Equations are derived for the approximate calculation of the air gap configuration of a hollow-rotor clutch

Yermolin, Y. A.

English

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NiASA A, 5 8 3
 
M~ORAND1M
NASA TECHnCAL 
A 1 YSTEREYSISGAP OFTHE AIRFIELD IN,AN'IC 

CYLINDRICALROTO
OLUTCH WITH 

YU. A.,erl]l
 
NA78-32428
 
AGNETIC FIELD IN THE 
(ASA-TN-7513) 

GAP OFA fYSTERESIS CLUTCH 
WITH CYLINDICAL
 
ROTOR- National Aeronautics 
and Space
 
Unclas
 
9 p HC A02/ME AO1 CSCL 131 
Administration) 

Vozdushm zaroryeTranslation of -Magnltnoye polye 
 63-66.
Rufty s tsiiridrichesklm rotorom
mis erizi noy
8, 1971, PP. 

ElektriChestvo'NO* 

A NONAD. l.05 AND SPACE 197 
MC 1978AY0)
NATIO~NA 
. c. 2054 
https://ntrs.nasa.gov/search.jsp?R=19780024485 2020-03-22T01:56:26+00:00Z
STANDARD TITLE PAGE 
I. Report No.NASA TM 7518 2. Govormont Accession No. 3. R.cipiena's Catalog No. 
4. Tit . nd Subti Magnet-ic Field in the Air Gap S. Repor Do,.a
 
of a Hysteresis Clutch,with Cylindrical' 6. P1r9ornngOrgonizono, Cod.
 
Rotor
 
7. Author(s) 8. Performing Organization Report No. 
Yu. A. Yermolin 10. Work Unit No. 
9. Pe.lorm.ng Orgo,zotuon Name and Add,.es 11. Contract or Giont No. 
.SCITRAN NASw-2791 
Box 5456 13. Typ. of Report and Period Coy.r.d 
Santa Barbara, CA 93108 Translation 
12. Sponsoring Agency Name and Address' 
National Aeronautics and Space Administration 14. Sponsoring Agency Cod.
 
Washington, D.C. 20546
 
15. Supplen.nary Note 
Translation o-f "Magnitnoye .polye v vozdushnom zazorye
 
gisterizisnoy mufty s tsilindricheskim rotorom",
 
Elektrichestvo)No. 8, 1971, pp. 63-66
 
16. Abstract 
Equations are derived for the approximate calculation 6f the
 
air gap configuration of a hollow-rotor qliitch. ..
 
i 1 14OIGINAL. y
Op? pooR QPAL 
17. Key Words (Selected by Auttior(s)) 18. Distributi n Statement 
Unclassified - Unlimited
 
19. Security Ciassil. (of this report) 20. Security Clossif. (of Iis pg. 21. No. of Pages 22. 
Unclassified Unclassified 9
 
h'ASA-IIQ 
MAGNETIC FIELD IN THE AIR!GAP-OF A HYSTERESIS CLUTCH /63*
 
WITH CYLINDRICAL ROTOR
 
Yu. A. Yermolin
 
Recently, electromagnetic clutches (EMC) have been finding in­
,creasing applications in various power-driven devices and in systems
 
'of automatic control. The principle on which these clutches operate
 
has been thoroughly-.deseibed--In"the, 'JAterat-ure--L, 21. Their design
 
is not complicated, and they posses's highly stable characteristics.
 
'In many instances, EMC compete successfully with other analogous de­
vices, e.g.,. with electromagnetic-particle clutches (EPC). Without
 
dwelling on the various applications of EMC, we shall show, however,
 
that in the case of pre-assigned overall dimensions, successful design,
 
lin most instances determines the'maximum moment transmitted by the
 
:clutch. From this.point of view, there is great practical interest
 
in EMC with a rotor in the form of -a fui cylinder. 'The schematicII 
diagram of this clutch is presented in Figure i.
 
The inductor 1 is made of cast steel. The complete rotor 2,
 
;made of maghetically-hard material is placed in the air gap of the
 
inductor and is seated rigidly on-the shaft 3. The control coil 4
 
receives d.c. current through the conductring 'ring 5, and generates
I
 
a magnetic flux, whose path is shown by broken-lines in Figure 1. Due
 
to the cylindrical slots 6, the magnetic flux, which passes across the
 
"air gap, splits up and forms poles. Thus, at each point of the air
 
gap within'the boundaries of the division into poles, induution oc­
'curs. The value and direction of this induction are functions of
 
the coordinates of each such point.
 
I When the distribution law for the induction in the air gap is /64
 
known, all characteristics of the EMC can be determined. However, it
I
 
is quite complicated to find this law for the gap configuration-under
 
'consideration(c.f. Fig 1). Successful solution of this self-contained
 
'problem can only be accomplished by approximation, using a number of
 
iNumbers in margin indicate pagination in foreign text.
 
1
 
.. .. . . A G ISassumptions. 

Op- PCIOR QUJALTY 
.1/i 
6A 
Figure 1
 
A AV,? :0a YU. |I I 'i 
jFigure 2 
 -
Since the axial dimension of Ithe gap is somewhat larger than the
 
iradial, the field can be considered as plane-parallel and a solution
 
1can be sought for the LaPlace equation relative'to the scalar magnetic
 
Ipotential U.on the plane
 
vU=O. (1) 
Let us consider the evolute of.the air gap having p poles within
 
!the boundaries of one polar divisidn (Figure 2).
 
Let us assume that to the upper boundary there correspond the
 
,potential U!, and to the.lower, the potential U2 , and let.us determine
 
the value of the magnetic potential on the straight-line DE. Analysis
 
of the results of modeling,on e'lectrically-conducting paper and consi­
'derations of symmetry make it possible to assume that-the field withif­
,the boundaries of a cylindrical slot can be described with sufficient
 
taccuracy by circular arcs of radius r, as shown in Figure 2.
 
As a firstapproximation, welshall assume that the magnetic po-,
 
tential at any point on the line DE differs from U by an amount which
 
2 
is proportinal to the length of the magnetic line of force 1 passing
 
through this point. From the triangles OBA and ABC, we obtain:
 
I-.,_ 2tP arc, 2p (2) 
where a is the radius of the cylindrical slot, and y is the moving co­
ordinate.
 
As shown above, we shall assume:
 
&U=k({p). .(3) 
Let us determine the coefficient k! from the following conside­
rations. It is obvious that between +E and -E when I + 0 (Figure 2),
 
we may regard the field as formed by two coaxial circular cylinders of
 
radii (R + a) and R with potentials U and U2, respectively. It is
 
well-known that the expression for the scalar magnetic potential of
 
such a field has the form [3]:
 
U=MI +N, Inp, (4) 
where E1 and N are constants, and ,p.is the moving radius.
 
To determine M1 and N let us solve the system:
 
Ut=1v1 N, ,n(Th-F-a);} (5) 
U,= + N, 111 ? ' 
We obtain
 
U2 III U1 I, R .1, (R, + -P 
V (6)(6
 
N= U, -- U, 
Then the potential U at the point 0 is determined by-substituting

-o 
.the values obtained for the constants M1 and M into (4) with p
 
Further,- we find that
 
In 
AU= U_ -UD=(U-- U) () 
and on equating (7) to the expression (3), whencp = 0, we obtain: 
L VAGI, 1ORIGIN 
QEPOOR QJJE 
al an 
On repeating the analogous arguments and calculations for the
 
lower boundary of the air gap, we have:
 
a 
a In R ­
(U --U.) R' (9) 
To simplify the further calculations, we shall approximate (2) by the
 
function
 
l(q) =acos pq. (10) 
The validity of this approximation is obvious from a comparison of the
 
graphs constructed from (2) and (10) (Figure 3).
 
Taking into consideration the assumptions made and the accompan­
ying calculations, it turns out to be possible to replace the actual
 
air gap by an annular gap with boundary conditions written in the form:
 
U R, -­
2p 2p?Ak ,cosrwhen
'1 

we 21, 2p' 
1 U2-j-ak~cosp? e.jj- Q '--; 
P 2p 
Thus the problem is reduced to that of finding a harmonic func­
tion inside the ring when the valu6s of the function are known on the
 
boundaries. This problem is known in literature as the Dirichlet Prob­
lem for a Ring [4].
 
Expanding the boundary conditions (11) in a Fourier series, and /6 
without sacrificing generality, placing UI = U and U2 0, we obtain: 
'2 
4 
--
-_ 
os,UQ=R,)=AU- ak, 
+ 2ak, ___) _2 
n - 1 -COS Upo;A 
,,cos.p. ( 12) 
UtpJ= ) = -- ±-% cospv-F 
U" n a2 h 
_-2 COS /IPf. 
- t4.6 n
Taking into account the form of the spectrum of the boundary con­
ditions in the expansion (12), we shall seek the solution of the La­
place equation (1) in polar coordinates in the form [41:

U (p,y)=A+IB Inp+ (C,pP+ C_,p)cospp+ 
+ (Cnpn,1+C.c_.n-,) cos (13) 
n=2,4,6,. 
To determine the constant coefficients in (1-3T, -itis sufficient to
 
solve the system: 
AU- ak, o, 2akX 
n00 
X OSnpcfA+BlnR-+
n=2,4.6,... 
+ (C1Rf+Cc_-RT :osp+ (CPR"-I­
+C_,,R i) Cos 11pT, 
_ okt -- x (14).akB 2ak 
-A, - ak- Cos pT+ 7 X 
(-i) ~ 
-A -l-ln R2 cosnpX 
n=2,4.6..., 
a- + (C1R2 + C C_ PTpC+osa arospq rC-, +(C R n=2.4,6.. 
if+C- 7; R2losz
 
9
 
0 3 
2p
 
Figure 3
 
Equating in (14) the amplitudes of the corresponding harmonics
 
on both sides of the equations and solving the system of two equations
 
thus obtained, we have
 
In-L1 Aio- a)rR, -
A B ORIGINAL PAGE IS 
AUa I ' . OF POOR QUALITY 
Inv 
______ k -~pSR '(p2--i) et-1_ 2 kl+k 2ppRFv(p2P-I) .(5 
.- (15) 
tp 
2a (-1)2 - k-- k "pr 
where 6 = 
. 7J 0.1 
Figure 4. 1 , 9 pp- r 
It should be noted that, in calculating the coefficients I and 
k2 from (8)-and (9), it is also necessary to set UI = AU and U = 0. 
Substitution of the coefficients (15) and (16)-into the expres­
sion for the magnetic potential (13) yields the solution of the prob­
lem. I 
The components of the induction with respect to the coordinate 
axes are connected to the potential U(p, y) by the familiar relation­
ships [31: . 
Bau B I-du 
V_-Io B!tET .A 
6-­
6 
To test the results obtainedfon the digital computer "Nairi", the
 
field in the air gap of a hysteresis clutch with the follqwing design
 
=
!parameters was computed: RI = 1.75 cm, R2 1.7 cm, a = 0.18 cm, p = 
15. Here in the series (13) under the summation sign five terms were
 
retained. The curves of the equip6tential .lines obtained -(Figure 4,
 
'broken lines) were compared with tie experimental curves obtained by
 
,modelling the field on conducting paper (Figure 4, the solid lines).
 
The results of the comparison makejit possible to 4raw a conclusion
 
'about the extent to which computation and experiment are in good agree­
'ment. 
In conclusion, it should be noted that the expression for the
 
:magnetic potential in the form (13) is valid only in the domain boun­
ided by the radii R and R2, i.e., in the region of greatest interest
 
1 2 , ergo fgraetitrs
 
from the point of view of possiblelfurther analysis of EMC, and cannot
 
be used to find the field in the cylindrical slots of the inductor.
 
This circumstance is explained by the fact that in the course of the
I 
,solution the problem was reduced to the Dirichlet Problem for a Ring.
 
REFERENCES
 
1. 	Vorob'eva, T. M.: Elektromagnitnyye mufty (Electromagnetic Clutches),
 
State Publishing House for Power Engineering, 1960.
 
2. 	Sotskov, B. S.: Osnovy rascheta i proyektirevaniya elektromekhani­
cheskikh elementov avtomaticheskikhA telemekhanicheskikh ustroy­
sty (The Bases for Designing and Planning the Electromechanical
 
Elements of Automatic and TelemechanicaliDevices). "Energiya",
 
Publishing House, 1965.
 
3. 	Neyman, L. R., Demirchan, K. S.: Teoreticheskiye osnovy elektro­
tekhniki (The theoretical bases of Electrical Engineering).
 
"Energiya" publishing House) 1967.
 
4. Smirnov, V. N.: Kurs vyshyey matematiki (Course in Higher Mathe­
matics), v-. 2, State Publishing House for Engineering and Econo-'
 
mics Literature, 1957. I
 
7 


Magnetic field in the air gap of a hysteresis clutch with cylindrical rotor

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