Study of the Inverse Magnetostriction Effect on Machine Deformation

O. A. Mohammed, Fellow IEEE, S. Liu, Member IEEE, and N. Abed, Student Member
Dept. of Electrical and Computer Engineering
Florida International University
Miami, Florida, USA
mohammed@fiu.edu

Abstract---- This paper studies the inverse magnetostriction effect on

the magnetization characteristics of electrical steel as well as the Excitation
magnetoelastic behavior of electrical machines. They include the winding

measurement of the inverse magnetostriction effect, the magnetoelastic B

coupled FEM analysis including the inverse magnetostriction effect, and Applied force Path Applied force
its application on a PM motor. The magnetic force on the stator irons with
and without IME are calculated numerically and compared. Conclusions B coil
on the magnetostrictive force calculated by the virtual work principle are
given. They show that the inverse magnetostriction effect is significant and
should be accounted.
A Hall probe

INTRODUCTION Excitation
winding

The inverse magnetostriction effects have been Fig. 1 Sample and measure method
recognized for many years. It means that the magnetic
properties of the magnetic materials are influenced by the
applied and internal mechanical stresses. It is described by COUPLED MAGNETOELASTIC FORMULAR INCLUDING THE
magnetization curves at different stress levels [1]. The changes MAGNETOSTRICTION EFFECTS
of the magnetization property will bring the changes of the
magnetic force acting on the machine stator iron. They reflect Below is the equation for the coupled magnetoelastic
in both the force magnitude and its distribution. For this reason, problem.
magnetostriction effects are considered to be one of the main
reasons for the machine noise and vibration. [S ][A] = [J e ] (1)
In this paper, first, our method of measuring the inverse [K ][U ] = [F ] (2)
magnetostriction effects is presented. Then, the FE formula,
used for the magneto-mechanical coupled analysis including Where, S and K are electromagnetic stiffness matrix and
anisotropy reluctivity tensor and inverse magnetostriction mechanical stiffness matrix respectively. J e represents the
effects, are reviewed and the implementation details are
explained. Last, the measured magnetization curves at various excitation of the magnetic field. F is the force applied on the
stress levels and the proposed formulas are applied to a surface stress field. A is vector potential; U is displacement.
mounted PM motor to investigate the effects of The magnetostriction effects should be considered both in
magnetostriction on the stator deformation. Both results the computation of S and F . When calculate S , the
obtained with and without magnetostriction effects permeability determination of the ferromagnetic materials
consideration are provided for comparison purpose. depends not only on the field strength H but also on the stress
level they are suffering.
MEASUREMENT OF THE INVERSE MAGNETOSTRICTION EFFECTS If the virtual work principle is adopted for the magnetic
OF ELECTRICAL STEEL SHEETS force calculation, an additional term of force needs to be added.
This is due to magnetostriction effects. The first reference of
According to the definition of the inverse magnetostriction this paper gives the detailed deduction of the force calculation.
effects mentioned in the introduction part, the measurement For convenience, the deduced element force formulation is
method is proposed by the authors, as shown in Fig. 1. rewritten below [2,3]:
A steel sheet sample is inserted at the center of the
excitation winding. Pulling forces are applied at the two ends B B (det(G ))

F e = [( H T ) det(G ) + ( 0 H
T
dB)
of the sample so as to produce the required stress level U U (3)
Re
everywhere of the sample. The measurement is performed
B
according to ASTM standard V3.04A341. B-coil is used for
flux density measurement. Hall probes are put next to the
+( 0
( H T ) dB)
U
det(G )]dudv

sample at different distances along the line perpendicular to the

sample to measure the field strength outside the sample. Where, B and H are the element flux density and the field
Extrapolation is used to determine the field strength inside the strength. is the element stress. G is the local Jacobian
sample. derivatives matrix and det(G ) is the determinant of G . The

0-7803-8367-2/04/$20.00 2004 IEEE 3

details on the implementation will be explained in the next
i
section.
One can find that the calculation of S and F are
interacted. Therefore, in order to reach the convergent point of e
the coupled system, a loop is needed for the coupled nonlinear i
magneto-elastic field calculation.
i +1
IMPLEMENTATION OF MAGNETOSTRICTIVE FORCE e

Equation (4) and (5) are the expansion of equation (3) for i +1
the diagonal reluctivity tensor case and the full reluctivity
B
tensor case respectively [4,5]. Be
Fig. 3 Derivative of element permeability to stress

R
e '
Fms = (( xx ( )B x2 + ' 2
yy ( ) BY )
e
(4) Lets assume that the element equivalent stress and the
E
det (G ))dudv element flux density obtained after performing one magneto-
(1 + )(1 2 ) mechanical coupled FEM analysis is e ( i < e < i +1 ) and
Be . i and i +1 are the permeability corresponding to Be at
stress level i and i +1 respectively. They can be evaluated

e '
Fms = (( xx ( )Bx2 + xy
'
( ) Bx B y
Re using any kind of nonlinear interpolation method. Linear
(5)
+ 'yx ( ) Bx B y + 'yy ( ) BY2 )
E
det (G )) dudv
interpolation is used to calculate the element permeability e .
(1 + )(1 2 ) Thus the permeability of this element e and the derivative of
the permeability component to the element stress can be
Where, xx , xy , yx and yy'
' ' ' calculated as follows:
are the derivatives of the
corresponding components of the reluctivity to the element i +1 e
stress, which is obtained from the coupled solution. i +1 + i
i e e e i
= i , e = (8)
e i +1 e i +1 e
A. e Calculation 1 + i +1
e i
Fig. 2 shows the magnetostriction effects on the
magnetization property of ferromagnetic materials. For diagonal reluctivity tensor,

e
B Highest stress e
xx yy e 0
If e 0 e 1 , then e
= e
= ;
e 0
Stress level (i+1) e
e
xx yy 1 e
Stress level (i) If e 0 > e 1 , then = =
e e 1 e
(6)
Without stress
Initial magnetic property Since the full tensor magnetization property of the steel
sheet is not available, the following formula is used as a
replacement for the full tensor reluctivity case [4,5].
H

Fig. 2 Magnetostriction effect on BH curve If e 0 e 1 , then

e
From this figure, one can find that the flux density at the xx e 0
= cos( B + ph ) (11)
same field strength is increased with the increasing of the stress e e 0
level. e
yy e 0
The permeability versus flux density curves shown in Fig. 3 = sin( B + ph ) (12)
can be obtained from the BH curves at stress level i and e e 0
i +1 ( i < i +1 ). e
yx e + 0
= sin( B + ph ) (13)
e e 0

4
 e
yy e + 0 (x4, y4) (fx4,fy4) (x3, y3) (fx3, fy3)
e
= sin( B + ph ) (14)
e 0
If e 0 > e 1 , then (xc, yc)
C4 C3
e
xx e 1
= cos( B + ph ) (15)
e 1 e
e
(fxe,fye)
xy e 1 C1
= sin( B + ph ) (16) C2
e 1 e
(x1, y1) (fx3, fy3)
e
yx + e
= sin( B + ph ) 1 (17) (x2, y2) (fx2 fy2)
e 1 e
e
yy 1 + e Fig. 4 Nodal force calculation
e
= cos( B + ph ) (18)
1 e
c1
( fx1 , fy1 ) = ( fxe , fy e ) (22)
c1 + c 2 + c3 + c 4
Where B are the inclination angle of the flux density, ph
c2
are the phase angle between the flux density B and the field ( fx 2 , fy 2 ) = ( fxe , fy e ) (23)
c1 + c 2 + c3 + c 4
strength H . c3
( fx3 , fy3 ) = ( fxe , fy e ) (24)
c1 + c 2 + c3 + c 4
B. det (G ) Calculation
c4
The local Jacobian derivatives matrix G and its determinate ( fx 4 , fy 4 ) = ( fxe , fy e ) (25)
c1 + c 2 + c3 + c 4
used for the triangular element can be found in the published
literatures [2]. The authors of this paper deduce the formula for
calculating G and its determinate of quadrilateral elements. Where, ( x c , y c ) and ( fxe , fye ) represent the coordinate
They are given below. and the element magnetostrictive force value at the element
physical center of one quadrilateral element, shown in Fig. 4.
x y ( fx1 , fy1 ) , ( fx 2 , fy 2 ) , ( fx3 , fy3 ) , and ( fx 4 , fy 4 ) are the nodal
det[G ] = G = u
u = x y y x (19) magnetostrictive force contributed by one element. c1, c2, c3,
x y u v u v and c4 are the distances between each node and the physical
v v center.
The total nodal magnetostriction force is obtained by adding
x y N1 N N N the elemental nodal magnetostriction force from all elements
= x1 + x2 2 + x3 3 + x4 4
u v u u u u around one node.
(20)
N1 N 2 N 3 N 4
y1 + y2 + y3 + y4 D. Coupled analysis procedure
v v v v
The element permeability is determined according to the
y x N1 N N N nonlinear magnetization properties as well as the stress level of
= y1 + y2 2 + y3 3 + y4 4
u v u u u u each element. Using the magnetization property without any
(21)
N1 N N N stress applied carries out the initial strong-coupled solution.
x1 + x2 2 + x3 3 + x4 4 Then the iteration for the proper element permeability and
v v v v
stress determination is performed until the energy of the whole
system keeps unchanged. Usually several times coupled FE
Where, N1 , N 2 , N 3 and N 4 are the interpolation functions
analyses are needed for the convergent iterations.
of quadrilateral element. (x1, y1 ) , (x2, y2 ) , (x3 , y3 ) and (x4 , y4 )
are the coordinates of the four nodes. RESULTS AND CONCLUSION

C. Nodal force summation A PM surface mounted motor is used for the application of
The calculated Fmse
is the magnetostrictive force acted on the proposed method and the procedure. With our results, the
following conclusions can be obtained.
one element. In order to know its contribution to the force at
each node, it should be distributed. Below gives the
1. If the virtual work principle is adopted for the magnetic
formulations used for this purpose.
force calculation and the magnetostriction effects are
considered, there are forces distributed inside the stator
iron. Fig. 5 and Fig. 6 show the force distribution along a

5
 round locus at the back iron with and without 3. If the full reluctivity tensor is considered, the magnetic
magnetostriction effects considered. No force appears in force profiles along any round locus of the stator iron will
the back iron in Fig. 6 is due to the application of the not be repeated. So are the displacements. This is already
virtual work principle. The result in Fig.5 shows that there shown in Fig. 5, 7 and can be further confirmed by the Fig.
exists force density in the back iron if the magnetostriction 8 below.
effects are taken into account.
9.E-07

8.E-07
2.E+01
7.E-07
2.E+01
6.E-07
2.E+01 5.E-07

2.E+01 4.E-07

1.E+01 3.E-07

9.E+00 2.E-07

1.E-07
6.E+00
0.E+00
3.E+00
0 25 50 75 105 130 155 180 205 230 255 280 305 330 355

0.E+00 A ngl e ( de gr e e )

3 27 53 78 102 127 153 177 203 227 253 277 303 327 353

A ngl e ( de gr e e )

Fig. 5 Magnetic force at the back iron at radius 0.100531m with Fig. 8 Displacement at the teeth shank at radius 0.084624m with (unrepeated
magnetostriction effects curve) and without (repeated curve) magnetostriction effects

1.E+00 Fig. 9 gives the stator deformation comparison with and

9.E-01 without magnetostriction effects.
8.E-01

7.E-01

6.E-01
Force (N)

5.E-01

4.E-01

3.E-01

2.E-01

1.E-01

0.E+00
3 27 53 78 102 127 153 177 203 227 253 277 303 327 353
Angle (degree)

Fig. 6 Magnetic force at the back iron at radius 0.100531m without

magnetostriction effects Fig. 9 Stator deformation with (right one) and without (left one)
magnetostriction effects
2. The magnitude of the magnetic force due to
magnetostriction effects depends on the flux density REFERENCES
magnitude. The bigger the flux density magnitude, the
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Applications of Magnetoelasticity, CRC Press, 1993
comparing the force at the teeth shank part, shown in Fig. [2] O. A. Mohammed, Coupled magnetoelastic finite element formulation
7 and the force at the back iron part shown in Fig. 5. including anisotropic reluctivity tensor and magnetostriction effects for
machinery applications, IEEE Transactions on Magnetics, vol. 37, No. 5,
1.E+02 pp. 3388-3392, September 2001.
1.E+02 [3] Coulomb, J. L., A Methodology for the Determination of Global
1.E+02
Electromechanical Quantities from Finite Element Analysis and its
Application to the Evaluation of Magnetic Forces, Toeque and Stiffness,
8.E+01
IEEE Transactions on Magnetics, Vol. Mag-19, No.6, Nov., 1983.
7.E+01 [4] Enokizono, M., Todaka, T. and Kanao, S., Two-dimensional Magnetic
Force (N)

6.E+01 Properties of Silicon Steel Sheet Subjected to a Rotating Field, IEEE

Transactions on Magnetics, Vol. 29, No. 6, November 1993.
5.E+01
[5] Enokizono, M. and Mori, S., A Treatment of the Magnetic Reluctivity
4.E+01 Tensor for Rotating Magnetic Field, IEEE Transcations on Magnetics,
2.E+01 Vol. Mag-33, No. 2, March 1997.
1.E+01

0.E+00
2 28 52 78 102 128 152 178 202 228 252 278 302 328 352
Angle (degree)

Fig. 7 Magnetic force at the teeth shank at radius 0.084624m with

magnetostriction effects