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7/30/2019 Introduction to Magnetic Bearings
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1
Introduction to Magnetic Bearings
Jagu Srinivasa Rao, (Research Scholar)
Department of Mechanical EngineeringIndian Institute of Technology Guwahati
December, 2008
Lecture presented in Quality ImprovementProgram (QIP08) at Indian Institute of
Technology Guwahati
Overview of the Presentation
Introduction
Design of Active Magnetic Bearings
Control Engineering of Magnetic Bearings
Control of Rotor by using Magnetic Bearings
Conclusions
Introduction
An active magnetic bearing (AMB) system supportsa rotating shaft, without any physical contact bysuspending the rotor in the air, with an electricallycontrolled (or/and permanent magnet) magneticforce.
It is a mechatronic product which involves differentfields of engineering such as Mechanical, Electrical,
Control Systems, and Computer Science etc.
Test Apparatus for rotor control
Eight-Pole Radial Magnetic-Bearing
Radial Magnetic Bearing
Horizontal shaft Vertical shaft
Rotor shaft
Upper AMB
Lower AMB
Rotor Disc
Coil WindingLeft AMB
Thrust Magnetic Bearing
Left AMB
Rotorshaft
Typical Actuator Controller unit of an AMB
Introduction to Active Magnetic Bearings
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Working principle of magnetic bearing
Electro magnet
Sensor
Controller
PowerPower
AmplifierAmplifier ffRotor
Introduction to Active Magnetic BearingsAdvantages of Magnetic Bearings
Magnetic Bearings are free of contact and can be utilized invacuum techniques, clean and sterile rooms, transportation of
aggressive media or pure media
Highest speeds are possible even till the ultimate strength ofthe rotor
Absence of lubrication seals allows the larger and stifferrotor shafts
Absence of mechanical wear results in lower maintenancecosts and longer life of the system
Adaptable stiffness can be used in vibration isolation,passing critical speeds, robust to external disturbances
Classification of Magnetic Bearings
According tocontrol action
Active Passive Hybrid
Forcing action Repulsive Attractive
Sensing action Sensor sensing Self sensing
Load supported Axial or Thrust Radial or Journal Conical
Magnetic effect Electro magnetic Electro dynamic
Application Precision flotors Linear motors Levitated rotors Bearingless motors
Contactless Geartrains
Applications of Magnetic Bearings
Turbo molecular pumps
Blood pumps
Molecular beam choppers
Epitaxy centrifuges
Contact free linear guides
Variable speed spindles
Pipeline compressor
Elastic rotor control
Test rig for high speed tires
Magnarails and maglev systems
Gears, Chains, Conveyors, etc
Energy Storage Flywheels
High precision position stages
Active magnetic dampers
Smart Aero Engines
Turbo machines
Fields of Applications of Magnetic Bearings
Semiconductor Industry
Bio-medical Engineering
Vacuum Technology
Structural Isolation
Rotor Dynamics
Maglev Transportation
Precision Engineering
Energy Storage
Aero Space
Turbo Machines
Electromagnetic field
Lorenz force
Electromagnetism
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Electromagnetism
When a charged particle is
at rest it wont emitelectromagnetic wavesrather it is surrounded byelectrostatic field
When the charged particle isin uniform motion (i.e. themotion with uniform velocityin a direction) theelectrostatic field isassociated withmagnetostatic field.
3d electrostatic fieldsurrounding a
charged particle
Magnetostatic field
Electromagnetism
When the particle is in
accelerated motion thenthe magnetic field will beoscillating.
In electromagneticwaves both the electricand magnetic fields areoscillating and harmonic.
The electric and magneticfields are generated byelectric charges
Charges generate electricfields
Movement of chargesgenerate magnetic fields
The electric and magneticfields interact only with eachother
Changing electric field acts likea current, generating vortex ofmagnetic field
Changing magnetic field
induces (negative) vortex ofelectric field
Feed back loop of electromagnetism
The electric and magneticfields produce forces onelectric charges
Electric force which isgenerated by the electric fieldand is in same direction aselectric field
magnetic force which isgenerated by the magneticfield and is perpendicular bothto magnetic field and to
velocity of charge
The electric charges move inspace
The electric charges move inspace when they are acted
upon by field forces
The electric and magneticThe electric and magneticfields are generated byfields are generated by
electric chargeselectric charges
The electric andThe electric andmagnetic fieldsmagnetic fields
interact only withinteract only witheach othereach other
The electric and magneticThe electric and magneticfields produce forces onfields produce forces on
electric chargeselectric charges
The electricThe electriccharges move incharges move inspace when theyspace when theyare acted upon byare acted upon by
field forcesfield forces
Feed back loop of electromagnetism
The four fundamental forces
Strong nuclear force
which holds atomicnuclei together
Weak nuclear force
which causescertain forms of
radioactive decay
The four fundamental forces
Electromagnetic force
Which is caused by
electromagnetic fields on
electrically charged
particles
Gravitational force
Which causes the
masses to attract
each other
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The four fundamental forces
All the other forces are derived from
these four fundamental forces
Electro-magnetic force is one of thesefour fundamental forces
1 2
3
04
cq qf
r= r
Force between two electrically charged particles
Coulomb force (Static)
c1q
r
2q
Lorenz force (Dynamic)
1 2 1 2
3 2 3
0 04 4
l
q q q qf
r c r
= +
r v rv
If q1=q then
( )q= + F E v B
2
2 3
04
q
c r
=
v rB2
3
04
q
r
=
rE
-7 2
02
0
1 = 410 N/Ac
=
Electric and magnetic componentsof Lorenz force
;=r r
12 28.854 10 C / J-m 0 =
( )2
1
1 /v c=
Electric flux; Magnetic flux;
Lorenz factor;
Magnetic permeability of vacuum;
Electric permeability of vacuum;
2
2 23
1
10
v
c
v B
E
Three conclusions: Magnetic component of Lorenz force is at least smaller by a factor of 1023!
But we dont face the effect of electric field in conductors because protonsand electrons are equal in number and generate equal and opposite electric fields
canceling each other
Protons have no motion with reference to conductor and there wont bemagnetic component from them. Thus the magnetic component observed isthe relativistic effect of electrons only
When the conductor is moving with reference to another frame both theprotons and electrons will move with the same velocity thus the relativisticeffects due to the velocity of conductor will be cancelled out
Comparison Electric and magneticcomponents of Lorenz force
Effective Lorenz force in macro calculations
For macro calculations Lorenz force isreduced to the form
( )q= F v BB
v
F
wB
Lorenz force acts perpendicular to both velocity
of charged particle and magnetic flux
Relations between E and B
0
q
=E
t
=
BE
0 0t
= +
EB J
0 =B
Gauss Law for linearmaterials
Gauss Law formagnetism
Faradays law of
magnetic induction
Amperes law andMaxwell'sextension
0
1
S Vqdv
= E ds
0S
= B ds
L St
= E dl B ds
0 0L S t
= +
E
B dl J ds
These relations are called simplified Maxwell's relations who formulated
the original relations from previous works
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Design of magnetic
actuator
Bearing magnet
Magnetic circuit
Coil
Designmethodology
of
magneticbearingsyste
ms
yes
Specifications
Mechanical design
Magnetic actuator design
Control system design
Simulation
Experimentation
Performance O.K?
Performance O.K?
End
Performance O.K?
yesno
yes
no
no
Magnetic bearingsystem design
Mechanicaldesign
Magnetic actuatordesign
Control systemdesign
Modalfrequencies
Bearing magnetdesign
Coil design Sensordesign
Controllerdesign
Poweramplifierdesign
Topology
Loadestimation
Magneticcircuitdesign
Admissible coiltemperature
Number ofturns
Windingscheme
Coil head
Positionsensing
Velocitysensing
Currentsensing
Fluxsensing
Stiffness
Damping
Balancing
Stability
Losses
Self sensing
Areas involved in the design of magneticbearing systems
Bandwidth
Magneto mechanical systems
According to the known technology tillAccording to the known technology till
now, magnetic bearings can be classifiednow, magnetic bearings can be classified
for their design according to the purposefor their design according to the purpose
of the levitated object asof the levitated object as
Precision flotors (precision stages,isolation bases, isolation springs)
Levitation force
Propulsion force
Magneto mechanical systems
A magnetic Precision Stage
Linear motors(Contactless sliders,maglev trains and
conveyors) Levitation force
Propulsion force
Levitation force Propulsion force
Principle of a linear motor
Magneto mechanical systems
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Levitated rotors
(gas turbines,energy storageflywheels, highspeed spindles,balancing andvibration controlof rotors)
Radial load
Thrust load
Magneto mechanical systems
Rotor levitated by Radial andAxial Active Magnetic Bearings
Bearingless motors
(canned pumps,compact pumps, bloodpumps, spindledrives,semiconductor
process)
Radial load
Thrust load
Torque
Magneto mechanical systems
Bearingless Motor
Contactless Gears andCouplersRegulated torque
transmission
Magneto mechanical systems Linear systems from rotary systems
Design of a thrust magnetic bearingMacro Geometry of Thrust Magnetic Bearing
Inner wall
Outer wall
Back-wall
Coil
Space for coil
Space for shaft
Figure 1: Parts of Thrust Magnetic Bearing
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Optimal design
Optimal design is carried outin two steps
Modeling the magneticcircuit
Determines the accuracy ofachieving the objective
Optimization of theparameters
Determines the efficiency ofthe achieving the objective
Magnetic circuit
aR
lR
gR
Ni
Equivalent electric (dc)circuit representation
Magnetic circuit
Ni
l
R
aR
gR
gap Levitated object
Actuator
Coil
0
fp fp
r
l l
A AR
==
Magnetic circuit analogywith electric circuit
Electro Motive
Force (EMF) or
Voltage (V)
Magneto Motive
Force (MMF)
Electric circuitMagnetic circuit
Resistance (R)Reluctance (R)
Electric Current (i)Magnetic Flux ( )
Ideal magnetic circuit model
( )Ampere's lawL SH dl J nda =
2g g a a s s
H l H l H l ni+ + =
or /B H H B = =
al
gl
sl0 0
2 a sg g a s
a s
B BB l l l ni
+ + =
0if is neglecteda s
a s
a s
B Bl l
+
0
2g
g
niB
l
=
H
J
Flux density is used to find the force exertedFlux density is used to find the force exerted
Extension of the ideal modela
l
gl
sl02 a g g iK B l K ni=
a 0
i
if K is added for
as core loss factor and K is added
as coil loss factor, then
a sa s
a s
B Bl l
+
0
2
ig
a g
K niB
K l
=
The model reduces toThe model reduces to0B B+
0B B
0i i+
0i i
0
0
2 gAB BF
=
Force by using flux density
Differential actuator
0
2( )g
Ni
A l xB
=
=
2
0
02
gBF A
=
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Linear Range
max satB
min satB
satB
Magnetic force, N
Magneticfluxdensity,
T
Hysteresis is assumed to be negligiblewhile setting the linear range
Linear range of flux density
0.1005
10.05
0.0010
1600
7.95e5 for air
3.97e4 for Fe
0.026
Magnitude
Wb-
turns
Magnetic flux
linkage
TFlux density
WbMagnetic flux
A-
turns
Magneto
motive force
Vs/AReluctance
Vs/AmPermeability
Vs/AmPermeability
of vacuum
UnitsFormulaSymbolQuantity
R0
fp fp
r
l l
A wl =
0 r
0 20
1
c
02 2( )g
Ni wlN i
R g x
=
B 02( )
Ni
A g x
=
ni n i
N
Terminology used inmagnetic circuit
7
4 10
19.84
804.2
804.2
0.0063
16e4
Magnitude
NMagnetic
force for diff
actuator
NMagnetic
force by flux
density
NMagnetic
force by
inductance
HNominal
inductance
H=Wb/AMagnetic
inductance
A/m2Current
density
UnitsFormulaSymbolQuantity
Different quantities used inmagnetic circuit
0L
2
0
0 2
xg
n wlL
l
=
=
L( )
2
0
2
g
n wl
i l x
=
F2
0
2g
L i
l
Ji i
A wl=
F2
0
02g
BA
F ( )2 202
gAB B
+
Design vector for optimal design
Known parameters areKnown parameters are
GapGap
Inner radius of the bearingInner radius of the bearing
Outer radius of the bearingOuter radius of the bearing
Free parametersFree parameters
Inner radius of the coil spaceInner radius of the coil space
Outer radius of the coil spaceOuter radius of the coil space
Height of the coil spaceHeight of the coil space
Current density suppliedCurrent density supplied
All the other parameters are dependantAll the other parameters are dependant
70mmMaximum height of bearing120mmMaximum outer radiusof bearing
820mm3Maximum allowable coilvolume
4.0A/mm2Saturation currentdensity
0.85Packing factor1.2TRemnant flux density ofbias magnets
0.840Flux leakage factor1.00TSaturation flux density
1.072Actuator loss factor10%Variation in the load1.394Coil mmf loss factor5%Variation in the gap
7.5g/cm3Specific gravity ofpermanent magnet materialneodymium-iron-baron
2025NOperating load
8.91g/cm3Specific gravity of thecopper
4.00mmOperating air gap
7.77g/cm3Specific gravity of thestator iron
25.00mmInner radius of thebearing
ValueParameterValueParameter
Input parameters taken for the design of thrust magnetic bearing Eight pole radial magnetic bearing
Eight Pole AMB
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Radial magnetic bearing
2
0
2
( )cos
4( ) 2
g i
a g
A K niF
K l
=
The component of force will beat an angle of half of the anglebetween two poles
Three pole radial magnetic bearing
Three Pole AMB
Magnetic Circuit forthree pole AMB
Coil design
Admissible coil temperature is determined bythe choice of insulation type
Number of turns are chosen such that itgenerates maximum admissible magneto
motive force at the maximum current suppliedby the power amplifier
Coil
Winding scheme
Permanent magnetic bearings Permanent magnetic bearings
rB
aH BH
B
cH maxBH
7/30/2019 Introduction to Magnetic Bearings
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MAGNETIC BEARINGS
CONTROL
Introduction
Control is the process of bringing asystem into desired path when it isgoing away from it
Earnshaw(1842) had shown that it isimpossible to hover a body in all sixdegrees of freedom by using permanentmagnets
But it is possible to maintain the body in
equilibrium condition by active control
Types of control systems
Open loop control systems
The control in which the output of the system has
no effect on input is called open loop control
Open loop control is used when the input is known
and there are no external disturbances
An example of open loop control is washing
machine which works on time basis rather than the
cleanliness of clothes
( )G s( )U s ( )Y s
Types of control systems
Closed loop control systems
If the control maintains aprescribed output and
reference input relation by comparing them and
uses their difference as controlling quantity, it is
called feedback or closed loop control
Temperature control of a room or a furnace is an
example of closed loop system
( )G s( )U s ( )Y s
( )H s
x
Classification of controllers
According to control action controllers are
classified as:
Two-position or on-off controllers
Proportional controllers
Integral controllers
Proportional-integral controllers
Proportional-differential controllers
Proportional-differential-integral controllers
Classification of controllers
Two-position or on-off controllers
The output of the controller will be a
maximum or minimum according to the state of
error as below:
are minimum and maximum values of
output0 1
andy y
0
1
( ) for ( ) 0
for ( ) 0
y t y e t
y e t
=
( )y t
( )e t
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Classification of controllers
Proportional controllers: The output of the controller is proportional tothe magnitude of the actuating error signal as
By Laplace transformation
is called proportional gain
( )y t
( )e t
( ) ( )py t g e t=
( )
( )p
Y sg
E s=
pg
Integral controllers: In integral control action, the value of thecontroller output is changed at a rate
proportional to the actuating error signal
By Laplace transformation
is called integral gain
( )y t( )e t
( )( )
i
dy tg e t
dt=
( )
( )
igY s
E s s=
ig
Classification of controllers
0( ) ( )
t
iy t g e t dt= (or)
Proportional-Integral (PI) controllers:
Control action is a combination of both
proportional and integral action
By Laplace transformation
( ) 11
( )p
i
Y sg
E s T s
= +
0
( ) ( ) ( )t
p
p
i
gy t g e t e t dt
T= +
Classification of controllers
proportional-differential (PD) controllers:
The control action is defined by
By Laplace transformation
( )(1 )
( )p d
Y sg T s
E s= +
( )( ) ( )
p p d
de ty t g e t g T
dt= +
Classification of controllers
proportional-Integral-differential (PID)
controllers:
It has the advantages of all three actions. So this is
the most common type of industrial controllers
Mathematical form of PID action is
By Laplace transformation
( ) 11
( )p d
i
Y sg T s
E s T s
= + +
0
( )( ) ( ) ( )
tp
p p d
i
g de ty t g e t e t dt g T
T dt= + +
Classification of controllers Control Design
An over all system
G(s)U(s) Y(s)
Transfer-function representation of a system
u(t) y(t)( )
( )
x t
x t
SystemInput Output
State-space representation of a system
( ) ( ) ( )Y s G s U s=
( ) ( ) ( )t A t B t = +y x u
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Control Design
An over all system
SystemInput Output
Studying the behaviour of a s ystem
KnownKnown unknown
UnknownKnown known
Studying the characteristi cs of a sys tem
UnknownUnknown known
Designing of a control system of required behaviour
Methods of design and
analysis of controllers
Methods of design and analysis
Transfer-function method State-variable method
Transient and
steady state
Response
analysis
Root locus
analysis
Frequency
response
analysis
Linear-
quadratic
optimization
Pole-placement
analysis
(Classical control) (Modern control)
Pole-placement method and Linear-
quadratic optimization are the main
methods of design and analysis.
Steady state and transient response
analysis, Root locus analysis and
frequency response analysis are the
main methods of design and analysis
Analysis consists of system ofn first
order differential equations.
Analysis consists of single higher
order differential equation
Time domain methodFrequency domain method
It is useful for nonlinear and
complex systems also.
It is useful for linear and simple
systems only
Used for multi input multi output
(MIMO) systems can be used for SISO
also
Used for single input single output
(SISO) systems
Modern control methodClassical control method
State-space methodTransfer-function
method
Mechanical and electro magnetic
stiffness
mf
mx
mg
Magnetic spring
Operating
position0x
Rotor
mechanical spring
Equilibrium
position
sf x
mg
0x
Mechanical spring stiffness
magnetic displacement stiffness
mf
mg
Magnetic spring
Operating
position0x
0iOperating
current
mi Instantaneous
current
0x
Magnetic Bearing Control
Equilibrium and Operating points
For a mechanical spring there will be an
equilibrium pointwhere the force resisted by the
spring is equal to the force applied on the spring
For electro magnets there will be a quantity of
current corresponding to position of the object and
force applied. At this point the gravity force and
magnetic force will be equal. A slight movement
form this point will cause indefinite movement of
the body. This point is called operating point
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Linearization of current Li nearization of displaceme nt
Linearization at operating point
0x
0img
0 mx x x=
0mi i i=
if k i=
xk x=
is the instantaneous currentmi is the instantaneous position
mx
Linearized formula around the operating point will be
( , )x i
f x i k x k i= +
xk is displacement stiffness
ik is current stiffness
x
i is the deviation of current
from operating current
is the displacement from the
operating position
where
f is instantaneous force
Linearized equation is suitable for most of the
applications of magnetic bearings
It is not valid in three occasions
When the rotor touches the bearing magnet
When there are strong currents such that magnetic
saturation of the material occurs
When or very small currents there wont be
levitation of the rotor because of very small
magnetic forces.
0x x=
0i i=
Magnetic Bearing Control
m
Rotor
xf
k c
spring mass dampersystem
Active magnetic
bearing system
x if k x k i= +
f mx=
By Newton's law
Combing above two equations we get
x imx k x k i =
If controlling current i is zero then
0x
mx k x =
Response of magnetic
bearing without control
And the response grows exponentially thus
the rotor may fall down or touch the magnet
Response of magnetic
bearing with control
If we supply controlling current i such that
then it becomes
( ) x
i i
k k ci x x x
k k
+= +
0mx cx kx+ + =
And the response is imitated to a spring mass damper
system by the magnetic bearing system
m
Rotor
xf
k c
spring mass damper system
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( ) x
i i
k k ci x x x
k k
+= +
PD controller model
The model is PD-controller with proportional
and differential feed back
In design of controller we choose the stiffness
and damping to ensure the system come to
steady state in optimum time.
The optimal stiffness suggested is
The range of damping ratio for better systems
suggested is 0.1 to 1
x
i
k kP
k
+=
i
cD
k=
xk k=
ci Pe De= +
Controller
cic
i i=i
ik ++
r
y
1/m
xk
f x x x
Amplifier
Sensor
y x=
Block diagram of PD controller with
current control
e
1
c
i
i Pe De
edtT
= +
+
Controller
ci
ci i=
ii
k ++
r
y
1/m
xk
f x x x
Amplifier
Sensor
y x=
Block diagram of PID controller with
current control
e
loadf
Control of rotors by usingmagnetic bearings
Topics to be covered
Rigid rotor model
Flexible rotor model
Differences between mechanical andmagnetic bearing models
Stiffness is very highthus the vibration of therotor will be transmittedto foundation
Damping is directlyobserved due tohydrodynamic effects
Stiffness is very low thusthe rotor can rotate freelyabout the principal axes ofinertia which results in avibration isolation system.
As the rotor is free in theair there is no coulombdamping acting on thesystem. The control lawwill have damping term.
Mechanical bearing model Magnetic bearing model
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Rigid rotor model
Rotor mechanical bearing system
Infinitesimal rotation about x axis
Infinitesimal rotation about y axis
d
dt
= d
dt
=
Angular velocity of shaft
Rigid rotor model
Angular velocity vector can be expressed as
0
0
0
cos sin
sin cos
x
y
z
t t
t t
+
= = +
z
y
x
O z
y
x
Oz
y
x
z'
y'
z'
x'
z
y
x
[ ] [ ]T T
1 2 3 4x x x x x y = = x
If the variable vector is chosen as
Motion aboutx- axis Motion abouty- axis
Rigid rotor model
Equations of equilibrium can be obtained as by using Lagranges pr inciple
i
i i
d T TF
dt x x
+ =
is the generalized force corresponding to variableth
iF i
( ) ( )2 2 2 2 2 20 0 0 0 0 01 1
2 2x x y y z zT m x y z J J J = + + + + +
Kinetic energy is expressed as
Rigid rotor model
Equations (1) can be expressed in matrix form by rearrang ing
( )M G C+ + =x x F
F can be expressed as
( )K N= +F x
)
is the gyroscopic matrix )
is the damping matrix )
is the inertia matrix (
( -
(
T
T
T
G
C
M M M
G G
C C
=
=
=
)
is non-conservative force matrix )
is conservative force matrix (
( -
T
TN
K K K
N N
=
=
Rigid rotor model
Conservative forces include
forces due to stiffness
Non-conservative or circulatory forcesinclude
Internal or structural damping
Steam or gas whirl in turbines Seal effects
Process forces such as in grinding
Unbalance, etc
Damping include
Coulomb damping due to hydrodynamic effects
Rigid rotor model
From Eq. (2) and (3) we get
If the non-conservative and gyroscopicforces neglected, we have
( ) ( ) 0K NM G C ++ + + =x x x
0KM C+ + =x x x
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Natural modes
The solution of the equations (5) givesfour modes, for there are four degreesof freedom considered
Translation mode Rotation mode
Natural modes
Forward whirl Backward whirl
Forward nutation Backward nutation
Magnetic bearing model
In a magnetic bearing if we neglect theconservative, non-conservative, anddamping effects, we will have
For small rotations gyroscopic effectscan be neglected and the equations in x
andy directions can be decoupled
GM + =x x F
M =x F
Weight considerations
mg
0ig mg k if = = 0cosig
mgk if = =
Imbalance considerations
( )2 cosme tf = +is the imbalance massm
is the eccentricity of
imbalance mass
e
e
is the angular position
of imbalance mass
Magnetic bearing model
It can be written as
wherec gk
mx f f f f = + +
( )
0
2 cos
x
i
i
k
c
g
k x
k i
mg k i
me t
f
f
f
f
=
=
= =
= +
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Magnetic bearing model
It will be
i at any instant will be
( ) ( )20 cosx ik x k i i me t mx += + +
( )2
0
cosx
i
k x me t i i
k
mx += +
Rigid rotor with magnetic bearing
Three steps involved:
Formulation with respect to centre of gravity Transformation with respect to the bearing
coordinates
Transformation with respect to the sensorcoordinates
z
x
y
O
Bearing
Sensor
Centre of gravity
Why with respect to sensor
coordinates
Sensors cannot bearranged directly in themagnetic actuator.
This requires certaingap between themagnet and the sensor.
The displacements withrespect to sensorcoordinates will betransformed to bearing
coordinates
With respect to centre of gravity
In slow rolex andy directions can be decoupled
y
mx f
I p
=
=
BA
z
x
y
O
a b
c d
ax bxf
p
x In matrix form as
where
0,
0 y
M
m xM
I
f
p
=
= =
=
x f
x
f
With respect to bearing coordinates
Forces are transformed as
ax bx
ax bx
f f f
p af bf
= +
= +
1 1,
f B
f B
ax
bx
T
fT
fa b
=
= =
f f
f
BA
z
x
y
O
a b
c d
axf bxf
p
x
With respect to bearing coordinates
1
1 1
B B
B
B
a
b
b a
T
x
b aT
x
x
=
=
=
=
x x
x
x
BA
z
x
y
O
a b
c d
ax bx
p
x
Displacement vector can betransformed as
f
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With respect to sensor coordinates
fBA
z
x
y
O
a b
c d
axf bx
p
x
S S S B BT T T= =x x x
S
d
cx
x
=x
1
1S
cT
d
=x
=x
0
0
s s b
S B
s
S B
S
S
T
T TT
T T
=
=
=x
x xx
dx
cx
State feed back
The control vector is found by using controllaw
We do not know the velocity components
directly from sensors. So a state observer isrequired to find the velocities
sF= u x
s SC=x xis the full state vector
is the vector from the sensor
s
S
x
x
1
s s b s
s b
A T A T
B B
=
=s s s sA B+= ux x
State space form with respect to sensor coordinates
State feed back
The whole closed loop system can be shown as
block diagram
s s s sA B+= ux x
sF= u x
S sC=x x
sB+ C s
x
sA
F
d
dt
u
( )s s s sA B F=x x
decides the closed loop
dynamics of the system
s sA B F
sx S
x
Model at high speeds
At high speeds the gyroscopic effects cannotbe neglected, thus the model becomes
The displacements inx andy directions no
longer decoupled, so four forces and fourdisplacements should be taken into
consideration simultaneously.
The same procedure is to be followed as for
the slow rotation
GM + =x x F
Model at high speeds
0 0 0
0 0 0
0 0 0
0 0 0
y
x
m
I
m
I
M
=
0 0 0 0
0 0 0 1
0 0 0 0
0 1 0 0
G
=
x
y
y
x
p
f
p
=fB
a
b
a
b
y
y
x
x
=x
Conclusions on rigid rotor model
There is an optimal design for each speed
The optimal design at higher speed may not
be stable at lower speeds, for the gyroscopiceffects are reduced.
The optimal design at zero speed may not be the
optimal at higher speeds
The gyroscopic effects will not destabilize the systemwhich is stable at lower speeds.
Further more the design at lower speeds is decoupledand easier to design. Decentralized designs for lowerspeeds can be implemented
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Conclusions on rigid rotor model
Thus for stability considerations and otheradvantages systems are designed forlower speeds and with decentralization
xa xa aF xu =
xb xb bF xu =
yb yb bF xu =
ya ya aF xu =
Decentralized control mode scheme
Flexible rotor model
Rigid rotor can bedefined by two
points
Flexible rotor hasinfinite degrees of
freedom. Onecannot define
uniquely by some ofthe points
Flexible rotor model
Equation motion ofan Euler-Bernoulli
beam is given by
The variable
separable form is
4 2
4 20
y yEI m
z t
+ =
L
( , ) ( ) ( )y z t Y z q t=
z
Flexible rotor model
By substituting we get
By rewriting we get
4 2
4 2
2
( ) ( )
( ) ( )
d Y z d q t
dz dt EI
m Y z q t
= =
4 24
4
( )( ) 0,
/
d Y zY z
dz EI m
= =
22
2
( )
( ) 0
d q t
q tdt + =
Flexible rotor model
By applying initialconditions and solvingwe get the naturalfrequencies
By substituting theEigen values in (29) weget the Eigen functionsor model functions
Lz
( )Y z
0
1
Rigid rotor modes
z
z
2
3
Flexible rotor modes
The mode shapes ormodal functions
depend on the end
conditions
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Actuator sensor location
Sensor should not be set at nodes
Sensor and actuator should not lie on
opposite sides of a nodeactuator
sensor
Actuator sensor location
We can conclude that the sensor can beset at a place where we can getinformation from each mode underconsideration
Modal reduction
While designing a flexible rotor system, wecan not consider all the modes of the system
for they are infinite
Thus we consider first n number of modes
corresponding to first n natural frequenciesand neglect the remaining modes
If we study the effect of the reduced modes
we can find the number of modes which we
can consider without destabilizing the system
Modal reduction (mathematical representation)
Mathematical model of the
full system
Divided system
Reduced system
A B
C
= +
=
x x u
y x
[ ]
M M MR M M
R RM R R R
M
M R
R
A A B
A A B
C C
= +
=
x xu
x x
xy
x
M M M M
M
A B
C
= +
=
x x u
y x
Modal reduction
The reduced modesgive three kinds of
effects on the system
called spillovers
Control spillover (By theinput)
Interconnection spillover(By the parameters of thesystem)
Observation spillover (onthe estimated output)
Input System Output
Controlspillover
Interconnectionspillover
Observationspillover
[ ]
M M MR M M
R RM R R R
M
M R
R
A A B
A A B
C C
= +
=
x xu
x x
xy
x
Modal reduction
Block diagram of effect of model reduction
MA
MB
MC+ +M
x yu
+ Rx
RA
RB RC
RMA
MR
A
Controlspillover
Interconnection
spillover
Observationspillover
Modeled modes
Unmodeledmodes
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Conclusion on flexible rotor control
Modal reduction is studied to considerthe number modes to be taken intoconsideration for having stable control
Mechanical design is studied for findingthe sensor actuator locations
Conclusions
Magnetic bearings advantages andapplications have been discussed
Electromagnetism and Control systemtechnologies have been introduced
Design of thrust and radial magneticbearings have been studied
Control of a rotor by rigid rotor andflexible rotor models have been studied
Schweitzer, G., Bleuler, H. and Traxler, A., 2003, ActiveMagnetic Bearings: Basics, Properties and Applications of ActiveMagnetic Bearings, Authors Working Group, www.mcgs.chreprint.
Chiba, A., Fukao, T., Ichikawa, O., Oshima, M., Takemoto,M. and Dorrell, D.G., 2005, Magnetic Bearings & BearinglessDrives, Newnes, Elsevier.
Maslen, E., 2000, Magnetic Bearings, University ofVirginia.
Groom N.J. and Bloodgood, V.D. Jr., 2000, AComparison of Analytical and Experimental Data for a MagneticActuator, NASA-2000-tm210328.
Bloodgood, V.D. Jr., Groom, N.J. and Britcher, C.P., 2000,Further development of an optimal design approach applied to
axial magnetic bearings, N ASA-2000-7ismb-vdb.
Further References
Anton, V.L. , 2000, Analysis and initial synthesis of anovel linear actuator with active magnetic suspension, 0-7803-8486-5/04/$20.00 2004 IEEE
Chee, K.L., 1999, A Piezo-on-Slider Type LinearUltrasonic Motor for theApplication of Positioning Stages,Proceedingsof the 1999IEEE/ASME.
Shyh-Leh, C., 2002, Optimal Design of a Three-PoleActive Magnetic Bearing, IEEE TRANSACTIONS ON MAGNETICS,VOL. 38, NO. 5.