Research Collection

Doctoral Thesis

Design and control of active magnetic bearing systems for highspeed rotation

Author(s): Larsonneur, René

Publication Date: 1990

Permanent Link: https://doi.org/10.3929/ethz-a-000578355

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

Diss. ETHNo. 9140

Design and Control ofActive Magnetic Bearing Systems

for High Speed Rotation

A dissertation submittedto theSWISS FEDERALINSTITUTEOF TECHNOLOGY

ZÜRICH

for the degreeofDoctor of TechnicalSciences

presented byRENE LARSONNEUR

Dipl. Mech.Eng. ETHborn June 6,1958

Citizen of Allschwil/BL

acceptedon the recommendationofProf. Dr. G. Schweitzer, examiner

Prof. Dr. W. Schaufelberger, co-examiner

Offsetdruckerei AGZürich1990

coniugiet consorticarissimae

ACKNOWLEDGEMENTS/ VORWORT

This thesis is the result of research I have made between 1984 and 1990 at

the Institute of Mechanicsat the Swiss Federal Institute of Technology(ETH)in Zürich. At this stage I would like to express my sincere thanks to each personwho supported me during this important period,and I would like to do that in mynative language!

All jenen, die die vorliegendeArbeit mitermöglicht haben, möchte ich an

dieser Stelle meinen aufrichtigenDank aussprechen. An erster Stelle gilt dieserDank natürlich meinem Doktorvater Prof. Dr. Gerhard Schweitzer. Er

gewährte mir bei der Bearbeitung des gestellten Themas grösstmöglicheFreiheit und unterstützte mich darüberhinaus stets mit guten Ideen,konstruktiver Kritik und freundschaftlichen Ratschlägen.Auch bei meinemKorreferentenProf. Dr. Walter Schaufelbergermöchte ich mich ganz herzlichfür die Durchsicht der Arbeit und für das ihr entgegengebrachte Interesse

bedanken.

Wesentlich erleichtert wurde mir der Einstieg in die Magnetlagertechnikdurch die vielfältigen Hilfestellungenmeines KollegenDr. Alfons Traxler,wofürichmich herzlich bedanke.Bedanken möchte ichmich auch bei Eduard Randakund Richard Hüppi, ohne deren unermüdlichen Einsatz der erheblicheHardwareaufwandwohl nicht zu schaffen gewesen wäre. Grosser Dank gilt in

diesem Zusammenhangauch Ruedi Kummer, dessen Engagementfür dieSache ihn auch vor ungezählten gemeinsamenNachtarbeitseinsätzennichtzurückschrecken liess.

Wesentlichen "Input" erhielt meine Forschungsarbeit auch durch diezahlreichen Diskussionenin praktischen und theoretischen Belangeninnerhalbunseres "Magnetlagerteams". Ohne dieses kreative Umfeld meiner KollegenDr. HannesBleuler, Dr. Jürgen Salm, Dr. DieterVischer, Dr. RolandSiegwartund Raoul Herzog wären wohl viele Ideen gar nicht erst entstanden: Ihnengebührtdaher ein ganz besondererDank.

Bei unserer Sekretärin Gertraud Junker und allen Kolleginnen und Kollegender Institute für Mechanikund Robotik möchte ich mich für die überausfreundschaftliche Arbeitsatmosphäre und die vielen aufmunterndenWortebedanken, die mir von grosser Hilfe waren, besonders dann, wenn die Dingemal nicht so liefen, wie sie sollten.

Nicht unerwähnt bleiben sollen auch die vielen Ingenieurkollegen undSachbearbeiter, die ich durch zahlreiche Firmenkontakte kennenlernen durfte.Ihrem wohlwollenden Interesse an Hochschulforschungsprojekten ist dasGelingen meinerArbeit sicher ebenfalls zu verdanken. Und nicht zuletzt denvielen Organen der Eidgenössischen TechnischenHochschule möchte ichdanken, die es mir ermöglichthaben, Einblicke in etliche Bereiche der Technikzu gewinnenund interessante Kontakte überallauf der Welt zu knüpfen.

EvelineSondereggerund Linda Wegen danke ich herzlich für das Korrektur¬lesen der vorliegendenArbeit und für die zahlreichen linguistischenTips.Ein ganz grosser und besonderer Dank gilt zum Schluss meinen Eltern,

meinerLebensgefährtin, meinen Kollegen und Freunden.Durch sie alle wurdemir stets, gerade auch in schwierigen Situationen, viel Verständnis und

Unterstützungzuteil. Danke für diese schöne Zeit!

Zürich,im März 1990 ^j&UZ sGo/Lö&svydJüt/k.

TABLE OF CONTENTS

Abstract v

Kurzfassung vü

Chapter 1: INTRODUCTION 1

1.1 Basic Functional Principleofan ActiveMagnetic Bearing 1

1.2 HistoricalReview and State of the Art 31.3 Motivationand Goal ofthe Present Contribution 51.4 Structure ofthe Thesis 8

CHAPTER 2: STRENGTHASPECTSOF HIGH SPEED ROTATION 11

2.1 General Calculation of Stress and StrainUnder Centrifugal Forces 112.1.1 Single Axisymmetric Ring 122.1.1.1 Ring with Concentric Borehole 152.1.1.2 Füll Ring 172.1.2 Two or More ConcentricAxisymmetricRings 20

2.2 Magnetic Bearingsand High SpeedRotation 252.2.1 Basic DesignConcepts 262.2.2 Material Requirements 31

CHAPTER 3: HARDWAREDESIGN 33

3.1 High SpeedRotor Design 333.1.1 Rotor Shaft 333.1.2 Rotor with StandardTransformer Sheets 353.1.3 AmorphousMetals in a First Magnetic Bearing Application 383.1.4 Shrink-FitofThin AmorphousMetal Rings 423.1.5 Rotor DesignOverview 45

3.2 Magnetic Bearing Design 473.2.1 Radial BearingDesign 473.2.2 Axial BearingDesign 49

3.3 Other Test Stand Components 513.3.1 Sensors 513.3.2 Motor 533.3.3 RotorHousingDesign 53

3.4 Electronic Equipment 563.4.1 Filtering 563.4.2 Switched PowerAmplifier 573.4.3 Hardware Monitoring System 57

3.5 DigitalController Hardware 583.5.1 Application ofa Digital Signal Processor (DSP) 583.5.2 A DSP/PC Concept 59

3.6 Test Stand DesignOverview 61

CHAPTER 4: MODELING 63

4.1 Modelingofthe Elastic High SpeedRotor 634.1.1 General LinearModelfor SymmetrieRotors 634.1.2 Finite Element(FE) Model 664.1.3 ModelReduction 694.1.4 ModelCorrection by ModalAnalysis 71

4.2 ModelingofMagnetic Bearings, Sensorsand Amplifier 734.2.1 Linearized Bearing Model 734.2.2 Sensoringand Filtering 75

m

4.2.3 NonlinearPowerAmplifierDynamics 764.3 LinearOverall Plant Description 78

CHAPTER 5: SPOC-D: A NEWAPPROACHTO THE DESIGNOFDISCRETE-TIMEDYNAMIC OUTPUT FEEDBACK....81

5.1 Dynamic Output Feedbackversus Observer-Based State Feedback ... 815.2 Predefinitionof ControllerStructure 82

5.2.1 Decentralized Controller of PredefinedOrder 845.2.2 Observer Structure based on a Reduced Plant Model 855.2.3 TraditionalLayout for Structurally ConstrainedControllers 865.2.3.1 Discrete-TimePD Approximation 865.2.3.2 Observer based on a Reduced Plant Model 87

5.3 Optimal ControllerLayoutwithSPOC-D 885.3.1 Optimal DynamicOutput Feedback 895.3.1.1 QuadraticPerformance Index for StructurallyConstrained

Systems 895.3.1.2 ConditionsNecessary for Optimality 915.3.1.3 Additional Parameter Interdependencies and Modified

Optimality Equations 925.3.1.4 Numerical OptimizationProcedure 965.3.2 Application Example: a Simple Elastic Rotor 975.3.2.1 Controller Layoutwith Discrete-TimePD Approximation 1015.3.2.2 SPOC-D Optimized Controller 102

CHAPTER 6: APPLICATIONOF SPOC-D TO THEHIGH SPEEDROTORCONTROLLERLAYOUT 105

6.1 Controller Structure and Parameter Interdependencies 1056.2 OptimizationResults 1086.3 Gyroscopic Effects 111

IV

CHAPTER 7: EXPERIMENTALRESULTS 113

7.1 Measurementsand Comparisonwith SimulationResults 1137.1.1 Step Response 1147.1.2 Frequency Responseto Harmonic Excitations 117

7.2 Operationat High Rotational Speeds 1187.3 Discussion 122

CHAPTER 8: UNBALANCEFORCESCAUSING INSTABILITY 123

8.1 Numerical Simulationofthe Effectsof

DynamicPowerAmplifierSaturation 1238.2 A Simple Adaptive UnbalanceForce Cancellation Method 126

8.2.1 AMB Rotor and "Force-Free" Rotation 1268.2.2 UnbalanceForce Cancellation with AMB 129

CHAPTER 9: SUMMARYANDOUTLOOK 135

CHAPTER 10:APPENDIX 141

10.1 Search ofthe MaximumReference Stress 14110.2 Stress and Strainin Long Cylindric Rotors 144

10.3 LinearMatrix Equation (2.17): An Example for TwoThin Rings 14610.4 Numerical Values for the High SpeedRotorModel 14810.5 Performance Index and Vector-Gradient 15010.6 SystemDescription ofthe Simple Elastic RotorExample 16010.7 Circle Property of Adaptive UnbalanceForce Cancellation Method .. 163

REFERENCES 165

ABSTRACT

Active magnetic bearings (AMB) provide a means of supporting a bodycompletely without any contact. The advantages of such bearings comparedwith traditional Solutions are: absence of mechanical wear and friction,lubricant-free Operation and therefore suitability for severe environments,active Vibration control, unbalancecompensation etc., to name only a few.

The rapid progress made in electronics and control during the last two

decades has promotedresearch in the field of AMB Systems, and led to quite anumberof industrial applicationsin domains such as turbomachinery,vacuumtechnology, machining and transportation.Today, there are two clear trends in AMB technology: on the one hand, high

speeds gain more and more importance, especially in the fields of machiningand vacuum technology.On the other hand, analog control is abandoned infavor of digital control, which offers much moreflexibility for taking füll

advantageof the possibilitiesprovidedby active magnetic bearings.In this thesis design and control of high speed rotor Systems are considered

with the general objective to analyze some of the most restricting problemsassociated with high speed rotation. Practical and feasible Solutions are

proposed and tested experimentally.The first aim is to gather all information necessary to discuss material

behaviorunder highcentrifugal load, in order to provide a general designtool for

determiningoptimalmaterial-geometrycombinations. A high speed test-rotor is

designed and built relying on these fundamentals. The test-rotor features

amorphous metals which are well suited for high speed AMB applications. Toour knowledgethis rotor is the first one incorporatingthis newmaterial.

vi

Special emphasis is put on the control aspects of high speed AMB Systems.The inherent problemassociated with the application of digital control to highspeed rotor Systems is the conflict between a most often high order plant(flexible rotor) and a Controller of lowest possible order aiming at short

computation times and thereforebeing realizable with today's micro- and signalprocessor technology. Differing from many model reduction or Controllerreduction techniques which cannot always guarantee stability of the finallyrealizedclosed-loop system, a so-calleddirect design Controller layout method -

SPOC-D - is proposed for this demanding control task. This method providesvarious features of practical importance:order and structure of the discrete-time Controller can be chosen freely accordingto practical needs, and aimed-atController properties such as static bearing stiffness, noise reduction at highfrequencies etc. can be included in the Controller layout process from the verybeginning by means of additional control parameterconstraints. Then, startingfrom a stabilizing but sub-optimal set of control parameters, a parameteroptimizationis performed taking into account the füll dynamicsof high order

plant and low order compensator and yielding stability of the resulting closed-

loop system. The SPOC-DController layout method closes the gap betweentraditional PD and familiärtype layout methodsfor low order compensators onthe one hand, and LQ/LQG methods for observer-basedhigh order controlschemeson the other hand.

The third goal of this contributionis the practical applicability of the ideas formechanical design and Controller layout. Importantresults are obtained from a

high speed AMB test stand especiallybuilt for this purpose. The test stand is

designed for a maximum rotational speed of lOO'OOO rpm and for a dynamicränge of the Controller up to 2 kHz, achieving sufficient damping for4 bendingmodes. The suitability of amorphous metals for AMB applications is proved,and the good Performance of the SPOC-D optimized Controlleris confirmed.

However, dynamic poweramplifier Saturationas a result of unbalance occurs

at the third critical speed (64'000 rpm), making a passing of this critical

impossible. As an outlook to solve this problem, a simple adaptive unbalanceforce cancellationmethod is proposed.

Vll

KURZFASSUNG

Aktive elektromagnetischeLagerungen stellen eine Möglichkeit zur völligberührungslosen Lagerung eines Körpers dar. Die Vorteile solcher Systemeverglichen mit herkömmlichen Lösungen liegen auf der Hand: VölligeVerschleiss- und Reibungsfreiheit, schmiermittelfreier Betrieb und darausresultierende Eignung für den Einsatz in aggressiver oder auch hochreiner

Umgebung, aktive Schwingungsbeeinflussung, Unwuchtkompensation usw.,

um nur einige wenigezu nennen.

Der rasante Fortschritt der letzten Jahre in Elektronik und Regelungstechnikhat die Forschung auf dem Gebiet aktiver Magnetlagerungensehr begünstigtund zu einer ganzen Reihe von industriellenAnwendungen dieser Technik

geführt, so z.B. in Bereichenwie Turbomaschinenbau, Vakuumtechnik,Metall¬bearbeitung und Transporttechnik.

Heute zeichnen sich zwei klare Entwicklungstendenzen für Magnetlager¬systeme ab: Zumeinen besteht der Wunschnach immer höheren Drehzahlen,vor allem in der Metallbearbeitung und der Vakuumtechnik,zum andern treten

mikroprozessorgestütztedigitale Regelungen gegenüber traditionellenana¬

logen Implementierungen immer stärker in den Vordergrund, dies nicht zuletzt

auch, weil digitale Regelungen eine beträchtlich höhere Flexibilität in der

Nutzung der vielfältigen Möglichkeiten berührungsloserMagnetlagerungenaufweisen.

VU1

In dieser Arbeit werden Hochgeschwindigkeits-Rotorsysteme auf mechani¬schen Aufbau und Reglerauslegung hin genauer untersucht. Dabei werden

Möglichkeiten und Grenzen der Anwendungaktiver Magnetlagerungenfürsolche Systemefestgestelltund experimentell überprüft.

In einem ersten Teil werden Grundlagenfür Materialverhalten unter hoher

Fliehkraftbeanspruchungzusammengetragen,und die Eignung verschiedenerMaterial-Geometrie-Kombinationen für magnetgelagerte Hochgeschwindig¬keitsrotorenwird untersucht. Aufbauend auf diesen Untersuchungenwird eineRotorkonstruktionvorgeschlagenund realisiert, welche sich - unseres Wissensals erste - durch die Anwendung amorpher Metalleauszeichnet.

Das Schwergewicht dieser Arbeit wird auf die Auslegung der digitalenRegelung gelegt. Ausgangspunktist die bei magnetgelagerten Rotorsystemenmeistens auftretende Diskrepanz zwischen hoherSystemordnungder Streckeeinerseits (elastischer Rotor) und möglichst niedriger Reglerordnungandererseits,um mit bestehenden Mikro- oder Signalprozessorengenügendhohe Abtastfrequenzen erreichen zu können. Anders als bei Modell- oder

Reglerreduktionsmethoden,bei welchen die Stabilität des Gesamtsystemsoftnicht a priori garantiert werden kann, wird eine zeitdiskrete Reglerausle¬gungsmethode- SPOC-D - vorgeschlagen, die es erlaubt, Ordnungund Strukturdes Reglers von vornherein völlig frei nach praktischen Gesichtspunkten zuwählen. Wichtige zusätzliche Anforderungenan den Regler, so z.B. statische

Steifigkeit, Reduktion von hochfrequentem Rauschen usw., können dabeiebenso in den Auslegungsprozess miteinbezogen werden. Ausgehendvoneinem stabilisierenden aber sub-optimalen Parametersatzwerden für die so

festgelegte Reglerstruktur die Reglerparameterdurch ein Stabilitäts¬erhaltendes Optimierungsverfahren bestimmt, wobei die volle Dynamik des(nicht reduzierten) Streckenmodells und des vorgegebenenReglers berück¬

sichtigt und optimal aufeinander abgestimmt werden. Die vorgeschlageneAuslegungsmethodeorientiert sich sehr stark an praktischen Gesichtspunkten,wo einfache Regler niedriger Ordnung oft eine grosse Rolle spielen; sie liefertaber im Vergleich zu herkömmlichen Auslegungsverfahrenoft bessereResultate. Durch die Berücksichtigung der vollen Streckendynamik unterBeibehaltung einer einfachen Reglerstruktur schliesst die vorgeschlagene

IX

Methode somit gewissermasseneine Lücke zwischen Auslegungsverfahrenfür Beobachterregelungenhoher Ordnung (LQ/LQGusw.) und herkömmlichenAuslegungsverfahren für zeitdiskrete Regler niedriger Ordnung (PD-Approximation usw.).

In einem dritten Teil dieser Arbeit werden die vorgeschlagenenkonstruktivenund regelungstechnischenIdeen experimentellüberprüft. Zu diesem Zweckwerden Messungen an einem eigens zu diesem Zweck realisierten

Hochgeschwindigkeits-Magnetlagerversuchsstanddurchgeführt. Die durch dieneue ReglerauslegungsmethodeSPOC-D erwartete gute Reglerdynamik bisetwa 2 kHz wird im Versuchbestätigt. Vier elastische Schwingungsmodes des

Hochgeschwindigkeitsrotorswerden zufriedenstellend gedämpft. Allerdingskann die vorgesehene Maximaldrehzahlvon lOO'OOO U.p.M. nicht erreichtwerden: Nach erfolgreichemDurchfahren zweier kritischer Drehzahlen stelltsich aufgrund der Unwucht bei der dritten kritischen Drehzahl (etwa64'000 U.p.M.) dynamische Stromsättigung im geschalteten Verstärker ein,was ein Touchieren der Fanglagerzur Folge hat. Zur Lösung dieses Problemswird als Ausblick eine einfache adaptive Unwuchtkompensationsmethodevorgeschlagen.

Chapter 1

INTRODUCTION

1.1 BASIC FUNCTIONALPRINCIPLEOF ANACTIVEMAGNETICBEARING

Magnetic bearings providesupport for any ferromagnetic body1-1 withoutanymechanical contact. Two basic bearing types can be distinguished: the active

magnetic bearing (AMB),and the passive magnetic bearing (PMB).

With passive magnetic bearings, the bearing forces result from pairs of

permanentmagnets with opposing field directions producing mutual repulsion.Althoughthis kind of contact-free support seems to be very simple, there are

two majordrawbackswhen compared with an active magnetic bearing: firstly,a complete six degree of freedom support of a rigid body by uncontrolled

ferromagnetic forces is impossible [Braunbek 39]. Secondly, the bearingcharacteristics of a PMB cannot be changedeasily during Operation, as is the

case with an AMB. However,hybridAMB/PMB Systems are applied to reduce

system complexityand cost.

- Diamagnetism and superconductivity are not considered in this contribution.

- no mechanicalwear and friction- low drag torque- no lubrication, and therefore non-polluting- low energy consumption- high circumferentialspeeds (more than 200 m/s)- Operation in severe environments- easily adjustablebearing characteristics (stiffness, damping)- active Vibrationcontrol and easypassing of critical speeds- on line balancing and unbalancecompensation- on line system parameteridentification- on line monitoringof safe Operation

In the early 1980ies, theoreticalinterest in the fundamentals of magneticbearing technology and in the new possibilities in rotordynamics offered byAMB grew quickly. Importantresearchworkwas done in Munich[Ulbrich 79],Duisburg [Pietruszka& Wagner 82], Darmstadt [Arnold85] and here in Zürich[Traxler & Bleuler 83], [Schweitzer, Traxler, Bleuler & Bauser 83],[Bleuler84], [Traxler85].

Recently, two clear trends in active magnetic bearingtechnology could benoticed: thanks to the fast progress in microprocessor development, analogcontrol is abandoned in favour of digital control, which offers much higherflexibility. As a second trend, very high rotor speedsare gaining importance inthe fields of machining,vacuum technologyand turbomachinery.In this domain

very important progress has been made here in Zürich lately by the

developmentof a high speed milling spindle [Siegwart 89]. Some of the ideas

presented in this contributionhave already been realizedin that milling spindleproject.

An overview on the State of the art in magnetic bearingtechnology can befound in the Proceedings of the First International Symposiumon MagneticBearings [Schweitzer 88].

1.3 MOTIVATIONANDGOALOF THE PRESENT CONTRIBUTION

As stated in the previous section, the interest in high speed rotors in various

applications is increasing quickly, and, thanks to magnetic bearing technology,the technical realization of such Systemshas becomefeasible.

But only the criterion of high rotor speed is not sufficient to determine the

complexity of the mechanical designand the magnetic bearing control. It is thecombination of high speed, large rotor geometry,the numberof critical speedsto run through and the severity of external disturbances as e.g. unbalanceforces that makes the design of an AMB high speed rotor system a very

demanding task.

The design of a high speed rotor system including magnetic bearingsunderlies various limitations and requirements. Some of the most importantones are shortly cited below:

limitations- mechanical

control

- power amplifier

requirements- material

- control

- power amplifier

- material strength- loss of contact betweenrotor and lamination- material fatigue due to Vibration- high order plant (flexible rotor structure)- numberof critical speedsto pass- few measurable Output signals- changing plant parameters

(gyroscopic effects)- amplifier Saturation due to unbalanceforces

- high yieldstrength- low electricaland magnetic loss- high frequencyresponsebandwidth- high samplingrate in case of digitalcontrol- high frequencyresponsebandwidth

The general objective of this contribution is an analysis of some of the mostrestricting problems associated with high speed rotation and a presentation offeasible Solutionsin order to reach as high rotational speeds as possible. For the

experimentalverification of the proposed ideas, a special high speed AMB test

stand has to be designed. A comparison has to be made between experimentaland theoreticalresults. In a last step, additional effects occurring during the

experiment, not known a priori or left out of consideration at first, shall bediscussed.Possible Solutions shall be proposed as an outlook.

Regarding a first group of already knownproblems in high speed rotation,special emphasis is put on mechanical material propertieson the one hand, andon the magnetic bearing control strategy on the other hand.

A first goal of the present workshall be to provide all information necessaryto determine material behavior of fastly rotating machine parts. Thefundamentalequations for stress, strain and displacementunder centrifugal loadshall be formulated. Different materials have to be compared, and their

suitabilityfor high speed rotation with magnetic bearings must be discussed. Allthese considerations shall providea tool to determineoptimal combinationsofmaterial properties and geometry when designing a high speed AMB rotor

system.

A second goal is to find a control strategy taking into account the füll

complexityof the high order plant, most often an elastic rotor, withoutresultingin a complicated high order Controller structure. This demand is especiallyimportantin the case of digital control in order to reduce Computingtime andthus to realize relatively high sampling rates. The basic proceduresto obtain a

reduced order Controller for a high order plant are shown in figure 1-2.

Very importantwork on the subject of Controllerorder reduction has beendone recently. A useful summaryof presently availablereduction techniques isfound in [Anderson& Liu 89].

High OrderPlant

LQG or H^ Design_

(HighOrder)

ModelReduction

.DirectJ)esign

LowOrderPlant

LQG or H^ Design(Low Order)

High OrderController ]

ControllerReduction

LowOrderController

Fig. 1 -2: Basicprinciplesofloworder Controllerdesign(figure from[Anderson & Liu 89])

In the field of magnetic bearing applications,the most followed Controller

designmethods Start from a low order plant, i.e. from a rigid body descriptionora model obtained by reduction techniques such as modal or quasi-modalreduction. Then, a low order Controller is designed which often results in a

decentralized Controller structure such as P(I)D or familiär type control

schemes[Bleuler 84], [Salm 88].

Decentralized low order Controller structures have turned out to be very well

applicable to AMB Systems, even if the plant is of ratherhigh order, which is

generally the case with high speed applications, in order to include all the

critical speeds to pass through. In case of digital control, the necessity to designlow order Controllers for high order plants is even intensified, since, up to now,

this seems to be the only way to achieve high sampling rates. However, theController layout approach which is found most often, a simple digitalapproximation of well known continuous-timelow order Controllers, often

results in unsatisfactorydynamicbehaviorofthe closed-loop system.

Hence, it is evident that a direct design Controller reduction method (seefigure 1-2) for discrete-timeSystems must be sought. Actually, very little work

has been done on this subject, especiallywork involving digitalcontrol. The low

8

order Controller structure must be freely predefinable according to practicalneeds. Additionalrequirementsespeciallyfound in AMB applications, such as

static bearing stiffness, low sensitivity to noise etc., shall be included in theController layout method as an integral part. Furthermore,an efficient algorithmto compute the Controller parametersmust be developed.The third goal of this work is to prove the practical feasibility of the ideas

described above. Important results are obtained from a special high speedAMB test stand, designed and built on the basis of the constructional conceptsaforementioned as a first goal. For the control, fast digital signal processors(DSP) shall be implemented, representing the latesttechnologyin this field. Onthe side of electronics hardware, new switched power amplifiers will beapplied. Also, experimental results are obtained from other AMB researchprojects currently going on at our institute, mainly from a high Performancemilling spindleproject [Siegwart 89].

Finally, additional problems occurring in high speed AMB Systems are

discussed. It will be shown that power amplifier Saturationeffects due to rotor

unbalances quite far below the cutoff frequency can be dominating. Thus,independentlyof the Controllercapacity to damp even high frequency elasticmodes of the rotor, it may be impossible to pass through the critical speedswithoutfurthermeasures if rotor unbalanceis too high. Since perfect balancingof elastic rotors is a difficultand costly task, a very simple adaptive unbalanceforce cancellation method, which can very efficiently be implemented withmagnetic bearings, will be described as an ouüook.

1.4 STRUCTURE OF THE THESIS

In chapter 2, the analytic calculation of stress, strain and displacement in

rotationally Symmetrie parts under centrifugal load is described.The suitabilityof different materials for high speed rotation is discussed. Various basic designconcepts for AMB high speed rotors are compared, and concludingrecommendationsregarding design are made. Although many of the ideas

presentedmay be found in literature,this chapter is meant to summarize all

important information, in order to serve as a design tool.

Chapter 3 describes all the hardware components used for the high speedAMB test stand. Regarding the rotor design, the concepts found in chapter 2

are followed. Designing an AMB rotor system is a rather interactive processand can never be done withoutconsideration of mathematical rotor model and

control theoretical aspects such as controllability and observability of the

resulting system. However, only the result of this interactive process is

described in this chapter.

In chapter 4 the linear model of the plant including elastic rotor, magneticbearing, sensor and poweramplifier dynamics is presented. Nonlinear aspectsare discussed.

In chapter 5 sl discrete-time Controller layout method (SPOC-D) is

introduced. Due to the fact that this method optimizes feedback parametersfordiscrete-timeControllers of any predefined order and structure, including the füll

high order plant dynamics, the SPOC-D method is perfectly suitable for highspeed AMB Systems.

Chapter6 shows the application of the SPOC-D Controllerlayout method to

the high speed rotor system. Numerical simulations of the closed-loop dynamicbehavior are presented. The influence of gyroscopiceffects on overall systemPerformance is discussed.

Experimental results follow in chapter 7. Measurements of the closed-loopbehavior, such as step and frequency response, are compared with Simulationresults. Measurements of a run-up test up to the third elastic critical speed(64'000 rpm) are presented. It is shown that, in the third elastic critical speed,power amplifier Saturation due to rotor unbalances causes instabilityleading to

a contact between rotor and retainer bearings.

Chapter 8 puts special emphasis on the effects of unbalances at very highfrequency critical speeds. The effect of power amplifier Saturation on rotor

stability is numericallydemonstratedby means of an expandedplant description

10

including nonlinearities. A simpleadaptive unbalanceforce cancellationmethodis proposed.

Chapter9 summarizesall results obtained in this thesis and puts them into a

general context. As an outlook, the importance of the results for future highspeed AMB projects is pointed out.

Finally, chapter10 mainly contains mathematicalderivationsas an appendix.In addition, numericaldata for the high speed rotor system obtained from finiteelementmodeling,modal analysis and other measurementsare given.

The references are attached to the end of this thesis.

11

Chapter2:

STRENGTHASPECTSOF HIGH SPEED ROTATION

2.1 GENERAL CALCULATIONOF STRESS AND STRAINUNDERCENTRIFUGALFORCES

Fast rotating machine elements are subject to very high centrifugal forces.The magnitude of the resulting stress in such high speed rotors depends on

rotationalspeed, size and shape of the rotor and its material properties. The

followingconsiderations are restricted to homogeneous isotropic axisymmetricrotors which,nevertheless,cover a wide field of applications.

The linear stress calculation for the above mentionedrotor class loaded bycentrifugalforces is done mostly under one of the following basic assumptions:planar stress or planar strain. The assumption of planar stress is correct for

parts of very short length, i.e. thin disks or rings, where the material contraction

along the rotation axis is free, and thereforeno axial stress may occur.

In case of very long cylindricalparts, however, a three-dimensionalstresscalculation(planar strain) must be done. The reason for this is that, for a zone

far away from the rotor'sfront ends, material contractionalong the rotation axis

is prevented(de St. Venant's principle). Hence the cross-sectionsremain plane,and consequentlyaxial stress occurs.

As it will be shown in section 2.2.1, thin rings play an importantrole in the

constructional concept of magneticallysuspended rotors, and, therefore, the

12

following stressand strain calculationis done for the case of thin rings, i.e. underthe assumption of planar stress. The corresponding equations for longcylindrical rotors, however, can be derived in a very similarway (see appendix10.2).

Some of the followingconsiderations can also be collected from literaturerelevant to the subject [Den Hartog 52]. The reason to summarize all thisinformation in this chapter is to providea helpful tool in order to facilitate the

design ofAMB high speed rotors.

2.1.1 Single Axisymmetric Ring

Starting with the basic geometry of an infinitesimallysmall ring elementshown in figure 2-1, the kinematicrelationship betweendisplacementand strainis derived. In a second step the equüibrium equations for the rotating ring are

formulated and, together with the linear-elastic material law, lead to thedifferentialequation for the ring's displacement. This is solved in the last step.

MQ

03d©

o CTt(r)Vu

fcCJt(

Or(r+drdFc = prft2dV

Fig. 2-1 : Infinitesimally small ring element under centrifugalforces

In the above figure, a thin ring element (thickness8, radial width dr, angle dq>)is rotating about the vertical axis with rotational speed Q. The resultingcentrifugal force dFc is proportional to the ring's density p, volume dV and

13

radius r and to the Square of the angular speed Q. Due to the centrifugalforce,radial and tangential stress o> and <Jt occur. For symmetry reasons, neither o>nor at can be functions of the angle q> and, therefore, only dependon the radiusr. The ring's symmetry also causes all shear stress components to be zero,

hence, radial, tangential and axial directions of the ring are simultaneouslyprincipal directions.

Any material point of the rotating ring mainly moves in the radial directionindicated by the displacementu which, again for symmetry reasons, is only a

function of the radius r. The axial displacementis not explicitly given in above

figure; as it will be shownin equation (2.4), the axial strain ez can be expressedin terms of radial and tangentialstress o> and Gt. An integrationover the rotor

length finally leads to the Solutionfor the rotor's axial contraction.

The kinematic relationshipbetween radial and tangential strain er and et on

the one hand and the radial displacementu on the other hand can be given bythe following equations:

£ _ lim u(r+dr)- u(r) _ du(r)<*->0 dr dr

= lim (r+u(r))d(p - rd(p _ u(r)dqh->0 rd(f> r

(2.1)

The formulation ofthe equüibrium in radial direction results in:

o{r+dr)8 (r+dr)d(p - G^r)8rd(p - 2(Jt(r)8dr^+ pr2Q28dr d(p = 0

(2.2)

After expanding of equation (2.2) and neglecting higher order terms, the

foUowingdifferentialequation for the radial stresscan be found:

dcjr= Of - or- pr- it

(2.3)dr

14

By solving the linear-elasticthree-dimensional material law (Hooke'slaw)

£r = l(o>-VO})E

£t = J_(o7-vo>)E

£z = -^(o>+o>)E

(2.4)for ar and ot and substituting into (2.3), the differential equation for the radial

stress can be expressed in terms of radial and tangential strain er and et. Thetwo material constants in (2.4) are Young'smodulusE and Poisson's modulus v.

FinaUy, using the kinematic relations (2.1), a simple differential equation ofEuler's type can be found for the radial displacementu:

(2.5)The general Solutionof the inhomogenuousdifferentialequation (2.5) is

r ZE H

(2.6)where a and b are integrationconstants depending on the specific boundary

conditionsofthe givenproblem.

Knowing the Solution (2.6) of the differential equation for the ring'sdisplacementu, it is possible to determineaü other interesting quantities such as

stress and strain by simply applying (2.6) to the kinematicrelations (2.1) and to

the material equations (2.4). In the next two paragraphsthis wiü be shownwithreference to two importanttechnical examples: a ring with a borehole in itscenterand a fuü ring.

15

2.1.1.1 Ring with ConcentricBorehole

The necessary boundary conditions to determine the constants a and b in(2.6) are found by a consideration of the radial stress at the ring's inner andouter edge rt and r0. Both stress componentsor(r,) and <7r(r0) have to be zero:

Gr(rö = 0

Gr(ro) = 0

(2.7)The boundaryconditions given in (2.7) result in the following values for the

integrationconstants a and b:

a = l^pß2(v+3) (rf + rl)aE

b = l±Y_pß2(V+3) (rfr})oJb

(2.8)The complete equations for the radial and tangentialstress in function of the

ring's radius r are then:

om = Ipß2(v+3) (rf + r% - &£- r* )8 rl

ot(r) = ipß2 ((v+3)(r/*+ rft + (v+3)^M.(i+3v)r2 j8 rl(2.9)

For technical applications it is most essential that under normal operatingconditions the yield strengthof the givenmaterial is not reached, i.e. a referencestress oref must be smaller than the yield strength o0 at any point of the givenstructure. This one-dimensionalreference stress has to be determined from thetwo-dimensional stress State, for example under the assumption of Tresca'sshear stress hypothesis.Thus, for the case of the rotating ring, one obtains:

16

<V (T) = max (| o, <r) - o> (r)\, | Gr (r)\, \ Gt <r)\ )(2.10)

It requires some analysis to show that the maximum of the reference stress

Gref is found at the inner edge rt of the rotating ring. This is an importantfact.The necessarytransformations are shown in appendix 10.1. The final result is

presented here:

°r*max = l<*N = \P& (d-v)r? + (v+3W)(2.11)

In figure 2-2 the graphs of radial, tangential and reference stress are

qualitatively shownin order to demonstratetheir fundamentalbehavior.

Q

örOt

Omfö

CTt

L^

Fig. 2-2: Radial, tangentialand reference stress in a rotating ringwith concentric borehole(Q^ 0)

17

2.1.1.2 Füll Ring

This case differs from the previous one by the boundaryconditions for thebasic equation (2.6). New values for the integration constants a and b are

obtained. The center-displacementw(0) in (2.6) must not be infinite, which is

only possible if the constant b is set to zero. The second boundarycondition at

the outer edge r0 remains the same, hence, one obtains the foUowingboundaryconditionsfor the rotating fiül ring:

M(0) * oo

GAr0) = 0

(2.12)

Although the boundary conditions (2.12) are different from those given in

(2.7), it can be shown that the integration constants a and b and the stress

components o> and Gt may stiU be expressedby (2.8) and (2.9) respectivelybysetting the inner radius rt equal to zero. As a consequenceof the completesymmetry in the ring's center, radial and tangentialstressfor rt=Q are equal.

The maximum of the reference stress Gref again occurs at the inner edge, i.e.in the center of the füll ring. Its value, however, cannotbe expressedby (2.11)by simplysetting rt to zero. For the fuü ring, one can find the interesting fact thatthe maximum reference stress is exacüy half as high as for the case of a ringwith a very smaU concentric borehole(r,-«0):

°nfmax = J- lim G^n) = lpß2(v+3)r22 r, ->0 8

(2.13)This is an importantproperty of rotors or cylinders under centrifugal load:

small concentric boreholes raise the resulting reference stress by a factor of

two and should thereforebe avoided.

The qualitative plots of radial, tangentialand reference stress for the fuü ringare shown in figure 2-3. Note that the stress peaks near the borehole shown in

figure 2-2 do not occur in case ofthe fuü ring:

18

Ci3Q.

Gr°t

öref

Gt

Fig. 2-3: Radial, tangentialand reference stress in a

rotatingfüll ring (£2 =£ 0)

Knowing the maximum reference stress in the center of a rotating fuU ring, itis possible to determine the maximum attainable circumferential velocity vmaxfor any kind of material. Replacing Gref in expression (2.13) by the

correspondingyield strengthg0, one obtains:

'max (r0Q\max. 8oo(v+3)p

(2.14)

Thus, the attainable circumferential speed only depends on yield strength,Poisson's modulus and density of the given material. The foUowing tablecontains a collection of technically interesting materials and correspondingvalues for the crucial speedv,^.

19

material Vmac/[f]

transformer sheet 240 - 360

yeüowbrass 280 - 380

bronze 300 - 380

steel 330 - 500

aluminum 500 - 610

titanium 520-750

amorphousmetal 715 - 830

carbonfiber 1000 -1200composite

Fig. 2-4: Technically interesting materials and their maximum

circumferential speeds

However, the results obtainedup to here for single rings under centrifugalforces cannot unconditionaUy be applied to the design of magneticallybornehigh speed rotors. The crucial requirementof magneticaUyborne rotors is thatthe rotor material shows a combination of good mechanical properties,i.e. a

high yield strength, and good magnetic and electrical properties, i.e. a highmagnetic flux, low magnetic and low eddy current loss. Therefore, a

magneticaUy borne rotor will pose designproblems similar to those known fromelectric motors, and it is not surprising to find an importantdesign feature in

common: a laminatedconstruction consisting of a massive rotor shaft and thinlamination sheets. Regarding the laminated construction, it is necessary to

discuss the behavior of two or more concentric rotor parts under centrifugalload.

20

2.1.2 Two or More Concentric Axially SymmetrieRings

The derivation of radial, tangential and reference stress in an assembly of nconcentric rings is very similar to that shownin section2.1.1 for one single ring.For each ring, identical considerations concerning radial displacement, strain,material law and equüibriumcan be made, resulting in the same differential

equation for the radial displacementUj for each ofthe n rings:

uj(r) = air + ^-^-pjrln2SEjj = l...n

(2.15)The integration constants o, and bj for each ring are again determined by

radial stress and displacementat the inner and outer edges rt. and r0.. For the

innermost ring or rotor shaft, either the radial stress at the inner edge Gr(rix)must be equal to zero if rtl*Q or the displacementw^O) must be finite if rtl=0.For the outermost ring the radial stress Gr(r0n) must vanish. The remainingequations are obtained via the transition conditions between the rings:displacementand radial stress of the inner ring must equal those of the outer

ring. Thus, the In necessary boundary conditions can be derived for the n

concentric rings:

ofrn) = 0 resp. Mi©) ^ °°

Grfr0j) = Grj+l(rij+1) j = 1...H-1

Uj(r0j) = uj+i(rij+l) j = 1...H-1

(2.16)

The boundaryconditions (2.16) can be combined into a system of 2n linear

equations. The constants aj and bj wiü then form a vector x of length 2n. The

resulting 2nx2n-matrix A will contain all geometric and material parametersfound in (2.16). Finally, with the 2n-vector/ of the external load, i.e. the

centrifugalforces, an inhomogenuouslinear matrixequation can be formulated:

21

Ax=f(2.17)

Anexample showingthe structure of the matrixand vectors in (2.17) is givenin appendix 10.3.

As the most importantresult of (2.16) and (2.17), one finds that the radial

stress in aU rings is positive, a fact which correspondsto general experienceand intuition.However, tensile radial stress cannotbe transmitted from one ringto the other if the rings are not shrink-fittedor otherwise tightly connected inradial direction. As a consequence,the n rings wül separate from one anotherwith increasing rotational speed ß, and the assumptions underlying thederivationof (2.16) and (2.17) are no longer true.

The classic way to prevent this described loss of contact is to shrink-fit the nconcentric rings onto one another in order to achieve a compressive pre-stress.This is done by providing a negative gap betweenthe rings, i.e. the outer radiusof the inner ring r0.andthe inner radius of the outer ring rij+1 are not the samebut differ by a small quantity. Thus, the radial displacements of inner and outer

ring at the transition surfaces are no longerequal, a fact which has to be takeninto accountwhen replacing the expression for the displacementsUj and uj+1 of

(2.16) by thenew transition condition:

r0j + uj(r0j) = rij+1 + uj+i(rij+l) j = l...«-l

(2.16a)To solve the linear matrix equation (2.17) with the boundary conditions

(2.16/2.16a) numerically for any number of rings and any material and

geometryparameters, an interactive FORTRAN program has been developed.All numerical results including the qualitative stress-diagrams in this sectionhave been obtained with this program.

In the next figure, radial, tangentialand reference stress in two shrink-fitted

rings are qualitatively plotted. Note that, due to the shrinkage, the radial stress

at the transition from inner to outer ring remains negative, and thereforeno lossof contact occurs for the given rotational speed.

22

e^Q

ViCf

öfef

Ö<*UliSäfeK

» m®m muttmmMttmum

°1ri12

»2

Fig. 2-5: Radial, tangentialand reference stress in two

shrink-fitted rings (Q * 0)

The peak of the reference stress at the inner edge of the outer ring is

explained by the fact that the stress due to shrinkage and the stress due to

centrifugalforces mainly act in the same direction, i.e. the tangentialdirection,whereas, in the inner ring, they counteract and partially cancel out each other.For this reason, the maximum reference stress wiü always occur at the inner

edge ofthe outer ring, even if the inner ring has a concentric borehole.

The need of shrink-fit in order to avoid loss of contact between the rotatingrings has a further consequence of crucial technical importance: the maximumcircumferential speed at the surface of a set of two or more shrink-fitted ringswith the same material properties can never reach those values derived insection 2.1.1.2 for the fuü ring (figure 2-4) with the same outer radius. There are

two reasons for this fact. First of aü, the maximum reference stress in a shrink-fitted assembly is always larger than that in a füll ring due to the radial pre-stress caused by the shrink-fit.As a second reason, disregarding the influenceof shrink-fit,the maximum reference stress in a ring with concentric boreholeis

always larger than that of a fuü ring with the same outer radius (see equations(2.11) and (2.13)).

23

The following considerations are made in order to determine the optimalmaterial exploitation for a given shrink-fit assembly. NumericaUy solving theUnear matrixequation(2.17) for two rings with arbitrary radii, it can be shownthat the maximum reference stress at the inner edge ofthe outer ring increaseswith the rotational speed, while the radial pressure between the two ringsdecreases.This is iüustratedin figure 2-6:

1.2

©leo

rtoooow

00

1.0

0.8

0.6

0.4

0.2

0

-0.2

-0.4

rCTref,

reaching theyield strength

amount ofshrinkage

loss of contact

°r

amount ofshrinkage

0 0.5SPEEDRATIO Q

1.0

Qmax

Fig. 2-6: Maximum reference stress and radial stress between two

shrink-fitted rings as afunction ofthe circumferential speedfor different amounts of shrinkage

Hence, for a set of two shrink-fitted rings with given geometry, the optimalmaterial exploitationand, therefore, the maximum rotational speed ß^^ can beobtained when loss of contact and reaching the yield strength g0 occur bothforthe same value of Q (curves 2 in figure 2-6 at Q=Qmax). This is achieved by a

properselection ofthe amount ofshrinkage.

24

The mathematical formulationof the above Statement can be found easily.Knowing the expression for the maximum reference stress of a rotating singlering with a concentric borehole (2.11) and replacing the reference stress by the

yield strength, one onlyhas to solve for the circumferentialvelocity:

'max = (rO0Q\/max 4Q-Q2

p2((l-v2)-|-+ (V2+3))

(2.18)

Figure 2-7 below shows a plot of the maximum surface speed v^^ achievablewith an optimalshrink-fit versus the thickness ratio ofthe shrink-fitted rings, i.ethe ratio between the transition radius ri2 and the outermost radius ror Speedvmax for the shrink-fitted rings is related to the maximum speed for the füll ringgiven by (2.14). Note that here again all calculations are made under the

assumption that the two rings have the same material properties(v=0.3).

.71

8s>

8E

o

Qww00

.70

.69

.68

.67

.66

.65

.64 ff^N0 0.5 1.0

THICKNESSRATIO y1r°2

Fig. 2-7: Dependency of v^^ on the thickness-ratio

ofthe shrunk rings

25

The most important conclusion one can draw from the plot in figure 2-7 is that

the thinner the outer ring is chosen, the lower the maximum reachable

circumferentialspeed will be.

The results obtained up to here, especiallythose for the maximum attainable

circumferentialvelocity, are of essential interest for the design of magneticallysuspended high speed rotors and form the basis for the study of a few rotor

design concepts introduced in the following section.

2.2 MAGNETIC BEARINGSAND HIGH SPEEDROTATION

Rotors for active magnetic bearing applicationsmost often feature the same

basic design: a laminated structure as shown in figure 2-8 for an exemplaryinternal rotor. For an external rotor the considerations are basicallythe same.

Q

u

VsWs

//

/

rotor shaft

Stator

lamination consisting ofthin iron rings withsoft magnetic properties(transformer sheets)

non-rotatingmagnetic force

Fig. 2-8: Basic design ofa rotor in active magnetic bearings (AMB)

The reasons for the use of a laminated structure for AMB rotors are

described in detail in [Traxler 85]. To summarize, thin lamination sheets

26

minimizeelectric loss (eddy currents), whereas their magnetic properties arechosen in order to achieve high magnetic induction and low hysteresis loss (seealso figure 2-12).

However, for the field of high speed rotation, a laminated structure alsocauses all the strength and contact problems described in section 2.1. In thenext section a few rotor design concepts are presented to show how one can

cope with the specific design problems ofhigh speed rotation.

2.2.1 Basic Design Concepts

The shrink-fit discussed above is not the only method to avoid loss of contactbetweenlamination sheets and rotor shaft. A few examples of other possibilitiesare shown in figure 2-9. The different methods are briefly discussed in thefollowing.

a) shrink-fit

b) adhesiveconnectionbetweenrotor shaftand lamination

c) wrapping withhigh strengthandlow mass composites

e) "positiveshrinkage gradient"

d) applicationof a

special geometry

Fig. 2-9: Connection types between rotor and laminationsheetsto avoid loss ofcontact

The shrink-fit (figure 2-9a), the most classic way of connecting rotor andlamination sheets, has been discussed in detail in 2.1. The method is simpleand

27

resulting stress and strain can be derived analytically. However, for some

materials the amount ofshrinkage is not limited by the yield strengthbut by the

maximum temperature to which the lamination sheets may be heated before

shrinking (see section 3.1.3). Thistemperature limit is given either by the Curie-

temperature or by the mechanical properties, which generally deteriorate

drastically over a certain temperature. For ordinary transformer sheets a

maximum temperature ränge of 200°C to 300°C has been found

experimentaUy.Thus, a total temperaturedifference of about 400°C, includingcooling ofthe rotor shaft, is a reasonable maximum value when determiningthehighest possible amount ofshrinkage.

As a second way of mounting rotor shaft and lamination, an adhesiveconnection is taken into consideration (figure 2-9b). This method is, like the

shrink-fit, rather simple. A certain drawback, however, is the fact that most

adhesives will change their properties with time in an uncontrollable way,especiaüywhen the environment conditions are unfavourableas for example inthe case of Operation at high temperatures or in aggressive media. Furthermore,it is knownthat adhesiveconnections behave optimally only under pure shear

stress, whereas, in the case of a rotating rotor, tensile stress in radial directionmust be transferred. Thus for the given stress configuration adhesiveconnections are not ideally suited.

Wrapping the lamination sheets from outside with compositematerials (figure2-9c) is anotherway to avoid loss of contact and is being done increasingly,especiallyin the field of composite flywheels. Glass, Kevlar®or carbon fibers,embeddedin a matrix of epoxy, are wound onto the lamination with a certain

pre-stress high enough to prevent Separationof the lamination from the rotor upto the nominal rotational speed. The calculation of resulting stress in rotor,lamination sheets and anisotropic composite wrapping is not as simple as

shown in section 2.1 for isotropic materials; however, under certain

assumptions, stress and strain can be derivedanalyticaüy [Holden-Day63],[Liu & Chamis 65]. In the next figure, the plots of the radial stress in a wrappedassembly is shownqualitatively for a nominal rotational speed Q and for a rotor

shaft with concentric borehole:

28

Gtlaminationsheets

rotorshaft

wrappinglayer

'

-r

mcreasingpre-stress in compositeloss of contact

Fig. 2-10: Radial stress in a wrapped assembly ofthe internalrotor

type for a nominal rotationalspeed Q

The thickness of the outside wrapping layer is in the ränge of one to severalmiUimeters.Thus, this method is in general not applicable to the internal rotor

type, since the magnetic field is strongly weakened by the large air gap. For theexternal rotor type, however, outside wrapping with composite materials is

perfectlysuited.

A further approach to the design of the rotor-lamination connection is the

forming of the lamination sheets in such a way that loss of contact is

geometricaUy inhibited(figure 2-9d). The clear drawbackof this method is theoccurrence of very high stress peaks formed at the geometric discontinuitieswith the consequence of a high risk of crack development or plasticdeformation. In addition an analytic stress calculation seems to be ratherdifficult, to say the least. However, a finite element approach - not worked out

here - might give more insights into this type ofrotor-laminationconnectionandshouldthereforebe investigatedin future.

The basic idea of the method of the "positive shrinkage gradient" (figure2-9e) is the foUowing: materials and geometryare chosen in such a way thatthe radial displacement of the inner ring (rotor shaft) grows faster with

increasing rotational speed than the radial displacement of the outer ring(lamination sheets). Thus, the two rings will "shrink" into each other with

29

increasing Q (hence the name "positive shrinkage gradient"). Themathematicalformulation for this behaviorcan be found easily. If rm denotes thetransition radius betweenthe two rings and ux and u2 the displacementsof ring1 andring 2 (see equation (2.15)), one obtains:

3 [ui(rm) - u2(rm)] > Q v Q>Q

(2.19)

Using expression (2.15) for the radial displacement u and (2.8) for the

integration constants, equation (2.19) can be brought into the followingformwhich is independentofß:

pi£2 yg(vi+3)+ r^(l-Vi) > xp2Ei r$2(v2+3)+ rm(l-v2)

(2.20)

Setting Poisson'smodulus v to the typical numerical value of 0.3, equation(2.10) simplifies to:

PlE2 3.3r?+0.7rmPiE\ 3.3r22+ 0Jr%

> 1

(2.21)A most simpleand intuitively obvious result is obtained if the influence of the

geometry in (2.21) is neglected.Theinner ring must be of smaller stiffness and

higher density than the outer ring:

P\E2 > 1 or E± < ^P2#l Pi P2

(2.22)The figure below shows the geometric parameters satisfying (2.20). The

inner ring is assumed to be of brass, whereas the outer ring's material is ironwith soft magnetic properties. For an outer radius r02 varyingfrom 20 to 60 mmand for a transition radius rm from 16 to 48 mmthe minimumof the innermostradius rtl is plotted.

m

30

0

Q

m

- ~ *y ¦"¦"-

/ &

9m

** w w ¦"*- >* *¥ /« w '^fmywiWiyi .«i...—y

o2

-*-*>

10 20 30 40 50 60mm

Fig. 2-11: Minimum of the innermostradius rt for the example ofbrass(innerring) and softmagnetic iron (outerring) andforvarying geometricparameters

It is very important to see that (2.20) generally cannotbe satisfiedfor massiverotor shafts (rt =0). Thus, the great drawback of this method is caused by the

property that a "positive shrinkage gradient" is only achievable for a veryrestricted geometry and a well-matchedmaterial pairing.

In conclusion, after considering the different methods of connecting rotor

shaft and lamination sheets, the classic shrink-fit seems to be the most

favourable method, especiallyfor the case of an internal rotor.

In the following section a short survey of the mechanical, magnetic andelectrical material propertiesnecessary for AMB applicationsshall be given.

31

2.2.2 Material Requirements

As it has been found in the previous sections, high yield strength and low

density combined with well matched geometry of rotor shaft and lamination

sheets are necessary for reaching highest possible circumferential speeds(equation (2.18), figure 2-7). In addition, the amount of shrinkage has to be

determinedproperly in order to achieve best material exploitation.Beside these mentioned mechanical qualities, a whole set of additional

(mainly magnetic and electrical) properties must be optimized for AMB

applications. The following table gives an overview of these propertiesand their

most desirable ränge.

propertiesdesirableränge typical values units remarks

mechanicalyield strength

density

Young''s modulus

high

low

high

200 - 2'000

7'000-8'100

150'000 - 210'000

N/ 2/mmz

kg/ 3

7mm2

high speed

high speed

outer ring

magneticmagnetic induction

magneticpermeability

coercive field

Cwn'e-temperature

hysteresis &

high

high

low

high

low

1.5-2.5

1*000 - 200'000

0.01 - 2.0

400 - 950

Tesla

A^cm°C

Saturation value

at 50 Hz

hysteresis loss

application

5-500 /kg at 1 Tesla &1kHz

electrical eddy currentloss

specific resistance

low

high 0.2-1.5 Qmm2/meddy currents

other coefficient ofthermal expansion

high 8- 13 10"6K_1 shrinkage

Fig. 2-12: Materialproperties necessaryfor AMB applications

33

Chapter 3:

HARDWAREDESIGN

3.1 HIGH SPEEDROTOR DESIGN

As already pointed out in the introductionto this contribution (see section 1.3)the design of an AMB high speed rotor underlies various limiting conditions.Consequently, the design procedure is not straightforward,on the contrary: it

basicaüy consists of a ratheriterativeprocess.

The constructionalconceptsfoüowed in this chapter are mainly based on the

ideas presented in chapter 2. An optimal compromise between designrequirementsand limitations (see chapter 1) shall be reached.

3.1.1 Rotor Shaft

The most important mechanical design specifications for the high speed rotor

are the foUowing:- internal rotor type for highest circumferentialspeedspossible- rotor designfor maximum rotational speed12^^= lOO'OOOrpm- five axes active magnetic bearing Suspension- rotor bending stiffness "reasonably" high

(not too many critical speeds to pass through)

34

High strength steel is chosen as rotor shaft material. The axial thrust bearingacts direcüyon the rotor shaft (see section 3.2.2); therefore, the ferromagneticpropertiesofthe rotor shaft material must also be taken into account.

The maximumattainable circumferential speed for a füll ring is given by(2.14). Although(2.14) is not exacüy applicable to a long cylindric rotor withthree-dimensionalstress State, the values derivedfrom (2.14) will not differconsiderably from those for long cylindric rotors (appendix10.2).

To calculate the maximum aUowable circumferential velocity, the yieldstrength of the rotor shaft material is divided by a safety factor 2, so that

possible material imperfections in the center of the rotor shaft are taken intoaccount. This results in the same value as if applyingequation(2.18) to a ringwith a very smaU concentric borehole(rz«0),and neglecting a safety factor.

The following table contains all necessaryinformationleading to the main

geometricdata for the high speed rotor shaft:

rotational speedQ lOO'OOOrpm

yield strengthGq 800^2densityp 7850kg/m3Poissonsmodulus v 0.3

vmaxfullring (2.14) 500nys

safety factor 2

vmaxS(^ety 350nys

maximumdiameter 66mm

Fig. 3-1: Mechanicaldata to determinethe maximumdiameter

of the rotor shaft

35

3.1.2 Rotor with Standard Transformer Sheets

The above calculatedmaximumdiameterof 66 miüimeters wül only occur at

the rotor's thrust disk (axial bearing) which is an integrated part of the rotor

shaft (figure 3-2). At the radial bearings shrink-fit between rotor shaft andlamination sheets is necessary, even if the maximum attainable surface speed isreduced drastically (see figure 2-7).

It is mainly the bending stiffnessofthe rotor shaft which defines the number ofcritical speeds to be overcome when running the rotor up to lOO'OOO rpm.

Although AMB are perfectly suited for reducing resonance amplitudes byactive damping, the numberof critical speeds to overcome shall not be higherthan 3 or 4 for this application (see also section 4.1). Therefore, to meet this

requirement, the overaU thickness of the rotor shaft shouldnot be too small. Forthis high speed rotor a rotor shaft diameterof 20 millimeters is chosen.

axial bearing zone

066 mma——

thrust disk

rotor shaft

= = :_—- lamination sheets

shrink-fit

020mm

Fig. 3-2: Basic geometry ofrotor shaft including thrustdisk

For the chosen rotor shaft the geometry of the lamination sheets (radii ri2 and

r02 respectively) is determinedin the following.

As it was pointedout in section 2.1.2 the maximum circumferential speed at

the surface of the shrink-fitted laminationsheets is attainable for the optimalcase when loss of contact on the one hand and reaching the yield strength on

36

the other hand occur for the same value of the rotational speed Q. This

important property is achievedby properselection ofthe shrinkage radii.

For the given values ofthe maximum rotational speed Ü, (lOO'OOOrpm) and ofthe inner laminationsheet radius ri2 (-10 mm, see figure 3-2), the outer radius

r02 can be obtained by applying (2.18). This results in expression (3.1) denotingoptimalmaterial exploitationregarding stress:

r^J—^l hVLr}p2n\v2+3) v2+3 ll

(3.1)As with the rotor shaft,yield strengthg02 of the lamination sheets is divided by

a safety factor2.

In order to determinethe exact value of the inner laminationsheet radius ri2,the loss of contact condition for the maximum value of Q. must be considered.This is achieved by means of the FORTRANprogram already mentioned insection 2.1.2. FinaUy, the total temperaturedifferenceAT=ATj+AT2 necessaryfor the optimal shrink-fit is obtainedby using (3.2). Au represents the radialdilatation due to heating of the laminationsheets and to cooling of the rotor

shaft, at denote the coefficients of thermalexpansion, and AT{ the temperaturedifferencesapplied to rotor shaft and lamination sheets.

Au = rol - r/2 = - r0l(X\AT\ + ri2a2AT2(3.2)

In the figure below all decision criteria leading to the maximum outer radius

,2 of the lamination sheets in order to i

velocity at the radial bearings are listed up:r02 of the lamination sheets in order to achieve a maximumcircumferential

37

rotational speedQ lOO'OOOrpm remarks

yield strength002 420 ^2densityp 2 8100Vm3PoissonsmodulusV2 0.3

safety factor 2

outer radius r0% 16 mm (3.1)

inner radiusn2 9.98 mm FORTRANprogram

coefficient o^ofthermalexpansion

lllO^K'1 rotor shaft

coefficient a2ofthermalexpansion 9.510'6K-1

temperature differenceAT for shrink-fit 211°C (3.2)maximumdiameteroflamination sheets 32 mm

Fig. 3-3: Decision criteria to determinethe maximumdiameter

of the laminationsheets

Concluding from figure 3-3 the maximum rotor diameterat the radial bearingsdoes not even attain half the value of that obtained at the rotor's thrust disk.Therefore, if Standard transformer sheets for the rotor laminationare used, themaximum circumferential speed at the surface of the radial bearings is

drastically reduced, compared to that for the rotor shaft itself (thrust disk). Thereason for this is the shrinkage necessaryto avoid loss of contact, but which,however, results in high pre-stress in the lamination sheets. The temperaturedifference AT, however, is relatively small (400°C allowed, see figure 2-12),and does not cause any realization problems.

38

3.1.3 AmorphousMetalsin a First Magnetic BearingApplication

Classic metallic materials have one property in common: their atomicstructure is always crystaüine.A process of cooling special molten alloys veryquickly may prevent the formation of a crystalline structure and allows thecreation of amorphousmetals (also caUedmetallic glasses) [Duwez,WiUens&Klement60].

Amorphousmetals [Pfeifer & Behnke 77], [BoU & Warlimont81] have latelybeen subject of increasing interest in various applications, mainly in the field ofelectronics: high power transformators, magneto-elastic sensors, recorderheads, shieldingetc.

MetaUic glasses do not only showvery good magnetic properties, but are alsowell suitable for high speed AMB applications because of their outstandingmechanicalproperties: some amorphous metals have extremely high yieldstrength values up to 2000 N/mm2. Additionally, hysteresis and eddy current

loss can be kept very low. The most interesting material properties for a

specific amorphous metal alloy are listed in figure 3-4 and compared with thesame properties of Standard transformer sheets (for the requirementsof AMBapplicationssee also figure 2-12).

However, two certain drawbackshave to be mentionedif amorphous metalsfor high speed AMB applications are used. Firstly, amorphous metals are

produced in very thin bands (0.02 - 0.05 mm) of variable width due to the

special cooling process. This form makes it somewhat difficult to build largerstructures such as radial magnetic bearing units on the rotor (see section 3.1.4).

Secondly, amorphous metals should not be heated to over 50 to 60% of the

crystaüizationtemperature so that the material propertiesremain unaffected bythe heating process. Therefore, utmost care regarding temperaturesshould betaken when shrink-fitting amorphousmetals.

Nevertheless, the procedure to determinethe maximum possible outer radiusof thin metallic glass lamination sheets is exactly the same as for ordinary

39

transformer sheets (see section 3.1.2). The corresponding data are listed in

figure 3-5.

property value unit remarks value for ordinarytransformer sheets

Saturation induction 1.5 Tesla 2.0

static coercive field 40 «Vcm l'OOO

max. permeabüity lOO'OOO - 50 Hz 50'000

Cwne-temperature 420 °C 950

hysteresis&eddv currentloss 6 w/kg at 1 Tesla &

1kHz 100

specific electricresistance 1.3 Qmm2/m 0.35

density 7100 Vms 8'100

Young'smodulus 150'000 /mm2 200'000

yield strength 2'000 /mm2 420

crystaüizationtemperature

500 °C -

Fig. 3-4: Magnetic, electrical and mechanical properties ofa specificamorphous metal alloy well suitedfor highspeed AMBapplications,and the correspondingvaluesfor ordinarytransformer sheets

As can be seen in figure 3-5, the calculated optimalvalue for the inner radius

ri2 of the amorphous metal lamination sheets is only achievable if a total

temperaturedifferenceof over 800°C is provided for the shrink-fit.It is evidentthat this temperature difference cannot be attained even if the rotor shaft iscooled down by liquid nitrogen or similar cooling agents, a fact which can besummarizedby the general Statement following figure 3-5.

40

rotational speedQ lOO'OOOrpm remarks

yield strengthG^ 2000^2densityp2 ™»ym3PoissonsmodulusV2 0.3

safety factor 2

outer radius r02 39 mm (3.1)

inner radius r;2 9.935 mm FORTRANprogram

coefficient «xofthermalexpansion

ll-lO^K"1 rotor shaft

coefficient a2ofthermalexpansion 8.0-10*K'1

temperature differenceAT for shrink-fit 818°C (3.2)maximumdiameterofamorphousmetals

?

Fig. 3-5: Decision criteria to determinethe maximumdiameter

ofamorphous metal lamination sheets

Optimal shrink-fit, i.e. reachingthe yield strength of the laminationmaterialand loosing contact between laminationand rotor shaft for the same value ofthe rotational speed Q, is normativ impossible if amorphousmetals are applied,since the extremely high yield strength of these materials would lead to an

excessive difference in shrink-fit temperature. Therefore, the maximumattainable circumferential speed at the surface of the laminationsheets is onlylimited by the allowedheating temperature of the amorphousmetals.

However, for the given application, using metallic glasses as laminationsheets still allows a highermaximum surface velocity than applying Standard

41

transformer sheets. Limitingthe heating temperature AT2 to 280°C (56% of the

crystallization temperature of amorphous metals) and cooling the rotor shaft bymeans of liquid nitrogen (AT^ -150°C), the minimum inner radius of the

amorphousmetalsis determinedby solving (3.2) for ri2.

1 + aiATi s 996mm (r0= 10mm)2 1 + CC2AT2 1 1

(3.3)

The corresponding maximum outer radius r02for the reduced shrinkageovermeasure given by (3.3) is again determined numerically solving(2.16/2.16a/2.17)under the condition that loss of contact occurs when the givenmaximum rotational speed Q (100*000 rpm) is reached. This results in the

foUowingvalue:

r^ = 30 mm (FORTRANprogram)(3.4)

Therefore, by applyingamorphousmetalsas lamination sheets for the presentAMB high speed rotor, a much higher circumferential velocity at the radial

bearings is attained(up to 87.5%)than by using Standard transformer sheets:

Vmax t, ,,= r0o Q = 314 m/s'""«¦amorphous metals "2 '>>

'^^transformersheets = V°2 Q = 168 m/j(3.5)

The yield strength (divided by the safety factor) is not yet reached when lossof contact occurs,thus providing an additionalsecurity margin.

However, the obtained result (3.4) for the maximum possible outer radiusshould be relativated considering the technical feasibility: shrinkageovermeasureestablished by the temperature difference (3.2) must include a

certain air gap to make sure that both parts can be shrunk safely onto each

other, before the cooling of the outer part Starts. Hence, a certain reduction ofthe shrinkage oversize can be achievedby providing a slightly higher value for

the inner radiusof the lamination sheets r,2 than givenby (3.3).

42

To conclude: To our knowledge, the present AMB high speed rotor is the firstone built with amorphous metals, whichallow for a 40% increase in maximumcircumferential speed at the radial bearings, compared with Standardtransformer sheets. However, a higherrotational speed of over 80% would bepossible theoreticaüy.

The following geometry is therefore applicable for the amorphous metallamination sheets:

inner radius r; 9.975 mm

outer radius r^ 22.5 mm

max. surface velocityat radial bearings 236 m/s

Fig. 3-6: Innerand outer radius and surface velocity at the radial

bearingsfor the highspeed rotor in AMB, built with

amorphous metal laminationsheets

3.1.4 Shrink-Fit of Thin AmorphousMetal Rings

As it has been already mentioned metallic glasses are usuaüyproduced in thin

long bands of variablebreadth, a form which is not very practical for shrink-fit.In this sectiona way of shrink-fittingthese amorphousmetal sheets is shown.

In a first step the single lamination sheets have to be punchedout ofthe band.Due to the very high hardness of amorphous metals a carbide tipped tool mustbe providedfor punching. This step is ülustrated in figure 3-7.

To preventbuckling of the punched out discs when shrinking them onto therotor shaft, it is necessaryto build an axiaUy pre-'stressed Stack out of hundredsof roundels, depending on the actual bearing length. This unit is shrink-fitted

ontothe rotor shaft as one whole part (figures 3-8 and 3-9).

43

Üü

m mm

*i,>6 tems

Qu*

thin amorphousmetalband

lamination sheetpunchedwith a

carbide tipped tool

0.025 mmtfV£50

Fig. 3-7: Punchingamorphous metal lamination sheets out

ofthe band

screwsto provideaxial pre-stress

bushingcap(high strengthsteel)

stapledamorphousmetallamination sheets

bushing(high strengthsteel)

^ m.KS

^S

J '—]cz=m>

S

^

^C25f J

radial bearing unit shrink-fit removalof the whole of the bushingbearing unit (by lathe tooling)

Fig. 3-8: Radial bearing unit to be shrink-fitted onto the rotor shaftas one whole part

44

m%käimms

Fig. 3-9: Completeradial bearing units before shrink-fit,amorphous metalband and punched lamination sheet

In a last step, part of the bushing is removed by lathe tooling so that theamorphous metal lamination sheets are exposed again. After removal of thelamination sheets buckling is stiU preventeddue to the remaining caps (formerbushing) at both front ends. Note that also the bushing material should have a

high yield strength to resist the stress from centrifugalforces, and that loss ofcontact between rotor shaft and front end caps should not occur before thelaminationsheets separate from the rotor shaft. However, there is no need ofgood magnetic properties for the bushing material. In the present application,the same high strength material has been provided for the bushing as for therotor shaft.

45

3.1.5 Rotor Design Overview

In the sections above, the design of rotor shaft and lamination sheets has been

presented. The remaining rotor parts are the sensor rings and the high speedmotor element.

The sensor rings are manufactured from the same high strength steel as the

rotor shaft is. Hence, the maximum outer radius r0 of the sensor rings is givenby(3.1) or figure 2-7 respectively. Including a safety factor 2 for the yield strengthone obtains:

r0 = 22.5 mm"sensor ring

(3.6)

This result also explains the geometry chosen for the amorphous metal

lamination sheets (figure 3-6): for constructional reasons it is not suitable to

choose different diameters for the different rotor parts (except for the thrust

disk), hence all outer radii of lamination sheets, sensor rings and motorelement

are made equal.

For the present high speed rotor no special motor element was developed.Instead, a commercially availablehigh speed motorelementmatching the outer

radius (3.6) of all other rotor elementsis used. However, there will be no safetyfactor regarding stress in the motor element, and, therefore, at the maximum

rotational speed Ü (lOO'OOO rpm), the yield strength of the motormaterial mightbe reached. In other words, the motor element will be the critical part of the

high speed rotor.

The next two photographs (figures 3-10 and 3-11) show all rotor parts before

shrinkage, respectivelythe finishedhigh speed rotor:

46

ifttffc

Fig. 3-10: Rotor shaft and parts to be shrink-fitted:a) rotor shaft with thrust disk

b) radial bearing units c) sensorringsd) motor element e) additionalpay load

0fi^

*m

*/,v<*>

A£M

Fig. 3-11: Finishedhigh speed rotor with axialbearing -

first application ofamorphous metals for AMB

47

3.2 MAGNETICBEARING DESIGN

3.2.1 RadialBearing Design

As it was shown in the introduction to the present work the most simpleactive magnetic bearing stator basically consists of two electric coils facingeach other. At least three coils are necessaryfor a complete radial magneticbearing in order to generate forces in two perpendicular directions. However,the most common design uses four independent coils. For more details

concerningmagnetic bearing design possibilitiesrefer to [Traxler 85].

For the present high speed application two radial magnetic bearings of the

following type and geometry are provided:

Ss*

r

*s*

io + i(t)

magnetic flux

?x

magnetic force

io - i(t)

Fig. 3-12: Radial magnetic bearingunit with fourpairs of electric coils

For each direction (x,y) two opposite pairs of electric coils are used. A

linearization of the quadratic force-currentrelationship is achievedby applyinga differential current control with the constant pre-magnetizingcurrent i0 and

the actual control current i(t) [Schweitzer & Lange 76], [Traxler 85]. The

following table contains the main characteristics of the realized radial bearings:

48

maximum force per directionmaximum current in each coilnumberof windingsper magnetmagnetic flow cross sectionAmaximum voltagenominal magnetinductivityouter diameterinner diameter

bearing lengthnominal air gap

400 N6A220 (2x110, totally4 magnets)550mm2150V24 mH122 mm45 mm50 mm0.7 mm

The linearized description of the force-current-displacement dependencyofthe radial magnetic bearing is given in section 4.2.1. A measurement of thisbehavioris shown in figure 3-13 for a nominal air gap betweenrotor and Stator

of 0.7 mm, and for a pre-magnetizingcurrent i0 of 2.5 ampere. A photograph ofthe radial bearing unit is shown in figure 3-14.

f N 400.0

200.0

0.0 1 1 1 1 II1 1 1 I

200.0

400.0

0.2

0.10.0

-0.1-0.2

Vmm

-2.0 -1.0 0.0 1.0 2.0 A

Fig. 3-13: Radial magnetic bearing: measuredstatic

force-current-displacementdependencyfor a nominal air gap of0.7mm between rotor and Stator,andfor a pre-magnetizingcurrent i0 of2.5A

49

2i8y^««^v.

^ ¥

Fig. 3-14: Radial magnetic bearingunit

3.2.2 Axial Bearing Design

For the axial direction only one pair of opposite coils is necessary. Since thereis no magnetic reversalduring rotation of the rotor's thrust disk, each of the two

axial magnetic bearing units (upper and lower unit for the vertically suspendedrotor) can be designedas a pot-shapedmagnet, as shown in figure 3-15.

The lower bearing unit (stator) for the present high speed rotor is brought into

its final position before all other rotor parts are shrink-fitted onto the rotor shaft.

Therefore, it cannot be removed from the rotor after shrinkage (see figures3-11 and 3-16). The reason for this special design: if the lower axial bearing unit

shall be removed from the rotor after shrinkage, either a larger value for the

inner diameterof the lower axial bearing has to be provided, or the axial bearingpot has to consist of two separable parts. Both Solutions would lead to a

decreased axial bearing force, and, therefore, the drawback of a non-

removable lower axial bearing unit was accepted.

50

>XN\ \ \\ \ \\ \\coüs ^

Ä

\\ \N \N

iÖS

upper bearing

rotorthrust disk

lower bearing

A

Fig. 3-15: Axial magnetic bearingacting on the rotor's thrust disk

The main data of the axial magnetic bearing are listedbelow:

maximum force 220 Nmaximum currentin each coil 6 Anumber of windingsper coil 200

magnetic flow cross sectionA 460mm2maximum voltage 150Vcoil inductivity 17 mHouter diameter 66 mminner diameter 25 mm

bearing height h 95 mmnominal air gap 0.7 mm

51

msjti

KU*

Fig. 3-16: High speed rotor

with non-removable lower axial bearing unit

3.3 OTHERTEST STAND COMPONENTS

3.3.1 Sensors

Ten eddy current sensors are provided for the contact-free displacementmeasurement. A pair of two sensors for each bearing and for each directionis

used in order to satisfy the collocation condition [Salm 88], and thus to avoid

52

destabilization due to spill-over effects [Salm & Larsonneur 85]. Therefore, foreach pair of sensors the actual displacementsignal is the arithmetic mean ofboth sensor signals. The two remaining sensors are used to measureaxial rotor

displacementand rotor dilatation at high rotational speeds.The eddy current sensors cover a displacementränge of 2 mm. They show

almost linear characteristicsand negligible drift, and are non-sensitive to

external disturbances.

Figure 3-17 shows the mounting unit for the sensors in the vacuumhousing.

HHK

Fig. 3-17: Mounting unit for the displacementsensors

53

33.2 Motor

An asynchronous motor is applied with the present high speed rotor. Asshown in section 3.1.5, the high speed motor element is shrink-fitted onto the

rotor shaft.The main technicaldata are:

nominal power at 60'000rpmfrequency rängevoltageouter diameter (stator)outer diameter (motor element)motorelement lengthnominal air gap

1.5 kW10Hz-3kHz220 V~ (3 phase)80mm45mm44mm0.5 mm

33.3 Rotor Housing Design

All parts for the present AMB high speed rotor are integrated in a specialvacuum-sealed housing. The reasons to provide a rotor Operation in vacuumare mainly preventionof air flow effects acting on the rotor at high rotational

speeds, and reduction of aerodynamic drag. However, the drawbackof vacuum

Operation is the fact that hysteresis and eddy current loss result in increasingrotor temperature which can only be dissipated from the rotor by radiation.Rotor temperatureis therefore continuouslymonitored by means of a contact-

free infrared thermometer.

54

A cross-section of the complete rotor housing including all internal parts isshown in figure 3-18.

Fig. 3-18: Cross-section ofrotor housing (see nextpage):a) rotor housing b) housingcover including

upper axial bearingunitc) vacuum ductfor d) lower axial bearing

electric supplye) high speed rotor f) radial bearingunitg) radial h) motor Stator

displacement sensori) variablepayload j) payloadhousingk) base plate l) axial

displacement sensorm) radiall axial n) vacuumductfor infrared

retainer bearing temperature measurement

o) outside radiationfin p) radial retainerbearingsq) sealing r) inspection glass

55

a)1*2 StIIb

:SsWi.^**,.

s mSc) ™^'

^ ^ iiivwlO^d) ß> ntvv>ss%\

UE&1.VVW*-'

533^JJlV^^.e)

o)^E S^v ? ^* :EV«""" Sg) IS s p)sss^h ^

süE? s^ q)i n

^m

) r

W» ^.^k

^^ TT S3wmz.A '66 @

O mm 100 mm

56

3.4 ELECTRONICEQUIPMENT

3.4.1 Filtering

With regard to the implementeddigital control for the high speed rotor, a

filtering of the displacement signals is necessary in order to avoid aliasingeffects. For each of the ten sensor signals a second order low pass filter of the

foUowing type is provided:

R3

C2Ri R2Uin

u

Ci

/ /

out

Fig. 3-19: Secondorder anti-aliasing lowpassfilter

The second order differential equation of this filter as well as the filtercharacteristic obtained from the settings of the resistances Rx to R3 and the

capacitorsC\ and C2 are pointed out expliciüy in section4.2.2.

3.4.2 SwitchedPower Amplifier

57

bearing mu X¦

KöoiUifc

+

'••••••••¦'¦•¦•¦•¦mf

power FETO

Uo

Fig. 3-20; Schematic circuit diagramofa switchedpoweramplifierincluding magnetic bearing load

Switched pulse width modulated (PWM) current amplifiers are provided for

all five AMB axes. The advantage of switchedpower amplifiers is the fact that,in comparison with Standard analog power amplifiers, loss is reduced

drastically. The switching rate of each FET (field effect transistor) is 125 kHz.

Bearing current i is limited to 6 A. Output voltage U0 is 150 V. Thus,maximumpower for each Channel is 900 VA. Coil inductivityL is 24 mH for the radial

magnetic bearing and 17 mH for the axial magnetic bearing. Coil resistance

Rcoü is approximately1 Q.

3.4.3 Hardware MonitoringSystem

Investigationson AMB failures have shown that tumbling of the rotor (figure3-21 a) after a magnetic bearing drop out has to be avoided by all means, since

very high retainer bearing forces will result [Szczygielski & Schweitzer 85]].This can be achieved by applying a constant force, so that the dropped rotor is

pulied steadily into a determined direction (figure 3-21b). The constant pullingforce can be applied by those magnetic bearings that are still operationable.

58

retainer bearingforce

retainer bearing

constant force ^"~ fapplied by AMB ^--

retainer bearingforce

a) tumblingrotor b) avoidanceof tumbling

Fig. 3-21 Rotor movements andforces after AMB failure

For the present high speed rotor the hardware monitoringsystem (watch dog)realized applies the described constant pulling force in each of the followingcases:

- drop out of Controller hardware (i.e. digitalController)- failure of the sensors

- failure of the amplifier- manual emergency breaking

Furthermore,sufficient energy dissipationin case of an AMB failure can beachieved if an additionalbreaking moment is applied by switching the motordrive to generator Operation.

3.5 DIGITAL CONTROLLERHARDWARE

3.5.1 Application of a Digital Signal Processor (DSP)

In order to make best use of the multitude of outstanding magnetic bearingproperties, a digital control scheme is realized. This also correspondsto latesttrends in AMB technology, as mentioned in the introductionto this contribution.

59

Highest possible sampling rates are a fundamental requirement when

implementing digital control for high speed rotors, since sufficient damping is

necessarywhen the rotor passes throughhigh-frequencycritical speeds.

Basically,the high sampling rates are reachedby two different measures: on

the one hand, the Controller structure is predefined in such a way that a simpleand fast control algorithmresults (decentralization, prescribed low Controller

order etc., see also section 5.2). On the other hand, the Controllerhardware is

adapted to the demandingsof high speed AMB Operation.Instead of a Standard

microprocessora muchfaster digital signal processor (DSP) is used.

3.5.2 A DSP/PC Concept

The DSP system used for this applicationhas been developed at the Institute

of Electronics of the Federal Institute of Technology (ETH) and is

commercially available [Compar 87].

The heart of the system features the Texas Instruments signal processorTMS 320C25, driven by a clock signal of 20 MHz. The DSP boards further

consist of 10 A/D and 10 D/A Converters,a single 16 bit digital I/O port and a

dual port memorywhich enables communicationwith an IBM compatible PC.The boards can be plugged directly into the free slots of the PC.

The PC acts as a Software developing and debugging area. Direct access to

all memory addresses and I/O ports is possible. A special DSP/PCcommunicationSoftware written in TURBO PASCAL is available with the DSP

hardware.

The described DSP/PC system is perfectly suitable for the control of active

magnetic bearings. Apart from the computational power of the signalprocessor, communicationwith the PC facilitates quick access to any systemparameters to adapt the magnetic bearing characteristics. Additionally, an

overaU PC-basedprocess control is possible.

60

A schematicdiagram of the signal processor boards including PC-interface isshown in figure 3-22.

<&,,w,,,,,,,,,w,.

IBMPC

^pcbusj«^

Sz.DigitalSignal

Processor(DSP)TMS320C25

IZ

Jb.? M Dual Port

^ Memory

00D I Digital

I/O PortLZ QT f^Arbiter

PQ 10 A/DConverters

(12 bit/12 us)>00Q

10 D/AJConverters«

(12 bit)

Fz'g. 3-22; SchematicdiagramoftheDSP/PCdigital control hardware

The DSP Software is written in assembler in order to achieve time optimalprogram code and thus to guarantee shortest sampling rates and time-delays.Program structure and a timing diagram of the program flow are presented in

[Siegwart 89].

Finally, figure 3-23 shows a photograph of the processor board as well as of

analog and digitalI/O boardsready to plug into an IBM PC bus.

61

JiBBHlffljß ;|

ü* rpsipKg

KW

*£*

Esä Si

«3

F/g. 3-23; Fexas Instruments TMS 320C25 signal processorboardas well as analog and digitalI/O boardsfor use with the IBMPC bus

3.6 TEST STAND DESIGN OVERVIEW

The next two figures show photographs of the complete high speed rotor test

stand. In figure 3-24, the vacuum housing including rotor, magnetic bearings,sensors and motor is presented (for the detailed description refer to figure3-18). Finaüy, figure 3-25 gives an overview over the whole experimentalSet¬up including electronics rack and PC with integratedsignal processor boards.

•spuvoq jossdooudpuSisSmpnpmjjpun

'(dAUp1010W'Wd?s£sSWUOJJUOIU'SDlUOUJDdJd

Suunsvdutpdddspun

uosuds'jdifijduiü

ddMod)yovd

SDjuoujDdp'Smsnoif

umnovA

tfjiMpuvjs

jsdjuojojpddds

y8iifdjdjdutoj:$ZS '%M

(8ISdunSifosp

dds)uojoupdddsyStifuofSmsnoifumnov\

:pz~£'%}d

ääsrapa

min

IIIWIM

I

^^

M3

63

Chapter 4:

MODELING

In order to analyze the dynamic behavior of the active magnetic bearingsystem and for the designof the controUer a mathematicaldescription of elastichigh speed rotor, magnetic bearings, sensors and electronic components suchas amplifierand filters is necessary.

The basic assumption for this modeling task and the following Controller

designis that a linear model reproducesthe most importantphysical effects ofthe actualplant in a satisfactory way. As wül be shownthroughoutthe foUowingsectionsthe assumption of a linear model is justified for all system components,exceptfor the power amplifierwhich, under certain operating conditions,showsnonlinear behaviorbecause of Saturation effects.

4.1 MODELINGOF THE ELASTIC HIGH SPEEDROTOR

4.1.1 General LinearModel for SymmetrieRotors

The dynamic behavior of numerous rotor types can be describedby linear

time-invariantpartial matrix differentialequations, if the foUowingassumptionsare made:

64

- rotor displacementsare small, comparedwith the rotor geometry- the rotor is rotationaüy Symmetrie (except for smaU unbalances)- rotational speed is constant- material behavioris linear-elastic

If, in addition, the rotor is considered an assembly of rigid elements withelasticjoints, or if a finite element (FE) approach is pursued, the mathematicalmodel wiU result in a set ofordinary,second-order differentialequations.

To obtain the rotor model all later considerations will be based on, some

further assumptionsmust be made:

- torsional vibrationsare neglected- lateral and axial rotor displacements are decoupled- axial rotor movementis left out ofconsideration

(simple rigid body motion)- all external forces act on the rotor at discrete locations- all measurementquantities result from discrete locations

Lateral displacement of an elastic Symmetrie rotor from the equilibriumposition can be described by equation(4.1). The derivation procedure is basedon the principlesof mechanics and will not be described here. For more detailsrefer to [Gasch & Pfützner 75], [MüUer 81], or to any rotordynamicstextbook.

M 0 **+

Di 0 lx+Q

0 G lx+K 0 lX

+QOMJl2x\ L 0 AJl2x\ L-GoJl2x\ Lo k\l2x\

=

.0 2B. i2f\ + Ü2 2fu\\ly) •co lx

[2y\ .0 2c. Ujj

0

¦Di

cos Qt+ Q2

Di0

-% sin Qt

(4.1)

65

The Symbols used in expression (4.1) are the positivedefinite mass matrixM,the positive semidefinite inner damping matrix Z)„ the positive semidefinite

gyroscopicmatrixG, and the positivesemidefinite stiffness matrixK. AU thesematrices are Symmetrie as weU. The column vectors lx and 2x respectivelycontain the rotor displacementsand angular rotations (xt and <pit see figure 4-1)at the chosen i discrete points, and are defined with respect to an inertial frame.

The influence of outer forces (e.g. magnetic bearing forces) is represented bythe column vectors !/and 2f acting on the rotor at the locations determined bythe matrices 1B and 2B. Unbalance forces are labeled by lfu and 2fu, andincrease by the Square of the rotational speed ß. Finally, the Output columnvectors ly and 2y result from measurements at discrete locations and are

characterized by the matrices !C and 2C respectively.

Indices l and 2 in (4.1) mark the rotor movements in two perpendicularplanes. These movementsare only coupled by the gyroscopic effects and bythe non-conservative forces resulting from inner damping. These non-

conservative forces may lead to instability above a certain rotational speed[Müller 81]. However,the rotors considered in this contributionshall showverysmaU inner damping and, therefore, the correspondingdestabilizing effect isassumedto be insignificant.

If symmetry is not only restricted to the rotor geometry but also applied to thelocation of outer forces and measurementsthe influence matrices lB and 2B

respectively lC and 2C wiU becomeequal. Furthermore,matrices 1>2B and 1>2CTare equal for collocated sensors and actuators [Salm 88].

In figure 4-1 some quantities used in expression (4.1) shaU be visualized,withoutconsidering unbalance forces.

66

discrete locationsV©

forces& moments

displacementsangular displacements&

®

xi Q<Pi/*2

q>2

q>3

rotor (M,Di,G,K)

Fig. 4-1: Rotor (M, Dt, G, K), displacementvector example (2x) andforce vector example (2f) correspondingto equation(4.1)

The numericalvalues in (4.1) for the elastichigh speed rotor are obtained by a

FE-modelingapproachwhichwiUbe examinedin the foUowingsection.

4.1.2 Finite Element (FE) Model

The theory of FE-modeling will not be gone into here. One may refer to

[Bathe82] for Standard reference. Various Software packages are available inthe field of finite element modeling. For the present high speed rotor theSoftware package MADYNhas been used, since it is especially adapted to

rotordynamicsapplications[MADYN82].

The first FE-modelingstep is the definition of nodes. The number of nodes

depends on the complexity of the rotor geometry and on the number of

eigenmodesto be reproduced properly in shape and frequency. For the presenthigh speed rotor 31 nodes have been defined.The location of the chosen nodesis given in figure4-2. For each node a set of4 displacementquantities is defined(two translatoric and two rotatoric degrees offreedom, see figure4-1).

67

m mmsmm vm mm ea «nnriiniKi mm

S B S S B S

Omm 100 mm

Fig. 4-2: Partitioning ofthe elastic high speed rotor into30 elementswith 31 nodes (black points):S: sensor locationB: radial bearingforce location

The effect ofthe shrink-fitted lamination sheets (magneticbearing and motor)and of sensor rings on rotor stiffness is the most critical point of FE-modeling.With a high degree of shrinkage shrink-fit leads to increased local rotor

stiffness, whereas,in the case ofa smaU degreeof shrinkage, only the influenceofadditionalmass has to be considered.

A certain experience is needed to determine by what degree a given shrink-fit increases local rotor stiffness. However, a rather exact FE-model can beobtained from an iterative model correction process, if modal analysismeasurements are taken into account (see section 4.1.3). A simple but verysuitable way to consider the stiffening effect by the shrink-fitted parts is to

consider a slightly increased rotorshaft diameter.

The results of FE-modeling in terms of the first 7 eigenmode shapes and

correspondingeigenfrequencies is presented in figure 4-3. A distinction is madebetween rigid body modes of the unsupported rotor (modes 1 and 2), elastic

eigenmodes important as critical speeds (modes 3,4 and 5), and additional

higher frequencyeigenmodes(modes6 and 7) to be included in the rotor model.The mode shapes are calculatedfor the uncoupled system (Q= 0).

68

Ül

S B S S ^B—^S

(\) rigid bodymode of translation: 0 Hz

(2) rigid body mode of rotation: 0 Hz

BB

(3) first elasticeigenmode: 202Hz

(4) second elasticeigenmode: 540 Hz

(5) third elasticeigenmode: 1063 Hz

S B S S B S

(ö) fourth elastic eigenmode: 1735 Hz

@ fifth elasticeigenmode: 2433 Hz

Fig. 4-3: FE-modeled mode shapes and eigenfrequenciesof the elastic high speed rotor (Q=0)S: sensor locationB: radial bearingforce location

69

4.1.3 Model Reduction

In the previous section the FE-modelingprocedure for the high speed rotor

was presented. The resulting eigenmodesand correspondingeigenfrequencieswereshown in figure4-3. As a further result matrices M, Db G, K, l2B and X^C

in the second-ordermodel description(4.1) were obtained (A= 0).

As the numberof defined nodes is 31 these matrices wül be of order 62, and

the total system order in equation(4.1) wül be 124 (4 degreesof freedomfor

each node). Thus the obtained FE-model includes the system dynamics for

each planeup to the 29thelastic eigenmode.

There are some obvious reasons for a subsequentmodel reduction: at first,this high system order is very impracticaland nearly impossible to handle in the

magnetic bearing Controllerlayout and in the dynamic behavior simulations.

Also, the FE-model will not reproduce the higher frequency dynamicscorrecüy. As a furtherreason it is not necessaryto achieve a dynamic ränge ofthe rotor model higher than 2 to 3 kHz due to the low pass characteristics of

sensors, filters and ampüfier and due to the sampling frequency of about 8.3

kHz.

Various model reduction techniques are available at present. In the field of

Vibration controlmosüy modal reduction techniques are favoured [Bucher 85],[Salm & Schweitzer 84], [Matsushita, Bleuler, Sugaya & Kaneko 88]. In the

next step, the reduction procedureshaü be brieffy described:

If one Starts from the homogeneous part of the differentialequation (4.1) inner

damping and gyroscopiceffects are left out of consideration.For each plane this

approach results in a second order differentialequation:

Mx + Kx = 0(4.2)

Equation (4.2) represents the free movement of the unsuspendedconservative elastic rotor. The eigenvectors 0/ resulting from (4.2) are real and

can be included in ascendingorder in the eigenvector matrix 0:

70

0 = L01 , 02,-,031J(4.3)

Both mass and stiffnessmatrices (M, AT) can be brought into diagonal form byusing 0 as transformation matrix. For this transformation the columns of 0 are

normalized so that an identity matrix/ results for the modaüy transformedmassmatrix. The diagonal elements of the transformed stiffness matrix are the

Squares of the eigenfrequencies (Oj (in radians). The following modaltransformation of (4.2) is obtained:

Mx+Kx=0

X = O X

M = 0TM0= I

K = 0TK0 = diag (6$ j = 1...31

(4.4)The reduction now consists in a restriction ofthe eigenvector matrix 0 to the

relevant eigenmodes, i.e. cutting off aU high frequency contributions.This isdone with 7 eigenmodes, including those for the rigid body (see figure 4-3). Thereduced eigenvector matrix 0R is then

0r = [xi,x2,..., Xi](4.5)

Finally, the complete reduced order plant model is given in (4.6). The order ofthe submatrices involved is seven. Note that only the mass and stiffnessmatrices are diagonalized, whereas, in general, neither the reduced inner

dampingmatrixnor the gyroscopicmatrixprove to be diagonal.

71

MR 0

0 MR .

1«.Xr2~L Xr J

+

+D 0IR

o ÄÄ& o

0 KR J

2~L */?

Xr

Mr = 0rtM0r (identity)

Kr = 0rK0r (diagonal)1,2

- /rTUiBÄ = <« ^BUc* = uc^

+ß

+ß

0 GR

V-Gr 0

0 DIR

-D

=

0

0 1%\ -2/J + ß2 JuR

2f. J"r J

\ly] lcR 0 [ %-

2y. .0 2Cr\ . 2S*

IR 0

L Xr

xXr2v„

cos Qt + Q'

DiR = 0k Di 0R

GR = 0&G0R

fuRl% = 4*f.

4.1.4 Model Correction by Modal Analysis

-2fJuRUR J

sin Qt

(4.6)

In order to obtain an improvedset of numerical values for the mathematical

system description (4.6) measurementson the real elastic rotor structure must

be made. A common approach is the modal analysismethod, throughwhich theexact rotor eigenmodes and the corresponding eigenfrequencies can bedeterminedand thenbe used for model correction.

For this purpose radial force impacts are provided along the rotor structure.

Applied force and resulting transient rotor response are measured and

subsequently Fourier-transformedin order to build transfer functions fromwhich the modal parameters (eigenmodes,eigenfrequencies) can be extracted[Ewins 84].

72

In a next step FE-modeling parameters are iteratively updated until thedifferences between measurements and FE-model become tolerably small.

Figure 4-4 comprisesthe final numbers for the FE-modeledeigenfrequencies ofthe present elastic rotor structure obtained after some model correctioniterations.Note that the calculatedeigenmodes do practicaUy not differ from themeasured ones. However, small differences remain between measured andcalculated eigenfrequencies.

FE modeled measuredeigen¬

frequencies [Hz]eigen¬

frequencies [Hz]measured

damping [Hz]

Ist rigid bodymode 0 0 02nd rigid bodymode 0 0 0Ist elasticmode 202 200 1.472nd elasticmode 540 525 4.853rd elastic mode 1063 1062 5.514th elasticmode 1735 1737 10.365th elasticmode 2433 2450 17.94

Fig. 4-4: Calculatedand measuredeigenfrequenciesof the elastic rotorfor the first seven eigenmodes4-1

For the last step of model correction,a very simple approach may be: thecalculated eigenfrequenciesin the diagonal of the stiffness matrix Kr are

replaced by the measured ones. The inner damping matrixDiR - being zeroed

by FE-modeling - is defined as a diagonal matrix containing the measured

damping values (structural damping).This results in:

4 1Eigenfrequencyand damping are the imaginaryand the real part resp. of the corresponding eigenvalue.

73

Kr = diag (arf+S2) j=l..J; coj, 8}measured

DiR = 2 diag (^) y=1...7(4.7)

With these corrected numerical values the calculated eigenfrequencies for

(4.6) will be the same as the measured ones. The eigenmodes will remain

unchanged. Thus, (4.6) and (4.7) represent a nearly exact description of the

dynamics of the non-rotating rotor for the first seven modes. However, the

gyroscopic matrix obtained from FE-modeling will stiU be subject to model

errors, since measuring these effects is very difficult. Furthermore, theunbalanceforces are still unknown (see also chapter 8).

The numerical values for the matrices in (4.6)/(4.7) for the high speed rotor

are expliciüy given in appendix 10.4.

4.2 MODELINGOF MAGNETICBEARINGS,SENSORSAND AMPLIFIER

The overaü system dynamics will be mainly characterized by the rotor andthe digitally controlled magnetic bearing characteristics. However, the anti-

aliasing filters necessaryfor digital control and the amplifier wiU alter these

dynamics significantly. Hence, the overall plant description should be

augmentedby taking both filters and amplifierinto account.

4.2.1 Linearized Bearing Model

The force-current-displacement dependencyfor a rotor in an activemagneticbearing with only one magnetic pole can be described by (4.8), if magneticSaturationeffects are not considered:

74

(4.8)The Symbols in (4.8) are a constant k, coil currentis i, the nominal air gap is s,

and rotordisplacementfrom the nominal air gap towardsthe magnet is x .

By applyinga constant pre-magnetizingcurrent to two opposite magnetsanda control current with different directions to each of the two opposing magnets(see figure 4-5), the magnetic force will dependquite linearly on control currenti and displacements [Traxler 85]:

(4.9)The two constants in (4.9) are the force-currentfactorkt and the "negative" -

i.e. destabilizing - bearing stiffness ks (open loop stiffness). These constants caneither be determinedanalyticaUy or measuredexperimentaUy.

io + i

io - i

Fig. 4-5: Magneticbearingwith constant pre-magnetizingcurrent ioand opposite control current i in order to achievelinearized

bearingcharacteristic(4.9)

A measurement of (4.9) for the high speed rotor is shown in section 3.2.1.Note that, due to magnetic Saturationeffects for high currents /, a differencebetweennominallinear behaviorand actual forcewiU occur (see figure 3-13).

75

The measured magnetic bearing constants ki and ks for both radial and axial

bearings are given in figure 4-6 for various pre-magnetizingcurrents:

radialAMB pre-magnetizingcurrent 1.0 1.5 2.0 2.5 3.0 A

force-currentfactor ki 61 92 122 152 181 »?Anegative bearing stiffness ks 92 207 368 575 828 M'mm

axialAMB pre-magnetizingcurrent 1.0 1.5 2.0 2.5 3.0 A

force-currentfactor ki 19 29 38 48 57 n/anegative bearing stiffness ks 22 50 89 139 200 ^ymm

Fig. 4-6: Radial and axial magnetic bearingconstants ki and ksfor different pre-magnetizingcurrents

4.2.2 Sensoringand Filtering

In section 3.3.1 the contact-freeeddy current sensors used for the high speedAMB test stand were shortly presented. They show a very linear characteristicand negligiblephase lag up to approximately10 kHz.

However,with regard to digital control with a sampling rate of about 8.3 kHz

(see chapter 6), anti-aliasing filters must be implementedfor each sensor. For

this application 2nd order low pass filters were used (see section 3.4.1). Theyare described by the foUowingdifferentialequation:

U,out + *l*2 + *l*3+ /?2/?3 C2Üout +R2R3ClC2Üout = -fltfKi Riin

(4.10)

Resistors Rt and capacitors Ct have been determined in order to achieve a

filter cut-offfrequencyof 3 kHz and only little phase lag below that frequency.The resulting filter transfer function is shown in figure 4-7, and the numerical

76

values for the first order matrix differential equation of (4.10) are given inappendix 10.4.

20 w

.SC3

0

-20101

^ 0«wo

3CO

ja

100

200IO1

filter transferfunction"i i i i i i i "i—i—i—1111

j i i i 11111

i i i—1111

j i i i ' 1111 i i i 111

102 10J

IO2frequency[Hz]

10J

IO4

IO4

Fig. 4-7: Low pass filter transferfunction

4.2.3 NonlinearPower Amplifier Dynamics

The power amplifier hardware was introduced in section 3.4.2. This switchedtype pulse-width-modulated (PWM) amplifier provides current i in (4.9) bymeans of an internal current control loop. The dynamicsof this internal controlloop can be neglected because ofthe high switching rate of 125 kHz .

However, two limitations producing nonlinear effects must be noted. On theone hand, control current i is limited to a maximal value of6 A. Therefore, anyController Output demanding currents above this value will lead to amplifierSaturation which changes the closed-loop system dynamicssignificantly.

Onthe other hand the limitation ofthe currentbuild-up time mustbe taken intoaccount. As shown in figure 3-20, maximum voltage at the coil of the

77

electromagnetis U0 (150 V), if coil resistance is neglected. Thus, the maximumcurrent slope imax is limited by the given values of amplifiervoltage U0 and coil

inductivity L, and can be determinedby the following expression:

•¦max-Uo

(4.11)With an assumed sinusoidal ControllerOutput acting as an input //„ for the

current amplifier,

iin = i sin(fl) i) iin = (Oicos(a)t)(4.12)

the amplifierwiU only be capable to produce the correspondingcurrent Outputiout if the foUowing conditionis met.

(öi<V* ^ i<V$-L eoL

(4.13)In figure 4-8 the nonlinear ampüfier characteristics described are graphicaüy

recapitulated:

out

' maximum currentlimitation

lmax/C0A =

Uolmax-L*lmax

|accurateamplifier[Operationränge

isignal distortion=Irangei

logco020 dB / decade

voltage limitation

?log co

Fig. 4-8: Operationrängefor a current amplifierwithmagnetic bearingas inductiveload

78

Hence, in a well-defined Operation ränge the switched type PWM current

amplifiers in conjunctionwith the magnetic bearing load work without anyremarkable signal distortion nor phase lag, whereas, depending on frequencyand magnitude ofthe current signalto be reproduced, important signal distortionmay occur. Althoughthese overmodulation effects may deteriorate the systemdynamics significantlythey cannot be considered in a linear plant model.Therefore, they can enter the linear controüer designprocess onlyindirectly.

In chapter 8, the consequences of amplifier Saturation on the AMB highspeed rotordynamicswiU be discussed.

4.3 LINEAR OVERALL PLANTDESCRIPTION

As pointed out in the previous sections the linear overall plant model willinclude rotor dynamics (4.6)/(4.7), magnetic bearing characteristics (4.9) andfilter dynamics (4.10).

In most of the subsequent investigations, especially in the Controllerlayout,the gyroscopicand non-conservativecouplings of the rotor movement planes(indices 1 and 2 in (4.6)) wiU not be considered. Therefore, plant dynamicsare

taken into account in only oneplane. Furthermore,for symmetry reasons inputand Output matrices of each plane in (4.6) are assumed to be equal and,therefore, no indices will be necessary. Finally, unbalance forces are not

included.

Thefirst order State space descriptionfor the overaU plant

x = Acx + Bquy = Ccx

(4.14)

79

with plant State je, continuous-timematrices Aq,Bc and Cq of order 22 (foursensors), vector of magnetic bearing currents u, and filter Output vector y wiUthen result as foUows4-2:

xr

xr

-Xfilters.

AC =

0

-Mr{Kr -BrBr ks] -MRÜiRBfiUer^R 0

BC =

0

MrBrh0

Cc = [ 0 0

0

0

Afiiu

'filters

ers

]

(4.15)

The corresponding discrete-time State space description with the systemState xk at the sampling times kT

Xk+\ = AoXk + Bd Uk .

yk = CDxk Xk = xikT)U (kT-T)

(4.16)wiU showthe matrices AD,BD and CD given in expression (4.16), if zero order

hold (ZOH) elements at the plant inputs and a time-delayTd are assumed

[Ackermann 85].

-nrhe arithmeticmean of the sensor signalsneighbouring each magneticbearing is taken (see section 3.3.1).

80

AD =

ArTeAc

0

T

/T-Td

eAcTBcdr

0

T-Td

eAciBcd%

BD= ; cD=[ cc o ]

(4.17)The realizedsampling timeT and time-delay Ta are 120 p.s and 36 p.s.

A value of 2.5 ampere has been chosen as pre-magnetizingcurrent for the

magnetic bearings. The corresponding values for ki and ks are shown in

figure4-6 (see also appendix 10.4).

81

Chapter 5:

SPOC-D:A NEWAPPROACHTO THE DESIGNOFDISCRETE-TIMEDYNAMIC OUTPUT FEEDBACK

As it was already pointed out in the introduction of this work, rapidlyincreasing interest in digital Controllerrealizations for mechatronic Systems isnoticeable. This trend has not left out the field ofAMB applications.

However, specific problems wül occur when discrete-time control schemesare applied to active magnetic bearing Systems.

This chapter aims at listing up some ofthe mostimportant difficultiesof digitalcontrol in conjunctionwith AMB applications,and to propose practical andfeasible Solutions. A new layout method (SPOC-D)for structuraüy constraineddiscrete-timecontroüers wiUbe presented.

5.1 DYNAMIC OUTPUTFEEDBACKVERSUSOBSERVER-BASEDSTATE FEEDBACK

A typical property of active magnetic bearing Systems is that only very few

system states, most often the displacements in the magnetic bearings, are

measurable.This is the case especiallywith flexible structures where the plantorder is muchhigherthan the number ofOutput signals available.

82

A further fundamental property of such Systems is the open-loop instabilitydue to the negative magnetic bearing stiffness that results from the pre-magnetizing current necessaryfor linearization (see section 4.2.1). In additionto this destabilizing effect, all system poles corresponding to nie elastic

eigenmodesare located on the imaginary axis, if inner damping ofthe structure

is neglected. These poles must therefore be shifted to the left half of the

complexplaneby sufficient damping generated by the magnetic bearings.

A well-known control theoretical consequence resulting from too fewmeasurable Output signals being available on the one hand, and from the

necessity of achieving enough outer damping up to high frequencies on theother hand, is that an observer-basedState feedback should be implementedincluding the fuü system dynamics. Furthermore,high sampling rates must berealizedin case of digital control(Shannon theorem).

However, with today's microprocessortechnology, such high sampling rates

for high order Controllers (observer structures) cannot be achieved, unless a

sophisticatedmulti-processor system is provided for this complex control task.Furthermore, observer-based State feedback Controllers do generally not

guaranteethe same robustness properties as direct State feedback Controllers.Thisusuaüy leads to stability problems if the plant model is not known exactly,or if the plant parameters change during Operation (e.g. gyroscopic couplingsfor fast rotating rotors)

Consequently, simpler low order discrete-time dynamic Output controlschemes must be sought that show satisfactory dynamic and robustness

properties, even in the high frequencyränge, and which are thus well suitableforAMB Systems.

5.2 PREDEFINITIONOF CONTROLLERSTRUCTURE

Many Controller design methods presently available do not feature one

specific property of practical interest: they do not allow Controller structure

83

constraintsto be specifiedprior to the Controller layout. In the case of classicobserver-basedState feedback for example, the resulting controüerstructure isan integral part ofthe controUer layout method, and is thereforenot changeable.The order of such Controllers is given by the consideredplant, and high order

plantsthus lead to high order Controllers (LQG,H^, pole placement etc.).

For the Controller layout methods mentioned, reduction of the Controller

complexity is only made possible by either a previous model reduction or by a

Controller order reduction subsequent to the Controller layout process (seefigure 1-2).

Many of these model or controUerreduction techniques can be consideredsocaüedopen-loop reduction methods. Thismeans that the stabilityof the closed-

loop system obtained cannot be guaranteed to be an integral part of thereduction method, buthas to be confirmedby additionalinvestigations.

However, there are so-calledclosed-loop reduction methods available which

yield the stability of the closed-loop system in the first place: on the side ofmodel reduction techniques with subsequent Controller layout, a method

avoidingthe destabüizing spill-over effects has been proposed by [Salm 88]. Aso-called hyperstability approach has been investigated by [Bals 89]. Acontribution to the stability guaranteeing Controller reduction has been pointedout by [Brunner88].

In the following considerations, a completely different type of Controller

complexity reduction, a so-called directdesign method (see figure 1-2) will belooked at:

Order and structure of the discrete-timecontroUerfor the nominal high order

(i.e. not reduced)model of the plant are predefined prior to the Controller layout.This predefinition can be made according to practical needs (physicalconsiderations, sampling time, decentralization, symmetry etc.) under thecondition that stabilizing sets of controUercoefficients exist. In the subsequentcontroUerlayout, a parameteroptimization is performed taking into accountthe

dynamics of the high order plant as weU as those of the low order controUer and

yieldingstability of the resulting fuU order closed-loop system.

84

The next two sections shortly present some frequent concepts of Controllerstructure predefinition.

5.2.1 Decentralized Controller of Predefined Order

Decentralizedcontrol schemes are often used if the plant consideredcan bedivided into several Subsystems. TheseSubsystems generaüy do not have to be

decoupled [Senning79].

In the field of AMB applications, there may be another motivation fordecentralized control: if the magnetic bearing is regarded as an active

representationof a classic spring-damperelement, its characteristic is achieved

by a decentralized feedback using only the local rotor displacement[Bleuler84].

A discrete-time description of such a decentralized feedback law of order n

and a block diagram of the correspondingcontroUerstructure are given below:

Uk = doyk + di yk-i +... + dn yk-n- C\ Uk-\ -

...

- Cn Uk-n

U,?o ?o ^9-^

(y (z) © ©-^4^TJ a

5LJL9

(5.1)

Fig. 5-1: Block diagramofa decentralizedHR compensator(feedback-canonical form [Ackermann 851)

85

The discrete-time «th-order transfer function (z-transform) g(z) of (5.1) is

givenby expression (5.2).

o(z) = M(z) = dp + d\z-1 +... + dnz-ny(z) 1+ciz-1 +... + cnz-n

(5.2)In expression(5.1), uk denotesthe ControllerOutput and yk is the Controller

input (a measurable Output of the plant). The coefficients dt and c, are thecontrol parameters.

DynamicOutput compensators ofform (5.1) are also caüedHR compensators(Infinite Impulse Response) in filter theory, since the Output sequence uk to an

initial input impulse will not reach the steady State value of zero. However,if aüinner feedback coefficients ct equal zero, the response sequence to an initial

input impulse shows a dead-beat characteristic. In this case, the result is a

FIR compensator(Finite Impulse Response).

5.2.2 Observer Structure based on a ReducedPlant Model

Another type of Controller structure predefinition with the aim of orderreduction is the implementation of a dynamic observer based on a reducedorder model ofthe plant5-1. A block diagram of the resulting closed-loop systemis given in figure 5-2.

In the next section, traditional layout methodsfor the structurally constrainedcontroüers mentionedabove are described shortly.

- This definition does not correspondwith the Standard reduced observer. The latter one is based on a

füll order model of the plant, but its order is reducedby the numberof Output signals available.

86

high order plant

uk

xfc*ii * x«B«^/*n >~ t -1 __£»*, p

H H<*S *** <?*««.

%*tB CIz

pf-nmmni

A

KltaW9SS^»«MSÄ«»«ttS»SS*W»»SB

nadboNtonkrobserver

Fig. 5-2: Block diagramofa closed-loop system with highorderplant(A, B, C) and reduced order observer(AR, BR, CR, H, K)

5.2.3 TraditionalLayout for Structurally ConstrainedControllers

5.2.3.1 Discrete-TimePD Approximation

Decentralizeddiscrete-time Controllers of the type introducedin figure 5-1are most often layed out by a so-called quasi-continuous approximationofwell-knowncontinuous-timedecentralized Controller structures. In the field ofAMB applications, approximations of continuous-timePID Controllers are a

very familiärapproach.

An exampleof the most simple PD algorithm uses the difference sequence oftwo consecutive Controller input values to approximate the non-measured

velocity signal. From this, the followingfirst order Controller with samplingperiod T results:

87

uk = (p + Ll)yic-D7yk-i(5.3)

In order to improve the system Performance, especially in the presence of

high frequencyplant modes to be damped, a PD approximation ofhigherordermight be used for the approximation of the velocity signal. A second order

examplecan be given as foUows:

uk = (p+ ^)yk-^yk-i+ j^yk-2(5.4)

An overview of decentralized discrete-timePD Controllerapproximationsofvarious Orders is found in [Hoflmann87].

There are two fundamental drawbacks of this type of Controller

implementationandlayout.

Firstly, the plant dynamicsare not involved in the controUerlayout process.Contrarily to continuous-time PD implementations, for which, at least in case of

AMB applications, the resulting closed-loop system wiü be stable [Salm 88], thediscrete-time PD approximation cannot guarantee stability of the closed-loopsystem (see examplein section 5.3.3).

Secondly, it is important to see that the classic discrete-time PD

approximation sets all inner feedback coefficients c, of (5.1) to zero,

independenüyof Controllerorder. Therefore, more general IIR compensatorscannotbe achievedand considerable designfreedomis lost.

5.2.3.2 Observer based on a Reduced Plant Model

The Standard open-loop layout method for this type of reduced orderobserver (see footnote5-1) is to determine the feedback matrices H and K

(figure 5-2) based onlyon the reducedsystem matrices AR, BR, CR by applying a

LQG or pole placement method. However, this procedure generally does not

88

guarantee stability of the closed-loop system due to the resulting spill-overeffects (an examplefor this can be found in [Seto & Takita 89]).

Nevertheless, it is possible to determinethe feedback matrices H andK with

a direct design Controller layout method involving both füll order plant andobserver State variables, and thus yielding stability of the closed-loop system.Such a method is presented in the foUowingsection.

53 OPTIMALCONTROLLERLAYOUTWITHSPOC-D

SPOC-D (Structure-PredefinedOptimalControlfor Discrete Systems) is a

direct design method to obtain optimal low order Controllers for high order

plants (see figure 1-2). For a linear Controller with arbitrarily chosen

structure5-2, the method optimizesfeedback coefficientsstarting from a sub¬

optimal initial guess5-3. Although this method was first developed with the

backgroundofmagnetic bearing applications, it is also suitable for other controltasks where the problemof Controller order reduction occurs.

Mainfeatures ofSPOC-D:- Controller order and structure predefinable (yieldingpractical needs)- determination of an optimaldiscrete-timedynamicOutput feedback

by minimizationof a quadraticPerformanceindex involving bothplant and controUer State variables

- analytical descriptionofPerformance indexand correspondingvector-gradientaUowing an efficientnumericalminimizationprocess

- optimization including linear or non-linearparameterconstraints(e.g. desired stiffness, symmetries,frequencydomain demands etc.)

The method was first presented in [Larsonneur & Herzog 88] and

[Larsonneur& Bleuler 89]. A detailed descriptionof the method is given here.

-However, the given plant must be stabilizable with the chosen Controllerstructure.

5 3A practical approach for finding a stabilizing initial guessis described in appendix 10.5.

89

53.1 Optimal Dynamic OutputFeedback

Continuous-time static and dynamic Output feedback optimization has been

subject of quite a number of investigations in the past two decades. Veryimportantworkhas been done by [Levine& Äthans 70], [Kosut 70], [Senning79], [Ly 82], [Bleuler 84] and [Nuss 87].

However, little work has been done in the field of discrete-time dynamicOutput feedback optimization. Here, SPOC-D shaU providea feasible approachallowing practical requirementsto be included in the controUer optimizationprocess, as for example a freely choosable Controllerstructure or the control

parameter interdependencies mentioned.

5.3.1.1 QuadraticPerformance Index for Structurally ConstrainedSystems

The linear discrete-time State space description of the overall plant (4.16) is

starting point for all subsequent considerations.This State space description is

augmentedby the controUer State, thus the vector xk in (5.5) contains both plantand controUerState variables:

m

XM = Axk+ Y, bi Uiki = \

yik = CiXk i = l.../n

(5.5)In equation(5.5) the global system matrixA describes the dynamic behavior

of plant and Controller. It also includes the predefined structuralinterconnections between (high order) plant and (low order) dynamiccompensator. The vector of the control inputs Uk in (4.16) is replaced by the

equivalent sum of m scalar control inputs uik, the so-called control stations.The m column vectors ft, determine the influence of each control Station on

plant and Controller dynamics. To each of the m control stations a specialObservation Stationyik with matrix Ct is assigned which contains measurable

plant Output signals and Controller State variables. Therefore, each of the m

90

control paths is a linear time-invariant combination of different sets ofmeasurable plant and controUerstates. This type of controUerpredefinitioncanbe regardedas a multiple control structure constraint if the definitiongiven in

[Kosut 70] is applied. Moreover, it is assumed that the dynamic system (5.5) is

completelycontroüable and observable.

Note that the somewhat unusual system description (5.5) is not restrictive.Itcan be applied to any given linear discrete-timedynamic system.

The structure-predefined control loop is closed by assigning a single rowvector di to each of the m control stations containing the unknown feedback

parametersthat are subject to optimization:

Uik = diyik = didxk i = l...m

(5.6)

If for example a decentralized feedback structure, as introducedin (5.1), is

predefined, the elements of the feedback vector d are coefficients dj and Cj ofthe nth order IIR compensator, and the matrixC provides the correspondingplant Output and Controller State variables.

The Performance index PI to be minimized for the closed-loop system(5.5)/(5.6) is formulatedas a quadratic form ofthe global State vector xk and the

scalar control inputs uik, involving the Symmetrieand positive semi-definite

weightingmatrixQ and the weightingfactors rt > 0:

m

PI = X (XfßX* + X M£r*'«k)k=0 *=l

m

= X *£(ß + X CfdfndiC^Xk -> min*=o *=i

(5.7)

The infinite sum in expression (5.7) exists if the dynamic system (5.5) is

stabüizedby feedback (5.6) (see appendix 10.5).

91

5.3.1.2 ConditionsNecessary for Optimality

In this section, an analytical description of the vector-gradient for the

Performance index (5.7) is found, and the conditions necessaryto minimizePIcan be formulated.

At this point, it must be noted that SPOC-Dprovides conditions that are

necessary for optimality. Hence, it cannot be guaranteed that a Performanceindex minimumfound is a global one. The answerto the question if only oneglobal minimum or if several local minima exist is obviouslynon-trivialanddepends on plant model and Controller predefinition. No proof concerning this

specific problemwiUbe providedin this contribution. However,no indicationof

multiple minimahas occurredin practice for the control problems investigatedup to now.

In a first step, an alternative formulationfor the Performance index (5.7) hasto be found that replaces the State vector xk by the (non-zero) initial State vector

x0 and the global closed-loop matrixAdsd:

xk+\ = Acisdxk => xk = A^sdXo ; *0*0m

Adsd = A + £ bi di Cii = l

(5.8)

Thisresults in the foUowingnewexpression for the Performance index:

PI = 4 X (ALdf(Q + % Cj'd!ridic}iAkcisd\ xo

(5.9)In order to simplify expression (5.9), the new matrices P and X0 are

introduced withthe foUowingdefinitions:

92

m

p = X (Acisdf\Q + X cidfndiCi\Akcisdk=0 V '=1 '

Xq = xox1(5.10)

Applying the trace Operator to (5.9) and using the definitions (5.10), one

obtains the simpleresult for the quadratic Performance indexPI:

PI = trace (P X0)(5.11)

In order to determine the vector-gradient corresponding to (5.11), small

perturbations are applied to the feedback row vectors d^ This procedure leads

to the perturbed Performance indexPl+dPI,and finaüy to the vector-gradient:

^ = 2{bIPAcM + ridiCi)XClddi

(5.12)

The exact procedure leading to expression(5.12) which contains the newlyintroduced matrix X is described in detail in appendix 10.5 (see also

[Geering76]). However,the final result of the past steps in terms of conditions

necessaryfor the minimum of the Performance index (5.7) is presentedin the

set of equations (5.13). The essential part of (5.13) are two Lyapunov equationsfor P and X which have to be solved simultaneously in order to computePerformance index and correspondingvector-gradient.

5.3.1.3 AdditionalParameter Interdependencies andModified Optimality Equations

SPOC-D is able to directly include parameter interdependencies in the

Controller optimization process. This ability is mainly motivatedby the fact that

achievingyielded closed-loop properties indirectly by an appropiate choice of

weightingmatrices in the Controller layout is rather difficultand not suitable for

practice. Important use of this SPOC-D-feature to include parameter

93

interdependencies has already been made in the AMB milling spindle project[Siegwart 89].

m

Adsd = A + ^ bidiCii = \

m

ALdPAdSd- P + Q + X cldfndid = 0i=\

Xo = XoXq

PI = trace (P X0)

AdsdXAdsd - X + Xq = 0

dPI= 2{bfPAdsd + ndiCijXCf = 0 i = \...m

ddi(5.13)

Conditionsnecessaryfor the minimum of the Performanceindex

Some practical aspects leading to additional constraints for the control

parametersin the feedback vectors rf, can be listed as foUows:

- postulationof specific magnetic bearing propertiessuch as e.g.

bearing stiffness- postulationofequalityof independentdecentralizedfeedback paths

(e.g. in presenceofplant symmetries)- realization of a specific frequencydomainbehaviorof a

decentralized feedback path (e.g. for noise rejection)- definition ofparameterinterdependencies in case of

overparametrizedControllerstructures

Such additional parameter interdependenciesare expressed by equationswhichmust be satisfied by the elements ofthe feedback vectors dt in (5.6), andwhich lead to modified expressions for Performanceindex and necessaryoptimalityequations.Note that no special numericalalgorithms for minimization

94

under constraints is necessary to include parameter interdependencies.However, the functions expressing parameter dependencies must bedifferentiable.

The way to introduce parameter constraints is to separate the m controlstations in expression (5.5) into mt independent and m; dependent controlstations uik and üjk. The correspondingdependent and independent Observationstations are then yik and yjk. Thus, the system description (5.5) is brought intothe foUowingnewform:

Xk+l = A Xk + X bi Uik + X bJMJk

yik = Q*k ; » =i'-wt#* = c7^ ; j=i..jnj

(5.14)In analogy to (5.6), the control loop for (5.14) is closed by a structurally

predefined Controller attributing independent feedback vectors dt to the

independent control stations and dependent feedback vectors fj to the

dependent control stations. As parameters, the mj dependent feedback vectors

fj contain linear or nonlinear functions of the residualindependentelements ofvectors dt:

Uik = diyik = diQxk i = l...m;

Üjk = fjVi* ^k = fj Cj *k J = L"m7(5.15)

For the modified Performance index the foUowing expressioninvolving the

dependent feedback vectors,/j can be obtained:

PI = X 4(Q + X CfdfridiCi + X c[f[fjfjCj)xk -> mink=0 i=1 7=1

(5.16)

95

The procedure to determine the modified optimality conditions for this

stracture-predefined control problem including parameter constraints is

basicaüy analogous to that outlined in the previous section. Concerningthevector-gradient of the Performance index, the derivativesof the elements infjwith respect to the independent control parameters in di must be includedasadditional terms. This explains the requirement of differentiable functions

governing the parameter interdependencies. The most importantmathematicalsteps leading to the modified optimalityconditions below are described in detailin appendix 10.5.

m,- "*>_ _

Acisd = A + X bi di Ci + X bjfj Cji=i 7=1

m'1 "*'_ _

ALdPAdsd - P + Q + X Cldfndid + X cffffjfjCj = 0i=l 7=1

Xq = *o*0

PI = trace (P X0)

Adsd XAdsd - X + Xq = 0

(bfP Adsd + n di G)XCf+J, [bjP Adsd+ fjfj C7)Xcj%Lj=i odi

= 0 ; / = l...m,(Hü

Conditionsnecessaryfor the minimum of the Performanceindexincluding additionalparameterconstraints

96

5.3.1.4 NumericalOptimizationProcedure

The necessaryoptimality conditions (5.13) and (5.17) are coupled and highlynonlinear.An analytical closed form for the minimumof the Performance indexPI does not exist. Therefore, a numericalsolving algorithm is necessary.

The Davidon-Fletcher-Powell(DFP) numerical minimization algorithm, a

special Quasi-Newton method, is applied to solve the given problem[Luenberger 84]. This powerful nonlinear programming techniquerequires anadditional one-dimensional minimum search algorithm which has been

implemented as a cubic fit. The DFP method makes use of the analyticaldescriptionof Performance index and vector-gradient,and it iteratively Updatesa guess for the Hessian matrix.

The numerical minimizationprocess for SPOC-D has been embeddedin the

very efficient and well-knownmatrix-handlingSoftware packages CTRL-C®and MATLAB® respectively. Figure 5-3 briefly shows the program flow

necessary to solve optimalityconditions (5.13)/(5.17):

During optimization, stability of the closed-loop system is checked byregardingthe eigenvaluesof the closed-loop matrix at each parameter pointwhere Performance index and vector-gradienthave to be evaluated.

The optimization process includes a non-negligible off-linecomputation effort

and may therefore, depending on the machine, require several minutes of

computation time. However, for well posed and well scaled problems no

convergence difficultieshave arisenup to now.

The problemoffinding a stabilizing initial guess for the controlparameterscanmosüy be solved using the foUowingpractical approach: a "shrink" coefficient ais introduced which slighüy modifiesexpressions (5.13) and (5.17) in order to

move aü eigenvalues of the closed-loop matrixAdsd into the unit circle of the

complex plane. By successive reduction of this "artificial" stabilization after

each optimization step, a parameter set which stabilizes the original plant can

be found and then be used as an initial guess for further optimization(seeappendix 10.5 for more details).

97

step 1

step 2

step 3

step 4

step 5

Determine stabilizing set of controlparameters(see also appendix 10.5 for the determination ofastabilizing initial parameterguess).

EvaluatePerformance indexand vector-gradient.Determine direction of steepest descent.

One-dimensionalminimumsearch (cubic fit) alongpreviously specifieddirection. Check closed-loopstability during minimumsearch.

Terminateprocess if gradientnorm smaller than a

specifiedtolerancevalue.

Else:UpdateHessianmatrix, determinenewdirection ofdescent (not steepest descent) asprescribed by theDFP method, and returnto step 3.

Fig. 5-3: Programflow to solve optimality equations (5.13)1(5.17)

The efficiencyofthe DFPnumericalminimizationmethod in conjunction withthe SPOC-D controUerlayout method is shown in the next section along with a

simple example. The application of SPOC-D to the high speed rotor in active

magnetic bearings is presented in chapter 6.

53.2 Application Example: a Simple Elastic Rotor

Figure 5-4 shows a simple AMB rotor system: a Symmetrieelastic rotor isassumed to move only in one plane. The order of the continuous-timeState

space description is eight (three translatoric, one rotatoric mechanical degreeof freedom). Two Output signals, i.e. the two rotor displacements in the

magnetic bearings, are available. The physical input signals are the electriccurrents in each AMB.

98

The physical plant parametersand equations of motionare given in appendix10.6. However, in order to give an idea of the dynamicbehaviorof the system,the open-loop eigenvalues of the plant including the destabilizing negativemagnetic bearing stiffness ks (see section 4.2.1) are noted shortly:

Ai>2 = ±47.540 + Oi [Hz]A3,4 = ±66.407 + Oi [Hz]^5,6= 0 ± 91.069 i [Hz]A7)8 = 0 ± 228.18 i [Hz]

(5.18)The discrete-time plant description for the given example correspondingto

(4.16) is presented in appendix 10.6. A value of 1.2 kHz is chosen for the

sampling frequency T, which is about 5 times higher than the highest planteigenfrequency. Time-delay is neglected.

The plant description is augmented by two decentralized dynamic HR

compensators of only first order, and can be described by (5.14). Note that a

Standard reduced observerwould be of order six and fully coupled. Therefore,the chosen controUerstructure predefinitionresults in a low order Controller fora high order plant. The numerical values correspondingto (5.14) are found in

appendix 10.6.

Each decentralized compensator can be described by (5.1) with the unknown

control parameters do, dj andcj. For symmetry reasons, both decentralizedcontroüers are described by the same coefficients.

As it is shownin figure 5-4, a total numberof 5 control stations is assigned to

the given system: two control stations are the physical inputs for the plant (ulk,ü2k), the inner feedback paths of each decentralized compensatorare definedas two further control stations (u2k, Ü3k), and a final control Station U\k is the

feedback path involving the feedback coefficient d0 of only one dynamiccompensator. The steps leading to these definitions are explained in the

following paragraphs.

99

m

activemagneticbearing (kiJes)

L,EI M,(

^bearingl/^plant

electriccurrent

<P

0

displacementsensor

m

\L,EI

O bearing2 /,ss y y v y s s s f .

\*Si2k

T

Vi* CdIst orderHR compensator

y2CtTi

yiicdUl

ZOH

Y3 U3Cl T^x N,ctr2ZOH

1v2U2 Ist order

HR compensator

Fig. 5-4: ElasticAMB rotor example with two decentralizeddiscrete-timeIst order HR compensators(samplingtime T, zero order holdZOH)

Only two control stations feeding back measurable Outputs from the

augmented plant are independent: the physical plant input ulk with feedbackcoefficient dj and the inner feedback path u2k with corresponding feedbackcoefficient cj. However, the feedback path ü\k involving coefficient d0 is

dependent, since a certain static bearing stiffness kst must generally be

prescribed for AMB Systems. This leads to a nonlinear interdependencyof thefeedback parameterswhich can be formulatedby simpleapplication of transferfunction (5.2) to the decentralized Controller in figure 5-4. The frequencyvariable z must be set to 1, since the static transfer behavior is considered.

100

Thus, one obtains the foUowing expression for the static bearing stiffnesscondition:

1 + c\ kt \ ki )

(5.19)

As mentioned before, the feedback parameters of the second decentralized

Controller must equal those of the first controUerfor symmetry reasons. Hence,the corresponding control stations u2k and üik must also be defined as

dependent control stations.

In order to summarize the chosen control structure constraints, the two

independent and the three dependent control stations for the given examplecorrespondingto (5.15) are formulatedin expression (5.20) below. Dependencyd0(d1,c1) is givenin (5.19).

u\k = d\ Ci xk = [di] C\ Xk

U2k = d2 C2Xk = [ci] C2 Xk

üik = fM)CiXk = [ d^dud), - doci ] Ci xt

Ü2k = f2(di)C2 Xk = [ d<£di,ci), di - doci ] C2 Xk

Ü3k = MdaCixk = [ci] C3 xk

(5.20)

Matrices C, and Cj in (5.20) are given numericaüy in appendix 10.6. Thederivation ^fj/ddi of the dependent feedback vectors fj with respect to the

independent control parameters necessaryfor the optimality equations (5.17)can be derived from (10.45) in appendix 10.6.

The next two sections show the Controller layout results for the givenstructure-predefined Controller. A SPOC-Doptimized Controlleris comparedwith a Controller layout achieved by using a Standard discrete-time PD

approximation approach.

101

5.3.2.1 Controller Layoutwith Discrete-TimePD Approximation

The first order discrete-time approximation of a continuous-time PDcontroUeris givenby (5.3). Note that this approach automaticaUysets the innerfeedback coefficient Cj to zero, from which a FIR behaviorresults (see section5.2.1). After selection of the static bearing stiffness kst (5.19) there wiUbe onlyone freely selectable control coefficient for the adjustment of the dampingparameter D.

step responsesfor varying damping D

<sx

time<$X

time

<5X

time

9

<$X

time

Fig. 5-5: Closed-loop step responsesbasedon discrete-timePD approximationfor a unitforce in bearing1 andforvarying dampingparameterD

102

Figure 5-5 shows the first 50 miüiseconds of the closed-loop step responsetoa unit force applied in bearing 1 at a varyingdamping parameter D. A plot foreach of the three mass displacements (jcj, x2, x3) and for the angle (p of thecenter disk is shown. The static bearing stiffnesskst is set to 75'000 N/m.

As it can be seen clearly from figure 5-5, the closed-loop behavior for a PD

approximated decentralized first order Controller does not show satisfactoryresults for any choice ofthe damping parameterD: forsmaUvalues ofD there isnot enough damping for the low frequency rigid body modes, whereas largervalues ofD tendto destabilize the 2nd elasticmode of the rotor (rotatoricmodefor center disk). However, muchbetter results can be achievedby performing acontrol parameter optimization for the structure-predefinedcontroUerusing theSPOC-D method.

5.3.2.2 SPOC-D Optimized ControUer

The SPOC-D method is now used to optimize controlparametersdQ, dj and cjin the present example. The static bearing stiffness condition (5.19) is includedas additionalcontrol parameterconstraint in the optimality conditions necessary(5.17). Weighting matrices Q andX0 are chosen to be diagonal. Small values are

attributed to aü weightingfactors rt and fj ("cheap control", see appendix 10.6).

The step response for the same unit force and static bearing stiffness as

shown in figure 5-5, but for the optimizedset ofcontrolparameters, is presentedin figure 5-6 and compared with a "good" result from the PD approximationapproach.

The control parameters obtained by optimization differ significantlyfromthose resulting from a discrete-time PD approximation: the inner feedback

parameter cj is no longer zero. Instead, it turns out to play an importantrole in

improving overallsystem Performance.

103

-5

15Xlv.x

\.::«!\

S 10 /; :

=t

X3

Phx2

-o

SPOC-DPD

0 10 20 30 40 50time [ms]

SPOC-D optimizedcontrol coefficients:

do = -1.8259 -IO4di = 1.3655 • IO4Cl= 8.4155 -10"1

Fig. 5-6: Closed-loopstep responsesbasedon a SPOC-D optimizedControllerofsame orderand structureas in the previousPDexample (figure 5-5). The improvedPerformanceis obvious.

Figures 5-7 and 5-8 show a three-dimensional plot of the Performance indexPI (5.16) and the correspondingcontourplot for the present example. Only twocontrol parameters (d\, c{) are freely selectablebecause of the static bearingstiffness condition (5.19). A stabilizing set of control parameters, is found in a

triangulär shaped area. There is only one minimumofthe Performance index.

The efficiency of the Davidon-Fletcher-Powellnumerical minimization

procedure (see also section 5.3.1.4) is illustrated by figure 5-8: startingoptimization from an initial parameter set which correspondsto a discrete-timePD approximation,the minimum of the Performance index is found after 5iterations. It is also seen that the DFP minimizationroutine makesuse of theHessian matrix, thus differing from a simplesteepestdescentmethod.In this contribution,SPOC-D optimized structure-predefinedcontroüers wiU

not be examined for robustness. As it is shown in figure 5-8, however,considerablylarger control parameter changes are possible for the SPOC-D

optimizedcontroUer than for any PD approximationlayout (c^=0). Furthermore,a numerical investigation for the given example has shown, that the PD

104

approximated Controller only yields stability for sampling rates down to 3%below the nominal value, whereas the SPOC-D optimized Controller can

tolerate sampling rates reducedby nearly 20%.

unstablePIset ofparameters("PI= oo")

minimum

Fig. 5-7: Three-dimensionalplot ofthe Performanceindex PI

0.540

0

0.5-10

dl 1.0-104

startingoptimization

1.5-10 h2'°'l°4 T optimal«-ir.4 set of parameters

3.0-10'

3

0.5 0 0.5ci

1.5

Fig. 5-8: Contourplot ofPI with path ofoptimization

105

Chapter6:

APPLICATIONOF SPOC-D TO THEHIGH SPEED ROTOR CONTROLLER LAYOUT

In this chapter, the SPOC-D controUerdesign approach presented in detail in

chapter 5 is applied to the high speed AMB rotor, so that good dynamicPerformance of the low order Controller in conjunction with the relatively highorder plant is realized. The plant model is described in chapter 4. Necessarynumerical data for continuous-timeand discrete-time description of the highspeed rotor system are found in appendix 10.4.

6.1 CONTROLLERSTRUCTUREANDPARAMETER INTERDEPENDENCIES

The high speed rotor model introduced in chapter 4 considers

eigenfrequencies up to nearly 2500 Hz. The third critical speed is situated at

about 1060Hz and shaU be passedthrough. Therefore, sufficient damping at this

frequencyhas to be providedby the Controller.

Continuous-timeState space matrixAc including plant and filter dynamics foraü four radial displacementsensors is oforder 22, but onlytwo input and Outputsignals are available. However, today's microprocessor technology does not

allow for the implementationof a füll order observerfor the estimation of theresidual non-measurable system states which reaches high enough sampling

106

rates. Therefore, the only feasible controUer design approach is to provide a

simple low order compensator structure.

Hence, for each magnetic bearing a decentralized low order dynamiccompensator is implementedfeeding backthe signalofthe correspondingradialdisplacementsensors. In addition, a "one-directional"couplingbetweenthe twodecentralized feedback paths is used. The reason for the application of this

coupled compensatorstructure is the fact that the third elastic bending mode

(~ 1060 Hz) cannot be controUed easily by a fully decentralized Controller,which is a consequence of the Vibration node locations near sensors and

bearings (figure 4-3). The chosen Controller structure is shown in figure 6-1:

S B B

-®--E <1>

2nd order dynamicIIR compensator

®Z +filter» £ + filter!

2nd order dynamicHR compensator

"one-directional" coupling ¦

filter 4th order dynamiccompensator

©r

Fig. 6-1: PredefinedControllerstructurefor the high speed rotor:

low order discrete-time decentralized

dynamiccompensators and "one-directional"couplingS: sensor location B: radial bearingforce location(A/D & DIA Converters and zero order hold are omitted)

In order to keep the computationaleffort for the DSP as low as possible, thedecentralized feedback paths are implemented as second order dynamiccompensators ofform (5.1) for each magnetic bearing Channel (five Channelsintotal for radial and axial bearings).The coupling feedbackacts as a fourth order

dynamiccompensator.The total Controller order is 8 instead ofover 20 for a fuüorder observer. Discrete-time controUer descriptionis given as follows:

107

Second order dynamic compensator for each decentralized feedback:

uk = d0yk + di yk-i + d2 yk-i- ClUk-l - C2Uk-2

o(z) = ^1 = dp + diz-i + d2z-2y(z) 1 + ciz-1 + c2z-2

Fourth order dynamiccompensator for coupling:Ss. SS. SS. SS. XS,

Uk = doyk + di yk-i + d2 yk-2 + d$ yk-3 + da, yk-A- Ci Uk-l - C2 Uk-2 - C$ Uk-2 - C4 Uk-4

(6.1)

^S. SS. SS. SS. ss.

o(z) = M(z) - dp + diz-1 + d2z-2 + fifez-3 + d^zy(z) 1 + CiZ"1 + C2Z"2 + CJZ"3 + C4Z"-4~

(6.2)The control parameters in (6.1)/(6.2) are subject to optimization through the

SPOC-D method. However, neither all five control parameters of eachdecentralized compensator nor those of the coupling feedback will be

independent (see section 5.3.1.3). For each decentralized control path, two

additionalconditions have to be satisfied:

- a prescribed static bearing stiffness kst has to be achieved- noise penetration at high frequencies has to be suppressed

These conditions can be formulatedmathematicaUyby evaluating z-transferfunction (6.1) at z=l for the static bearing stiffness condition, and at z= -1

(- 1/2 sampling frequency)for the noise rejection condition.This results in the

foUowingadditionalcontrolparameterinterdependencies:

to| = d0 + di + d2 = _£t±Az^1 1 + ci + c2 ki

i#(')U-i _ dp - di + d2 _ £ kst + ks £s=1\-ci + c2 ki

(6.3)

108

For the couplingfeedback path, control parameterconstraintsare introducedfor the subsequentpurposes:

- vanishing static response(no additionalstiffnessthroughcoupling)- noise rejection at high frequencies- drastic reduction of additional computationaleffort for coupling

The correspondingconstraintequations can be formulatedas follows:

\o(7\\ - dp + di + d2 + d3 + d* _ ori 1 + Ci + C2 + C3 + C4

_ dp - di + d2 - dj + a\ _

1SS. SS. SS. SS.

-Ci + C2-C3 + C4isoou = ar"i+°2~?™4 = o

di = d2 = d3 = ci = c2 = C3 = C4 = 0 =» dp + d$ — 0

(6.4)Ss Ss

Hence, for the couplingfeedback only controlparametersdp and d* satisfyingexpression (6.4) are used.

As shown in section 5.3.2 for the simple elastic rotor example, additionalcontrol parameter constraints (6.3) and (6.4) are included in the Controller

optimization process by applying optimalityconditions(5.17). Formulationof the

augmented system description (plant and Controller) and definition of

independent and dependent control paths are completely analogous to the

procedure in that example. Optimization results are presented in the next

section.

6.2 OPTIMIZATIONRESULTS

SPOC-D optimization for the seven independentcontrol parameters resultingfrom the aforementionedcontrol structure predefinitionis performed.Diagonalmatrices are chosen for weighting matrices Q and X0 in optimality conditions

(5.17), and smaU values are attributed to the weightingfactors rt.

109

A static bearing stiffness kst of 500'000 N/m is provided for each magneticbearing. Sampling time T for the füll five-axes magnetic bearing control is120 (is, achieved with one single digital signal processor. Control parameteroptimization results as foUows:

independent control parameters:radial bearing 1 radial bearing 2 couplingd2 = 3.3162-104 d2= 3.1778-104 24 = -4.1264-103 [A/m]ci = 0.10702 ci = 0.08565

c2 = -0.38688 c2= -0.39290

dependent control parameters:radial bearing 1 radial bearing 2 couplingdp =-3.7498-IO4 dp = - 3.6072-IO4 dp = 4.1264-103 [A/m]di =- 7.5685-102 di = - 6.0575-102 [A/m]

(6.5)A transfer function bode plot for the decentralized dynamic compensator

(bearing 1) with optimizedcontrolparametersfrom (6.5) is shown in figure 6-2(transfer functions for bearing 1 and bearing 2 are similar). For comparisonreasons Standard PD Controller (expression (5.4)) is chosen with equal static

bearing stiffness and same damping of the third elastic eigenmode as achieved

by theSPOC-D optimizedcontroUer (P = 500'000N/m, D = 600 Ns/m).

A comparison of the SPOC-D optimized compensator transfer function withthe Standard PD transfer function shows the remarkable gain reduction of theSPOC-D optimized Controller in the high frequency band. Because of the

vanishing phase lead of any compensator in this frequency ränge, plantdynamics cannot be positivelyinfluenced. Therefore, high gain is useless and

only generates noise, as it is the case with the Standard PD approximateddynamic compensator. The SPOC-D optimized compensator,however, showslow gain at high frequencies and thereforereduces noise considerably.

110

decentralizedcompensatortransfer function140

.903

120

"i—i i i i ii n 1—i i 11 ii ii 1—i i i 11 T I I I I I II

static bearingstiffness

\i i i i i i i 11 i i i i i i i 11 Tt

i i i

Snoise rejecuonli i i i i i n i

TTTD

io- 10° IO1 IO2 103 IO4

rzk -100 rn00 M>T3

-200

äPh

30010 10 IO1 IO2

frequency[Hz]10J l(f

Fig. 6-2: Bodeplot oflnd orderSPOC-D optimizedController(1) andPD Controller(2) of same order (P=500N/mm,D=600Ns/m)

100

_

80

£ 60 h03Oh

.1 40C3

a•~ 20 h

0

rigid body closed-loopeigenfrequencies

1200

1000N

£ 800 hC3

600 -

w> 400 -w

100 -50real part [Hz]

0

200

0

<*

elastic closed-loopeigenfrequencies

o x

ex

20 -10 0real part [Hz]

10

Fig. 6-3: Closed-loopeigenvalues of thefirstfive modesfor theAMBhigh speed rotor (inner damping neglected)"o": SPOC-D optimizedController"x": SPOC-D Controllerwithout coupling"*"; PD approximated Controller(without coupling)

111

A further advantage of the SPOC-D optimized Controllerover the StandardPD approximated controUeris the largerphase lead in conjunction with highergain in the frequency mid ränge (figure 6-2), offering better dampingcharacteristics up to the third elasticbending mode(figure 6-3).

6.3 GYROSCOPIC EFFECTS

Gyroscopic couplings of the rotormovementsin x- and y-directionhave not

been consideredso far. Instead, the model description(4.15), representing onlyrotormovementsin one plane, has been used. Furthermore,no cross-couplingof x- andy-planes is providedby the controUers.

However, the influence of gyroscopiceffects can be investigatedby using thefuü coupled rotor system description(4.1) together with the controUerstructuredescribed above, which is separately implemented for each plane. Thenumerical values for the gyroscopic couplings are taken from appendix 10.4.

Figure 6-4 shows the eigenvaluebehavior of the gyroscopically coupled highspeed rotor system for varying rotational speed.

On what regards the imaginaryparts ofthe rotoreigenfrequencies, figure 6-4illustratesthe well-knowngyroscopic effect of Splitting the rotor eigenmodesinto nutation modes (ascending imaginary parts) and precession modes

(descendingimaginary parts).

However, the real parts of the eigenvalues differ significantly from the usualwell-known behavior of conservative and Symmetrie rotor Systems, i.e.

ascending, descending or constant real parts remaining negative for any valueofthe rotational speed [MüUer 81].

There are mainly two reasons for this difference. Firsüy, the high speed rotor

does not have collocated sensors and actuators. Instead, the arithmetic meanof two neighbored sensors is considered for each decentralized control path(figure 6-1). Thus, concerningthe elastic bending modes, actual and measured

112

modal rotor position in the magnetic bearings are not equal. The result is, thatthe rotor system is no longerconservative. The second reason is that, not even

for Symmetrie and conservative rotor plants with collocated sensors andactuators (plants with "positive-real"transfer functions), stability of the

digitally controlled rotor system can be guaranteed in the presence of

gyroscopic couplings. Positive real continuous-time Controllers, however,would not cause instability of such plants. This importantdifference betweendigital and continuous-timeController implementations arises due to the factthat the property "positive real" of continuous-timeplants of form (4.1) is lostwhen zero order hold elementsfor digital controlare implemented[Herzog 90].

g 1000

ts03

.503

1400

1200

800

600

400

200

0 500 1000 1500 2000

rotational speed [Hz]

20

0

-20f ' \

Nffi -40artp* -60139>U -80

¦100

¦120

instability

0 500 1000 1500 2000

rotational speed [Hz]

Fig. 6-4: Calculatedeigenfrequencies of thefirstfive eigenmodesfor the gyroscopically coupled and digitally controlledhigh speed rotor systemandfor varying rotationalspeed

Nevertheless, with the high speed AMB rotor, gyroscopicinstabüitywill not

occur at rotor speeds below lOO'OOO rpm. Therefore, additional precautionssuch as cross-coupling of x- and y-plane by the active magnetic bearings to

prevent gyroscopic instability [Okada, Nagai & Shimane 89], are not

considered in the present high speed rotor application.

113

Chapter 7:

EXPERIMENTALRESULTS

In this chapter, the theoretical results based on the SPOC-Dlayout for the

very high speed AMB rotorpresented in chapter 6 are experimentaUy verified.Measurements of the closed-loop dynamic system behavior at standstill are

compared with Simulation results. Special effects occurring during Operation at

high rotational speeds are discussed.

7.1 MEASUREMENTSANDCOMPARISONWITH SIMULATION RESULTS

In order to judge quality and appUcability of the SPOC-D Controller layout to

the high speed rotor system, two types of measurements at standstiU are

performed: step and frequency response to external excitations in each of the

magnetic bearings. The SPOC-D optimized Controller is compared with theStandardPD approximated Controller. Control in the x- and the y-plane isidentical. Therefore, rotor behavioris measuredin one plane only. Comparisonswith Simulation results are made.

The characteristics of the axial magnetic bearing and of the correspondingrotor dynamics are not consideredhere. It shall be mentioned, however, thatthe axial magnetic bearing is very uncritical and robust, aüowing a great varietyof different controlparametersto stabilizethe axial rotormovement.

114

7.1.1 Step Response

In figures7-1 and 7-2, step responses of radial rotor displacementare shown.External force steps of 50 N are applied separately in each of the two magneticbearings by adding the corresponding voltage signal (0.55 V) to the current

amplifier input. The rotor displacementsat the sensor locations next to both

magnetic bearings are measured.

bearing 1 bearing 2

sensorI ,1 4

„isor 1 I sensor 2

force input _ixr^™

force step in bearing 1

sensor 3 T

force input

sensor 4

[J50Nforce step in bearing 2

¦¦¦I—'" ! I t . ¦|iiii..i..»iii.iT —,i.,T.iii,— i.T-

-I I L

" T'™'™""T I

measurement167 167measurementsensor 1

=L

a 83 83*"V^ kMA

sensor2 sensor3Simulation SimulationOh

¦-B

SPOC-D¦ ¦

50 0

¦T"" -'"1 11"

sensor4

SPOC-D_J t i I I I L_

25time [ms]

25time [ms]

50

Fig. 7-1: Measuredand simulatedstep responsesofradial rotor

displacementwith SPOC-D optimizedController.Force steps (50 N) are appliedseparatelyin each magnetic bearing.

115

force step in bearing 1 force step in bearing 2

o

sensor4measurement measurement167 167

sensor 1

=L

a 83 83

sensor2Ph

Simulation Simulation¦<B

-i i i i_

| time-discrete PDapproximation

_1 I L_ _l I I L-

sensor3

time-discrete PD.approximation

25time [ms]

50 0 25time [ms]

50

Fig. 7-2: Measuredand simulatedstep responsesofradial rotordisplacementwith StandardPD approximated Controller.Force steps (50 N) are appliedseparatelyin each magnetic bearing.

As it is clearly shown in the above figures measurementresults coincide weüwith simulations. Measured steady State values of the step response as weü as

low frequency (i.e. rigid body) dynamic behavior almost perfectly correspondwith the computationalresults.Such correspondenceunderlines both the goodmodeling of magnetic bearing characteristics and the lower frequencyrotordynamics.

However, slight differences between measurements and numericalSimulation can be noted regarding higher frequency dynamics, and especiallyregarding the first elastic eigenmode with an eigenfrequency near 200 Hz

(= 5 ms). This mode shows lower damping than the one resulting from thetheoretical model. The reasons for this divergencecan be mainly due to the factthat this eigenmodeis notweü controllable, since one of the two Vibrationnodesis located in bearing 2 (see also figure 4-3). Another reason leading to

differencesbetweenmeasurementsand simulations is the non-ideal behavior of

important test stand components such as sensors and power amplifier.As a

matter of fact, switched power amplifiers often generate noise, a problemwhichmust be investigatedcarefuUyin any applicationof this type.

116

Damping of the low frequency rigid body eigenmodes (figure 7-1) with theSPOC-D optimized Controller is significantly better than with a simple PDapproximated control algorithm (figure 7-2). That results from the SPOC-D

optimized compensator transfer function featuring more gain and betterphaselag in the low and middle frequency ränge than the quasi-continuous PDcompensator(figure 6-2). It is pointed out again here that the PD Controller

layout was made in order to reachsame static bearing stiffness and damping ofthe third critical eigenmode (near 1060 Hz) as with the SPOC-D optimizedcontroUer(see section 6.2).

One shouldthink that a better low frequencydamping could be achievedby a

higher damping coefficient D, although a PD approximateddiscrete-timecompensatoris used. However, this procedure will not be satisfactory sincenoise generation of the PD Controller (high gain at high frequencies) will beintolerable. The SPOC-D Controller, however, is much more silentthan the PDController used for the measurementsin figure 7-2 - a fact which is not reflected

by the above measurements,but which is clearly audible. Also, amplifierSaturation tending to destabilize the AMB system mightoccurwhen high valuesof D are used. Finally, filtering of the high frequency components in order to

reduce noise is a non-feasible approach for high speed AMB applications, since

high frequency critical speeds have to be passed through and must be dampedsufficiently.

From the above measurementsit may be concluded that the SPOC-Dlayoutfor the structure-predefinedcontroUer (figure 6-1) is muchbetter adapted to the

present control task than the simple quasi-continuous PD approximationlayout.Low frequencies are damped better, noise is rejected by introducingcorrespondingparameter interdependencies prior to optimization, so that low

gain at high frequencies results (see section 6.1), and sufficient damping of thethird eigenmode is achieved through considering the one-directional couplingfeedback path (figure 6-1) in the optimization process.

117

7.1.2 Frequency Response to Harmonic Excitations

In figures 7-3 and 7-4, measurementsof the frequency responses(dynamiccompliances) of radial rotor displacement at all four sensor locations (seefigure 7-1) for a harmonic force input in magnetic bearing 1 are shown and

compared with numericalresults.

frequencyresponseforharmonicforce in bearing 1

3.0I^pÜc-d

sensor 1 fo)2.5 *

0 !

<d 2.0

sensor2 (x);1.5 * X-iG

1.0 *-+

sensor3 f*

ö 0.5^ sensor4 [+)T3

100010 100frequency[Hz]

Fig. 7-3: Measured (ojc*,+)and calculated dynamiccompliancewith SPOC-D optimizedController.Harmonicforce input is applied in bearing1.

Figures 7-3 and 7-4 confirm the correspondence between Simulationandmeasurementresults already pointed out in the above section. Furthermore,theobvious superiority of the SPOC-D optimized Controller over the quasi-continuousPD Controlleris underlinedby the smaller resonance peaks of the

dynamic compliance.

118

o

Booo

¦I

frequencyresponseforharmonicforce in bearing 1

PD

+ J..........J...+.

sensor! 1 (oi)

sensQrj2..(xj)sensori3 Ct)

(-*senson

10 100frequency[Hz]

1000

Fig. 7-4: Measured (ojc,*,+) and calculated dynamic compliancewith discrete-timePD approximated Controller.Harmonicforce input is applied in bearing1.

7.2 OPERATIONAT HIGH ROTATIONALSPEEDS

Run-up tests under vacuum conditions with the high speed rotor using abovedescribed SPOC-D Controller were performed. As elaborated in chapter 3,mechanical design ofthe rotor would aUow rotational speedsup to lOO'OOOrpm.This value correspondsto maximum circumferentialvelocities of over 340 m/sat the axial bearing thrust disk and of over 230 m/s at the radial bearings.However, it should be pointed out already here that, during the run-upexperiments, difficulties arose in exceeding rotor speeds of 64'000 rpmcorrespondingto a surface speed of 220 m/s at the thrust disk. The reasons forthese difficulties will be explainedand suggestions for remedy will be given.Measurements during high speed Operation of rotor displacementat sensors 1and 3 and ofelectric current in magnetic bearing 2 are displayed in figure 7-5.

119

Q = 0 rpm Q= lO'OOO rpm1

§oo

OhCO

ä1OhCO

[Ii sensor 1

1

i sensor3

Sl—i—i—i—i—

sensor 1

sensor3

0 1 2 0 25 50Q = 20O00 rpm Q = 30'000 rpm

[Isensor 1

sensor3i

sensor 1

sensor3

0 10 20 0 5 10Q = 40O00 rpm Q = 50'000 rpm

g 8«I

O

I-8 äTI 81

i current 2 J <!t

+

sensor 1

_J I L_

s-,\r/Y^y^t^v/^rlY\

current 2

smsnr 1sensor 1

j i i i : i 1 i 1_

5 100 5 10 0Q = 60'000rpm Q = 63O00 rpm

< *|—i c

8 8

CA

53

51B

8 Jj-J r»^fK^-*Ai^^^»Vyir*y i^T pi^wrA^vv^vvW^^vv^U h-L. | sensor 1 . S-L | sensor 1

<T^^VVVVVr^'V^/1,V^<T ^rSWVsM4rWUA^WKcurrent 2 . —|-L ! current 2

S

10 10time [ms] time [ms]

Fig. 7-5: Measured rotor displacementat sensors 1 and 3 andelectric current in magnetic bearing2 duringOperation at

high rotationalspeeds.SPOC-D optimizedControlleris used (fixed gain).

120

First and second elastic critical speeds located at 12'000rpm and 32'000 rpmrespectively could be passed through. Therefore, sufficient damping of thesemodes is providedby the active magnetic bearing control.

Concentricity error of the rotormovement above the second critical speed israther small (~ 30 um), which underlines the good balancing State of the elasticrotor. If ISO 5343 and ISO 5406 for balancing quality criteria are applied, thepresent high speed rotor belongs to class 3c, since the rotordynamics are

significantly affected by more than the first and second mode unbalance

[Kellenberger 87]. For this class of flexible rotors, recommendations

concerning balancingare not given expliciüy.The balancingquality has to be

judged according to tolerable Vibrationamplitudes.

Magnetic bearing current is well away from maximum current limitationwhichwould be at approximately6A. However, the current signalin the higherfrequency ränge shows steep flanks corresponding to very short building-uptimes.

As it was discussed in section 4.2.3, maximum current slope imex, andtherefore shortest possible current building-up times for a PWM current

amplifier, is limited by the constant amplifier voltage Uq and the magneticbearinginduetivityL (expressions (4.11)/(4.12)). Therefore, dynamic current

amplifier Saturation occurs for a current slope given by the followingexpression:

lmax ""

j

(7.1)In case ofharmonic excitations with angular frequencyco, condition (7.1) can

be formulatedby using maximumharmonic currentamplitudeimax •*

• =Up_1 max TCOL

(7.2)

121

Concluding from (7.2), only small current amplitudes are possible andtherefore small rotor unbalances must be postulated for high frequencyOperation,in order to avoiddynamiccurrentamplifier Saturation.

However, dynamic amplifier Saturation may be reached, even with very smaUrotor unbalances, if a very high frequency critical speed should be passedthrough. This is iüustrated by the measurement in figure 7-6 for a rotational

speed of64*000 rpm nearthe third elastic critical speed:

Q, = 64'000 rpmdynamic amplifier Saturation current 2

/ XI<=L

< 3a b

maximumyicurrent slope

T3 low frequency \ /mode ^

isensor 1

600 Hzi

10time [ms]

Fig. 7-6: Rotor displacementmeasuredat sensor1 andelectric current in magnetic bearing2 during Operationvery near to the third elastic critical speed (64V00 rpm).SPOC-D optimizedControlleris used.

Dynamic amplifier Saturation, i.e. maximum current slope, is clearly reachedat 64'000 rpm. The consequences are amazing: it is not the third elastic

eigenmode the Vibration amplitudesof whichreach intolerably high values, butit is a very low frequency component and one of approximately 600 Hz thatdominate the scenery. The latter frequency component might belong to the

122

second elastic nutation mode whose eigenfrequency has almost reached600 Hz due to the gyroscopic effects (see figure 6-4).

Increasing rotational speed over the value shown in figure 7-6instantaneously caused instability. In fact, the high speed rotor touched its

housing. At 58'000 rpm approximately, the magnetic bearings caught the rotor

and re-established stable rotor movement. Unfortunately,this incident at

64*000 rpm had already caused non-negligible rotor damage.

7.3 DISCUSSION

Measurement results have proven the practical feasibility of the SPOC-DcontroUerlayout. Accordancewith Simulation results is very good. With theSPOC-D approach, low order discrete-time Controller structures can be

implemented according to practical needs and possibilities, and the controlparameterscan be optimizedtaking into accountthe füll order plant dynamics.Resulting dynamic behavior of the closed-loop AMB system is significantlysuperior to controüers obtained by traditional low order layout methods such as

discrete-timePD approximation.

However, nonlinear effects can drasticaUy deteriorate system Performance.Dynamic current amplifier Saturation, which seems to be one of the most

important non-lineareffects,is difficultto consider with a linear controUer layoutapproach. Although Controllerproperties such as significantly reduced gain at

high frequencies can be specified with SPOC-D, those measures have turnedout to be insufficient for high speed Operation in presence of high frequencycritical speeds. Since perfect balancing of high speed elastic rotors is almost

impossible, different measures must be found so that dynamic amplifierSaturation can be avoided. A possible approach is described in the next chapter.

123

Chapter 8:

UNBALANCE FORCES CAUSING INSTABILITY

8.1 NUMERICAL SIMULATIONOF THE EFFECTSOFDYNAMIC POWERAMPLIFIERSATURATION

In figure 7-6 of the previouschapter a measurement of the high speed rotor

behavior at sensor location 1 and the electric current in bearing 2 for a rotor

speed of 64'000 rpm was shown. This rotor speed nearly corresponds to thethird elastic critical speed. Therefore, already small unbalance forces cause

considerable displacementand current amplitudes. Althoughmaximum current

is not reached in the power amplifier, dynamic Saturation occurs, i.e. a

maximum current slope determinedby amplifier voltage and magnetic bearinginductivity is achieved. The consequenceof this dynamic power amplifierSaturation was rotortouch-downat the high speed mentioned.

However, it could be noted in figure 7-6 that the reason for such an

instantaneous rotor failure was not an expected intolerably high value of thethird critical speed amplitude, but a very lowfrequency component tending to

destabilize rotor movement.

To obtain a theoretical confirmation of this effect ofdynamicpower amplifierSaturation, a nonlinear Simulation of the rotor behavior including the füllnonlinear amplifier dynamics (described in section 4.2.3) is performed. Sinceunbalance forces are not known exactly, those quantities must be assumed.

124

Therefore, the results can reflect the real Situation only in quality, not in

quantity.

Results of such a nonlinear Simulation for small rotor unbalances are

presented in figure 8-1:

sensor 1

sensor 1

ControllerOutput\

ÖWNPPple0 0.01current 2

time [s]

7 0.08

Fig. 8-1: Simulation ofhigh speed rotor behaviorat 64V00 rpmincludingfüll nonlinearitiesofswitchedpoweramplifier.A small value for the unbalanceforce is assumed.

Apart from the beating effect, which may be due to the transient responsetothe initial condition,no particular effects occurduring the Simulation timeof0.08seconds. A frequency component correspondingto 64*000 rpm dominates bothin the sensor and the current signal. Magnetic bearing current accuratelyfollows the signal given by the discrete-time Controller. The current rippleresulting from the alternating voltage that is caused by the switched poweramplifier is well visible. The value of this ripple depends on the constant

amplifiervoltage and on magnetic bearing inductivity.

For larger assumed values of the unbalance force, however, nonlinearSimulation results tum out to be completelydifferent.This is shownin figure 8-2for a Simulation time ofonly0.02 seconds:

125

sensor 1

i 1 i

instabülty

x-Controller output v ysensor 1

flVII^ current 2

0.02

time [s]

Fig. 8-2: Simulation ofthe highspeed rotor behaviorat64V00 rpmincludingfüll nonlinearitiesof switchedpoweramplifier.A large valuefor the unbalanceforce is assumed.

Figure 8-2 confirms the measuredresults. Due to the larger unbalance force

assumed, maximum current slope and therefore dynamic power amplifierSaturation is reached: magnetic bearing current cannot follow the Controller

Output fast enough. The Simulation also confirms that amplifier Saturationcauses low frequency modes of the rotor behavior to grow and to become

unstable, whereas the amplitudeof the rotation-synchronous signal componentremains nearly unaffected.

From the measurementspresented in chapter 7 and from above nonlinearsimulations it can be concluded that high frequencycritical speeds which are to

be passed through are the most limiting factor when magnetic bearings forSuspensionand active Vibration control of an elastic high speed rotor are used.Even if sufficient damping at high frequencies and, therefore, nominal stability is

providedby the linear controUer, dynamicamplifierSaturation in the presenceofunbalance forces can destabilize the system.

In the next section a rather simple unbalance compensation method isdescribed: it allows reduction of power amplifier Saturation effects without

perfect rotor balancingbecoming necessary.

126i

8.2 A SIMPLEADAPTIVEUNBALANCE FORCECANCELLATIONMETHOD

;' W'.j

8.2.1 AMB Rotor and "Force-Free" Rotation

All subsequent investigationsare made along with the simple elastic AMBrotor exampleused in chapter 5 (figure 5-4). For this purpose, the descriptionofthe rotor movementin only one plane (appendix10.6, expression 10.47) must beaugmented by the gyroscopic terms and the unbalance forces, so that the

general rotor description (4.1) is matched. Figure 8-3 shows the assumptionmade for the unbalance distribution: a small mass excentricityAm is added to

the center disk, producingunbalance force and an unbalance moment.

w

M,e,c

£L

Am«M

O

u

Fig. 8-3: Center disk ofthe elastic rotor examplefrom chapter5:assumption madefor unbalance distribution

From figure 8-3 unbalance force vectors lfu and 2fu result as foUows (indices 1

and2 determinethe planes of rotor movement, see notation in section 4.1.1):

lfu =

0

0

Me

KL L

2fu =

0000

e = Am_u ; e = AmuwM

(8.1)

127

Gyroscopic matrixG which couples rotormovementsin both l and 2 planeshas the foUowingform, wherein C denotes the axial momentumof inertia of thecenter disk:

0 0 0 0

G =

0 0 0 0

0 0 0 0

0 0 0-£-L2J

(8.2)In order to obtain aü quantities necessaryto apply the overall description of

the rotor dynamics (4.1), magnetic bearing inputs lf and 2f have to be

determined. Identical control for each plane is implementedand, for the sake of

simplicity, an ideal proportional-differentialOutput feedback with constants PandD is assumed for each magnetic bearing. The following expressionfor thefeedback vectors results:

h2f _lf = -PC^x - DC1**

All numericalquantities for this exampleare given in appendix 10.6.

(8.3)

As it was shown in the previous section, feedback (8.3) may tend to cause

dynamic power amplifier Saturation for high frequencies in the presence ofharmonic unbalance force vibrations. Hence the question arises whether it is

basicaüy possible to "cancel" feedback (8.3) only for the rotation-synchronousharmonic signal components, althoughretaining the feedbackfor other transient

signal components, and thus to obtain a so called "force-free"rotormovementeven for an elastic rotor. This fundamental question can be answered bytransforming inertially fixed description (4.1) into a rotation-synchronousreference frame. By only considering the constant Solution in this referenceframe (l'2x°°) and by canceling transformedfeedback (8.3) one obtains:

128

K + Q\G-M) 0

0 K + Ü^G-M)lx°°

2x°°= n1

(8.4)For rigid rotors, stiffness matrixK vanishes, and for most rotor geometries

(G * M) matrix on the left side of (8.4) is regulär and can thereforebe inverted.A constant Solution for the displacementin the rotating reference frame thenexists independentlyof the rotational speed Q. For this case, expression(8.4)exactly matches the definition ofthe principal axes ofa rigid rotor.

For elastic rotors, however, stiffness matrixK is not zero. Hence, the matrixon the left of (8.4) is only regulärif the foUowing condition is satisfied:

det(# + ß2(G-M)) *0(8.5)

If the determinant(8.5) is zero, the exact condition for an eigensolutionisencountered, hence (8.5) cannot be satisfied for those rotational speeds Q that

correspondto the eigenfrequencies of the free and unsuspendedflexible rotor.

One can conclude as follows:

At a given rotational speed £2, "force-free" AMB rotor movement in the

presenceof arbitrary unbalance forces by cutting off the harmoniccomponentoffeedback (8.3) is possiblein the foUowingcases:

- for a rigid body at any rotational speedQ. The rotorthen rotates aroundits principalaxes whichremain the same for any value ofQ.

- for an elastic rotor, if jQ does not correspondto an eigenfrequencyofthe free and unsuspendedrotor. The shape of the "force-free" rotor

movementdepends on the value of Q and can be obtained by solvingexpression(8.4).

129

8.2.2 Unbalance Force Cancellationwith AMB

There are basicaüy three possibilities for the technical realization of a "force-free" rotationat a given rotational speed, though feedback for transient signalcomponentsis maintained:

- filtering the feedback signal with a rotation-synchronous notch-filter- dynamicobserverfor the on-linedetermination ofa harmonic

compensation signalfor each magnetic bearing, so that the totalharmonicsignalcomponent (normalfeedback and compensation) ineach bearingvanishes

- simpleadaptive determination of such a compensation signal

Notch-filters described as a first unbalance force cancellation method are

already industriaüy applied.They provide a rather simple way of cutting-offrotation-synchronoussignal components. In order to guarantee "force-free"rotation at various rotational speeds, the filter must be synchronizedwith therotation speed. Sophisticated hardware is required here - one of the drawbacksof this method. Furthermore, critical speeds corresponding to flexible rotor

eigenfrequenciescannotbe passed through if notch-filters are used (as it wasshown in above section). Trying to do this would be equal to a cancellationofthe damping necessaryfor passing the correspondingelastic critical speed.The second possibiüty is very similar to techniques the aim of which is to

compensate Vibration amplitudes due to unbalance [Burrows, Sahinkaya,Traxler & Schweitzer 88], [Nonami & Kawamata 89]. In order to obtainsmallest harmonic bearing force components possible, a dynamic observerrepresenting the plant dynamics estimates the unknown unbalance forces andthe resulting harmonic bearing forces which are then subtracted from thecontrol signal. This approach requiresan exact descriptionofthe plant includinghigher frequency elastic modes. Furthermore, for the case of a digitalimplementation,the on-line computationaleffort is significant. Finally, if therotational speed corresponds to an elastic eigenfrequency of the free rotor,convergenceproblems might occur, since unbalance force cancellation is

physicaüy impossible at these speeds (see section above).

130

The third approach shaü be investigatedmorein detaü. Is it possible to find a

simple adaptive method to cancel or to reduce unbalance harmonicsin the

magnetic bearing force signal?

The basic idea is to providean additionalharmonicand rotation-synchronousinput signalfor each magnetic bearing:

f = /feedback + AuCOSQt - JBMSÜ1 Qt

/ = /feedback + BucosQt + Ausin Qt

(8.6)SS. SS,

Vectors Au and Bu are unknown, normal feedback vectors 1>2f contain theharmonic response to the unknownunbalance forces as well as transient

components to stabilize the AMB rotor system. These transient signalcomponents are neglectedin the further considerations, so that the followingharmonicfeedback component results:

1/S. 1/V SS. SS.

f = /feedback + AwcosQt - Busin Qt

2/\ 2Ä *** •"¦

/ = /feedback + BucosQt + AMsin Qt

(8.7)The goal of an adaptive unbalance force cancellationsystem is obviouslythe

determinationof vectors Au and Bu, so that total harmonic parts / of thefeedback vectors !»2/ vanish:

u/=i-Ts/i

Ss

/2= 0 <=> Au =

SS

auSs.

a"2= ?;#„= y«i

b'"2 J

= ?

(8.8)

This determinationcan be performed adaptively without any informationofthe plant description and the bearing parameters being available, and at anyrotational speed, exceptfor those speeds correspondingto the eigenfrequenciesofthe free and unsuspended rotor (see above). It is shown in appendix 10.7 that

SS.

the scalarcompensation parameters aUl 2and bUl2 of both magnetic bearings 1

and 2 are easily found for any linear rotor system described by (4.1):

131

ss.

Compensationparameters aUl 2 and bUl 2of each magnetic bearing input are

located on circles*-1 around the minimumreachable harmonicforce amplitudein each magnetic bearing (\fi\ and \f2\ respectively). In the optimal case

harmonic force amplitudesare zero in both bearings.

Figure 8-4 visualizes this fundamental characteristic along with the givenexample (figures 5-4 and 8-3, appendix 10.6). It is representativefor any linearAMB rotorsystem ofform (4.1).

If an additionalharmonic force signalwith properly chosen gain and phase ineach magnetic bearing is provided, so that the total harmonic bearing force

including harmonicfeedback component vanishes, the same result is obtainedas with an ideal notch-füter or with a compensation signal from a dynamicunbalance force observer. However, advantages of this adaptive unbalanceforce cancellationmethod are:

- no information ofthe plant descriptionis necessary as it is the casefor methodsusing a dynamic observer

- the compensation signals are exacüyharmonicand do not containtransient componentsas it is the case for a dynamic observer

- a very simple search algorithm in conjunction with digital controlcanbe implementedin order to determinethe harmonic compensationsignals, since the compensation parametersare located on circles

(figure 8-4)- instabilityat specific rotational speeds, as it occurs using rotation-

synchronousnotch-filters,can be avoided if the compensationamplitudesare watchedby the digital controUer and limitedto

specifiedvalues- significant reduction of the total magnetic bearing force amplitude

can be achieved, even if compensation signals are "frozen" when

passing through critical speeds (see figure 8-6)

8 1Magnetic bearing characteristics are considered to be equal in both 1 and 2 planes.

132

bearing 1

B

withoutcompensation(2Ul=N= 0)

cQ

0343

70

60-

50-

40-with

30 "

compensation(40 Hz)

10

0

r-, 70

t^ 60 h

bearing2

o 50Q[Hz]

contourlines of I fi-2P^

100

/\

I——I

N

4 -

fi minimum

"\-6

an = 0°z

b.. = 0Hz\10-4 -2024

tUl [N] —*-

50-

40-with

swithoutcompensation(aU2=bU2= 0)

^ 30 "

compensation(40 Hz)0u2 u2ä 20

1043

10050ß[Hz]

contourlines of |f2-2

iL -4minimum

Z -6

<«= 0ui

b„.= 0ui^10

bU2 [N] —«-

Fig. 8-4: Compensationofharmonicforce componentdue to

unbalance is made independently in each magnetic bearingat a rotationalspeed Q of40 Hz.

SS.

ParametersaUl 2 and bUl 2 ofthe harmoniccompensationsignalsin each magnetic bearingare always located oncircles aroundthe minimumreachableharmonicforceamplitudes\fi\and\f2\ respectively.

133

By independently compensatingthe harmonic force component in each

magnetic bearing using the drc/e-propertypresented, harmonicbearing forcesin bothbearings will nearly equal zero after someiterations.A Simulation of thischaracteristic is shown in figure 8-5:

Q=40Hz

withoutcompensationIst iteration(bearing 1)

2nd iteration(bearing 2)

3rd iteration(bearing 1)

4th iteration(bearing 2)

5th iteration(bearing 1)

total harmonic bearing force components

3Ist iteration(bearing 1)

fi [N]

5.7615«10 l

2.2781 «10"

8.0503*10 o

2.3489-10-3

6.8133-10 o

2.173710-3

2.635710-3

f2 [N]

5.812M0

8.7491 »10o

1.751240-3

7.4037-10 o

1.7790-10"

6.2663 10*

8.442010

Fig. 8-5 Simulation of an iterative unbalanceforce cancellationmade independently in each magnetic bearingusing the circle-propertymentioned

Although reduction of the total harmonic force amplitude is alreadyapproximately 85% after the first iteration, convergence for subsequentiteration steps is rather poor. The reason is the fact that an independent

134

unbalance force cancellation in one magnetic bearing partiy deterioratestheresults obtainedfrom the previous adaptation in the other bearing. To date it isnot yet investigated how this effect could be taken into account to achieve a

significant convergence accelerationwithoutusing any knowledge of the plantmodel. Future investigations in that direction are necessary.

However, this method of adaptive unbalance force cancellation provides a

technicaüyfeasible means of drasticaüy reducingharmonicforce amplitudesinthe magnetic bearings, and a way to avoid dynamic amplifier Saturation,especiaüyin the case of high resonancefrequencies.

In figure 8-6 this process is ülustratedalong with the elastic rotor exampleat

the 4th critical speed. Harmonic force amplitudes are independentlycompensatedand iterated as described above at a rotational speed Q of 233 Hz.The resulting compensation parameters are "frozen". Harmonic bearing forceand displacementamplitudesare displayed for varyingrotational speed.

Z 140

120

100

•83

43

60 -

8 80«SDOa•c

Jß 40o

20

0200

without_ Kompensation -

withcompensation

_ (233 Hz)

-^\sA bearing2

_ i l bearing 1

j\ \ bearing2

-j\ ssA bearing 1

250

Q[Hz]300

withoutU compensationwithcompensation(233 Hz)

5 -

TS3 LCPtx

3 -

Dhx,,x1' "2•O

200 250Q Hz

300

Fig. 8-6: Unbalanceforce cancellation "frozen" at 233 Hz(4th criticalspeed of elastic rotor example).Reduction offorce and displacementamplitudes resultswithin the wholeresonanceränge.

135

Chapter 9:

SUMMARYAND OUTLOOK

Active magnetic bearing (AMB) Systems, typical mechatronicsproducts, findmore and more industrial applications thanks to their outstandingcharacteristics: complete contact-free support, no wear, no friction, activeVibrationcontrol etc., to name onlya few. The rapid progress in electronics and

microprocessingtechnology has made it possible to take fuü advantage of thetechnicalpossibilitiesof AMB Systemsby using digitalcontrol.

Of specialinterest in this contributionwas the applicationof AMB Systemstohigh speed rotation: the main task was to discuss problems of high speedrotation in general, and especially in conjunction with AMB, in order to

determinethe limitations of practical applicability of such Systems.

For this purpose, three main directions of research were defined:

- What are the mechanical requirementsnecessaryfor materialsused in high speedAMB rotor Systems, and what basic

design rules mustbe foüowed ?- What specific problems occurwhen using digital control for

high speed AMB Systems, and what Controllerdesign methodspresenüy availablecan be applied ?

- What additionalquestions arise in conjunction with a practicalrealization of a high speed AMB rotor system and duringhigh speed Operation ?

136

Concerning the first thematic complex, special emphasis was put on themechanical behavior of linear elastic materials under centrifugal load. Thefundamental equations for stress and strain were derived. Different materialswere compared, and their suitability for high speed rotation was discussed. In

addition, the laminatedconstruction as an important aspect of magnetic bearingrotor design was taken into account. Consequencesof this fundamental AMB

system requirement on high speed applicability of various materials were

discussed.A practical definition ofoptimal shrink-fitwas given.

Some ofthe ideas presented in this first part ofthe present contributioncan befoundin literature. The basic intentionof that importantsurvey is to provide a

design tool for high speedAMB Systems.

Key issue of this contributionwas the second group ofquestions listed above.As a first point it was stated that most high speed rotorplants mustbe described

by high order mathematicalmodeisso that elastic eigenmodescan be takenintoaccount. However, most of the discrete-time Controller design techniquesavailable at present would result in high order and complex control schemessuch as dynamic observer since only very few plant states are available formeasurement,which is typical for AMB Systems. Therefore, they must bereconstructed by the Controller. These high order Controllers cannot be

implementedby using today's State of the art in microprocessor technology,since much too low sampling rates would result. Reduction of Controller

complexityis thereforeimperative for high speed AMB applications.There are basically three different approaches to obtain low order control

schemesfor high order plants. Firsüy, various model reduction techniques priorto Controllerlayout are available. Secondly,"post-layout" Controller reductionmethods exist. Among both groups, there are many approaches which do not

guarantee stability of the resulting closed-loop system (so-called open-loopreduction techniques). Stability guaranteeingmodel or Controller reduction

techniques (closed-loop reduction), however, are foundvery rarely and mostoften rely on fundamentals of time-continuously controUed Systems which are

lost if digital control is implemented [Herzog 90]. For this reason, a newmethod

137

named SPOC-Dbelongingto the third type of reduction approach, the so-caüeddirect design techniques, was presented: the method allows to determine an

optimal set of control parametersfor a low order controUertaking into account

the fuU dynamics ofthe highorder plant.

Direct designreduction techniques are often applied in practice. Theyconsistin a predefinition of a low order Controllerstructure which is then applied in

order to control the high order plant. In the case of discrete-timecontrol, loworder decentralized compensators are a frequent concept of Controllerstructure predefinition. For this type of controüers, the direct designtechniqueconsists in finding controlparameters for the predefined Controller providing the

necessaryPerformance. Parameters for decentralized compensators are oftenfound by means of a quasi-continuous approximation of well-knowncontinuous-timeControllers such as P(I)D and familiär types. Hence, sincethose control parameters are suboptimal and do not make use of the füll

dynamicränge offered by the structure-predefinedcontroUer, the main target ofthe new Controller design method presented was to make best use of anypredefinedlow order Controller structure, and thus to achieve best dynamicPerformance.

The SPOC-D method (Structure-Predefined Optimal Control for Discrete

Systems) presented in this contribution has proven to be weU applicable in

practice. Control parameters are computed off-line for any linear structurallypredefineddiscrete-timeController by minimizationof a quadratic Performanceindex including both low order Controller and high order plant dynamics.Therefore, a good mathematicalmodel of the plant must be available. Control

parameter interdependencies, a feature being of basic importance for practicalimplementation,can be included in the optimization process to achieve specialproperties: prescribed static bearing stiffness, low Controller gain at highfrequencies for noise rejection and robustness, to name only a few. Introducingthese parameter constraintsdirectly into the optimization process is much moreconvenientthan achieving the correspondingController properties indirectly byan appropriate choice ofweightingmatrices in the Performance index.

138

In the third andlast part of this contribution, practical results ofthe applicationof above described investigations were discussed. Measurements with an

especiaUy built large high speed AMB test rotor were performed. Five-axescontrol with a single digital signal processor(DSP) was implemented, and a

sampling frequency of 8.3 kHz was achieved. The test rotor featuresthe latelydeveloped amorphous metals as lamination sheets - a consequenceof the

designrules obtained in the first part of this contribution. This material seems to

be perfectlyappropriate for high speedAMB applicationsdue to its outstandingmechanical and magnetic properties. The tooling of amorphous metals,however, is a somewhat difficultand complicatedtask.

Measurement results have confirmed the Performanceachieved with theSPOC-D Controller layout method, its practical applicability and its robustnessto gyroscopic effects, inevitable model errors and external disturbances.Sufficient damping could be provided so that two critical speeds at

approximately 12'000 rpm and 33'000 rpm could be passed without furtherdifficulties.

Circumferential velocities up to 220 m/s could be reached without

encountering any material limitations. Actually, the high speed rotor was

designedmechanicaUy, so that it shouldbe possible to reach a maximum speedof lOO'OOO rpm correspondingto a maximum surface speed of 340 m/s. At thethird elastic critical speed located at approximately64'000 rpm, however, the

problem of current Saturation in the switched PWM amplifiers occurred and

finaüy led to rotor instabilitywithsubsequenttouch-down.

The effects of dynamic amplifier Saturationwere shortly investigated.Sincethis Saturationis a strongly nonlinear effect, it can be taken into account onlyindirectly in the Controller layout process. Therefore, a simple approach was

sought by providing additional measures, so that amplifier Saturation can beavoided in future. A simple adaptive unbalance force cancellation method

making use of a circle property common to all linear rotor Systems was

presented.

139

Lately, significant progress has beenmade in the field of AMB applications.Thanks to the newly formedInternational Symposiumon MagneticBearings,world-widecommunication and productive exchange of experience in this

special domain has been activated. Therefore, further rapid progress can be

expected for the Coming years.

Many questions remain to be answered though.

Due to the clear trend towards digital control, discrete-timeController designmethods have to be examined more deeply from the aspect of control theory,and special attentionmust be paid to robustness properties and sensitivity to

external disturbances. Furthertheoreticalresearch must be performedin thedomain of nonlinearcontrol strategies,so that nonlinear problems as describedabove can be handled more easily. FinaUy,the neuralnetwork technologycouldfind its applications in the fieldofAMB.

More practical research must be made into hardware. New material

technologiesmust find their way into AMB Systems and related fields such as

high speed high power motor drives. Amorphous metals, high temperaturesuperconductingmaterials etc. are important steps in that direction. Significantprogress can also be expected in the field of sensoring.New trends in AMB

sensoring are expected from flux measuring and from the completely voltage-controüed "sensorlessmagnetic bearing" [Vischer 88].

FinaUy, safety and reliabilityof AMB Systems must be augmented. Digitalcontrol offers new possibilities even in this field [Diez 89]. It provides more"intelligence"by new concepts such as integrated safety design, data basedfailure diagnosis or redundancy management for recovery action, thus

rendering AMB Systems a mechatronic product with a high degree of built-inSoftware.

140

141

Chapter 10:

APPENDIX

10.1 SEARCH OF THE MAXIMUMREFERENCE STRESS

In section 2.1.1.1 reference stress (Thunder Tresca's shear stress assumptionwas introduced for the two-dimensionalcase:

Gref(r) = max(|cT((r)-CTr(r)|,|(Jr(r)|,|o-f(r)|)(10.1)

The maximum of the reference stress and the radius where that stress occurs

are ofmost interest:

<Vmo* = °p (<M»0) = max [max (I o* <r) - <*r (r)\, IO (r)|, | ct (r)|)](10.2)

Single Ring with ConcentricBorehole (rt > 0):

Radial and tangentialstress are given by (2.9). Evaluatingthe expression forthe radial stress at the inner and outer edges r, and r0, one comes to the weU-known result that radial stress vanishes in both cases.

oün) = GM = 0

(10.3)

142

Furthermore,the derivativeof o> withrespect to radius r is givenby:

-?,. 2|S-W(vf3)<!M-r)dr 4 rsn < r < r0

(10.4)

Settingthe derivativeequal to zero

(r*) =0 <=> r* = Vr^"dcrr , *s_

dr(10.5)

one finds only one extremum of the continuously differentiable function a/r) inthe ränge rt<r<r0. That occurs at the geometric mean r* of inner and outer

radius, and its correspondingvalue is positive:

OXr*) = lpß2(v+3)(r0-rf)2 > 08

(10.6)

Hence, the radial stress ar must be positiveor zero for aU values of the radiusr in the givenränge.

Similarly,by evaluating (2.9) for the tangentialstress er,, it can be shown for

compressiblematerials (v < 0.5) that the values at the inner and outer edges are

bothpositive,

Gtin) = lpß2((l-v)r2+ (v+3)r02) >0

ct(r0) = LpQ2((l-v)r2+ (\*3)r?) >0(10.7)

whereas the derivative of at with respect to the radius r remains negative in

the wholerängeri<r<r0:

^ = -Ipß2 [(V4-3) r-^-+(l+3v) r ] < 0 V rt<r <r0dr 4 r3

(10.8)

143

Consequently, tangential stress is positive for all values of the radius r

aüowed,and its maximum occurs at the inner edge r,.

For compressible materials, the difference between tangential and radialstress is positive for any radius r betweenrt and r0. Therefore, tangentialstressis always higherthan radial stress:

ot -or = Ipß2((v+3)r-&+ (1-v) r2) > 0 V n<r < r04 r2

(10.9)

Knowingthat

- tangentialstress is always positiveand that its maximum occurs at the inner edge rt

- radial stress is zero at both edges and positivein between- the difference between

tangentialand radial stress is always positive,

one concludesthat the maximum reference stress orefmax (10.2) must occur at

the inner edge rt where it equals the corresponding value of the tangentialstress:

<V»« = OM = Jpß2((l-v)r?+ (v4-3)r2)(10.10)

SingleFüll Ring (rt = 0):

The procedureto determinethemaximum reference stress for the füll ring is

analogousto that examinedin the previous case. The basic equations for thestress components(2.9) are still valid if inner radiusr, is set to zero.

144

The final results are:

- tangentialstress is always positive, and its maximum occurs

at the inner edge rt- radial stress equals tangentialstress at the inner edge /*,;

it is zero at the outer edge r0 and positivein between- the differencebetweentangentialand radial stress is zero at the

inner edge rt andpositive elsewhere.

Therefore, for the fuü ring, one also concludesÜiat maximum reference stress

Grefmax (10.2) occurs at the inner edge rt where it equals the correspondingvalueof the tangentialstress:

<Vm« = °to) = l-pß2(v+3)r2(10.11)

10.2 STRESS AND STRAIN IN LONG CYLINDRICROTORS

As mentionedin 2.1, the cross-sectionsof a long cylindric rotor remain planefar away from the front ends, and the assumption of planar strain is therefore

justified.

The kinematic relations (2.1), the equüibrium equation (2.2) and thedifferential equation(2.3) are stiU valid for long cylinders, whereas Hooke's law

(2.4) must be formulated for the more general three-dimensionalcase (axialstress ctz9£0):

£r = L(ar -v(cF,+<xz))

£t= L(cjt - v(Or+Gz))E

£z=^{gz -v(CJr+(Jt)) = 0E

(10.12)

145

Combining (10.12) with the kinematic relations (2.1) and the differential

equationfor the radial stress (2.3), one finds the following differential equationfor the radial displacementu:

E (1-v) / 2d2u , du \ „ 1n2* '-—\rL +r—--u) = -pr5Q(l+v)(l-2v) v dr2 dr J

(10.13)Its general Solutionis:

u(r) = ar + b-(l+vX1-WprlQ2r 8£(l-v)H

(10.14)with a and b as integration constants determined by boundary conditions.

Theseboundaryconditions do not differfrom (2.7) or (2.12) for a cylindric rotor

with a concentric borehole or a füll cyhnder respectively. The expressions for

radial, tangentialand axial stress for each of the cases mentionedwiU be givenpresently.

SingleLong Cylindric Rotor with ConcentricBorehole (rt > 0):

am = WfiM(r?+r?-lM-r2)8 (1-V) rz

am = \P&m^irl^»+^^.0^,2)8 (1-v) (1-v) r2 (1-v)

<7z(r) = ipß2-y-((3-2v)(r/2 + r2) - 2r2 )4 (1-v)(10.15)

146

Single Füll CylindricRotor (rt = 0):

*r) = ±PQ2^(r2-r2)8F v(l-v) (1-v)

}

Gz(r) = \pQ2-y—{(3-2v)r2 - 2r2 )4 (1-v)(10.16)

Reference stress under Tresca's shear stress assumption for the three-dimensionalcase is givenby a more general form of (2.10):

Gnf(f) = max (| CT, (r) - Or(r)\, | Gt (r) - Gz (r)\, | (Jr(r) - <JZ (r)\ )(10.17)

It can be shown that the maximum of the reference stress occurs at the inner

edge r„ as is the case formin rings. The value ofthe maximum reference stress,however, is somewhat more difficult to determine analyticallythan shown in

appendix 10.1 for thin rings. In most applicationsthough, stress occurring in an

assembly of more than one cylinder or ring (e.g. shrink-fit) is of interest. Forthese cases, a numericalstress investigationis inevitable.

10.3 LINEARMATRIXEQUATION(2.17):AN EXAMPLEFOR TWOTHINRINGS

Two thin rings shaU be shrink-fitted onto each other. The inner ring is a fuU

ring, hence rtl=Q. The boundary and transition conditions necessary to

determinestress and displacementin n rings are givenby (2.16/2.16a).

147

For n=2one obtains:

Wl(0) * oo

Gn(r0l) = Gri(ri2)r0l + ui(r0l) = ri2 + u2(ri2)

Gr2(r02) = 0

(10.18)Along with the kinematicequations (2.1), the inversion ofHooke's law (2.4)

and the expression for displacement u (2.6), the boundary and transitionconditions (10.18) are combinedin the linear matrix-equation of order 2«=4:

Ax=j(10.19)

0 10 0

A =

/ =

£i(l+vi) £i(l-vi) £2(1+V2) £2(1-V2)(1-v?) (l-v2)r2 (l-v22) (l-vjV?

ro,

0

(WHr0l

0

0

n

d+v2) (1-V2)r2Iq2

X -

ai

bi

L62J

£- (pitvi-^ - p2(v2+3)r2)

Q2 /Pl(l-V2) , P2(l-Vbr3\8 l Ei rBl E^~l2)'rox + ri2

p2X22(l-v22)(v2+3) ,,

(10.20)Matrix A contains both geometry and material data. Vector / contains all

"forces" resulting from the rotation on the one hand, and from the radiusdifferencenecessaryfor shrinkage on the other hand. FinaUy,vector x containsaü unknown integration constants for both rings.

148

10.4 NUMERICALVALUESFOR THE HIGH SPEEDROTOR MODEL

Reduced OrderPlant Description (4.6)1(4.7):

MR = diag (

GR =

UBr =

8.82-10-2-2.3810-23.3110-2

-2.9010-2-1.75-10-2-1.0310-3-1.5610-2

-5.969

-8.1252

-3.1953

5.6202

2.0328

2.685

6.4608

); DiR = diag ( ); KR =diag(

Ol0

1.5792106

1.0882-107

4.4527-1071.1912-108

2.3698 108_

00

18.4760.9569.24130.19225.44J

-2.38-10-2 3.31-10-2 -2.9010-2 -1.75-10"2 -1.03-10"3 -1.56-10"2"1.0110-1 2.12-10-2 -1.86-10-2 9.82-10"3 -1.88102 7.18-10"32.1210-2 6.60-10-2 -1.04-10-2 -2.57-10-2 1.48-10-2 2.97102-1.86-10-2 -1.04-10-2 1.10-10"1 2.70-10-2 -3.2110-2 -5.78-10-29.82-10-3 -2.57-10-2 2.70-10-2 1.04-10-1 -6.46-10"2 -8.51-10-2-1.88-10-2 1.48-10-2 -3.2110-2 .6.46-10-2 1.02-10"1 9.24-10-27.18-10-3 2.97-10-2 -5.78-10-2 -8.51-10"2 9.24-102 1.78-10-1.

io-i

IO"2IO4

IO4

10

10-

10"

1

-2.6315-io-i

6.1686 io-i

-1.0798 IO"2

-4.7579-io-i

5.1772 io-i

1.89 io-i

3.201 io-i_

149

1,2Cr =

-6.79-10-1 -2.5410-1 2.77-10-2 5.08-10"1 7.55-10"1 -5.81-101 -3.78-10-1

-5.1410"1 9.18-10-2 -5.36-10"1 2.15-10"1 -5.75-10"1 4.45-10"1 -2.97-10"1-3.47-10-1 4.43-10-1 -3.91-10-1 -7.0410-1 4.65-10"2 -8.19-10"1 3.50-10"1

-1.8-10-1 7.9110"1 4.49-10-1 1.47-10"1 1.59-10"1 5.08-10"1 -S^l-lO-1.(10.21)

RadialBearing Characteristicsfor (4.9):

pre-magnetizingcurrentforce-current factor ktnegative bearing stiffness ks

2.5 A152 N/A

575*000 N/m

State SpaceDescriptionofFilter Dynamics (4.10):(one singlefilter)

AfiUer ~

Bfilter ~

-7.5398-103 -3.6952108

1 0

10

; Cfiiter = [ o 3.6952-108]

Discrete-TimeOverallState SpaceDescription (4.17):

(10.22)

sampling time Ttime-delay Td

120 ns36 jis

150

10.5 PERFORMANCEINDEXAND VECTOR-GRADIENT

"Integral"-Property of the Discrete-TimeLyapunov Equation:

LetA be the system matrixof a stable discrete-time dynamic system, i.e. aU

eigenvalues ofA are within the unit circle of the complexplane. The discrete-timeLyapunov equation involving arbitrary matrix Q

Adsd PAdsd - P + Q = 0

(10.23)then has the unique Solution P whichis equal to the infinite sum givenby

P = £ (ATfQA**=o

(10.24)This important"integral"-property(the name taken from the corresponding

continuous-timecase [Senning 79]) will be referred to during the followingconsiderations.

Controller Structure Predefinition,Closed-Loop Stability andPerformanceIndexMinimum:

Starting point for all subsequent considerations is matrix P necessary to

determinethe Performance indexPI and introduced in expression (5.10):

00/ m \

P = X (AciSdf\Q + 2 CJdJndid\Akclsdk=P V /=! '

PI = trace (P X0)

Xp = xpxfi(10.25)

151

The predefinedcontroUerstructure is reflected by the structure ofthe closed-

loopmatrixAdsd which contains the dynamicsofboth plant and controUer.Thecontrol parameters subject for optimization are includedin the control vectors

dt. It is evident that a Solution of the infinite sum in (10.25) onlyexists in the caseof a stable system matrix Adsd (see above "integral"-property). However,existence of stabilizing sets of control parameters yielding a stable systemmatrix for any arbitrary Controllerstructure is difficultto prove, and, in some

cases, it may even happen that no stabilizing set of control coefficients can befound.

Thus, a certain amount of experience is required when choosing (low order)Controllerstructures for a given (high order) plant, a fact which is known fromother dynamic Output feedback techniques as well. Helpful tools for the task of

predefining Controllerstructures can be obtained from physical considerations,from considering only the dominantterms in an alreadyknown stabilizing (highorder) control scheme or also from model or Controller reduction techniques.

The shape of the Performance index defined by (10.25) and especially theexistence of one or severalminima is a next question of crucial importance forthe optimizationmethod proposed in this thesis. An exact answer to this

question, though, cannot be given and remains subject for further and deepertheoretical considerations. However, it can be noted that, for the numerous

sample control problems investigated using SPOC-D as well as for quite a

number of technical realizations of AMB Systems (miüing spindle, very highspeed rotor, very flexible rotor) with various controUerstructure predefmitions,a minimumofthe Performance index could always be foundand no indicationfor more than one minimumwas encountered(see also figure 5-7).

Hence, for most "well-posed"problems with an "appropriate" choice ofdiscrete-time Controller structures, it can be assumed that a minimum of the

Performance index (10.25) can be found, and that the SPOC-D optimizationmethod provides a feasible and practical tool to find Controllers featuring thePerformance required for the givenproblem.

152

Finding a Stabilizing Set ofControlParameters:

In order to simplify the search for a stabilizing initial controlparameterguess,the following approachmay be useful:

A "shrink" coefficient a is introduced in (10.25) in order to move all

eigenvalues of the closed-loop matrixAdsd into the unit circle of the complexplane10-i. Hence, all unknown Controllercoefficients can be set to zero for a

first optimization step, and the problemofa stabilizing initial guess is eliminated.This results in the modifiedexpression formatrix P:

°°

I m \

P = 2 (aA&flQ + 2 CldhidiCMaAdsdf ; 0 <a < 1*=0 V '=1 '

(10.26)It is obvious that this coefficient a has to be set to 1 during Controller

optimization so that stability of the closed-loop system is achieved. That is

generaüy possible since, after the first few optimization iterations,a suboptimalset of control parameters dt is found which already stabilizes the closed-loopsystem for a=l. This sub-optimalset of controlparameters can then be used as

an initial guess for further optimization steps.

Vector-GradientwithoutAdditionalParameterConstraints:

The definition ofthe matrixP in (10.26) by an infinitesum can be replaced bythe following Lyapunov equation for P using above mentioned "integral"-property:

m

(aAcisdfP(aAclsd) - P + Q + 2 Cjdjndid = 0i=i

(10.27)

- For the eigenvalues of(aA) can be written: eig(aA) = a eig(A)

153

In a next step, smaU perturbations ddt are applied to the feedback vectors dt,which result in perturbedvalues for closed-loop matrixAdsd and matrixP and

finaUy lead to the foUowingperturbed Lyapunov equation:im \T i m \

cc\Acisd+ 2 hddidj(P+dP)Udsd+ 2 biddiCAcc - (P+dP)m

+ ö + 2 CKdi+ddifrifa+ddiJQ = 0

(10.28)

Combining expressions (10.27) and (10.28) and neglecting terms of higherthan first order, one can obtain:

al 2 biddidjPAcisdCc+aAdsddPAcisdCc+ccAdsdP 2 hddiQlam

- dP + 2 Cjddlndid +Cfdhiddid= 0

(10.29)After some simple algebraic transformations the following Lyapunov

equation for matrixdP is obtained:

aAcisd dPAclsda - dP + Q + Q = 0 , with Q givenby

Q = ^CfddficxbTpActsdCC+ridiCi)i = i

(10.30)If expression (5.11) is applied, the perturbationterm of the Performance index

can be written as foUows:

dPI = trace (dP Xp)(10.31)

Using the "integral"-propertyfor (10.30) the equivalent infinite sum notationfor matrixdP is obtained,

154

oo

dP = 2 (ocA^f(Q + QT)(aAcisdfk= P

and the perturbed Performance index dPI results as

(10.32)

dPI = trace

oo

2 (aAlisdf(Q + QT){ccAdsdfLk= P

Xp

(10.33)~T

Expandingexpression (10.33) to two terms involving matrices Q andQ , and

applying the rules for the trace Operator, i.e. trace(A2?C) = trace (BCA) =

trace(CA-B), a more simple expression for the perturbedPerformance indexresults:

dPI = 2- trace 2 (vAlisdfQ(aAclsdfLk = P

Xp

(10.34)In order to separate matrix Q containing the perturbation terms ddt from

matrices Adsd andX0, the rules for the trace Operator are applied again, and the

following newnotation for the perturbed Performance index is obtained:

dPI = 2- trace \Q 2 (<xActsdf Xp(aAjisdf\k = P

(10.35)

Making further use of the "integral"-propertythe infinite sum in (10.35) is

replaced by the matrixXbeing the Solutionof the following Lyapunov equation:

AdsdXAdsd - X + Xp = 0

(10.36)

Using (10.35) and (10.36) the perturbed Performance index dPI can berewritten as

155

dPI = 2-trace (QX)(10.37)

Ss*

The foUowing expressionresults from replacing matrixQ in (10.37) by itsdefinition (10.30) and from changing the sequence of trace Operator andsummation:

m

dPI = 2-2 trace (cfddf(a bJPAdsdCX + n di Ct)x)(10.38)

Makingfinal use of the trace Operator rules, the total perturbation dPI of the

Performance index can be direcüy expressedin terms of the m perturbationsddiof the feedback vectors d{.

m

dPI = 2-2 trace ((abfPAdsdCX + n di Ct)XCfddf)i=l

(10.39)

Recalling that input vectors bt are single column vectors, whereas feedbackvectors dt and corresponding perturbations ddt are single row vectors (seedefinition equations (5.5) and (5.6) for the scalar controlstations), each Operandfor the trace function is a scalarquantity. Hence, the trace Operator in (10.39)can be omitted:

m

dPI = 2-2 [ablPAcisdCc+ridiCl)xclddjr;=i

(10.40)

By introducing the single row vector-gradient of the Performance index with

respect to the /th single row feedback vector,

dPI . ,i = \...mddi

(10.41)the perturbedPerformance index can also be formulatedas:

156

dP/= T—ddftri ddi

(10.42)

FinaUy,the result for the vector-gradientofthe Performance index is obtainedwhen one comparesexpressions (10.40) and (10.42):

dPI= 2 (a bJPAdsda + n di Ci)XCj i = \...m

ddi(10.43)

Hence, at each parameter point given by the m feedback vectors dt, the

vector-gradientof the Performance index with respect to the feedback

parameters in the /th control path can be analytically calculated usingexpression (10.43).

It is pointedout here again, that only a value 1 for the "shrink" coefficient a

guarantees stability ofthe closed-loop system upon Controller optimization.

Vector-Gradientincluding ParameterConstraints:

The procedure to determine the Performance index vector-gradient includingcontrol parameter constraints introduced in expression (5.15) with the

dependent feedback vectors fj is rather similar to that one describedabove.

Therefore, the modified terms correspondingto expressions (10.26) to (10.43)are just noted, and the equation numberingis marked with an additionallabel a.

The derivation steps are the same as shown above and a comment to themodified expressions is only addedwhere necessary.

Matrix P as starting point is necessaryto determine the Performance index

applying (5.11):

157

P = £ MUß+ 2 CjdJndid+ 2 cJfJrjfjCMaAcisdfk=P \ i=l 7=1 /

0 <a < 1

(10.26a)

AiPAcM- P + Ö + 2 CjdJndid + 2 cJffr-jfjCj = 0i = l 7=1

(10.27a)

«Ucw+2 ^diftCi+2 *y*}C/ (P+dP) AcW+2 ^drffCz-+2ä/d/yC/aV /=1 7=1 / V i=l 7=1 /

m; m/_ _

- (P+dP) + Q + 2 C^^+dJj^-^+ckfjQ+ 2 Cllfj+dftfTjfadfjjCj = 0/=1 7=1

(10.28a)/ m,- ^

_ _\r«2 **" ^ c*+2 *!/d-£ C7 p Acfada + «AlsddP AdsdccV=l 7=1 /

+ «Aä«,P 2 ^d*Ci+_2*;4öCyb - dP

+ 2 Cjddfndid+ Cjdfnddid + 2 tfvf'TjfjCj+ Cj'ffrjVjCj = 0;=i 7=i

(10.29a)

aA5*,dPActe/a - dP + ß + QT+ Ö + ß = 0»1;

ö = 2 tf<W(a bfPAdsda + n dt d)i=l

~ "*_ _ _

Ö = 2 cfdfj[abJPAcisda+fjfjCj)7=1

(10.30a)

158

At this stage it is necessaryto define perturbation dfj more precisely. As itwas pointed out in section (5.3.1.3), the dependent single row feedback vectors

fj contain elements whichmust be differentiable functions of arbitrary elementsof the independent feedback vectors dt. Assuming that the jth dependentfeedback vector contains / dependent elements,

fj = \fh(dd ,fj2(di), ... ,fji(dd] i = l...m,-, j = l...my(10.44)

its derivative with respect to the /th independent feedback vector dt can bedenoted as foUows:

ddi

ddi i ddi i ddi i

ddi2 ddi2 ddi2

% Vi? Jfh_

\_ddik. *ddik.' '9^.

/ = l.../n/, 7 = \...mj

(10.45)In expression (10.45) the assumption is made that the single row vector dt

contains kt elements.

Hence, the total perturbation dfj occurredin above equations can be writtenas follows:

m;

dfj = 2 Mi ¥i1/=1 ddi j = 1.../W/

(10.46)The further process of deriving the vector-gradient including additional

parameter constraintsis again analogous to that one described in the previoussection (expressions (10.31) to (10.37)). The only difference is that instead of

159

matrixQ the sum of Q andQ must be applied. Modified expression (10.38) thusresults in:

mi

dPI = 2-2 trace (cfddl(a bfPAdsdOt + n di Ct)x)i=l

"*_ _ _

+ 2-2 trace [cjdff[abjpAdsda + fjfjCj)x)7=1

(10.38a)

Combining expressions (10.46) and (10.38a) and changing the sequence ofsummation (indices / andj), the perturbation of the Performance index can be

expressedby the perturbations ddt of the independentfeedback vectors:

m;

dPI = 2-2 trace ((a bfPAdsda + n di C/)XCfddf)

+ 2-2 trace £ (ccb[PAdsda + TjfjC]\XCj&\ ddfiZ\ \y=i \ddij

(10.39a)

By applying expressions (10.41) and (10.42) to (10.39a), the final resultfor the

vector-gradient of the Performance index including additional parameterconstraints is obtained:

^ = 2\(ablPAcisdCC+ridiC!jXCl+2 (ab- PAdsda +TjfjCJ)xcJ'^ddi \ j=i ddii = \...rm

(10.43a)

160

10.6 SYSTEM DESCRIPTIONOF THESIMPLEELASTIC ROTOREXAMPLE

PhysicalPlant Properties:

mass m 0.25 kgmass M 1.0 kgmass excentricityAm 0.01 kgradial momentumof inertia& 0.04 kgm2axial momentumof inertiaC 0.0008 kgm2rotor lengthL 0.7 munbalance excentricityu 0.006 munbalance excentricityw 0.07 mrotor stiffnessEI 8350 Nm2force-current factor ki 50 N/Anegative bearing stiffness ks 50'000 N/mprescribed static bearing stiffness kst 75'000 N/m

Equations ofMotion in One Plane:

m 0 0 0

0 m 0 0

0 0 M 0

0 0 o-ö-L2J

XiX2X3Lip

+

3E*-ks 0L30 3E/

- kL3 '

3EI _3EIL3 L33EI 3EI

L L3 L3

3E7L33E7L3

6ELL30

M/L33ELL30

6£ZL3

xiX2X3

L(p

\kt 010 ki 110 0 L/2J

Lo oj(10.47)

161

Discrete-TimeState Space Description:

The discrete-time State space description corresponding to expressions(4.16)/(4.17) can be obtained by using aforementionedplant properties as weüas mass, stiffness and input matrices from the equations of motion.

Samplingtime T of 833 |is correspondsto a sampling frequencyof 1200 Hz.

Time-delay is neglected.

Discrete-TimeDescription ofthe AugmentedPlant (5.14):

Xk =

An =

A12 =

A21 =

A22 =

A31 =

Optant An Au 0

Xctr\ » A = A2i A22 0Xctr2 k

. A31 0 0.

1.0311 0 0 00 1.0611 0 00 0 0.88846 00 0 0 0.36726-

8.4197-104 0 0 0

0 i5.5023-10"4 0 0

0 0 8.021M0-4 0

0 0 0 6.4876-10^75.123 0 0 0

0 148.02 0 00 0 -262.61 00 0 0 -1333.5.

1.0311 0 0 00 1.0611 0 00 0 0.88846 00 0 0 0.36726.

-6.4747-IO"1 fI.9438-10"1 6.1444-10"1 -1.4560-10"1-6.47-47-10"1 -i5.9438-10"1 6.1*14-10"1 1.4560-10"1

162

-3.0442-10"5~

5.1962-10"52.4134-10"5

00

ih =

-2.5245-10"5-7.3437-IO"21.259610-15.6821-10"2-5.3204-IO"2

0

; h =

000000-1_0_

0~

-3.0442-10"5~ ~

-3.0442-10"55.1962-10"5 -5.1962-10"52.4134-10'5 2.4134-10"5-2.5245-10'5 2.5245-10"5

h =

-7.3437-IO'21.259610-15.6821-IO"2-5.3204-IO"2

0

0

; h =-7.3437-IO"2-1.2596-10-15.6821-IO"25.3204-IO"2

0

0

Ci = ;o o 0 0 0 0 0 0 1 0]c2 = [o 0 0 0 0 0 0 0 1 0]

Ci =

c2 =

c3 =

b3 =

000000000

-6.4747101 5.9438-10-1 6.1444-10"! -1.4560-101 0 0 0 0 0 0

0 0 0 0 000010.-6.474710-1 -5.9438-101 6.1444-10"! 1.456aKT1 0 0 0 0 0 0"

0 0 0[OOOOOOOOOl]

0 0 0 0 0 0 1

(10.48)

163

Weightingmatrices andfactors:

Q = diag(

n = r2 = n =

MO2 MO1MO2 MO1

1 MO11 MO1

MO"6MO"6

); Xp--= diag(MO"6MO"6

MO"6 MO"6MO-6 MO"6MO-6 MO-6

.

MO"6_ _

MO"6

f2 = r3 = ][-10-10

(10.49)

Proportionaland differentialfeedback parameters used in chapter 8:

proportional feed backPdifferentialfeedbackD

125*000 N/m25 Ns/m

10.7 CIRCLE PROPERTYOF ADAPTIVEUNBALANCE FORCECANCELLATIONMETHOD

The total harmonicforce signal in each magnetic bearing including feedbackand unbalance (expressions (8.7)/(8.8)) can be considered a constant

SS.

synchronous rotating force / for a given rotational speed Q. This scalar

quantity is interpretedas the sum of two parts, one resulting from unbalances

164

(fu) and the other from the additional harmoniccompensation input (Ü). ThefoUowingexpression can be formulatedwith complexnotation :

Ss, ss, s*+.

f = glfu + g2 U

(10.50)

Complex numbers gi and g2 in (10.50) represent the transfer characteristicsof the linear plant given by description (4.1) and are constant for a givenrotational speed Q. Complexnumber U includes the compensation parametersintroduced in section8.2.2:

U = au+ ibu

Since only the absolutevalue of / is of interestone obtains:

(10.51)

1^1 ss. s**.\f\= glfu + g2U <=> u -

SS.

-glfu\S2g2

(10.52)

Expression (10.52) is the equation of a circle in the complex 3u,öM-plane.Hence, the contourlines of \f \ are circular with radius r and centerC givenby:

r.V\g2

c =glfug2

(10.53)

165

REFERENCES

[Ackermann 85]Ackermann J.: Sampled-Data Control Systems, Communications andControlEngineeringSeries, Springer-Verlag, 1985

[Anderson & Liu 89]Anderson B.D.O and Liu Y.: Controller Reduction: Concepts andApproaches, IEEE Transactionson Automatic Control, Vol. 34, No. 8,pp. 802-812,1989

[Arnold85]Arnold W.: Beitrag zu Entwicklungund Einsatz aktiv magnetgelagerterHochgeschwindigkeits-Frässpindeln,Thesis TH Dannstadt, 1985

[Bals 89]Bals J.: Aktive Schwingungsdämpfungflexibler Strukturen, ThesisUniversitätFridericianaKarlsruhe, 1989

[Bathe 82]Bathe K.J.: Finite ElementProcedures in Engineering Analysis, Prentice-HaU, 1982

[Beams, Young & Moore46]Beams J.W., Young J.L. and Moore J.W.: The Production of HighCentrifugalFields, Journal ofAppliedPhysics, Vol. 17, pp. 886-890,1946

166

[Bleuler84]Bleuler H.: DecentralizedControl of Magnetic Rotor Bearing Systems,Thesis ETHNr. 7573, Swiss Federal Instituteof Technology,Zürich, 1984

[BoU&Warlimont81]BoU R. and WarlimontH.: Applications of AmorphousMagnetic Materialsin Electronics, IEEE Transactionson Magnetics, Vol. Mag.-17, No. 6,pp. 3053-3058,1981

[Braunbek 39]Braunbek W.: Freischwebende Körper im elektrischen und magnetischenFeld, Z. Phys., Vol. 112, pp. 753-763,1939

[Brunner 88]BrunnerU.A.: Ein Beitrag zur Vereinfachung linearer Regler, ThesisETHNr. 8519, Swiss Federal InstituteofTechnology,Zürich, 1988

[Bucher 85]Bucher Ch.: Contributions to the Modeling of Flexible Structures forVibration Control, Thesis ETH Nr. 7700, Swiss Federal Institute ofTechnology,Zürich, 1985

[Burrows,Sahinkaya,Traxler & Schweitzer88]Burrows CR., Sahinkaya N., TraxlerA. and Schweitzer G.: Design andApplication ofa Magnetic Bearingfor Vibration Control and Stabilization ofa FlexibleRotor, in [Schweitzer 88]

[Compar 87]Systembeschreibung zu CSP 320 25 Signalprozessorkarte, Compar AG,Richterswil, Switzerland, 1987

[Den Hartog 52]Den Hartog J.P.: Advanced Strength of Materials, McGraw-HülBookCompany,New York/Toronto/London, 1952

[Diez 89]Diez D.: Sicherheitstechnisches Entwerfen eines Mechatronik-Produktesam Beispiel eines Magnetlagersystems,Proceedings of the VDI-

Getriebetagung, Bad Homburg, Germany, November29/30, 1989

167

[Duwez, Wiüens& Klement60]Duwez P., WillensR.H. and KlementW. jr.: Journal ofApplied Physics 31,p. 1136,1960

[Ewins 84]Ewins O.J.: Modal Testing, Theory and Practice, Research Studies Press,1984

[Gasch & Pfützner 75]Gasch R. andPfützner H.: Rotordynamik, Eine Einführung, Springer-Verlag,Berlin/Heidelberg, 1975

[Geering76]Geering H.P.: On Calculating Gradient Matrices, IEEE TransactionsonAutomatic Control, Vol. AC-21, August1976

[Habermann 82]HabermannH.: Le palier magnetique actifActidyne,AGARD ConferenceProceedings No. 323, Problems in Bearings and Lubrication, Ottawa,Canada, May 1982

[Herzog 90]Herzog R.: Positiv reeüe Funktionenund H^-Optimierungzur RegelungvonelastischenStrukturen, Thesis ETH, Swiss Federal Institute of Technology,Zürich, to appearin 1990

[Hoffmann87]Hoffmann G.: Digitale Approximation kontinuierlicher Uebertragungs-systeme (QTA-Verfahren), Automatisierungstechnik (at), 35. Jahrgang,Heft4,pp. 148-155,1987

[Holden-Day63]Theory of Elasticity of an Anisotropie Elastic Body,Holden-Day Series inMathematical Physics, Holden-Day, Inc., San Francisco, 1963

[Kellenberger 87]KeUenbergerW.: ElastischesWuchten,Springer-Verlag, 1987

[Kemper 38]Kemper H.: SchwebendeAufhängung durch elektromagnetischeKräfte:Eine Möglichkeit für eine grundsätzlich neue Fortbewegungsart,ETZ 59,pp. 391-395,1938

168

[Kosut70]Kosut R.L.: Suboptimal Controlof Linear Time-InvariantSystems Subjectto Control Structure Constraints, IEEE Transactions on AutomaticControl,Vol. AC-15,No. 5, Oct. 1970

[Kuhn 85]Kuhn U.: Neue Entwurfsverfahren für die Regelung LinearerMehrgrössensysteme durch optimale Ausgangsrückführung, Thesis,Technische Universität München, Fakultät für Elektrotechnik, München,1985

[Larsonneur& Herzog 88]LarsonneurR. and Herzog R.: Optimal Design of Structure-PredefinedDiscrete Controlfor Rotors in Magnetic Bearings, in [Schweitzer 88]

[Larsonneur& Bleuler 89]Larsonneur R. and Bleuler H.: A New Approach to the Designof DynamicOutput Feedbackfor Active Magnetic Bearings (AMB), Proceedingsof theInternational Symposium on Advanced Computers for Dynamics and

Design(ACD&D),Tsuchiura, Japan, Sept. 6-8, 1989

[Levine& Äthans 70]Levine W.S. and Äthans M.: Onthe Determinationofthe Optimal ConstantOutput Feedback Gains for Linear Multivariable Systems, IEEETransactions on Automatic Control, Vol. AC-15,No. 1, Feb. 1970

[Liu & Chamis 65]Liu Y. and Chamis CC: Residual Stresses in FilamentWoundLaminatesand OptimumProgrammedWinding Tension, proceedings, section 5-D ofthe 20th Annual Meeting ofthe Society of the Plastics Industry, 1965

[Luenberger 84]LuenbergerD.G.: Linear and Nonlinear Programming, Addison-WesleyPublishing Company, 2nd edition, Massachusetts, 1984

[Ly U.L. 82]Ly U.L.: A Design Algorithm for Robust Low Order Controllers, Ph.D.dissertation, Dep. of Aeronautics and Astronautics, Stanford University,CA, 1982

169

[MADYN82]Program Descriptionfor MADYN, Ingenieur Büro Klement, Darmstadt,1982

[Matsushita, Bleuler, Sugaya& Kaneko88]MatsushitaO., Bleuler H., SugayaT. and KanekoR.: Modelingfor FlexibleMechanical Systems, in [Schweitzer 88]

[MüUer81]MüUerP.C.: Allgemeinelineare Theorie für Rotorsysteme ohne oder mitkleinen Unsymmetrien,Ingenieur-Archiv51, pp. 61-74, Springer-Verlag,1981

[Nonami & Kawamata 89]Nonami K. and Kawamata S.: FeedforwardControl to Unbalance ForceCanceUation for Flexible Rotor Systems, Proceedings of the InternationalSymposiumon Advanced Computersfor Dynamics and Design(ACD&D),Tsuchiura, Japan, Sept. 6-8,1989

[Nuss 87]Nuss U.: Optimierungdes Störverhaltens linearer Ausgangsregelungen,Automatisierungstechnik(at), 35. Jahrgang, Heft 7, pp. 290-295,1987

[Okada,Nagai& Shimane 89]Okada Y., Nagai B. and Shimane T.: Cross Feedback StabiUzation of theDigitally ControUed Magnetic Bearing, ASME, Conference on Vibrationand Noise, Montreal, September1989

[Pfeifer & Behnke 77]Pfeifer F. and Behnke W.: Magnetic Properties of Amorphous AlloyToroids based on Iron and Nickel, Journal of Magnetism and MagneticMaterials 6, pp. 80-83,1977

[Pietruszka& Wagner 82]Pietruszka W.D. and Wagner N.: Aktive Beeinflussung desSchwingungsverhaltens eines magnetisch gelagerten Rotors, VDI-BerichtNr. 456,1982

172

173

CURRICULUMVITAE

I was born in Cologne/Germanyon June 6, 1958. In 1961, my parents movedto Allschwü/Switzerland where I entered primary school in 1965. After

receiving the general certificate of education (Matura) in 1977 at the

GymnasiumOberwil/BL, I studied one year at the Technical University ofDortmund/Germany.In 1978,1 started my studies at the Swiss Federal Instituteof Technology(ETH) in Zürich. They were interruptedby one year ofmilitaryService in 1981. In autumn 1983, I graduated from the ETH with a diplomaproject in rotordynamicsand control. Since spring 1984 I have been working as

a teaching and research assistant at the Institute of Mechanics (ETH) underthe conduction of Prof. Dr. G. Schweitzer. During this time I was involved inresearch projects mainly in the field of active magnetic bearing Systems. Apartfrom my workat the ETH, I have been a staffmemberof MECOSTraxler AGin Winterthur/Switzerlandsince autumn 1989, where I have the opportunity to

contributeto industrial realizations of active magnetic bearing Systems.