Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 1, pp. 99-117, Warsaw 2007

OSCILLATIONS OF ROTOR SUPPORTED ON MAGNETIC

BEARINGS WITH IMPACTS IN RETAINER BEARINGS

Ladislav Půst

Institute of Thermomechanics, Academy of Sciences of Czech Republic, Prague, Czech Republic

e-mail: pust@it.cas.cz

Dynamic properties of a rotor supported on two passivemagnetic bearings
are investigated by means of the numerical solution to the mathemati-
cal model of a prototype developed at the Institute of Thermomechanics
ASCR. Magnetic supports always have to include the so-called retainer
bearings in order to prevent the rotor from dangerous increase of oscilla-
tions due to damage in magnetic bearings. Retainer bearings are rolling
bearings, the inner ring of which rotates after an oblique impact by the
rotor journal. This rotation introduces an additional degree of freedom to
the mathematical model of the rotor support. The main aim of this study
is to gain the basic knowledge about the properties of such a system, the-
refore, a new model of impact motion with large amplitudes, introducing
radialHertz stiffness,material contact damping, tangential dry friction and
viscous damping is developed. Dynamic properties of the system described
by 6 differential equations containing strongly nonlinear terms are investi-
gated, and the results are presented in form of response curves. Frequency
intervals of periodic, quasi-periodic and chaotic motions are ascertained
and effects of various parameters on the dynamic behavior of the studied
system are determined.

Key words: space oscillations, passive magnetic bearing, rotor, retainer
bearing, impact

1. Introduction

Magnetic bearings have beenwidely useddue to a great variety of their advan-
tages compared to conventional types of oil and roller bearings. Their unique
advantages are: non-contact operation, lubricant-free working, possibility of
high speed revolutions, zero drag against rotation, etc. There are two basic
principles of magnetic bearings:

• ActiveMagnetic Bearings (AMB),which always require feedback control
of the current and magnetic force for stable levitation of a rotor in the



100 L. Půst

given position. The very important advantage of the AMB is the tuning
possibility for stiffness and damping in the radial direction by means
of which dynamic properties of the whole rotor system can be easily
changed, so that classical limitations can be overcome and reliability
increased.

• PassiveMagnetic Bearings (PMB) using strong permanentmagnets are
simpler than AMB because they do not need any feedback loops, any
power amplifiers and no energy supply, but they have fixed stiffness and
damping properties which cannot be controlled during operation. They
have also very low radial damping and are more prone to exhibit unsta-
ble motions. In spite of this, PMB are very suitable for small machine
aggregates, particularly for those with high angular velocities.

For safety reasons, all types of magnetic bearings have to contain central
safety components, which are realized by emergency retainer bearings with a
sufficient clearance between the shaft and inner ring of the rolling bearing.
They protect magnetic bearings from direct contact with the rotor and retain
amplitudes of vibrations within safe limits after their undesirable increase.

The present paper is oriented particularly on the investigation of dynamic
behaviour of a rotor supported by a PMB after contact with the inner ring of
the retainer bearing.

The rotor/stator rub is a very important problem of rotor dynamics and
it has been investigated by lots of authors for many years, but mainly for
the cases of rotor contact with the rigid, non-rotating inner surface of the
stator (GoldmanandMuszynska, 1994; Isaaksson, 1994;WeiYang andKikuan
Tang, 1994). This occurs in rotating machinery like turbines, compressors,
pumps,generators, etc., supportedonconventional bearings.However, retainer
bearings are usually ball or roller bearings, whose inner rings are set into
rotation after oblique impacts with the rotor journal. The rotary motion is
then damped by dry Coulomb friction in combination with viscous forces.
Deformation of the radial contact is given by deformation between the shaft,
inner ring, balls and the outer ring of the retainer bearing. Therefore, Hertz’s
law of the dynamic contact is used for description of radial forces in the PMB
retainer bearings.

2. Passive magnetic bearing

A prototype of a rotor supported on two passive magnetic bearings has been
designed andmanufactured at the Institute of Thermomechanics, ASCR.The
rotor has a mass of approximately 7kg, and is provided on both ends with



Oscillations of rotor supported on magnetic bearings... 101

three permanent magnetic rings M1 of a diameter D=60mm (Fig.1a) ma-
gnetized in the axial direction. These rings are inserted into slots andmatched
with clearances between four similar magnetic rings M2 fastened in the sta-
tor (Půst et al., 2003). The cap d1 = 0.5mm between the rings is constant
during relative radial dislocation of the rings against each other. Deformation
of magnetic lines produces the returning force F proportional to the radial di-
splacement x (Fig.1b).The retainer bearing Rbwith a clearance rh =0.5mm
limits amplitudes of the shaft Sh in the resonance or at accidents. Because
the rotor cannot be supportedbypermanentmagnets in all directions, another
type of support is used in the axial direction z. The stiff axial support As is
drawn in Fig.1a, however the axial activemagnetic bearing is used in the real
IT prototype.

Fig. 1. Passivemagnetic bearing with rolling retainer bearing

3. Spatial motion of the rotor

The first phase of investigation of rotor motion with impacts between the
shaft and the inner ring was limited to one bearing only. The influence of
various values of Hertz stiffness, combined with three types of damping of the
tangential velocity of the ring was studied numerically by P̊ust and Kozánek
(2002), Půst (2003). The obtained results are presented in form of response
curves and trajectories of motion in the XY plane. Various kinds of impact
motion were found – periodic with single or multiple periods, quasi-periodic
and chaotic oscillations.



102 L. Půst

Thispaperpresents amoreexact dynamicmodel of rotormotion supported
on two passive magnetic bearings and furnished with two retainer bearings.
Motions in both bearings influence each other due to inertia of the rotor.
Vertical and horizontal oscillations of the rotor differ by the weight mgwhich
acts in the vertical direction.Motion in these two planes (x,z and y,z) is also
connected by nonlinear properties of magnetic fields and nonlinear impact
forces.

4. Differential equation of motion of the rotor

A stiff rotor in space has in general six degrees of freedom. In our case, the
displacement in the z-axis is restricted by the axial bearing and canbeneglec-
ted. Velocity of rotation around axis the z is assumed to be constant: ωz =ω.
The remaining 4 DOFs are described by displacements x1, x2, y1, and y2,
in the points 1, 2 separated by the distance l. Both bearings are identical
and they act on the rotor in points 1, 2 by forces Fx1, Fx2, Fy1, Fy2, (Fig.2).
Inertial properties of the rotor are defined by its mass m [kg] andmoment of
inertia I [kgm2] in the point T . The latter can be substituted by threemasses
m1,m2,m3 in points 1, 2, 3.

Fig. 2. Model of rotor

There are two main external forces which influence motion of the system.
The force of gravity mg acts in the point T at a distance a measured from
geometric centre 3.The exciting force (unbalance) can act in arbitraryposition
on the axis z. Let us suppose that the resultant meω2 of the centrifugal forces
acts at thedistance a1 fromgeometric centre 3.The substituted inertiamasses
are (Brepta et al., 1994)



Oscillations of rotor supported on magnetic bearings... 103

m1
m
=−
a

l
+
2a2

l2
+
2I

l2m

m2
m
=
a

l
+
2a2

l2
+
2I

l2m
(4.1)

m3
m
=1−

4a2

l2
−
4I

l2m

Equations of rotor motion are then

m1ẍ1+
1

4
m3(ẍ1+ ẍ2)=−Fx1+

(1

2
−
a1
l

)

meω2cosωt

m2ẍ2+
1

4
m3(ẍ1+ ẍ2)=−Fx2+

(1

2
+
a1
l

)

meω2cosωt
(4.2)

m1ÿ1+
1

4
m3(ÿ1+ ÿ2)=−Fy1+

(1

2
−
a1
l

)

meω2 sinωt+
(1

2
−
a

l

)

mg

m2ÿ2+
1

4
m3(ÿ1+ ÿ2)=−Fy2+

(1

2
+
a1
l

)

meω2 sinωt+
(1

2
+
a

l

)

mg

where Fxi, Fyi (i= 1,2) are the reaction forces in bearings, containing both
magnetic and impact components. The subsystem of retainer bearings is pre-
sented in the plane x1y1 in Fig.4. The displaced journal of the rotor touches
the inner ring of the bearing in point A.

5. Rotor motion without impacts – small vibration

When displacements of the rotor journals are smaller than the clearance rh,

that is ri =
√

x2i +y
2
i ¬ rh, i=1,2, the only shearing component of the ma-

gnetic force acts on the rotor in its support.This force is centrally symmetrical
and shows a softening characteristic with a weak cubic non-linearity

F = kr−k3r3

Components of this force in Cartesian coordinates are

Fxi =(k−k3r2i )xi Fyi =(k−k3r
2
i )yi i=1,2 (5.1)

The damping of passive magnetic bearings itself is very low, close to zero.
However, there are external damping forces from air drag, eddy currents, etc.
Therefore, small lineardamping bẋ1, bẋ2, bẏ1, bẏ2mustbeaddedto thebearing
forces Fxi, Fyi in the mathematical model of the bearings. The differential
equations of rotor motion with 4 DOF in the case when the journals do not
touch the retainer bearings are



104 L. Půst

(

m1+
m3
4

)

ẍ1+
m3
4
ẍ2+bẋ1+[k−k3(x21+y

2
1)]x1 =

(1

2
−
a1
l

)

meω2cosωt

(

m2+
m3
4

)

ẍ2+
m3
4
ẍ1+bẋ2+[k−k3(x22+y

2
2)]x2 =

(1

2
+
a1
l

)

meω2cosωt

(

m1+
m3
4

)

ÿ1+
m3
4
ÿ2+ bẏ1+[k−k3(x21+y

2
1)]y1 =

=
(1

2
−
a1
l

)

meω2 sinωt+
(1

2
−
a

l

)

mg (5.2)

(

m2+
m3
4

)

ÿ2+
m3
4
ÿ1+ bẏ2+[k−k3(x22+y

2
2)]y2 =

=
(1

2
+
a1
l

)

meω2 sinωt+
(1

2
+
a

l

)

mg

Inertial properties of the rotor (m, I, a) have been determined very easily
from the real structure by a simple identification procedure or from its dra-
wings. The prototype made at the Institute of Thermomechanics is shown in
Fig.3. The left pedestal contains also an axial active magnetic bearing. Axial
motion of the rotor is measured at the right pedestal. The rotor prototype
has a mass m=7.379kg, moment of inertia with respect to the axis x going
through the centre of gravity T is I =945.7kgcm2 and the shift of the centre
of gravity along the axis z is a = 0.5mm. The distance between the centres
of magnetic bearings is l=352mm.

Fig. 3. Experimental test rig at Institute of Thermomechanics (ASCR)

Measurements of radial vibration can also be used for the identification of
unknown stiffness and damping parameters of the permanentmagnetic radial
supportafter its assemblement.Twocomplex eigenvalues of the stiff rotorwere
identified from the response

−2.4+28.5iHz and −2.5+46iHz (5.3)



Oscillations of rotor supported on magnetic bearings... 105

The corresponding linear stiffness of magnetic bearings is k=1.25 ·105kgs−2

and the averaged damping coefficient is b=45.8kgs−1.

6. Impacts in retainer bearings

The aim of this Section is to gain basic information about the type of rotor
motionwhen the rotor is subjected to impacts in the retainer bearings.Becau-
se parameters of impacts (kh, bh, f, b4, etc.) are unknown so far, numerical
simulations have been carried out in dimensionless values. Such formof results
is most general; it describes behaviour of a wide class of rotor systems. Pro-
perties of a real structure can be determined very easily from dimensionless
results by simple multiplication.

During operation of the rotor, particularly in resonance, at transient mo-
tion etc., impacts in the retainer bearing can occur, and therefore, a strongly
nonlinearmathematical model is required to describemotion of the rotor sys-
tem. The beginning of contact is shown in Fig.4, where the rotor (radius R1)
touches in point A the inner ring of the retainer bearing. The cause of the
damping of the inner ring (mass m4) duringmotion against the outer ring of
the retainer bearing is the rolling resistance of balls and the viscous resistan-
ce of the lubricant. Therefore, the coefficient b4 of viscous damping and the
constant dry friction force Fd is introduced into the mathematical model.

Fig. 4. Safety retainer bearing with deflected journal



106 L. Půst

The inner ring of the retainer bearing rotates with the velocity w after
an oblique impact and the mathematical model must respect both radial de-
formation of the bearing and tangential forces arising at the contact between
the rotor rotating with an angular velocity ω and the inner ring having ano-
ther velocity. Mathematical models of the supports must contain differential
equations of inner rings.
A dynamic Hertz’s contact force

Fr = kh(r−rh)
3

2

[

1+
bh
r
(ẋx+ ẏy)

]

is supposed to express the radial component of the force. Dry friction with
the coefficient f in contact describes the tangential component of the force
between the journal and inner ring

Fτ =Frf sgn
[1

r
(ẏx− ẋy)+R1ω−w

]

r=
√

x2+y2

where kh, bh are Hertz’s contact coefficients, wr = ẋcosϕ + ẏ sinϕ =
= (ẋx+ ẏy)/r and wt = ẏcosϕ− ẋsinϕ = (ẏx− ẋy)/r are radial and tan-
gential velocities of the rotor, w [ms−1] is the circumferential velocity of the
inner retainer bearing ring.
In this case, the support forces Fxi,Fyi (i=1,2) in equations (4.1) contain,

except for linear andweakly nonlinear terms bẋi+[k−k3(x2i +y
2
i )]xi, . . . (see

(6.1)), also strong nonlinear impact forces with vertical Fximp and horizontal
Fyimp components (i=1,2)

Fximp =Ai
(xi
ri
−
yi
ri
Bi
)

(6.1)

Fyimp =Ai
(yi
ri
+
xi
ri
Bi
)

where for i=1,2

ri =
√

x2i +y
2
i

Ai =H(ri−rh)Fri =H(ri−rh)kh(ri−rh)
3

2

[

1+
bh
ri
(ẋixi+ ẏiyi)

]

Bi = f sgn
(1

ri
(ẏixi− ẋiyi)+R1ω−wi

)

The velocities w1, w2 are given by additional two equations (6.3) describing
motion of the inner rings of retainer bearings.
While solving, Heaviside’s functions H(ri−rh) switch-on or switch-off the

multiplied expressions of impact forces into equations in question. Heaviside’s
function is H =1 if ri >rh and H =0 if ri ¬ rh.



Oscillations of rotor supported on magnetic bearings... 107

The complete set of equations describing spatial motion of the stiff rotor
supported on passive magnetic bearings with the retainer bearings is

m1ẍ1+
m3
4
(ẍ1+ ẍ2)+ bẋ1+(k−k3r21)x1+A1

(x1
r1
−
y1
r1
B1
)

=

=
(1

2
−
a1
l

)

meω2cosωt

m2ẍ2+
m3
4
(ẍ1+ ẍ2)+ bẋ2+(k−k3r22)x2+A2

(x2
r2
−
y2
r2
B2
)

=

=
(1

2
+
a1
l

)

meω2cosωt

(6.2)

m1ÿ1+
m3
4
(ÿ1+ ÿ2)+ bẏ1+(k−k3r21)y1+A1

(y1
r1
+
x1
r1
B1
)

=

=
(1

2
−
a1
l

)

meω2 sinωt+
(1

2
−
a

l

)

mg

m1ÿ2+
m3
4
(ÿ1+ ÿ2)+ bẏ2+(k−k3r22)y2+A2

(y2
r2
+
x2
r2
B2
)

=

=
(1

2
+
a1
l

)

meω2 sinωt+
(1

2
+
a

l

)

mg

and

m4ẇ1+ b4w1+Fd1H(w1)−A1B1 =0
(6.3)

m4ẇ2+ b4w2+Fd2H(w2)−A2B2 =0

where for i=1,2

Fdi = f4Fri

By introducing non-dimensional variables and parameters for linear and
weakly nonlinear parts of the rotor system

α=
2a

l
α1 =

2a1
l

δ=
mg

2krh
Xi =

xi
rh

Yi =
yi
rh

ec=
e

2rh
τ = t

√

k

m
η=ω

√

m

k

B=
b
√
km

κ=−r2h
k3
k

ρ2 =
4I

ml2

µ1 =
m1
m
=
1

2
(−α+α2+ρ2) µ2 =

m2
m
=
1

2
(α+α2+ρ2)

µ34 =
m3
4m
=
1

4
(1−α2−ρ2)

(6.4)



108 L. Půst

and for impacts and retainer bearings

κh =Kh

√
rh
k

Bh = bh
√
km µ4 =

m4
m

B4 = b4
√
km ρ0 =

R1
rh

Φdi =
Fdi
hrh

ρ1 =
r1
rh
=
√

X21 +Y
2
1 ρ2 =

r2
rh
=
√

X22 +Y
2
2 v=

w

rh

√

m

k

H(r1−rh)=H(ρ1−1)

(6.5)

we get a simpler form of differential equations of rotor motion, convenient for
numerical solution

(µ1+µ34)X
′′

1 +µ34X
′′

2 +BX
′

1+X1(1+κρ
2
1)+

+H(ρ1−1)κh(ρ1−1)
3

2 [1+Bh(X
′

1X1+Y
′

1Y1)] ·
·[X1−Y1f sgn(Y ′1X1−X

′

1Y1+ρ0η−v1)]= (1−α1)ecη
2 cosητ

(µ2+µ34)X
′′

2 +µ34X
′′

1 +BX
′

2+X2(1+κρ
2
2)+

+H(ρ2−1)κh(ρ2−1)
3

2 [1+Bh(X
′

2X2+Y
′

2Y2)] ·
·[X2−Y2f sgn(Y ′2X2−X

′

2Y2+ρ0η−v2)]= (1+α1)ecη
2 cosητ

(6.6)

(µ1+µ34)Y
′′

1 +µ34Y
′′

2 +BY
′

1 +Y1(1+κρ
2
1)+

+H(ρ1−1)κh(ρ1−1)
3

2 [1+Bh(X
′

1X1+Y
′

1Y1)] ·
·[Y1−X1f sgn(Y ′1X1−X

′

1Y1+ρ0η−v1)]= (1−α1)ecη
2 sinητ+(1−α)δ

(µ2+µ34)Y
′′

2 +µ34Y
′′

1 +BY
′

2 +Y2(1+κρ
2
2)+

+H(ρ2−1)κh(ρ2−1)
3

2 [1+Bh(X
′

2X2+Y
′

2Y2)] ·
·[Y2−X2f sgn(Y ′2X2−X

′

2Y2+ρ0η−v2)]= (1+α1)ecη
2 sinητ+(1+α)δ

and

m4v
′

1+B4v1+Φd1H(v1)−H(ρ1−1)fκh(ρ1−1)
3

2 [1+Bh(X
′

1X1+Y
′

1Y1)] ·
·sgn(Y ′1X1−X

′

1Y1+ρ0η−v1)= 0
(6.7)

m4v
′

2+B4v2+Φd2H(v2)−H(ρ2−1)fκh(ρ2−1)
3

2 [1+Bh(X
′

2X2+Y
′

2Y2)] ·
·sgn(Y ′2X2−X

′

2Y2+ρ0η−v2)= 0

Numerical solution of the set of six equations (6.6) and (6.7) has been carried
out with the help of the Runge-Kutta integration method. Numerical simu-
lations enable one to analyse spatial motion of the rotor, particularly in the
cases of contact between the stiff rotor and one or both retainer bearings.



Oscillations of rotor supported on magnetic bearings... 109

As an example, let us show response curves of a rotor with parameters

α=
2a

l
=0.01 α1 =

2a1
l
=−0.3 ρ2 =

4I

ml2
=0.96

R1
rh
=100 κ=−0.001

(6.8)

The retainer bearings are characterised by the following dimensionless data

κh = kh

√
rh
k
=10 Bh =

bh√
km
=1.5 f =0.2

f4 =0 µ4 =
m4
m
=0.01 B4 =

b4√
km
=0.1

(6.9)
The effect of different eccentricities e of the rotormass is given by the following
values of the dimensionless parameter ec= e/(2rh)= 0.005; 0.01; 0.015.

Vibrations of a well-balanced rotor with the eccentricity ec= 0.005 does
not reach the value of the retainer bearing clearance rh, and the dimensionless
maximum amplitudes Xm, Ym are always less than 1, see Fig.5. The system
behaves as a linear one.

Fig. 5. Response curves of well balanced rotor; ec=0.005

Two times higher unbalance ec=0.01 causes larger amplitudes that result
in impacts in the retainer bearings (Fig.6 andFig.7). In thesefigures, response
curves of motion in the horizontal direction x (amplitudes Xm), at slowly
increasing frequency η (15000 periods in ∆η = 0.3) are shown in the upper
half of the pictures, while the vertical amplitudes Ym are displayed in the
bottomhalf.Thick lines representamplitudesofbearingNo.1, thoseofbearing
No. 2 are indicated by thin lines. The frequency step is very low, the increase
of η by 1% lasts 500 periods. Therefore, the oscillations can be considered as
quasi-stationary.

The response curves in the upper (horizontal) and bottom (vertical) parts
differ very moderate by due to a very low value of the static deformation δ.



110 L. Půst

Fig. 6. Response curves of rotor with eccentricity ec=0.01. Increasing frequency η

Fig. 7. Response curves of rotor with eccentricity ec=0.01. Decreasing frequency η

Largedifferences are seen in the courses ofmotions of the inner rings v1(η) and
v2(η), characterized by black areas near the axis η. The scale of velocities v
is on the right side of the diagram.



Oscillations of rotor supported on magnetic bearings... 111

Several types of rotor oscillations can be ascertained during the increase
of the excitation frequency η. The range A (1.300<η< 1.389), D (1.425<
η < 1.430), and F (1.464 < η) in Fig.6 correspond to oscillations without
impacts in the retainer bearings. In the range B (1.389<η< 1.393), impacts
in bearing No. 2 arise. Impacts in bearing No. 1 only occur in the range
E (1.430<η < 1.464). The range C (1.393<η < 1.425) is characterized by
impacts inboth retainer bearings.These impacts cause interrupted revolutions
of the bearing rings.

The impacts also change forms of the response curves Xm(η), Ym(η). In
the range C, impacts in both bearings change periodic oscillations into quasi-
periodic ones or chaotic motion. A similar type of oscillations exists also in
the range E, but impacts occur only in bearingNo. 1, while in bearingNo. 2,
only irregular impact-less oscillations are observed.

The response curves at the decreasing frequency η (Fig.7) are very diffe-
rent fromthe response curves at the increasing frequency.Theareas of periodic
impact-less motions A,D,F are wider, and the areas of irregularmotions C,
E are very narrow. Also, interrupted revolutions v1, v2 of the bearing rings
have lower velocities.

Further increase of eccentricity ec=0.015 (Fig.8) results into enlargement
of the area C with quasi-periodic and chaotic motions, where impacts occur
in both bearings. For such a large eccentricity, the impact-less area A does
not change considerably, the area F changes a little more, but the area D
disappears completely.

Fig. 8. Response curves of rotor with eccentricity ec=0.015. Increasing frequency η



112 L. Půst

Irregularmotion in the area E exists only at the left margin, but then the
motion stabilizes on periodic oscillations, which in bearing No. 1 is given by
revolutionwith continuous contact between the rotor and the inner ring of the
bearing.

No contact appears in bearing No. 2, and the amplitudes Xm2 and Ym2
decrease along the curves G. At a frequency of η=1.507, this type ofmotion
jumps into regular periodic impact-less vibration F.

Fig. 9. Response curves of rotorwith eccentricity ec=0.015.Decreasing frequency η

The response curves at the decreasing frequency η and for the high eccen-
tricity ec = 0.015 are shown in Fig.9. Intervals of η labelled C and E are
smaller than those in Fig.8, andno oscillations with continuous revolution v1,
v2 of the inner rings of the retainer bearings occur. Impact-less oscillations F
convert into chaotic oscillations E with impacts in bearingNo. 1 after the di-
mensionless amplitude Ym1 reaches value 1. At the same time, the amplitude
Ym2 jumps on this boundary to an approximately three times higher value and
continues along the curve G (it is overlapped in Fig.9 by records of irregular
oscillations Ym1). This amplitude reaches value 1 on the boundary between
C and E regions and turns into irregular oscillations connected with the in-
terrupted revolutions v2 (see the bottom part of the region C). This type of
oscillations extends up to the field B, where the inner ring of bearing 1 does
not rotate (v1 = 0). Only ring 2 rotates intermittently, but with a very low
velocity. Large viscosity forces in the retainer bearings were supposed in that
example, and therefore the case of the rotor rotating together with the inner
ring did not occur.



Oscillations of rotor supported on magnetic bearings... 113

Motion of such a complicated system is influenced by many parameters.
There are 12 parameters in expressions (6.8) and (6.9). Because the radial
stiffness of passive magnetic bearings is not too high, it is worth to examine
the effect of weight on oscillations of the rotor.

Suppose that the rotor weight acts in the centre of a symmetric system
(a=0) and causes a vertical shift yT =mg/(2k), where k is the stiffness of
one magnetic bearing. This shift reduces the cap in the retainer bearing and
can cause undesired impacts of the rotor journal and the inner ring.Therefore,
the magnetic bearing box is separated from the stator in which the retainer
bearing is fastened. Then the vertical shift yT can be eliminated by a suitable
inverse shift ofmagnetic bearings pedestals so that the equilibriumposition of
the rotor axis coincides with the axis of the retainer bearing. In consequence,
the centre of magnetic field is shifted by yT =−mg/(2k) with respect to the
rotor (and retainer bearing) axis. Thisweight-shift in the prototype IT-ASCR
is 0.3mm,which is 60% of the clearance rh =0.5mm.Due to errors in setting
up to the exact position of the magnetic bearing support, some residual shift
can remain between the axis of the retainer bearing and the rotor axis. The
situation (Fig.10) is described by

yd = yT −ys

where
yT – static deflection due to weight, yT =mg/(2k)
ys – shift of the bearing support, ys =αyT
yd – resultant deflection of the rotor axiswith respect to the retainer

bearing axis.

Fig. 10. Elimination of weight

After adjustment elimination of the weight-shift, the axis of the retainer
bearing coincides with the rotor axis, but the magnetic field axis is moved
upwards.

The response curves of the maximum amplitudes Xm1, Xm2, Ym1, Ym2
versus the excitation frequency η=ω

√

m/k and curves of velocities v1, v2 are



114 L. Půst

plotted in the upper part of Fig.11 for the following parameters R1 = 40rh,
e = 0.03rh, kh = 50k

√
rh, bh = 3

√
km, f = 0.15, b4 = 0.03

√
km (other

parameters are the same as in (6.8)) and for 90% elimination of the weight
effect (α=0.9).

The left part of the upper half of Fig.11 consists of 6 sections; the first
three describe the properties of left bearing 1, the last three correspond to
right bearing 2.

Fig. 11. Upper half: response curves and trajectories of journal centres S and polar
diagrams with radius r= rh+vi, i=1,2, where vi – velocity of inner rings,

α=0.9. Lower half: response curves and trajectories of journal centres S and polar
diagrams of velocity v, but for α=0.6



Oscillations of rotor supported on magnetic bearings... 115

There are several types of rotor oscillations during a very slow increase of
the excitation frequency. Oscillations without impacts are in the range A,D.
In the range B, impacts in bearing 1 appear. Impacts in both bearings and
chaotic motion occur in the range C together with interrupted revolution
of both retainer bearing rings. In the second half of the range C, interrupted
motions v1, v2 change intomonotonous rotation, and the rotor is inpermanent
contact with the retainer bearings. At η = 1.483, the motion jumps into
free contact-less rotation in the range D. The response curves at the reverse
frequency trend are drawn in grey colour. In the frequency zones, where the
records for increasing and decreasing frequency overlap, the grey record is
interrupted (see grey strips in the range C).

Trajectories of the journal centres S at the frequency η = 1.45, where
impacts in both bearings occur, are in the upper right corner. Polar diagrams
with the radius r= rh+vi, i=1,2 in the lower part describe velocities of the
inner rings of the retainer bearings.

Plots in the bottom half of Fig.11 describe properties of rotor motion for
the large effect of weight: α=0.6. The response curves xm, η are similar, but
the curves for vertical oscillations ym, η are shifted downwards. The trajec-
tories on the left side show this shift, the impacts occur particularly on the
bottom part of the bearing rings.

7. Conclusion

• A mathematical model of spatial oscillations of a stiff rotor supported
on two passive radial magnetic bearings has been derived at first for a
weakly nonlinear system.

• The newmathematical model of rotor motion with impacts in the reta-
iner bearings has been derived in form of six differential equations con-
taining strong nonlinear terms – Hertz’s dynamic contacts, dry friction,
etc.

• Two equations of revolution of the inner rings of ball bearings have been
added to the four equations of rotor motion.

• Examples of response curves for various excitations defined by the rela-
tive mass eccentricity ec= e/(2rh) (rh is the clearance in the retainer
bearings) proved that horizontal and vertical oscillations differ due to
the effect of weight.



116 L. Půst

• Numerical solution enables one to find regions of the frequency ηwhere
the rotor oscillates without impacts, with impacts in the left bearing,
with impacts in the right one or in both bearings.

• The inner rings of the retainer bearings rotate according to the intensity
of impacts either with interruptions or continuously.

• Amplitudes of quasi-periodic impactmotion of the rotor are determined
and limited by clearances in the retainer bearings.

• Due to the large effect of weight on the equilibriumposition, elimination
of this shift is very important.

Acknowledgement

This work was carried out in the frame of project GACRNo. 101/06/1787.

Note: This article is extended paper [6].

References

1. Brepta R., Půst L., Turek F., 1994, Mechanical Vibrations (in Czech,
TP71), Sobotáles, Prague, pp. 589

2. GoldmanP.,MuszynskaA., 1994,Chaotic behavior of rotor/stator systems
with rubs,Transactions ASME, J. of Vibration and Acoustics, 116, 692-701

3. Isaaksson J.L., 1994, Chaotic response of rotor/stator rubs, Proc. 4. Inter-
national Conference on Rotor Dynamics, Chicago, 91-95

4. Kudra G., Awrejcewicz J., 2003, The triple pendulum with barriers as a
first step in modelling of the piston-connecting rod-crankshaft system, Proce-
edings 7th Conference on Dynamical Systems, Theory and Applications, Lodz,
Awrejcewicz J., OwczarekA., Mrozowski J. (edit.), 1, 199-206

5. PůstL., 2003a,Dynamicalproperties of passivemagneticbearing,Engineering
Mechanics, 10, 1, 1-16

6. Půst L., 2003b, Space oscillations of rotor supported on magnetic bearings
with impacts in retainer bearings, Proceedings 7th Conference on Dynamical
Systems, Theory and Applications, Lodz, Awrejcewicz J., Owczarek A., Mro-
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7. Půst L.,Kozánek J., 2002,Vibration of rotor supported inmagnetic bearing
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Oscillations of rotor supported on magnetic bearings... 117

8. Půst L., Kozánek J., Veselý J., 2003, Rotor supported on magnetic be-
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Oscylacje wirnika podpartego w łożyskach magnetycznych wywołane

obciążeniem udarowym łożyska ustalającego

Streszczenie

Dynamiczne właściwości wirnika podpartego w dwóch pasywnych łożyskachma-
gnetycznychzbadanowdrodze symulacji numerycznychmatematycznegomodelupro-
totypu zbudowanego w Instytucie Termomechaniki Akademii Nauk Czeskiej Repu-
bliki. Podpory magnetyczne zawsze muszą zawierać w swej strukturze tzw. łożyska
ustalające, które zapobiegają przed gwałtownymwzrostemdrgańwywołanychuszko-
dzeniem łożysk magnetycznych. Łożyska ustalające są łożyskami tocznymi, których
wewnętrzny pierścień zaczyna się obracać po ukośnym uderzeniu czopa wirnika. To
z kolei wprowadza dodatkowy stopień swobody do modelu. Głównym celem analizy
jest uzyskanie wiedzy na temat właściwości takiego układu, dlatego też zbudowano
nowy model odzwierciedlający ruch po udarze o dużej amplitudzie i uwzględniający
promieniowąsztywnośćHertza, tłumieniemateriałowewstrefiekontaktu, tarcie suche
styczne i tłumieniewiskotyczne.Właściwości dynamiczne układu opisano 6 równania-
mi różniczkowymi ruchu zawierającymi członynieliniowe.Rozwiązanie zaprezentowa-
no w formie wykresów odpowiedzi układu. Stwierdzono występowanie przedziałów
częstości odpowiadających ruchowi okresowemu, quasi-okresowemu oraz chaotyczne-
mu. Określono też wpływwielu parametrów układu na jego dynamikę.

Manuscript received August 3, 2006; accepted for print October 18, 2006