Recently magnetic bearings were proposed by several researchers and shown to be viable on a variety of rotor assemblies. The design and construction of such a bearing, which employs features hitherto not used by other workers is examined. These include an original approach to the design of the electromagnets and their amplifiers, and to software in a digital control system, to condition the control signals so as to make the magnets appear to be linear and uncoupled. The resulting system is used to control a rotor-bearing assembly, whose speed range covers two flexural-critical speeds

Gondhalekar, V.

Holmes, R.

English

NASA Technical Reports Server

DESIGLP OF EILECTJXOHAGWETIC BEBRING FOR VIBRATION 
CONTROL OF FLEXIBLE TRAZVSMISSION SIIAF’I? 
V. Gondhalekar 
E.T.H. Ziirich, Switzerland 
R. Holmes 
Department of Mechanical Engineering 
University of Southampton, U.K. 
Recently magnetic bearings have been proposed by several 
researchers and shown t o  be viable on a var ie ty  of rotor 
assemblies. T h i s  paper is concerned with the design and 
construction of such a bearing, which employs features h i the r to  
not used by other workers. These include an or ig ina l  approach 
t o  the  design of the electromagnets and t h e i r  amplifiers,  and 
t o  software i n  a d i g i t a l  control system, t o  condition the 
control s ignals  so as t o  make the magnets appear t o  be l inear  
and uncoupled. The  resul t ing system is used t o  control a 
rotor-bearing assembly, whose speed range covers two flexural-  
c r i t i c a l  speeds. 
INTRODUCTION 
T h e  f e a s i b i l i t y  of using magnet forces t o  control shaf t  
vibrat ions has recently been a subject of a t ten t ion  from a 
number of researchers and successful application of bearings 
and dampers i n  rotat ing assemblies has been reported [ 1 , 2 ,  3 1 .  
A f lex ib le  transmission shaf t  is a special  case of a rotat ing 
assembly which needs some form of control t o  maintain i t s  
vibrat ion amplitudes within reasonable l i m i t s  w h e n  passing 
through i t s  c r i t i c a l  speeds. Squeeze-film bearings have been 
shown t o  be capable of reducing vibration amplitudes but these 
can normally only react  t o  the rotor  displacement a t  the 
bearing locations.  A controlled electromagnet (CEM) located a t  
a point along the length of a transmission shaf t  has a l so  been 
shown t o  be e f fec t ive  i n  reducing vibration amplitudes [21.  
The CEM of fers  the poss ib i l i t y  of applying a force which is a 
acement and/or velocity a t  the CEM location or 
o t e  from it. This of fers  the. poss ib i l i t y  of 
implementing a control s t ra tegy which responds t o  one signal or 
t o  the weighted average of more than one signal measurement 
remote from the CEM location. 
Some magnetic bearing configurations proposed by past  
workers are  shown i n  Fig.1. The s t a t i c  m e m b e r  of t ha t  shown i n  
Fig. la  is the simplest t o  construct but i n  an attempt t o  combat 
eddy c u r r e n t  e f f ec t s  the ro ta t ing  sleeve mus t  be made from 
*The authors gra te fu l ly  acknowledge the Science & Engineering 
Research Council, U.K. ,  for  t h e i r  f inancial  support. 
351 
https://ntrs.nasa.gov/search.jsp?R=19850005827 2020-03-20T19:41:36+00:00Z
rectangular laminations. Any lamination completing the flux 
path only does so for a short time increment and other 
laminations sequentially perform this function as the shaft 
rotates. A rela vely long axial to provide a 
given magnetic f on of Fig.lb 
makes lamination of both static and rotating members easy, 
the latter now bei from flat discs. A 
disadvantage, however, is inefficient radial space utilization. 
A l s o  the long narrow limbs lead to high magnetic saturation, 
that is high flux concentration at the magnet bases. 
Magnetic bearings are best igned to ate in an 
attraction rather than a repuls mode, but either case the 
basic inverse square-law relationship between magnetic force 
and gap has to be overcome since this can lead to instability. 
Schweitzer [l] describes a design similar to that of Fig. lb in 
which stability is achieved, together with the linearity of the 
force-current characteristic by a differential circuit, in 
addition to a circuit ensuring premagnetisation with a constant 
current. Thus operation takes place about a non-zero steady- 
state condition. Easier analysis and controller design are 
assured by the improvement in linearity but bulky magnets are 
necessary to dissipate the consequent heat produced by the 
premagnetisation current. 
A logical next step is to integrate all four magnets into 
a single structure as shown in Fig. IC. However, there is here 
a high degree of interaction between the four poles, and 
consequently the magnet performance has to be improved by 
special techniques requiring bidirectional drive amplifiers to 
alter the direction of the flux as required [2]. This 
disadvantage can be overcome by using the four active/four 
passive pole configuration of Fig. Id C31, in which four 
unidirectional amplifiers can be used. 
Fig. le shows a 3 pole active/3 pole passive configuration 
as used in the present work, as it gives a better space 
utilization than does that of Fig. Id, as well as reducing 
magnet interaction. However, since the magnets do not lie on 
mutually perpendicular axes, xy, cross-coupling is introduced 
between these axes. This is counteracted by suitable software 
using a microprocessor interface which in any case is essential 
for adaptive control purposes. Also it was decided to d 
for operation about a point of zero magn xcitation as 
obviates the need for an extra coil to supply a constant 
magnetising current. Thus resistive heating of the magnet coils 
is reduced and consequently their weight and size. The whole 
device is required to exert dynamic rather than static force 
components and hence there is no need from that point of view 
for a non-zero quiescent operating point. One of the main 
objectives.of the present work is to design software to 
condition the control signals so as to make the magnets appear 
linear (and uncoupled) in spite of using this zero-excitation 
352 
operating point. 
The rotor system controlled by the electromagnets is shown 
in Fig.2, and the final electromagnet configuration in Fig.3. 
The overall system including the microprocessor and associated 
peripherals is shown in Fig.4. The electromagnets, their drive 
electronics and the computer system are described in some 
detail and some results obtained from the rig to date are 
presented. 
THE ELECTROMAGNETS AND THEIR DRIVE ELECTRONICS 
A block diagram of the electromagnet system designed for 
the present work is shown in Fig.5. Each power amplifier, A, 
which drives a magnet coil, C, is of the switching type, 
employing pulse-width modulation to reduce power losses. It is 
designed to operate at a maximum A.C. input of 220 v rms and as 
a result approximately 300 v D.C. is available to drive each 
coil, with a current limit of 10 amp, but with available 
capacity up to 15 amp. 
The electromagnet configuration of Fig. 3 consists of six 
radial poles - 3 active and 3 passive - the latter acting as 
return flux paths. The electromagnets and the rotor sleeve are 
constructed from 0.1 mm disc laminations to eliminate eddy 
currents up to approximately 300 Hz and the 3 active poles were 
suitably energised to generate forces along the x and y axes. 
This particular configuration was chosen because it utilizes 
the available radial space more efficiently in terms of force 
than a similar structure consisting of four active poles, and 
the number of drive amplifiers is reduced from 4 to 3 .  One 
djsadvantage is the force interaction introduced between the 
orthogonal axes, but as mentioned above, software is employed 
to overcome this problem. 
An electromagnet exerts a force approximately proportional 
to the square of the magnet flux present at the pole face. It 
is also unstable in an open loop mode since, as the deflection 
increases towards a magnet, so does the attractive force from 
that magnet. The system can, however, be made stable by 
feeding back a signal VH produced by a Hall probe, which is 
proportional to the flux at the pole face (Fig.5). 
Since the electromagnets can only operate effectively in 
an attraction mode, with no repulsion, then in order to 
overcome this unidirectional forcing feature in creating 
restoring forces to counteract displacements, X and Y, the 
excitation signals, SI, S2, S 3 ,  to the three drive amplifiers A 
(Fig. 5 ) would be constituted as follows : 
353 
- O.5CXl0 T/4  + 0.5Cxl 3T/4 - 0-5cX13T/4 T 
T/ 4 
T s1 = 
where T is the periodic t i m e .  For  displacements X and Y as 
shown i n  Fig. 6a, the s ignals  S thus appear as  shown i n  t h a t  
f igure  . 
I t  w i l l  be noted t h a t  a l l  th ree  s igna ls  contain both X and 
Y components. This is because magnets 2 and 3 have t o  exert  
forces t o  counter X and Y displacements ,  b u t  i n  the process 
introduce unwanted X-dependent forces i n  the Y di rec t ion .  
These are n e u t r a l i s e d  by addi t ional  X-component forces 
generated i n  magnet 1 by s ignal  SI. Assuming t h a t  a l i nea r  
re la t ionship  e x i s t s  between the s igna ls  and the a t t r a c t i o n  
forces  produced by the magnets, then, for  some amplifier gain, 
c,  
I n  f ac t ,  the  force produced a t  each pole face is 
proportional t o  the square of the f lux a t  t h a t  face and hence 
t o  the square of the c u r r e n t  supplied t o  each magnet c o i l .  
Thus, i f  there  is a l i nea r  r e l a t ion  between any s ignal  S and 
i t s  c o i l  current then each magnet force w i l l  be proportional t o  
S2 and not t o  S as assumed above. However, t h i s  can be 
p a r t i a l l y  accommodated by replacing the mul t ip l ie r  0.5 i n  the 
f i r s t  of equations (1) by 0 .5 ,  the  other two equations 
remaining unaltered. This i s  followed by wri t ing Fy and F, as 
FY = [ - ( s l ) 2  + 
As a r e s u l t ,  the forms of the forces produced i n  the x,y 
d i rec t ions  a re  shown i n  Fig. 6b. When combined t o  produce a 
354 
polar  force diagram a s ignf icant  3rd Harmonic is revealed 
( F i g . 6 ~ ) .  
ach 
S, 
Linearisat ion 
ieved by using 
t o  each magnet 
of each magnet device can, however, be 
the square root of the control  s ignal  input,  
amplifier.  This input s ignal  then produces a 
proportional f lux,  the approximate square of which is 
proportional to  the force exerted on the ro tor .  The control 
s igna l  S ,  vs force output thus follows an approximately l i nea r  
re la t ionship  and allows the  s ignal  and hence force 
contr ibut ions of the three magnets t o  be added l i nea r ly .  
The square root of a number can be found using software by 
computing a f i n i t e  number of terms of a s e r i e s  expansion, b u t  
t h i s  can be time-consuming i n  a real-time appl icat ion.  A 
piece-wise l i nea r  approximation t o  the function y = K J S  was 
therefore  used, where S is the control  s igna l ,  and K is a 
constant. The above function is divided i n t o  32 segmen t s  of 
progressively grea te r  length, since a square root function 
va r i e s  rapidly around zero, and the software computes the 
square root i n  96 usec on a 28002 cpu. The square-rooting 
procedure, when applied t o  the control  s ignals  of a l l  3 magnets 
r e s u l t s  i n  the polar force diagram shown i n  F ig .  6d, indicat ing 
considerable improvement over t h a t  shown i n  F ig .  6c. The 
software can be adapted, i f  required, to  compensate for  the 
sa tura t ion  of the magnet core material  a lso.  
A s  w e l l  as being influenced by the square-law non- 
l i n e a r i t y ,  the magnet forces a re  a l so  subject t o  current-rate  
sa tura t ion  i n  the  dr ive amplifiers.  This occurs i f  more than 
the  maximum drive voltage of the amplifier i s  demanded. 
Fortunately, i n  prac t ice ,  the current r a t e s  demanded are  low 
enocgh t o  be unaffected by any l imi ta t ions  i n  V, pa r t i cu la r ly  
when the restor ing forces provided by the magnets are  kept 
within reasonable l i m i t s .  
CONTROL STRATEGY 
Assuming negl igible  in t e rna l  damping, the ro tor  vibrat ion can 
be represented by the following set of equations: 
i = 1 t o 4  Yi - 1 a i j ( P .  + mw y . )  - ai4Kdy4, 2 
j=3 
j= l  
- 
3 J 3  
where a i  a re  the influence coef f ic ien ts  ( m / N )  r e l a t ing  t o  the 
mass s t a t ions  1, 2 and 3 ,  w is  the ro tor  speed i n  rad/s,  P j  a r e  
t he  forces due t o  mass unbalance, y j  a r e  displacement 
amplitudes a t  the  mass s t a t ions  and Kd i s  the  s t i f f n e s s  
introduced by the  CEM a t  s t a t i o n  4. The negative dynamic 
s t i f f n e s s  curve a t  s t a t i o n  4 for  t h i s  ro tor  i s  shown i n  Fig.7. 
Dynamic s t i f f n e s s  is the ro tor  dynamic force a t  s t a t ion  4 
355 
divided by dynamic displacement, and the frequencies at which 
the negative stiffness equals the dynamic stiffness of the 
support at station 4 are the natural frequencies of the 
complete system. Thus the intersections with a horizontal line 
Kd parallel to the w-axis give the natural frequencies wn (that 
is critical speeds) of the rotor system. A speed-dependent 
controller can ensure that the natural frequencies of the rotor 
system are sufficiently removed from the mass-unbalance 
disturbance frequency (that is the rotational speed) by 
automatically adjusting the stiffness Kd. 
It is also possible to introduce both stiffness and 
damping at station 4. Fig. 8 shows some typical response 
curves taken from the rotating assembly of Fig.2, sample values 
of stiffness and damping provided by the controlled 
electromagnets being indicated. These serve to indicate how 
the critical speeds can be altered and the peak responses 
brought down to acceptable levels. 
CONCLUSIONS 
The results of this research indicate that efficient and 
compact electromagnets have been designed to effectively 
control the vibrations of a supercritical rotating assembly. 
By the use of a microprocessor, software has been designed to 
overcome any problems of cross-coupling between orthogonal axes 
and of inherent non-linear force-def lection characteristics in 
the magnets. By the same microprocessor a considerable degree 
of control can also be achieved, as indicated in Fig. 8. 
REFERENCES 
1. Schweitzer, G. and Ulbrich, H. 'Magnetic Bearings - a novel 
type of suspension'. 1.Mech.E. Conference on Vibrations in 
Rotating Machinery, Cambridge, Sept. 1980. 
2. Nikolajsen, J.N., Holmes, R. and Gondhalekar, V. 
'Investigation of an electromagnetic damper for vibration 
control of a transmission shaft'. Proc. 1.Mech.E. Vol. 193, 
No.31, 1979. 
3. Habermann, H. 'Le Palier magnetique actif "ACTIDYNE"' . AGARD 
Conference Proceedings, No. 323, 'Problems in Bearings and 
Lubrication', Ottawa, Canada, May 1982. 
356 
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363 


Design of electromagnetic bearing for vibration control of flexible transmission shaft

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