Introduction to Magnetic Bearings

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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2

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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3

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


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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4


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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5

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


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



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6

Levitated rotors

(gas turbines,energy storageflywheels, highspeed spindles,balancing andvibration controlof rotors)

Radial load

Thrust load


Rotor levitated by Radial andAxial Active Magnetic Bearings

Bearingless motors

(canned pumps,compact pumps, bloodpumps, spindledrives,semiconductor

process)

Radial load

Thrust load

Torque


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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8

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


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10

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


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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11


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


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

( ) ( ) ( )t

p

p

i

gy t g e t e t dt

T= +


proportional-differential (PD) controllers:

The control action is defined by


( )(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


( ) 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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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

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


15/21

15

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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16

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


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

=

=

= =

= +


17/21

17


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


18/21

18

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


19/21

19

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


Rigid rotor can bedefined by two

points

Flexible rotor hasinfinite degrees of

freedom. Onecannot define

uniquely by some ofthe points


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


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 + =


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


20/21

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 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


21/21

21

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.

Documents

Introduction to Magnetic Bearings