<br />
( s<br />
E′<br />
)<br />
H dS<br />
S<br />
⎤<br />
+ ∫ δu P −δA<br />
n<br />
⎥<br />
= 0. (9)<br />
⎦
Introducing the following relations<br />
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
S = Lu u , B = ∇ × A = LA A (10)<br />
where Lu and LA are differential operators, and substituting Equation (10) into Equation (9) results in the following<br />
equation<br />
t2<br />
t1V (<br />
⎡<br />
T T T T T<br />
dt ⎢ −ρδuu − ( Luδu) T−( ∇ × δ A) H+ δ uP+ δAJ<br />
dV<br />
⎣<br />
∫ ∫ b<br />
T T<br />
( E′<br />
∫ s )<br />
+ δ −δ<br />
S<br />
⎥<br />
⎦<br />
Note the following relations for some terms in Equation (11)<br />
and<br />
H dS ⎤<br />
u P A n = 0. (11)<br />
∫ ∫ ∫<br />
T T T T<br />
V<br />
u<br />
V<br />
u<br />
S<br />
( L δ u) TdV =− δ u ( L T) dV + δu<br />
( NT ) dS<br />
∫ ∫ ∫<br />
T T T<br />
V V V<br />
( ∇×δ A) HdV = ∇ ( δ A× H) dV + δA ( ∇× H ) dV<br />
∫ ∫<br />
T T<br />
= − δ A H′ ndS + δA ( ∇× H ) dV.<br />
(12)<br />
S V<br />
In Equation (12) N is defined in cartesian coordinates as<br />
⎡nx 0 0 ny n 0 ⎤ z<br />
⎢ ⎥<br />
N = ⎢ 0 ny 0 nx 0 nz⎥<br />
⎢ 0 0 nz0 nx n ⎥<br />
⎣ y⎦<br />
where nx, ny, and nz are the components <strong>of</strong> the surface normal n, and H′ is defined as in Equation (6). Equation (11) is<br />
rewritten as<br />
t2<br />
⎡ T T T<br />
∫ ⎢ ∫ δu ( −ρ u + uT<br />
+ Pb) + ∫ δA ( −∇× H + J)<br />
dt L dV dV<br />
⎣<br />
t1V V<br />
T T<br />
( N ) dS ( HEH ) dS<br />
S S<br />
⎤<br />
+ δ − − δ ′ − ′<br />
∫ u Ps T ∫ A n<br />
⎥<br />
= 0. (14)<br />
⎦<br />
Hence, the following equations must be satisfied with appropriate boundary conditions on the corresponding surfaces<br />
−ρ u + L T + Pb=<br />
0<br />
T<br />
u<br />
−∇ × H + J = 0.<br />
(15)<br />
The above equations are the equations <strong>of</strong> motion for the mechanical field and the Maxwell’s equilibrium equation for the<br />
magnetic field, respectively [13].<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 129<br />
)<br />
(13)
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
FINITE ELEMENT FORMULATION<br />
For the <strong>finite</strong> <strong>element</strong> formulation, u and A are chosen variables for the mechanical and magnetic fields, respectively.<br />
The <strong>finite</strong> <strong>element</strong> approximations are written as<br />
ue = Nu ui<br />
Ae = NA Ai<br />
where the subscript e and i respectively stand for the <strong>element</strong> and nodes <strong>of</strong> the <strong>element</strong> and N’s are the appropriate shape<br />
function matrices. Noting the following relations<br />
and<br />
Se = Lu ue = [Lu Nu] ui = Bu ui<br />
Be = LA Ae = [LA NA] Ai = BA Ai<br />
and substituting Equations (1), (16), and (17) in Equation (11) yields the <strong>finite</strong> <strong>element</strong> equations as<br />
Muu u + Kuuu − KuA A = F<br />
−KAuu+ KAA A = M (18)<br />
where u, A, F, and M are the global vectors <strong>of</strong> mechanical displacement, magnetic potential, applied mechanical, and<br />
magnetic excitations, respectively. The <strong>element</strong> matrices and applied excitations in Equation (18) are found as<br />
[Muu]e =<br />
[KuA]e =<br />
Fe =<br />
Me =<br />
∫<br />
Ve<br />
e<br />
T<br />
u u<br />
ρ N N dV , [Kuu]e =<br />
T<br />
∫ Bu eBAdV , [KAA]e =<br />
Ve<br />
∫ ∫<br />
∫<br />
∫<br />
Ve<br />
Ve<br />
T<br />
u e u<br />
T T T<br />
Nu Pbe dV + Nu PsedS + Nu<br />
P ce<br />
Ve Se<br />
∫ ∫<br />
T T<br />
A e − A Ae<br />
Ve Se<br />
B c B dV<br />
T −1<br />
Aµ e A<br />
B B dV<br />
N J dV B H′ n dS<br />
(19)<br />
where KAu= KuA T and the concentrated force Pce is added to the definition <strong>of</strong> Fe.<br />
CASE STUDY<br />
A cantilever steel beam shown in Figure 1 is taken as an example to illustrate the use <strong>of</strong> the derived <strong>magnetostrictive</strong><br />
equations. A <strong>magnetostrictive</strong> (CoFe2O4) layer is bonded to the structure on its top surface and the distance <strong>of</strong> the layer<br />
from the fixed end <strong>of</strong> the beam is denoted by d. The layer is acting as a sensor in response to excitations applied to the<br />
beam. The material properties for the steel and CoFe2O4 are listed in Table 1. The length, width (out <strong>of</strong> plane) and<br />
thickness <strong>of</strong> the beam are taken as 0.12 m, 0.02 m, and 0.001 m, respectively. For the <strong>magnetostrictive</strong> sensor, the<br />
assumed dimensions are 0.004 m for the length, 0.02 m for the width, and 0.002 m for the thickness. Two-dimensional<br />
rectangular <strong>element</strong>s are used for the <strong>finite</strong> <strong>element</strong> formulation <strong>of</strong> the <strong>magnetostrictive</strong> sensor and beam <strong>element</strong>s with<br />
axial degrees <strong>of</strong> freedom for the beam structure. The sensor is divided into two <strong>finite</strong> <strong>element</strong>s and the division <strong>of</strong> the<br />
beam into <strong>finite</strong> <strong>element</strong>s is tuned according to the location <strong>of</strong> the sensor from the fixed end <strong>of</strong> the beam. More<br />
specifically, the number <strong>of</strong> beam <strong>element</strong>s varies from 8 <strong>element</strong>s to 13 <strong>element</strong>s. In all the simulations below, the<br />
system is assumed to have a damping ratio <strong>of</strong> ζ = 0.01, which is typical for a physical system like this.<br />
130 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004<br />
(16)<br />
(17)
Transient Response<br />
d<br />
Magnetostrictive Layer<br />
Beam<br />
Figure 1. Magnetostrictive system.<br />
Table 1. Properties <strong>of</strong> Materials.<br />
Magnetostrictive Ceramic (CoFe 2O 4)<br />
c 11 (N/m 2 ) 1.54×10 11<br />
31 (N/Wb or A/m) 2.86×10 8<br />
µ 11, µ 33 (H/m) 20π×10 −7<br />
ρ (kg/m 3 ) 7500<br />
Steel<br />
c 11 (N/m 2 ) 2.07×10 11<br />
ρ (kg/m 3 ) 7800<br />
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
The beam is subjected to a unit step force acting upward at its tip. The variable d for the location <strong>of</strong> the sensor is<br />
assumed to be taking on three values: 0 m, 0.03 m, and 0.06 m. The vertical tip deflection <strong>of</strong> the beam as well as the<br />
vertical magnetic potential (Ay) <strong>of</strong> the sensor at its top-right corner are calculated. The time domain and frequency<br />
spectrum plots <strong>of</strong> the system are shown in Figures 2 through 5. It is evident especially from the frequency spectrum plots<br />
that the sensor output tends to fade away as the sensor is placed further away from the fixed end <strong>of</strong> the beam. This is due<br />
to the fact that the first natural frequency is the dominant mode and the beam is strained highest at its fixed end. Hence it<br />
seems reasonable to place the sensor close to or at the fixed end <strong>of</strong> the beam. This is advantageous for applications such<br />
as rotating beams where it would be beneficial to place the sensor at the fixed end <strong>of</strong> the beam to protect it from damage<br />
and at the same time to get the most reading from the sensor. As expected, the fundamental (1 st ) natural frequency <strong>of</strong> the<br />
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<br />
57.07 Hz at d = 0.06 m. The variations <strong>of</strong> these natural frequencies are clearly seen in the zoomed magnetic plot <strong>of</strong><br />
Figure 5, which also shows the higher magnetic output at d = 0 m.<br />
Steady-State Response<br />
The steady-state response <strong>of</strong> the system is calculated for a unit sinusoidal force acting upward at the tip <strong>of</strong> the beam.<br />
The forcing frequencies <strong>of</strong> the sinusoidal force are taken as f = 50 Hz, 100 Hz, and 200 Hz and the sensor locations are<br />
assumed at the two limit positions <strong>of</strong> d = 0 m and 0.06 m. The tip deflection and vertical magnetic potential plots are<br />
shown in Figures 6 to 11. It can be seen from these figures that the <strong>magnetostrictive</strong> sensor is able to track the response<br />
<strong>of</strong> the beam at the two extreme locations <strong>of</strong> the sensor. However, higher magnetic peaks at the frequency spectrum are<br />
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<br />
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<br />
fixed end for the steady-state response monitoring <strong>of</strong> the system. In order to observe the response <strong>of</strong> the <strong>magnetostrictive</strong><br />
sensor to a sinusoidal force acting at the system’s fundamental natural frequency, the forcing frequency <strong>of</strong> the sinusoidal<br />
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<br />
observed in the frequency spectrum.<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 131
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
3<br />
2<br />
1<br />
x 10-3<br />
0<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
5<br />
4<br />
3<br />
2<br />
1<br />
x 10-5<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
-8<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
132 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
2<br />
0<br />
-2<br />
-4<br />
-6<br />
2<br />
1.5<br />
1<br />
0.5<br />
x 10-5<br />
x 10-8<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Figure 2. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit step input.<br />
4<br />
3<br />
2<br />
1<br />
x 10-3<br />
0<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
6<br />
4<br />
2<br />
x 10-5<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
2<br />
0<br />
-2<br />
-4<br />
x 10-5<br />
-6<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
x 10-8<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Figure 3. Vertical tip deflection and magnetic potential (MP) at d = 0.03 m for unit step input.
PSD <strong>of</strong> Vertical MP<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
4<br />
3<br />
2<br />
1<br />
x 10-3<br />
0<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
6<br />
4<br />
2<br />
x 10-5<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
-3<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 133<br />
0<br />
-1<br />
-2<br />
5<br />
4<br />
3<br />
2<br />
1<br />
x 10-5<br />
x 10-9<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Figure 4. Vertical tip deflection and magnetic potential (MP) at d = 0.06 m for unit step input.<br />
4.5<br />
4<br />
3.5<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
x 10-9<br />
d = 0 m<br />
d = 0.03 m<br />
d = 0.06 m<br />
0<br />
0 20 40 60<br />
Frequency (Hz)<br />
80 100 120<br />
Figure 5. Sensor output spectrum at various sensor locations.
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
5<br />
0<br />
x 10-3<br />
-5<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
8<br />
6<br />
4<br />
2<br />
x 10-3<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
134 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
1<br />
0.5<br />
0<br />
-0.5<br />
3<br />
2<br />
1<br />
x 10-4<br />
x 10-6<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Figure 6. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 50 Hz.<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
0.01<br />
0.005<br />
0<br />
-0.005<br />
-0.01<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
0.015<br />
0.01<br />
0.005<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
5<br />
0<br />
x 10-5<br />
-5<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
8<br />
6<br />
4<br />
2<br />
x 10-7<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
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)<br />
PSD <strong>of</strong> Tip Deflection<br />
1<br />
0.5<br />
0<br />
-0.5<br />
x 10-3<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
4<br />
2<br />
x 10-4<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
-4<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 135<br />
4<br />
2<br />
0<br />
-2<br />
3<br />
2<br />
1<br />
x 10-5<br />
x 10-7<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Figure 8. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 100 Hz.<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
1<br />
0.5<br />
0<br />
-0.5<br />
x 10-3<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
3<br />
2<br />
1<br />
x 10-4<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
1<br />
0.5<br />
0<br />
-0.5<br />
x 10-6<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
3<br />
2<br />
1<br />
x 10-10<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
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<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
1<br />
0.5<br />
0<br />
-0.5<br />
x 10-4<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
x 10-6<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
136 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
1<br />
0.5<br />
0<br />
-0.5<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
x 10-5<br />
x 10-8<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Figure 10. Vertical tip deflection and magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 200 Hz.<br />
Tip Deflection (m)<br />
PSD <strong>of</strong> Tip Deflection<br />
1<br />
0.5<br />
0<br />
-0.5<br />
x 10-4<br />
-1<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
1.5<br />
1<br />
0.5<br />
x 10-6<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
Vertical MP (Wb/m)<br />
PSD <strong>of</strong> Vertical MP<br />
5<br />
0<br />
x 10-6<br />
-5<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Time (s)<br />
8<br />
6<br />
4<br />
2<br />
x 10-9<br />
0<br />
0 100 200 300 400 500<br />
Frequency (Hz)<br />
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)<br />
PSD <strong>of</strong> Vertical MP<br />
CONCLUSIONS<br />
2<br />
1<br />
0<br />
-1<br />
x 10-3<br />
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
-2<br />
0 0.01 0.02 0.03 0.04 0.05<br />
Time (s)<br />
0.06 0.07 0.08 0.09 0.1<br />
1.5<br />
1<br />
0.5<br />
x 10-3<br />
0<br />
0 10 20 30 40 50 60 70 80 90 100<br />
Frequency (Hz)<br />
Figure 12. Vertical magnetic potential (MP) at d = 0 m for unit sinusoidal input with f = 60.87 Hz.<br />
The linear constitutive equations <strong>of</strong> magnetostriction are presented. The differential equations governing dynamic<br />
behavior <strong>of</strong> a <strong>magnetostrictive</strong> material are also given. The <strong>finite</strong> <strong>element</strong> method is applied to the material to obtain the<br />
coupled <strong>finite</strong> <strong>element</strong> equations, which are used in <strong>modeling</strong> and analyzing the illustrative system consisting <strong>of</strong> a<br />
cantilever beam and a <strong>magnetostrictive</strong> sensor. The location <strong>of</strong> the sensor is varied on the beam to observe its effect on<br />
the sensor’s response.<br />
The magnetic potential outputs <strong>of</strong> the sensor indicate that it can be utilized to monitor the dynamic response <strong>of</strong> the<br />
system. It is noticed from the simulation results on this cantilever beam structure that the <strong>magnetostrictive</strong> sensor<br />
produces magnetic outputs <strong>of</strong> higher magnitude when it is close to or at the fixed end <strong>of</strong> the beam. The location <strong>of</strong> the<br />
sensor has also an effect on the fundamental natural frequency <strong>of</strong> the system. This effect can be minimized by keeping<br />
the size <strong>of</strong> the sensor to be as small as possible. Hence, the conclusion is that it is advantageous to place the<br />
<strong>magnetostrictive</strong> sensor at the fixed end <strong>of</strong> cantilever beam <strong>structures</strong>. Similar conclusions were reached in the literature<br />
for piezoelectric sensors and actuators [11, 12]. It is hoped that this preliminary study will pave way for other research<br />
works that will include both <strong>magnetostrictive</strong> sensors and actuators for dynamic system sensing and control.<br />
ACKNOWLEDGMENT<br />
The authors gratefully acknowledge the support provided by King Fahd University <strong>of</strong> Petroleum & Minerals through<br />
Fast-Track Project # FT/2001-11.<br />
June 2004 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. 137
J.M. Bakhashwain, M. Sunar, and S.J. Hyder<br />
REFERENCES<br />
[1] E.F. Crawley, “Intelligent Structures for Aerospace: A Technology Overview and Assessment,” Journal <strong>of</strong> Aircraft,<br />
32(8) (1994), pp. 1689–1699.<br />
[2] M. Sunar and S.S. Rao, “Recent Advances in Sensing and Control <strong>of</strong> Flexible Structures via Piezoelectric Materials<br />
Technology”, ASME Applied Mechanics Reviews, 52 (1999), pp. 1–16.<br />
[3] C. Body, G. Reyne, and G. Meunier, “Modeling <strong>of</strong> Magnetostrictive Thin Films, Application to a Micromembrane”, Journal <strong>of</strong><br />
Physics III France, 7 (1997), pp. 67–85.<br />
[4] M. Kaltenbacher, S. Schneider, and H. Landes, “Nonlinear Finite Element Analysis <strong>of</strong> Magnetostrictive Transducers”, Smart<br />
Structures and Materials 2001 – Modeling, Signal Processing and Control in Smart Structures, 4326 (2001), pp. 160–168.<br />
[5] K. Delaere, W. Heylen, R. Belmans, and K. Hameyer, “Strong Magnetomechanical Coupling Using Local Magnetostriction<br />
Forces”, EPJ Applied Physics, 13 (2001), pp. 115–119.<br />
[6] O.A. Mohammed, “Coupled Magnetoelastic Finite Element Formulation <strong>of</strong> Anisotropic Magnetostatic Problems”, IEEE<br />
Southeast Conference, 2001, pp. 183–187.<br />
[7] B. Bhattacharya, B.R. Vidyashankar, S. Patsias, and G.R. Tomlinson, “Active and Passive Vibration Control <strong>of</strong> Flexible<br />
Structures Using a Combination <strong>of</strong> Magnetostrictive and Ferro-magnetic Alloys”, SPIE – International Society for Optical<br />
Engineering, 2000, pp. 204–214.<br />
[8] Y. Yamamoto, H. Eda, and J. Shimizu, “Application <strong>of</strong> Giant Magnetostrictive Materials to Positioning Actuators”,<br />
IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 1999, pp. 215–220.<br />
[9] M. Sunar, “Modeling <strong>of</strong> Coupled Mechanical, Electrical, Thermal and Magnetic Fields”, International Thermal Energy<br />
Congress, July 8–12, 2001, Cesme, Izmir, Turkey.<br />
[10] M. Sunar, A.Z. Al-Garni, M.H. Ali, and R. Kahraman, “Finite Element Modeling <strong>of</strong> Thermopiezomagnetic Smart Structures”,<br />
AIAA Journal, 40(9) (2002), pp. 1846–1851.<br />
[11] M. Sunar and S.S. Rao, “Distributed Modeling and Actuator Location for Piezoelectric Control Systems”, AIAA Journal,<br />
34(10) (1996), pp. 2209–2211.<br />
[12] M. Sunar, “Thermopiezoelectric Sensor Design and Placement”, Recent Advances in Transport Phenomena, The 12 th<br />
International Symposium on Transport Phenomena. ed. I. Dincer and M.F. Yardim. Paris, France: Elsevier, 2000, pp. 665–669.<br />
[13] M.N.O. Sadiku, Elements <strong>of</strong> Electromagnetics, 2 nd edn. New York: Oxford University Press, 1995, ch. 9.<br />
Paper Received 16 November 2003; Revised 23 May 2004; Accepted 13 October 2004.<br />
138 The Arabian Journal for Science and Engineering, Volume 29, Number 1C. June 2004