introduction to magnetic bearings - qip08-handouts · • magnetic bearings are free of contact and...

1

Introduction to Magnetic Bearings

Jagu Srinivasa Rao, (Research Scholar)

Department of Mechanical EngineeringIndian Institute of Technology Guwahati

December, 2008

Lecture presented in Quality Improvement Program (QIP’08) 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 supports a rotating shaft, without any physical contact by suspending the rotor in the air, with an electrically controlled (or/and permanent magnet) magnetic force.

• It is a mechatronic product which involves different fields 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

Ro

tor s

haft

Typical Actuator – Controller unit of an AMB

Introduction to Active Magnetic Bearings

2

Working principle of magnetic bearing

Electro magnet

Sensor

Controller

Power Power

AmplifierAmplifier ff

Roto

r

Introduction to Active Magnetic BearingsAdvantages of Magnetic Bearings

• Magnetic Bearings are free of contact and can be utilized in

vacuum techniques, clean and sterile rooms, transportation of aggressive media or pure media

• Highest speeds are possible even till the ultimate strength of

the rotor

• Absence of lubrication seals allows the larger and stiffer

rotor shafts

• Absence of mechanical wear results in lower maintenance costs 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 to •control 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

3

Electromagnetism

When a charged particle is

at rest it won’t emit electromagnetic waves rather it is surrounded by

electrostatic field

When the charged particle is in uniform motion (i.e. the

motion with uniform velocity in a direction) the electrostatic field is

associated with magnetostatic field.

3d electrostatic field surrounding a

charged particle

Magnetostatic field

Electromagnetism

When the particle is in accelerated motion then

the magnetic field will be oscillating.

In electromagnetic waves both the electric and magnetic fields are

oscillating and harmonic.

The electric and magnetic fields are generated by electric charges

Charges generate electric fields

Movement of charges generate magnetic fields

The electric and magnetic fields interact only with each other

Changing electric field acts like

a current, generating vortex of

magnetic field

Changing magnetic field

induces (negative) vortex of

electric field

Feed back loop of electromagnetism

The electric and magnetic fields produce forces on

electric charges

Electric force which is

generated by the electric field and is in same direction as

electric field

magnetic force which is

generated by the magnetic field and is perpendicular both

to magnetic field and to

velocity of charge

The electric charges move in space

The electric charges move in

space when they are acted

upon by field forces

The electric and magnetic The electric and magnetic fields are generated by fields are generated by

electric chargeselectric charges

The electric and The electric and magnetic fields magnetic fields

interact only with interact only with each othereach other

The electric and magnetic The electric and magnetic fields produce forces on fields produce forces on

electric chargeselectric charges

The electric The electric charges move in charges move in space when they space when they are acted upon by are acted upon by

field forcesfield forces

Feed back loop of electromagnetism

The four fundamental forces

Strong nuclear force

which holds atomic

nuclei together

Weak nuclear force

which causes

certain forms of

radioactive decay


Electromagnetic force

Which is caused by

electromagnetic fields on

electrically charged

particles

Gravitational force

Which causes the

masses to attract

each other

4


All the other forces are derived from these four fundamental forces

Electro-magnetic force is one of these

four fundamental forces

1 2

3

04c

q qf

rπε= r

Force between two electrically charged particles

Coulomb force (Static)

cf1q

r

2q

Lorenz force (Dynamic)

1 2 1 2

3 2 3

0 04 4l

q q q qf

r c r

γ γ

πε πε

×= + ×

r v rv

If q1=q then

( )q= + ×F E v B

2

2 3

04

q

c r

γ

πε

×=

v rB2

3

04

q

r

γ

πε=

rE

-7 2

02

0

1= 4π×10 N/A

cµ

ε=

Electric and magnetic components of Lorenz force

; = r r

12 28.854 10 C / J-mε −0 = ×

( )2

1

1 /v c

γ =−

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 don’t face the effect of electric field in conductors because protons

and 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 won’t be magnetic component from them. Thus the magnetic component observed is

the relativistic effect of electrons only

• When the conductor is moving with reference to another frame both the

protons and electrons will move with the same velocity thus the relativistic

effects due to the velocity of conductor will be cancelled out

Comparison Electric and magnetic components of Lorenz force

Effective Lorenz force in macro calculations

For macro calculations Lorenz force is

reduced 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

ε∇ ⋅ =E

t

∂∇ × = −

∂

BE

0 0t

µ ε∂

∇ × = + ∂

EB J

0∇ ⋅ =B

Gauss’ Law for linear

materials

Gauss’ Law for

magnetism

Faraday’s law of magnetic induction

Ampere’s law and

Maxwell's extension

0

1

S Vqdv

ε⋅ =∫ ∫E ds

0S

⋅ =∫ B ds

L St

∂⋅ = − ⋅

∂∫ ∫E dl B ds

0 0L S t

µ ε∂

⋅ = + ⋅ ∂

∫ ∫E

B dl J ds

These relations are called simplified Maxwell's relations who formulated

the original relations from previous works

5

Design of magnetic

actuator

– Bearing magnet

– Magnetic circuit

– Coil

De

sig

n m

eth

od

olo

gy o

f m

ag

ne

tic

be

ari

ng

sys

tem

s

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 bearing system design

Mechanical design

Magnetic actuator design

Control system design

Modal frequencies

Bearing magnet design

Coil design Sensor design

Controller design

Power amplifier design

Topology

Load estimation

Magnetic circuit design

Admissible coil temperature

Number of turns

Winding scheme

Coil head

Position sensing

Velocity sensing

Current sensing

Flux sensing

Stiffness

Damping

Balancing

Stability

Losses

Self sensing

Areas involved in the design of magnetic bearing systems

Bandwidth

Magneto mechanical systems

According to the known technology till According to the known technology till

now, magnetic bearings can be classified now, magnetic bearings can be classified

for their design according to the purpose for 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


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


6

Levitated rotors(gas turbines, energy storage flywheels, high speed spindles, balancing and vibration control of rotors)

– Radial load

– Thrust load


Rotor levitated by Radial and Axial Active Magnetic Bearings

Bearingless motors (canned pumps, compact pumps, blood pumps, spindle drives, semiconductor

process)

– Radial load

– Thrust load

– Torque


Bearingless Motor

Contactless Gears and Couplers – Regulated 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

7

Optimal design

Optimal design is carried out in two steps

• Modeling the magnetic circuit

– Determines the accuracy of achieving the objective

• Optimization of the parameters

– Determines the efficiency of the achieving the objective

Magnetic circuit

aR

lR

gR

Ni

Equivalent electric (dc) circuit representation

Magnetic circuit

Ni

φl

R

aR

gR

gap Levitated object

Actuator

Coil

φ

0

fp fp

r

l l

A AR

µ µ µ==

Magnetic circuit analogy with 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 ( )φ

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 Bµ µ= =

al

gl

sl0 02 a s

g g a s

a s

B BB l l l niµ µ

µ µ

+ + =

0if is neglecteda sa s

a s

B Bl lµ

µ µ

+

0

2g

g

niB

l

µ=

H

J

Flux density is used to find the force exertedFlux density is used to find the force exerted

Extension of the ideal modelal

gl

sl0

2a g g i

K B l K niµ=

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 lµ

µ µ

+

0

2

ig

a g

K niB

K l

µ=

The model reduces toThe model reduces to0B B+ ∆

0B B− ∆

0i i+ ∆

0i i− ∆

0

0

2 gAB BF

µ∆∆ =

Force by using flux density

Differential actuator

0

2( )g

Ni

A l xB

µφ=

−=

2

0

02

g

BF A

µ=

8

Linear Range

max satBα

min satBα

satB

Magnetic force, N

Magnetic f

lux d

ensity,

T

Hysteresis is assumed to be negligible while 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 wlµ µ µ=

µ0 rµ µ

0µ

2

0

1

cε

φ 0

2 2 ( )g

N i w lN i

R g x

µ=

−

B 0

2( )

Ni

A g x

µφ=

−

ni n i×

λ Nφ

Terminology used inmagnetic circuit

74 10π −×

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 in

magnetic circuit

0L

2

0

0 2

xg

n wlL

l

µ=

=

L( )

2

0

2

g

n wl

i l x

µλ−

=

F2

0

2 g

L i

l

Ji i

A wl=

F2

0

02g

BA

µ

F ( )2 2

02

gA

B Bµ

+ −−

Design vector for optimal design

Known parameters areKnown parameters are

••Gap Gap

••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 radius

of bearing

820mm3Maximum allowable coil

volume

4.0A/mm2Saturation current

density

0.85Packing factor1.2TRemnant flux density of

bias magnets

0.840Flux leakage factor1.00TSaturation flux density

1.072Actuator loss factor±10%Variation in the load

1.394Coil mmf loss factor±5%Variation in the gap

7.5g/cm3Specific gravity of

permanent magnet material

neodymium-iron-baron

2025NOperating load

8.91g/cm3Specific gravity of the

copper

4.00mmOperating air gap

7.77g/cm3Specific gravity of the

stator iron

25.00mmInner radius of the

bearing

ValueParameterValueParameter

Input parameters taken for the design of thrust magnetic bearing Eight pole radial magnetic bearing

Eight Pole AMB

9

Radial magnetic bearing

2

0

2

( )cos

4( ) 2

g i

a g

A K niF

K l

µ α=

The component of force will be at an angle of half of the angle between two poles

α

Three pole radial magnetic bearing

Three Pole AMB

Magnetic Circuit for three pole AMB

Coil design

• Admissible coil temperature is determined by

the choice of insulation type

• Number of turns are chosen such that it

generates maximum admissible magneto

motive force at the maximum current supplied

by the power amplifier

Coil

Winding scheme

Permanent magnetic bearings Permanent magnetic bearings

rB

aH BH

B

cH maxBH

10

MAGNETIC BEARINGS

CONTROL

Introduction

• Control is the process of bringing a system into desired path when it is

going away from it

• Earnshaw(1842) had shown that it is impossible to hover a body in all six

degrees of freedom by using permanent

magnets

• 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 a prescribed 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


• 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 and y y

0

1

( ) for ( ) 0

for ( ) 0

y t y e t

y e t

= <

= >

( )y t

( )e t

11


• Proportional controllers:

• The output of the controller is proportional to

the 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 the

controller output is changed at a rate

proportional to the actuating error signal


•

• is called integral gain

( )y t

( )e t

( )( )

i

dy tg e t

dt=

( )

( )

igY s

E s s=

ig


0( ) ( )

t

iy t g e t dt= ∫(or)

• Proportional-Integral (PI) controllers:

• Control action is a combination of both

proportional and integral action


( ) 11

( )p

i

Y sg

E s T s

= +

0

( ) ( ) ( )

tp

p

i

gy t g e t e t dt

T= + ∫


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= +


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

System Input Output

State-space representation of a system

( ) ( ) ( )Y s G s U s=

( ) ( ) ( )t A t B t= +y x u

12

Control Design

An over all system

SystemInput Output

Studying the behaviour of a system

KnownKnown unknown

UnknownKnown known

Studying the characteristics of a system

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 of n 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 point where 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

13

Linearization of current Linearization of displacement

Linearization at operating point

0x

0img

0 mx x x= −

0mi i i= −

if k i=

xf k x=

is the instantaneous currentm

i is the instantaneous positionm

x

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 won’t 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 damper

system

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

14

( ) 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

ci

ci i=

ii

k ++−

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 using

magnetic bearings

Topics to be covered

• Rigid rotor model

• Flexible rotor model

Differences between mechanical and

magnetic bearing models

• Stiffness is very high

thus the vibration of the rotor will be transmitted to foundation

• Damping is directly observed due to

hydrodynamic effects

• Stiffness is very low thus the rotor can rotate freely

about the principal axes of inertia which results in a vibration isolation system.

• As the rotor is free in the air there is no coulomb damping acting on the

system. The control law will have damping term.

Mechanical bearing model Magnetic bearing model

15

Rigid rotor model

Rotor mechanical bearing system

α Infinitesimal rotation about x axis

β Infinitesimal rotation about y axis

d

dt

αα = d

dt

ββ =

Ω 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

ω α β

ωω α β

ω

Ω + Ω

= = − Ω + Ω Ω

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 yβ α= = −x

If the variable vector is chosen as

Motion about x- axis Motion about y- axis

α

β

Rigid rotor model

Equations of equilibrium can be obtained as by using Lagrange’s principle

i

i i

d T TF

dt x x

∂ ∂+ =

∂ ∂

is the generalized force corresponding to variableth

iF i

( ) ( )2 2 2 2 2 2

0 0 0 0 0 0

1 1

2 2x x y y z zT m x y z J J Jω ω ω= + + + + +

Kinetic energy is expressed as

Rigid rotor model

Equations (1) can be expressed in matrix form by rearranging

( )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 forces include

– 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 gyroscopic

forces neglected, we have

( ) ( ) 0K NM G C ++ + + =x x x

0KM C+ + =x x x

16

Natural modes

• The solution of the equations (5) gives

four modes, for there are four degrees

of 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 the

conservative, non-conservative, and

damping effects, we will have

• For small rotations gyroscopic effects

can be neglected and the equations in x

and y directions can be decoupled

GM + =x x F

M =x F

Weight considerations

mg

0ig mg k if = =0

cosig

mgk if

α= =

α

Imbalance considerations

( )2 cosme tfω θ= Ω Ω +

is the imbalance massm

is the eccentricity of

imbalance mass

e

e

Ω

is the angular position

of imbalance mass

θ


It can be written as

wherec gk

mx f f f fω−= + +

( )

0

2 cos

x

i

i

k

c

g

k x

k i

mg k i

me t

f

f

f

fω θ

=

=

= =

= Ω Ω +

17


• It will be

• i at any instant will be

( ) ( )2

0cos

x ik x k i i me tmx θ− Ω Ω += + +

( )2

0

cosx

i

k x me ti i

k

mx θ− Ω Ω += +

−

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 sensor coordinates

z

x

y

O

Bearing

Sensor

Centre of gravity

Why with respect to sensor

coordinates

• Sensors cannot be

arranged directly in the magnetic actuator.

• This requires certain

gap between the magnet and the sensor.

• The displacements with respect to sensor

coordinates will be transformed to bearing coordinates

With respect to centre of gravity

• In slow role x and y directions can be decoupled

y

mx f

I pβ

=

=

BA

z

x

y

O

a b

c d

axf

bxf

p

xβ

fIn matrix form as

where

0,

0 y

M

m xM

I

f

p

β

=

= =

=

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

xβ

f

With respect to bearing coordinates

1

1 1

B B

B

B

a

b

b a

T

x

b aT

x

x

β

−

=

=

−=

−

=

x x

x

x

BA

z

x

y

O

a b

c d

ax

bx

p

xβ

• Displacement vector can be

transformed as

f

18

With respect to sensor coordinates

fBA

z

x

y

O

a b

c d

axf

bxf

p

xβ

S S S B BT T T= =x x x

S

d

cx

x

=x

1

1S

cT

d

=x

β

=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 control

law

• We do not know the velocity components

directly from sensors. So a state observer is

required to find the velocities

sF= −u x

s SC=x x

is 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 Sx

Model at high speeds

• At high speeds the gyroscopic effects cannot

be neglected, thus the model becomes

• The displacements in x and y directions no

longer decoupled, so four forces and four

displacements 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

f

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 gyroscopic

effects are reduced.

– The optimal design at zero speed may not be the optimal at higher speeds

– The gyroscopic effects will not destabilize the system which is stable at lower speeds.

– Further more the design at lower speeds is decoupled and easier to design. Decentralized designs for lower speeds can be implemented

19

Conclusions on rigid rotor model

– Thus for stability considerations and other

advantages systems are designed for

lower 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 be

defined by two

points

• Flexible rotor has

infinite degrees of

freedom. One

cannot define

uniquely by some of

the points


• Equation motion of

an 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


• By substituting we get

• By rewriting we get

4 2

4 2

2

( ) ( )

( ) ( )

d Y z d q t

dz dtEI

m Y z q tω

= − =

4 24

4

( )( ) 0,

/

d Y zY z

dz EI m

ωβ β− = =

22

2

( )( ) 0

d q tq t

dtω+ =


• By applying initial

conditions and solving we get the natural frequencies

• By substituting the Eigen values in (29) we get the Eigen functions

or model functions

ω

Lz

( )Y z

0ω

1ω

Rigid rotor modes

z

z

2ω

3ω

Flexible rotor modes

• The mode shapes or

modal functions

depend on the end

conditions

20

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 be set at a place where we can get

information from each mode under

consideration

Modal reduction

• While designing a flexible rotor system, we

can 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 frequencies

and 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 modes

give three kinds of

effects on the system

called spillovers

– Control spillover (By the input)

– Interconnection spillover

(By the parameters of the system)

– Observation spillover (on

the estimated output)

Input System Output

Control spillover

Interconnection spillover

Observation spillover

[ ]

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+ +Mx yu

∫+ Rx

RA

RB RC

RMA MRA

Control spillover

Interconnection spillover

Observation spillover

Modeled modes

Unmodeled modes

21

Conclusion on flexible rotor control

• Modal reduction is studied to consider

the number modes to be taken into

consideration for having stable control

• Mechanical design is studied for finding

the sensor actuator locations

Conclusions

• Magnetic bearings advantages and applications have been discussed

• Electromagnetism and Control system technologies have been introduced

• Design of thrust and radial magnetic bearings have been studied

• Control of a rotor by rigid rotor and flexible rotor models have been studied

Schweitzer, G., Bleuler, H. and Traxler, A., 2003, “Active Magnetic Bearings: Basics, Properties and Applications of Active

Magnetic Bearings”, Authors Working Group, www.mcgs.chreprint.

Chiba, A., Fukao, T., Ichikawa, O., Oshima, M., Takemoto, M. and Dorrell, D.G., 2005, “Magnetic Bearings & Bearingless

Drives”, Newnes, Elsevier.

Maslen, E., 2000, “Magnetic Bearings”, University of Virginia.

Groom N.J. and Bloodgood, V.D. Jr., 2000, “A

Comparison of Analytical and Experimental Data for a Magnetic Actuator”, 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”, NASA-2000-7ismb-vdb.

Further References

Anton, V.L. , 2000, “Analysis and initial synthesis of a

novel linear actuator with active magnetic suspension”, 0-7803-8486-5/04/$20.00 © 2004 IEEE

Chee, K.L., 1999, “A Piezo-on-Slider Type Linear

Ultrasonic Motor for theApplication of Positioning Stages”, Proceedingsof the 1999IEEE/ASME.

Shyh-Leh, C., 2002, “Optimal Design of a Three-Pole

Active Magnetic Bearing”, IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 5.

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