Online proceedings - EDA Publishing Association

Fig. 8. Central finite difference method applied to the magnetic flux
density.
The nodal magnetic force is
,
,,
∆
11-13
May 2011, Aix-en-Provence, France
(17)
where L, in the case considered.
The magnetic force acting on each slice of the proof mass
is related to the real volume of that portion of graphite.
Depending to the number of nodes present on the
longitudinal axis of one slice, each nodal force is referred to
a small volume centered on the same node. For each
graphite portion, it is related to the unit volume by the
ratio
(18)
where is the number of nodes on the longitudinal axis.
The magnetic force acting on the -th portion of the proof
mass is
∑ ,
(19)
where , is the magnetic force per unit volume
calculated in the node , . The total magnetic force acting
on the entire proof mass is
2·∑ ∑ , . (20)
VII.
RESULTS
The levitation height was measured by the laser sensor [9]
on the configuration with 1, 2, 3, nominal graphite side
10mm and nominal graphite thickness
0.3, 0.5, 0.7, 0.9, 1.0mm; the results, referred to the midplane
of the proof mass, are reported in Fig. 9.
the experimental levitation distance L, 1.0687mm in
the configuration with 1 and 1mm; the measured
thickness of the proof mass is 0.9117mm. The half
graphite mass was divided in 7 portions ( 1, … ,7) and
the coefficient for the volume correction was calculated for
every portion. The magnetic force acting on each slice of
graphite was calculated and compared to the corresponding
value of the gravity force acting on the same portion. The
mesh size used was 0.5mm that gives the number of nodes
along the longitudinal axis of each portion. The results are
listed in Table II.
Portion
Number
of nodes
TABLE II
NUMERICAL CALCULATION RESULTS
Volumetric
coeff.
(α)
Magnetic
force
[mN]
Gravity
force
[mN]
1 27 0.52 0.211 0.261 0.81
2 23 0.52 0.269 0.221 1.22
3 19 0.53 0.090 0.181 0.50
4 15 0.54 0.106 0.141 0.75
5 11 0.56 0.102 0.100 1.01
6 7 0.60 0.082 0.060 1.37
7 3 1.00 0.016 0.020 0.80
The gravity force acting on the entire graphite mass is
1.970mN. The numerical calculation of the magnetic
force was conducted at the levitation height of the graphite
mid-plane 1mm L, . Due to the high uncertainty
about the magnets coercive force, two values of
representative of its range of variability were considered.
For 750 kA⁄ m, the total magnetic force results
1.751mN, corresponding to the error with the gravity
force of about 11.1%. For 900 kA⁄ m, the total
magnetic force results 2.409mN, corresponding to
the 22.3% error with the gravity force. Following the first
approximation of linear interpolation, the actual value of
coercive force for the current magnets is about 800 kA⁄
m
In conclusion, the force equilibrium is demonstrated by the
discrete model, with relative small errors that are
addressable to (a) the experimental error in the evaluation of
levitation height, (b) the approximations introduced by the
discretization, (c) the uncertainty about the material
properties and the magnetization curve of the magnets.
Further investigations will provide more detailed results in
terms of resolution of the levitation height and material
parameters evaluation; different configurations of the
levitating system will be considered as well.
Fig. 9. Experimental levitation height of the proof mass mid-plane in
different configurations: N=1 (◦), N=2 (), N=3 ().
The calculation of the magnetic force was conducted at
VIII. CONCLUSIONS
The numerical model developed and presented in this
paper was used to calculate the magnetic force acting on a
proof mass of diamagnetic material levitating on permanent
magnets in the ‘opposite’ configuration. Starting from the
magnetic field distribution obtained with a commercial
FEM simulator, the nodal magnetic force on the proof mass
was calculated at the vertical position corresponding to the
experimental levitation height. The resulting magnetic force
was compared to the gravity force acting on the proof mass
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