Coupling between eddy currents and rigid body rotation: analysis, computation, and experiments Page: 3 of 5
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The last term in (4) can generally be dropped
■since 0 and C2 are both small. Then the currenc t'
can be elirainaCed Erora (3) and (4) Co give Che Ch.lrd-
>rder differencial equation:
■* + C3^1 + C4) — + C30
dt'3 dt'2 dt'
= - CjCjCjC^ exp [-C jC ') . (5)
In addition to the particular solution*
a (c') = f je C lt , (6)
P 1
where
C1C2C3C,
* C3](C1 - l) + C^C,,
generally has one real exponenctat solution [e^i j
and two complex conjugate exponential solutions
el»jui)c j . The homogeneous solution can Chen be
restated in terms of real variables, cos mt' and
sin ut ':
B^tf') » e'1*’ [f2 cos ut' +■ f3 sin ut' ]
where gj, g;l g3, gkS and g= are constant coeffi-
cients. Equations (8) and (11; 6how that Che oscilLa-
Clons occur in both Che mechanical and electrical
solutions. The frequency of oscillation is dominated
by Che coefficient of the third term in (5):
u - [C3(l + Cj]l/2 • (12)
Results
In the analysis we combine the nondlraensional
parameters Cj, C2, Cj, and C4 in Che forms they appear
In Che governing equations, (3) and (4). The new
parameters are C., C12 (- C1C2), C,, and C34 (»
C3C4). Equations (8) and (11) were solved over a wide
range of parameters within the small angle limit. Tor
practical applications, Che values of peak rotation
9ra, and frequency of oscillation u are important
quantities and are emphasized In this analysis. A
prior prediction of these values would allow a
designer to design the components suitable to the
mechanical tolerances.
Because of the vast combinations among the four
parameters Cj, C12, C3, and C34, efforts were made to
find a tractable characterization of 0ra foe easy com-
prehension and predictions. The following may be seen
from the results of our study.
♦ f^e8c' . (7)
Equations (6) and (7) combine to give the general
solution for the angle of rotation
-C.t" ot'r i
0( l ) - f je 1 + e [ f 2 cos mt +f3sinwi|
- . (8)
The constants t 2* fand f4 are determined from
the Initial conditions for 0 and l':
U 0
0 - -i and - - 0
dt '
At Z '
= 0
(9a)
l ' = U
at t'
= 0 .
(9b)
From (a >.
^ . 0
at c'
= 0 .
(9c)
it
So Luc ion for nondlraensional cucent l" coraes from
U ):
... -t-fr'
l U ) = e I
J 0
One can readily recognize In (10) two components
Jt Induced currents. The first component (terms In-
volved C ^0 2e-C lc ) is due to the decay of the time
.hanging magnetic field, and the second component
(terms involved dB/dt") arise from the rotation of
the conductor. The angular velocity d9/dc' may be
obtained from (8) and Integrated to give the following
expression for 1':
-C ,t' Bt' -t'
i (t ) = g je * + g2e + g3e
-C.t"
CjC2e 1 +
dt"
■ (10)
l. For any combination of Cj, C3, and C34, 0ra
increases linearly with Cj2 at a rate that depends on
the former three parameters as illustrated in Fig. 2,
l.e., 0ra may be written in the form
0m(Cl>CI2’C3>C3j ■ C12f(CI-C3-C3J • (l3)
o io o
g o 125
0 1001
0 075
0 050
0 025
Fig. 2. Maximum deflection for various combinations
of C j, C 3 , and C3(, <
2. For constant C34/C4(- C4), f is almost Inde-
pendent of C,. Therefore, the variation In 9m may be
understood from the relationship among f, Cj, and
C3./C3. Figure 3 shows contour curves of f versus C3
and C34/C3 at a contour increment of 0.25. The fol-
lowing observations may be drawn from this figure.
(1) For large Llxed values of C3(C3 > 1), f Is
generally small (f < 0.75) and Increases weakly with
C34/C3, approaching a constant limit when C34/C3 > 10
(not shown in figure).
at' at'
+ g4e cos tut + g3e sin mt
(11) For C3 < 0.5 and C34/Cj > 1, f Increases
(11) rapidly with C34/C3.
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Hua, T. Q. & Turner, L. R. Coupling between eddy currents and rigid body rotation: analysis, computation, and experiments, article, January 1, 1985; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc1093579/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.