FINAL REPORT 

EIGENVALUE ASSIGNMENT STRATEGIES 

IN ROTOR SYSTEMS 

John N. Youngblood 

Professor of Mechanical Engineering 

The University of Alabama 

and 

Kenneth J. Welzyn 

Graduate Assistant 

Department of Mechanical Engineering 

The University of Alabama I 

Prepared for 

National Aeronautics and Space Administration 

Marshall Space Flight Center 

NASA Grant NAG8034 

October 1986 

BER Report No. 389-177 I 

(NASA-Ct- 179906) E IG ENV ALUE ASS 1GN PlENT N 87- 1 22 12 

ZIBATEGIES IrJ RUlCR SYZ'IEMS Findl Lieport 

{Alacarna Univ., university.) 55 F CSCL 12A 

Uxla E 

G3/64 44879 
THE UNIVERSITY OF ALABAMA 

COLLEGE OF ENGINEERING 

The College of Engineering at The University of Alabama has an undergraduate enrollment of more than 2,300 students and a graduate enrollment exceeding 180. There are 

approximately 100 faculty members, a significant number of whom conduct research in 

addition to teaching. 

Research is an integral part of the educational program, and research interests of the 

faculty parallel academic specialities. A wide variety of projects are included in the overall 

research effort of the College, and these projects form a solid base for the graduate 

program which offers fourteen different master’s and five different doctor of philosophy 

degrees. 

Other organizations on the University campus that contribute to particular research 

needs of the College of Engineering are the Charles L. Seebeck Computer Center, Geological Survey of Alabama, Marine Environmental Sciences Consortium, Mineral Resources 

Institute-State Mine Experiment Station, Mineral Resources Research Institute, Natural 

Resources Center, School of Mines and Energy Development, Tuscaloosa Metallurgy 

Research Center of the U.S. Bureau of Mines, and the Research Grants Committee. 

This University community provides opportunities for interdisciplinary work in pursuit of 

the basic goals of teaching, research, and public service. 

BUREAU OF ENGINEERING RESEARCH 

The Bureau of Engineering Research (BER) is an integral part of the College of Engineering of The University of Alabama. The primary functions of the BER include: 1) identifying 

sources of funds and other outside support bases to encourage and promote the research 

and educational activities within the College of Engineering; 2) organizing and promoting 

the research interests and accomplishments of the engineering faculty and students; 

3) gssisting in the preparation, coordination, and execution of proposals, including 

research, equipment, and instructional proposals; 4) providing engineering faculty, 

students, and staff with services such as graphics and audiovisual support and typing and 

editing of proposals and scholarly works; 5) promoting faculty and staff development 

through travel and seed project support, incentive stipends, and publicity related to 

engineering faculty, students, and programs; 6) developing innovative methods by which 

the College of Engineering can increase its effectiveness in providing high quality educational opportunities for those with whom it has contact; and 7) providing a source of timely 

and accurate data that reflect the variety and depth of contributions made by the faculty, 

students, and staff of the College of Engineering to the overall success of the University in 

meeting its mission. 

Through these activities, the BER serves as a unit dedicated to assisting the College of 

Engineering faculty by providing significant and quality service activities. 
E I G E N V A L U E A S S I G N M E N T S T R A T E G I E S IN 

ROTOR SYSTEMS 

John N. Youngblood 

Professor of Mechanical Engineering 

The University of Alabama 

and 

Kenneth J. Welzyn 

G r a d u a t e A s s i s t a n t 

Department of Mechanical Engineering 

The University of Alabama 

Prepared for 

National Aeronautics and Space Administration 

Marshall Space Flight Center 

NASA Grant NAG8034 

October 1986 

BER Report No. 389-177 
PREFACE 

This report documents the work done on NASA Grant NAG8034 

to establish the control and direction of effective eigenvalue 

excursions of lightly damped, speed dependent rotor systems 

using passive control. Both second order and sixth order 

bi-axis, quasi-linear, speed dependent generic models were 

investigated. In every case a single, bi-directional control 

bearing was used in a passive feedback stabilization loop to 

resist modal destabilization above the rotor critical speed. 

Assuming incomplete state measurement, sub-optimal control 

s t r a t e g i e s w e r e used to define the preferred location of the 

control bearing, the most effective measurement locations, and 

the best set of control gains to extend the speed range of 

stable operation. Speed dependent control gains were found by 

Powell’s method to maximize the minimum modal damping ratio 

for the speed dependent linear model. An increase of 300 

percent in stable speed operation was obtained for the 

sixth order linear system using passive control. 

Simulations were run to examine the effectiveness of the 

linear control law on nonlinear r o t o r m o d e l s with bearing 

deadband. The maximum level of control effort (force) 

required by the control bearing to stabilize the rotor at 

speeds above the critical was determined for the models with 

bearing deadband. 

ii 
TABLE OF CONTENTS 

LISTOFTABLES ................. 

LIST OF FIGURES ................. 

CHAPTER 

I . INTRODUCTION ............. 

I1 . BACKGROUND .............. 

I11 . THE TWO DEGREE-OF-FREEDOM MODEL .... 

Introduction ............ Model Development ......... Characteristics of the 

Uncontrolled Rotor ....... The Control Actuator ........ Stabilizing the Rotor ....... Control Performance ........ Summary .............. 

IV . THE SIX DEGREE-OF-FREEDOM MODEL .... 

V . 

Introduction ............ The Model .............. Characteristics of the 

Uncontrolled Rotor ........ Control Structure .......... Control Determination ....... Control Application ........ Control Performance ........ Summary .............. 

CONCLUSIONS 

LIST OF REFERENCES . * * * v t * * * * 

iv 

V 

1 

3 

7 

7 

7 

18 

22 

23 

27 

29 

36 

36 

37 

42 

50 

52 

55 

68 

69 

82 

85 

ii i 
LIST OF TABLES 

3.1 Dimensionless Parameter Definitions. . . 17 

3.2 S y s t e m P a r a m e t e r Values. . . . . . . . . 19 

4.1 Dimensionless Parameter Definitions. . . 43 

4.2 Rotor Parameter Values . . . . . . . . . 45 

iv 
LIST OF FIGURES 

3.1 Two Degree.of.FreedomMode1 ......... 

3.2 Force Diagram for Rotor Model ........ 

3.3 Support Bearing Force-deflection Curve ... 

3.4 Uncontrolled Rotor Pole Trace ........ 

3.5 Unstable Response of Nonlinear System at 

R = 2.3 ................... 

3.6 Rotor Response at R = 5. = 0 ....... 

3.7 Maximum Required Control Force Vs . Spin Speed ................. 

3.8 Rotor Response to Impulse at R = 5. 

~=0 .................... 

3.9 Rotor Response to Impulse at R = 5. 

g=1 .................... 

3.10 Rotor Response to Impulse at R = 5. 

g=10 ................... 

3.11 Power Spectral Density for Response of 

Fig . 3.10. ................. 

4.1 Six Degree.of.FreedomMode1 ......... 

4.2 Uncontrolled Rotor Pole Trace ........ 

4.3 Modal Orbit Shape of First Unstable 

Mode at R z 1.775. ............. 

4.4 Modal Orbit Shape of Second Unstable 

Mode at R = 4.101. ............. 

4.5 Modai Orbit Shape of Tkird Vnstable 

Mode at R = 4.65 .............. 

4.6 Effect of Maximizing {RD .......... 

4.7 Control Gains Vs . Spin Speed ........ 

8 

11 

11 

20 

21 

30 

31 

32 

33 

34 

35 

38 

46 

47 

48 

49 

54 

59 

V 
4.0 

4.9 

4.10 

4.11 

4.12 

4.13 

4.14 

4.15 

4.16 

4.17 

4.18 

4.19 

4.20 

4.21 

4.22 

4.23 

4.24 

Controlled Rotor Pole Trace with 

Speed Dependent Control. . . . . . . . . . . 

Controlled Rotor Pole Trace with Speed 

Dependent Gains, Section A Enlarged. . , . . 

Controlled Rotor Pole Trace with Speed 

Dependent Gains, Section B Enlarged. . . . . 

Controlled Rotor Pole Trace with Speed 

Dependent Gains, Section C Enlarged. . . . . 

Controlled Rotor Pole Trace with 

Constant Gains . . . . . . . . . . . . . . . 

Controlled Rotor Pole Trace with 

Constant Gains, Section A Enlarged . . . . . 

Controlled Rotor Pole Trace with 

Constant Gains, Section B Enlarged . . . . . 

Controlled Rotor Pole Trace with 

Constant Gains, Section C Enlarged . . . . . 

Maximum Required Control Force Vs. 

Spin Speed . , . . , . . . . . . . . . . . . 

Rotor Response at Mass 1 for R = 2, 

~=l .................... 

Rotor Response at Mass 2 with R = 2, 

g'= 1.. . . . ... , . . . . . . *. . . . . 

Rotor Response at Mass 3 with R = 2, 

g=1 .................... 

Rotor Response at Mass 1 with R = 6, 

~=l .................... 

Rotor Response at Mass 2 with R = 6, 

g=1 .................... 

Rotor Response at Mass 3 with R 2 6, 

g=1 ........,........... 

- 

Power Spectral Density of Response of 

Fig.4.20 ..,.... . .......... 

Rotor Response at Mass 1 with R = 6, 

gz10 ................... 

60 

61 

62 

63 

64 

65 

66 

67 

71 

72 

73 

74 

75 

76 

77 

70 

79 

vi 
4.25 Rotor Response at Mass 2 with R = 6, 

g=iO ................... 8C 

4.26 Rotor Response at Mass 3 with R = 6, 

~=lO ................... 81 

vii 
CHAPTER I 

INTRODUCTION 

This study investigates the stabilization of 

self-excited rotor vibrations by active control. Rotordynamic systems under the influence of self-exciting 

mechanisms exhibit unstable behavior above some threshold speed. Quasi-linear, speed dependent models of 

these systems show eigenvalue excursions into the 

unstable zone as this threshold speed is approached. 

Nonlinear models with bearing deadband exhibit unstable 

oscillations above this same threshold speed. By the 

application of control forces as can be produced by 

magnetic bearings, the eigenvalues of the linear models 

are re-positioned s u c h t h a t the systems are stable. The 

control is then applied to the linear and nonlinear 

models by simulation. The effectiveness of the control 

is judged by the resulting stability characteristics of 

the nonlinear models and by the control bearing force 

levels required to m a i n t a i n r o t o r stability. 

Two rotor models are used in this study. The 

first of these is a two degree-of-freedom model resembling a modified Jeffcott model. Control bearing forces 

1 
2 

are sir-iictiired to counteract the self-exciting forces, 

resulting in stable operation at any rotor spin speed. 

By the use of a speed dependent control strategy, the 

eigenvalues of the closed-loop system are maintained at 

fixed locations. The control is chosen such that these 

locations match those of the original system at its 

optimally damped speed. 

The second rotor model used in this study is a six 

degree-of-freedom model. It is shown to possess three 

natural modes, each becoming unstable at a different 

rotor spin speed. The stabilization of all three modes 

using one magnetic bearing set is investigated. The 

ability of the control bearing to stabilize each mode is 

dependent upon such factors as bearing location, feedbacK signals, and control gains. The magnetic bearing 

location is determined by examining the uncontrolled 

r o t o r ’ s m o d e shapes, with placement attempting to maximize the bearing’s effect on e a c h m o d e . FeedbacK is 

provided through combined output signals with incomplete 

state information. The combination of the output signals and the control gains are determined by using a 

search method, which maximizes the damping of the least 

damped mode. Control strategies using constant and 

speed dependent control gains are compared Sased upcn 

the effect that each has on the excursions of the linear 

systems’ eigenvalues towards the unstable zone. 
CHAPTER I1 

BACKGROUND 

The first published work dealing with the dynamics 

of rotating shafts was by Rankine [l] in 1869. At that 

time Rankine showed that rotating systems were unstable 

above the critical speed (the speed at which synchronous 

resonance occurs). Because of this, for many years 

manufacturers designed rotating systems to operate below 

the critical speed. In 1919, Jeffcott [23 identified 

the critical speed, not as a speed of instability, but 

as a natural frequency of lateral vibration. By including damping in his analysis, Jeffcott showed that stable 

operation was possible above the critical speed. 

The increased demands on rotor systems for more 

power and lighter weight soon led manufacturers to 

design rotors for operation above the first critical 

speed. This led to instability problems, as rotors 

reached speeds above which stable operation was impossible. Several mechanisms of rotor instability have 

been identified and are the subject cf works by 

Ehrich [3, 43. These include hydrodynamic bearings and 

seals, internal rotor friction, and turbine blade 

3 
4 

effects. Each of the mechanisms of instability possess 

a common characteristic in that they produce forces 

tangential to radial deflections. This cross-coupling 

effect generally increases with speed until it overcomes 

the restoring and dissipative forces and the rotor 

becomes self-excited, Violent whirling or "whipping" of 

the rotor occurs as the stability threshold is reached. 

While most instability mechanisms are nonlinear 

phenomena, they can generally be modelled as linear 

systems with variable coefficients. A significant nonlinearity occurs, however, in bearings with radial 

clearances or lfdeadbandsl*. Although these are not 

usually considered mechanisms of instability, their 

effect is significant on rotor performance. This effect 

has been examined by Childs [5] and others [6,7] whose 

results show that these nonlinearities often result in 

subsynchronous oscillations at frequencies exactly 

one-half of rotor spin speed. It has also been shown 

that bearings with radial clearances do not affect the 

overall stability of rotor systems. 

The need for still greater performance from rotor 

systems requires operation at speeds above what is now 

the limit of stability. For this reason, the focus of 

much recent work has been on the improvement of rotor 

stability characteristics. This area has appeared more 

promising with the advent of active magnetic bearings. 
5 

Haberman and ilai;d [8j preseiii ar; exc2!!ent discussi~r! 

on the practical applications of these bearings. 

Schweitzer and Lange [9] present a more detailed 

description of magnetic bearing characteristics, and 

show how they can be implemented in a closed-loop 

control system. Gondhalekar and Holmes [lo] discuss the 

various designs of magnetic bearings for controlling 

rotors. They show, both analytically and experimentally, how suitable configurations of magnetic bearing 

systems can lead to linear, uncoupled, completely 

determined control forces. 

For stabilizing rotor systems, Burrows and 

Sahinkaya [Ill examine the use of magnetic bearings to 

control oil-whirl. They apply pole-placement techniques 

to a single-mass, rigid rotor supported by hydrodynamic 

journal bearings. They determine the contol gains 

neccessary to position the unstable pair of eigenvalues 

sufficiently far from the imaginary axis, and they 

discuss the constraints which affect reachable pole 

zones. More recently, Stanway and Burrows [12] examine 

rotor stabilization by applying control to the rotor’s 

support structure. They find that, with full state 

feedback, it is possible to obtain some degree of stabilization. They discuss the use of observers to construct 

the full state vector when it is not directly obtainable. Schweitzer 1131 examines stabilization, using 
6 

magnetic bearings, of a linear multi-body rotor with 

internal damping. He uses one magnetic bearing set and 

defines an approach to locate the bearing based on the 

rotor's mode shapes. He determines an 8*optimal" control 

by maximizing the lowest damping of the system's modes. 

In a later work [in] he applies magnetic bearing control 

to a low order model obtained from a higher order modalbased model. Schweitzer then develops the control for 

the lower order model and examines the 88spillover8t 

effects on the higher order system. 

The purpose of this study is to investigate the 

stabilization of self-excited rotor systems by active 

control. In particular, the application of linear 

control to nonlinear rotor systems under the influence 

of bearing deadband is examined. Its importance resu1t.s 

from the fact that nearly all physical rotors supported 

by radial rolling element bearings are affected by 

deadband. While this factor is usually neglected in 

most rotor studies, it is one which must be understood 

for the full benefits of stability enhancement to be 

achieved. 
CHAPTER I11 

THE TWO DEGREE-OF-FREEDOM MODEL 

Introduction 

The first phase of this study deals with the 

stabil ization of a two degree-of -f reedom model which is 

similar to a modified Jeffcott model. The equations of 

motion for the model are developed and converted to 

dimensionless form. The control strategy for rotor 

stabilization is developed for the linear rotor and 

applied to the linear and nonlinear models. Maximum 

control bearing forces required to stabilize the rotor 

are taken from dynamic simulations of linear and nonlinear models. Finally, the frequency spectrum of the 

nonlinear simulation is analyzed to observe the characteristic behavior of the nonlinear system. 

Model Development 

The two degree-of-freedom model is shown in 

Fig. 3.1. Although simple, this model is actually a 

very useful tool in dealing with realistic rotor problems. It is widely used for conducting analytical rotor 

studies, as well as gaining insight into various rotor 

phenomena. More significantly, this model exhibits many 

7 
8 

x 

E 

0 

a 

al 

al 

b 

Fr 

I 

cu 

0 

I 

PI 

al 

& 

M 

aJ 

n 
9 

of the types of behavior found in more complex rotor 

systems. 

As shown in Fig. 3.1, the model consists of a uniform disk of mass m suspended by a massless rigid shaft. 

The bearings have a radial clearance g which provides a 

"deadband" in the f orce-def lection curve. The shaft is 

supported on each end by massless radial bearings having 

a combined linear stiffness Kb. Damping 1s introduced 

by hydrodynamic seal type elements which provide stiffness and damping Ks and C,, respectively, and crosscoupled stiffness and damping as and Cq, respectively. 

Although for true rotor systems most parameter values 

are functions of rotor spin speed, the most significant 

is the cross-coupled stiffness Qs. Since this term is a 

source of self-excited vibration, it is sufficient to 

introduce speed dependency with this term only. Thus, 

it will be given the familiar approximate form for 

hydrodynamic seals, 

The rotor is unbalanced with the mass m at a radius e. 

The rotor is assumed to be symmetric, both radially and 

laterally. The rotor spin speed o is Constant, aithough 

different speeds will be examined. Motion occurs in the 

Y and Z directions only. 
10 

A force diagram is shown in Fig. 3.2, where Fbz 

and Fby are the restoring forces from the support bearings, FsZ and Fsy are the "seal1* forces, 

unbalance force. Fig. 3.3 shows a force-def lection 

curve for the support bearings. The magnitude of the 

radial force produced by the bearings is given by 

and mew2 is the 

with the force acting in the direction of radial 

displacement. Resolving this force into the Y and Z 

directions yields 

and 

and 

Fbz 

The forces due to the seals are given by 

Fsy = Ksy + C,? + Q,Z + C,i 

(3.4) 

(3.5) 
11 

The differing signs on the cross-coupling terms 

indicates a mechanism of instability. 

Fig. 3.2. Force Diagram for Rotor Model 

It g -1 Radial Deflection (r) 

Fig. 3.3. Support Bearing F o r c e - d e f l e c t i o n C u r v e 
12 

Combining the force reiationsnips in their respective directions yields the equations of motion for the 

two degree-of -f reedom rotor, 

Irl 

= meu2cos (ut 

I’l 

= meds in (ut 

Division by m yields 

QS 

(1 - - )y + - Y + - i+- Z 

Eb g KS cq 9.- CS i;+- 

m m m 

m 1’1 m 

= eu2cos(ut) 

and 

(3.7) 

(3.9) 

QS 

m m m m m 

j,-- Y .. CS Kb g KS cq z+- i + - (1 - - )z + - z-- 

PI 

= ewZsin(wt) . (3. IO) 

To effect a more general study, these equations 

are nondimensionalized. First, a system frequency is 

defined as the undamped, uncoupled frequency given by 

Ks Kb 

m 

wo2 = (3.11) 
13 

By defining the dimensionless time parameter T as 

T = wot 

and using the chain rule it follows that 

dy dy dr dY i=--.=--- - wo- = way'. 

dt dT dt dt 

S imi 1 ar ly, 

z = woz', 

and 

(3.12) 

(3.13) 

(3.14) 

(3.15) 

(3.16) 

Substituting for t, using (3.13) - (3.16) and dividing 

by wo2 yields 

Kb g KS 

y' + - (1 - - )Y + - y + - z' 

CS 

y" + - 

mu mu 2 mu 2 mw 

0 0 Irl 0 0 

(3.17) 

QS 

mwo2 

+- 

and 
14 

gb 8 KS 

z, + - (1 - - )z + - Z--y# 

ZS 

2" + - 

mu 2 mu 

0 0 

mw mw 2 

0 0 

(3.18) 

Now make the following definitions: 

CS cq 

w 

5s=-, 54 =-' - R, 

00 2mw0 2mw0 

Substituting these yields 

+ ~,RZ = R~~COS(RT) (3.19) 

and 

- CsRy = R2esin(Rr). (3.20) 

Finally, division by a characteristic length, say 

e, yields the dimensionless model 
(3.21) 

- CsRT = R2sin(Rr). (3.22) 

Table 3.1 summarizes the dimensionless parameter definitions. 

Equations (3.21) and (3.22) represent the nonlinear, uncontrolled rotor. Defining the state vector _x 

as 

the system in first-order form becomes 

- X* = Ax + 

where 

0 

0 

A= 

0 1 0 

0 0 1 

(3.23) 

(3.24) 

(3.25) 

-25 

S 

-5 R -25 

-(R + R ) 25 

Kb s 

S KS kb 9 
16 

and 

d= (3.26) 

Note that the vector 4 contains both the unbalance 

forces and the nonlinear contributions from the support 

bearings. The same restrictions apply as in equations 

(3.3) and (3.4) to the nonlinear terms, i.e., 

Also note that, by definition, Rks + RKb : 1, so the 

s y s t e m m a t r i x becomes 

A= 

0 0 1 0 

0 0 0 1 

-1 -5 R -25 -25 

S S q 

(3.27) 
17 

Table 3.1 

Dimensionless Parameter Definitions 

Nomenclature Parameter Equ iva 1 en t 

Frequency - K +K 

m 

0 

0 

C 

s 

2mw 

Damping 

S 

s 

St iff ness K 

R 

kb K +K 

K 

S b 

R 

ks 

L 

K +K 

Cross-Coupled 

Damping I 

C 

5 

9 2mw 

5R 

S 

Cross-Coupled 

St iff ness 

5R 

S 

w 

- 

0 

w 

Y 

- 

E 

Spin Speed R 

Di s p 1 ac ements - 

Y 

Z 

- 

E 

Z 

De adband 

Time 

0 

wt 
Characteristics of the Uncontrolled Rotor 

This study concerns stabilization of rotors for 

speeds at which the uncontrolled rotor is unstable. For 

nonlinear systems, however, stability and instability 

are amplitude dependent. When examining the instability 

of a rotor with deadband, it is apparent that the effect 

of the deadband is diminished as the rotor’s amplitudes 

of vibration begin to grow. Global stability can be 

determined by neglecting the deadband, producing a linear system compatible with various linear analysis techniques. Therefore, this study deals with analyzing linear systems obtained by neglecting deadband. The 

results of these analyses are applied to the nonlinear 

system, which is examined by simulation. 

The eigenvalues of the linear system can be 

readily obtained and will provide nearly all of the 

information required, both for examining the stability 

of the rotor and for determining the desired control 

laws. Examination of the system matrix A shows a speed 

dependency in the term gsR. Thus, for any given speed 

R, a different linear system is obtained. At some 

speed, the system becomes unstable. That speed is the 

onset speed of instability or the instability threshold 

speed. Using the parameter values shown in Table 3.2, 

the system eigenvalues over the speed range 0 5 R I 5 
19 

are shown in Fig. 3.4. The system is at the threshold 

of instability for a speed of R = 2.2. The frequency of 

the unstable mode ("whirl" ) is 1.1, approximately onehalf that of the rotor spin speed, characteristic of 

self-excited rotors. As expected, the nonlinear system 

with deadband becomes unstable above R = 2.2. This is 

shown by simulation results depicted in Fig. 3.5, using 

a deadband = 1 for R = 2.3. 

P ar ame t e r 

Table 3.2 

Value 

S 

q 

5 

s 

0.5 

0.1 
E 

0 

c 

bErn V 

cla 

wv 

-lo 

-l' 

0 

CK 

t7 

0 

0 

7 

3 

I I I I I I I I I I I I I I 

u! 

0 

2 

To 

F 

0 

I 

'? 

I 

0 

u? 

0 

I 

2 

I 

? 

0 

I 

7 

F 

I 

'? 

r 

I 

? 

r 

I 

0000000000000000 

"~"~0000000000 

"tt?q79S9?9"t?cyTO 
21 

2.1 

# 

A 

1 .f 

1 

-1 

-1 .5 

-2 

-2.5 

ROTOR CENTER POSITION 

Uncontrolled Rotor at R = 2,3 

I 

I 

I I I I 1 I I I I I 

0.5 1.5 2S -2,s -1.5 4.5 

husonds) 

Y 

Fig. 3.5. Unstable Response of Nonlinear System at R = 2.3. 
22 

- The Control Actuator 

The problems of applying completely definable 

forces to high speed rotating machinery have been the 

focus of much recent study 18, 9, IO]. The result of this 

effort is a versatile class of devices widely known as 

magnetic bearings. These bearings utilize electromagnetic interaction to produce forces on shafts suspended 

within their housings. Many experimental studies and 

several practical applications attest to the feasibility 

of these bearings for use in rotating machinery. 

The force produced by a magnetic bearing can be 

made relatively proportional to the driving current. The 

use of two bearings grouped together results in a device 

which can produce two independent forces in arbitrary 

directions. By structuring the input currents to be 

functions of the states of the system, a closed-loop 

control system is produced. The forces can be made to 

resemble stiffness, damping, combinations of these, or 

other suitable forms. 

For the purposes of this study, the magnetic bearing forces are assumed to act in orthogonal directions 

coinciding with the Y and Z rotor coordinate directions. 

The transients associated with the actuator and control 

loop are assumed to be negligible compared with those of 

the rotor. The input current is proportioned to the 

state vector ,x through a gain matrix G so that 
23 

- u= [:: 1 -Gx 

where 

g21 g22 g23 g24 

G = [ 

(3.28) 

(3.29) 

Stabilizing the Rotor 

Modifying equation (3.24) to include the control 

forces acting through a control distribution matrix B 

yields 

- X0 = A3 + B_u + 4. (3.30) 

Again, since system stability is of interest, 

bearing deadband is t e m p o r a r i l y n e g l e c t e d . The stability 

of linear systems is independent of input, so the vector d is neglected temporarily as well. The resulting 

linear system is 

- X0 = Ag + B_u (3.31) 

or 

- x0 = (A - BG)x. (3.32) 
24 

Since there are independent] unconstrained control 

forces acting in each direction, it can be shown that 

all states are accessible by the control, and the system 

is controllable. Thus, by selection of the gain matrix 

GI the eigenvalues of (A - BG) can be arbitrarily positioned. 

A simple method for stabilizing the rotor can be 

developed by examination of the closed-loop system matrix (A - BG). From the definition of the state vector g 

and the form of the input g, the control distribution 

matrix B has the form 

(3.33) 

This results in the closed-loop s y s t e m m a t r i x 

(3.34) 

Since the rotor is radially symmetric, it foiiows 

that the control should conform to the same symmetry. 
Theref =re 

g21 = -g12 

g22 = g21 

g23 = -g14 

(3.35) 

(3.36) 

(3.37) 

and 

e24 = g13 . 

Thus, 

0 0 

0 0 

1 

0 

25 +gi3 

S 

(3.38) 

0 

1 

25 +g14 

q 

(3.39) 

Examination of this matrix shows a speed dependency only with the term (gsR + g12). By selection of 

g12 as a linear function of R, this term can be held 

constant for any spin speed. Furthermore, by setting 

g11 = g13 = g14 = 0, the above matrix at an arbitrary 

spin speed can be made equal to the open-loop matrix A 

at some speed, say Ro. In other words, making 

will make 

(3.40) 
26 

(A - BG)R = AR~ . (3.41) 

This essentially cancels the effect of the cross-coupled 

stiffness, which is the mechanism of instability in this 

model. Furthermore, the eigenvalues of (A - BG) at any 

spin speed R remain constant and equal to those of A at 

the speed Ro. Thus, the problem of control determination is reduced to choosing the speed Ro for which the 

system has the desired characteristics. 

One logical choice of the "reference speed" R, can 

be made by examining the pole-trace of Fig. 3.4. At the 

speed R = 0, the system has the highest relative damping 

of any speed. Relative damping associated with the ith 

mode is defined as 

-*i 

5RD = Jim (3.42) 

where Xi = di +, jwi represents the ith eigenvalue of the 

system. As an eigenvalue moves towards the unstable 

zone, its relative damping decreases. The minimum relative damping thus becomes a useful measure of the stability of the system. At R = 0, this value is maximum 

(~RD = 0.497), with both modes being equally damped. 

For comparison, the minimum relative damping at the 

uncontrolled first critical speed of R = 0.78 is 

~RD = 0,323. Thus, the choice of Ro = 0 results in an 
27 

"optimally" stable system with a 54X increase in critical speed damping. ('*Optimalii of course, refers only to 

the control approach chosen here.) 

Using equation (3.40) with Ro = 0 yields 

and 

(3.43) 

(3.44) 

This keeps the poles of the closed-loop system at 

Xi = -0,55723 ,+ j(O.97366)' and 

A2 = -0.44272 ,+ j(0.77366) 

for any rotor spin speed R. 

Control Performance 

While knowledge of the system poles gives an indication of the degree of stability and damping of the 

system, it gives no insight into other factors relative 

t o t h e performance of the control. For example, while 

the implemented gains may cause a stable eigenvalue 

assignment, t h e f o r c e levels required to accomplish the 

stabilization may be higher than the physical actuators 

can produce. Also, the control of the n o n l i n e a r s y s t e m 
28 

is of concern. These questions can be addressed by 

dynamically simulating the controlled rotor. 

Using the control developed in the previous section, n u m e r o u s s i m u l a t i o n s w e r e c o n d u c t e d for various 

spin speeds and bearing deadband values. Fig. 3.6 shows 

a typical simulation result. For each case, the maximum 

control force required during the steady-state rotor 

response was computed. These are plotted in Fig. 3.7 

versus spin speed. The three cases include the linear 

system (g = 0) and two nonlinear systems (2 = 1, and 

= IO). This figure shows a nearly linear dependence 

of t h e m a x i m u m control force on the spin speed R. 

Furthermore, the variation between linear and nonlinear 

cases is slight, if not negligible. Another interesting 

performance indicator is how well the rotor responds to 

an impulsive disturbance. Fig. 3.8 through Fig. 3.10 

s h o w t h e rotor response to an impulse velocity in one 

direction. Again, the cases include the linear system 

and two nonlinear systems, showing t h e m a x i m u m control 

force for each. While for these cases the maximum 

forces are higher and increase with deadband, this is to 

be expected. Loss of stiffness due to deadband causes a 

greater excursion from the steady-state orbit, producing 

a higher control force. 

Finally, the frequencies of a rotor orbit are 

often of interest. Sub-synchronous whirls cause cyclic 
29 

flexure of the rotor shaft and are often destructive. 

They also indicate the presence of self-excited vibration mechanisms which lead to instability as the speed 

is increased. Fig. 3.11 shows the frequency composition 

of the rotor response of Fig. 3.10. it is predominantly 

synchronous, as the self-exciting forces are counteracted by the control, and subsynchronous components are 

not detectable. 

Summary 

In this chapter, a two degree-of freedom model was 

developed and stabilized for an arbitrary speed . While 

the control approach utilized was fairly simplistic, it 

was shown to be quite effective. More importantly, it 

allowed for an evaluation of the control performance. 

While such indicators as control force levels and the 

response of the nonlinear rotor may change quantitatively for different control schemes, the trend of each 

would probably not differ substantially. The results of 

this chapter will be applied to a more realistic, complex rotor system in chapter four. 
30 

ROTOR CENTER POSITION 

N 

-1.20 4.80 4.40 0.00 0.40 0.80 1.20 

Y 

Fig. 3.6. Rotor Response at R = 5, = 0. 
31 

5 

E 

d 

X 

I 
32 

ROTOR RESPONSE TO IMPULSE 

- 

g = 0, R = 5.0, Fmox = 9,3777 

4.00 

N 

Y 

Fig. 3.8. Rotor Response to Impulse at R = 5, = 0. 
4.00 

300 

200 

N 1.00 

0.00 

-1 .00 

-200 I I I I I I 

ROTOR RESPONSE TO IMPULSE 

- 

g = 1, R = !io, Fmox = 9,7128 

I I 

-3.50 1.50 -2.50 -1 850 -0sQ 0.50 

Y 

Fig. 3.9. Rotor Response to Impulse at R = 5, = 1. 
34 

500 

4.00 

300 

N 200 

I .00 

a00 

-1.00 ' 

ROTOR RESPONSE TO IMPULSE 

- 

g = 10, R = 5,0, Fmox = 11,3646 

-2.00 0.00 2.00 

Y 

Fig. 3.10. Rotor Response to Impulse at R = 5, = 10. 
35 

9 

I 

d 

I 

IO 

In 

K 

r 

x 

0 

C 

U 

3 

U 

LL 

e 
CHAPTER IV 

THE SIX DEGREE-OF-FREEDOM MODEL 

Introduction 

This chapter deals with stabilizing a six degreeof -freedom rotor using one active magnetic bearing set. 

In the previous chapter, a two degree-of-freedom rotor 

was stabilized by using the control bearing to counteract 

self-exciting forces. The result was a rotor that was 

stable for any given spin speed. In many physical 

rotors, however, the locations of self-exciting mechanisms dimensionally restrict the positioning and number 

of control forces that may be applied. Full state feedback for large rotor systems would also be impractical. 

For these reasons, the s ix degree-of -f reedom rotor model 

is examined. This model enables the introduction of 

several self-exciting mechanisms and nonlinearities 

located away from the point of control application. To 

investigate the stabilization of this rotor, the model 

is presented in dimensionless form. Its uncontrolled 

characteristics are studied to determine such factors as 

control location and signal measurement. An "optimal" 

stability criterion is defined using the eigenvalues of 

36 
37 

the linear system. The control gains are determined by 

a search method. Finally, simulations are conducted to 

determine required control force levels for the linear 

and nonlinear systems. 

-- The Model 

The six degree-of-freedom model is shown in 

Fig. 4.1. This rotor has three uniform lumped masses 

mi, m2, and m3 connected by two massless, flexible 

shafts of stiffness Kf1 and Kf2. Acting at each mass 

are support bearings K1, K2 and K3, respectively. These 

are radial type bearings, each with a deadband gi, 

(i = 1, 2, 3). Also at each mass are elements which provide cross-coupled stiffness and damping Qsi and Cqi, 

as well as support stiffness and damping Ksi and Csi. 

Each mass is "unbalanced" by its mass mi at a radius €j, 

where all three unbalanced radii are assumed to lie in 

the same plane. The rotor is assumed to be radially 

symmetric operating at a constant spin speed w. Only 

motion in the Y and Z directions is considered. Gyroscopic effects, which can be shown to enhance rotor stability, are neglected. 

The support bearings are initially assumed to have 

zero deadband. This simplifies the form of the equations of motion. The non:inear contributions from deadband are included during the simulations by a 
38 

0 

I 

aJ 

X 
39 

disturbance vector. The resulting system is linear with 

the following equations of motion: 

By defining the dimensionless parameters shown in 

Table 4.1, the equations are nondimensionalized and 

become: 
40 
the system in first-order form becomes 

- X' = Ax + 1 

where 

A= 

and 

K= 

(R +R 1 R 

R R 

A 

mi mi 

0 

R (R +R +R ) R 

2- 22 

R R 

m2 mi? I 0 

R 

2 

Rm3 Rm3 

(4.14) 

(4.15) 

(4.16) 

(4.17 ) 
42 

and 

- d = disturbance vector of unbalance and nonlinear 

bearing forces. 

0 

5R 

2 

R 

m2 

0 

(4.18) 

(4.19) 

Characteristics of the Uncontrolled Rotor 

This section describes the characteristics of the 

rotor model developed in the previous section. Table 

4.2 shows the parameter values used for this model. 

As in the previous chapter, stability can be 

determined by examination of the poles of the linear 

system, Fig. 4.2 shows a trace of the rotor poles for 

0 5 R 55. The first mode is unstable above R 1.78, 

the second above R = 4.1 and the third above !? = 4.65. 

Fig. 4.3, Fig. 4.4, and Fig. 4.5 show modal "orbit" 
43 

Table 4.1 

Dimensionless Parameter Definitions 

Nomenclature 

Frequency 

Mass 

Damping 

St iff ness 

Cross-Coupled 

Damping 

Cross-Coupled 

Stiffness 

Spin Speed 

Di s p 1 ac ements 

De adb and 

Time 

Parameter 

w 

0 

R 

mi 

R 

K 

R 

f 

5R si 

R 

- - 

Yt 2 i i 

Equivalent 

I 

m 

-A 

t 

M 

1 

C 

- 2m u 

K +K. 

Sl 

K 

t 

K 

-fi, 

t 

K 

C 

ai 

2m.w 

10 

si 5R 

w 

w 

0 

Y 2 

-1 -A 

e e 

t t_ 

g 

2 

e .b 
44 

shapes for each of these modes at their respective 

unstable speeds. For the chosen mode, the modal orbits 

are obtained by combining the complex eigenvector and 

eigenvalue into the form of a solution and retaining the 

real part. These orbits show that the (mainly) rigid 

body "bounce" mode is the first to become unstable, followed by the rigid body "pitch" mode, and finally the 

flexible mode. This is the expected order, as the lower 

frequency (rigid) modes are most easily excited. 

.I 
45 

Table 4.2 

Rotor Parameter Values 

Parameter 

R 

R 

mi 

m2 

R 

m3 

R 

IC1 

R 

k2 

R 

R3 

R 

f2 

R 

R 

R 

€1 

€2 

Value ~~ 

0.25 

0.50 

0.25 

0.25 

0 

0.25 

0.50 

0 

0.50 

0.50 

0.50 

0.05 

0 

0.05 

0 

1.00 

0 
46 

00 000 000 00 000 000' 

??9Yc??9?9t?9?9tc??9 m@J@J@4@J@JrrrrroOooo 

0 

I I I I I I I I I I I I I I 0 

'9 
I I I I I I I I I I 

47 

rn 

h 

h 

4 

II 

U 

a 

al 

rl 

.. s 

a 

U 

3 

2 

al 

a 

(d 

U 

d 

P 

k 

0 

m 

U 

M 

d 

k4 
48 

r 

F 

0 

4 

l 

a 

Y 

0 

u 

U 

0 

I 

u 

3 

0 

3 

0 

C 

U 

v) 

Y 

E 

8 

I I I I I I I I 

U 

(d 

Q) 

a 

2 

Q) 

rl 

P 

.. a 

L1 m 

C 

3 

a 

C 

0 

0 

Q) 

v) 

rcc 

0 

Q) 

a 

a 

s 

v) 

U 

rl 

P 

$4 

0 

U 

U 
49 

w 

I I I I I I I I 
50 

Control Structure 

In this section, the control structure is developed and implemented on the linear rotor system. 

Disturbances are neglected. Taking the system without 

disturbances, the control y is added so that 

- X' = A2 + By. (4.20) 

Now let 

- U = -Gy 

where 

y = cx. 

(4.21) 

(4.22) 

The above control form is chosen because full state 

feedback is impractical on physical rotor systems. 

The closed-loop system becomes 

- X' = (A - BGC)_x. (4.23) 

To produce the two control forces 

(4.24) 

the gain matrix must be based on the form of the 

output y. The output y is chosen to take the form 
where 

- 

abcOOOOOOOOO 

OOOdefOOOOOO 

OOOOOOabcOOO 

0 0 0 0 0 0 0 0 0 d e f i A 

Yc is a linear combination of the Y-direction 

displacements y1, y2, y3, 

Vyc is a linear combination of the Y-direction 

velocities vyl, vy2, vy3, 

2, is a linear combination of the Z-direction 

displacements 21, 22, 23 , and 

V,, is a linear combination of the 2-direction 

velocities vzi, vz2, v23. 

Obviously, there are many ways to structure the output 

y. The above choice is logical, however, in that it 

simplifies the structure of both the output matrix C, 

and the resulting gain matrix G. Recalling the radial 

symmetry of the rotor, it is apparent that the Y and Z 

directional measurements of the output should be symmetric. This results in C having the form 

c= (4.26) 
52 

Also from symmetry and the results of the previous 

chapter, the gain matrix G has the form 

(4.27 ) 

gl 1 g12 J -43 14 

The apparent difference in equation (4.27) and the gain 

matrix of the previous chapter results from the order of 

the state vector. The control distribution matrix B 

depends upon the chosen point of application of the control forces. It is assumed that both forces will be 

applied at one mass only. 

Control Determination 

To stabilize the rotor represented by the closedloop system of equation (4.23), it is sufficient to 

position all of the poles in the left half plane. It is 

not necessary, however, to place all the poles to 

achieve stability. Thus, the system of equation (4.23) 

need not be completely controllable. By using one magnetic bearing set in the absence of full state feedback, 

the control is suboptimal. 

The various analytical techniques available for 

dealing with the 1 inear suboptimal control problem generally influence a portion of the system in a specif ic 

way, and minimize the influence of the control on the 
53 

remainder of the system. Modal control, for example, 

"fine-tunes** a specific mode or modes, while the "spillover" of the control to other modes is not directly 

controlled. The use of this technique for this application limits the number of directly controllable modes to 

one. While this allows the rotor to operate above 

previously unstable speeds, the influence of the control 

on the two remaining modes would be questionable. 

For these reasons, a direct search method for 

determining the control gains is used. Using Powell's 

method 1151 a function is minimized by iteratively 

searching an independent vector, in this case, the control gains. As discussed in the previous chapter, a 

useful measure of the stability of the system is the 

relative damping of the systems's modes. The relative 

damping ~RD of the ith mode is defined as before to be 

where Xi = di 2 jwi is the eigenvalue associated with 

that mode. It is the minimum relative damping which is 

critical to the stability of the system. Maximizing 

that minimum relative damping is equivalent to maximizing the shaded region of Fig. 4.6. The minimum relative damping of the closed-loop system is determined by 

a computational function. The negative of this function 
54 

is minimized using Powell's method to determine the Control gains. The applications of this approach are discussed in the next section. 
55 

Control Application 

Before the control gains are determined, two 

factors must be specified concerning the control: 

(1) Point of application of the bearing, 

(2) Combination of measurements for the output 

signal. 

To best determine the location for applying the 

control, it is useful to examine the uncontrolled 

rotor's modal orbit shapes. Recalling Fig. 4.3 through 

Fig. 4.5, it can be seen that, for affecting the first 

unstable mode (Fig. 4.3), the bearing can be located at 

either three mass positions. This is because that mode 

is very nearly a rigid body translational (or "bounce") 

mode. Examination of the next unstable mode (Fig. 4.4) 

clearly shows that a bearing located at the center mass 

location offers no stabilizing effect for this mode. 

Since this mode is unstable at a speed lower than the 

third mode, controlling it takes priority over modes 

that follow. Therefore, the magnetic bearing should be 

located at an end mass location, say that of mi. 

The decision of where to measure and how to combine the signals for the output y is not as simple as 

the location of the control. It is helpful to make a 

restriction regarding this factor. Since full state 

feedback is impractical for most rotor systems, it is 

assumed that the information from all three mass 
56 

locations will not be available. The output can contain 

information from two locations, at most. An examination 

of the uncontrolled rotor's modal orbit shapes shows 

that if information is provided f r o m t h e c e n t e r m a s s 

location only, the second ('*pitch") mode cannot be 

affected. Thus, an end location must be measured. For 

stabilization of the third mode, information from the 

c e n t e r m a s s is required as well. To gain more insight 

into the question of measurement, the search program was 

run for varying combinations of measuring locations. 

For each trial, the stabilization capabilities were 

observed. The results of these show that the best degree 

of stabilization is obtained from signal measurement at 

the center mass location and the end location where the 

magnetic bearing is to be located. Furthermore, by 

appropriate weighting of the measurements an even higher 

degree of stabiliztion is achieved. 

Using the end mass mi as a location for the magnetic bearing and measuring the displacement and velocity signals at the end and center masses mi and m2 with 

O1 

21000000000 i 

O1 

10 0 0 0 0 0 0 0 0 2 1 
57 

the search program was used to determine the control 

gains. For various spin speeds the gains were computed 

to maximize the relative damping. For the chosen output 

and bearing location, the rotor was stabilized for 

speeds up to approximately R = 7.2. Fig. 4.7 shows the 

control gains obtained over the speed range 0 5 R 5 8. 

This figure illustrates that for any given speed, there 

are numerous solutions which optimize the damping. 

Using the gains obtained from a prior speed as starting 

values and incrementing the speed slightly, it was possible to "tracK" a particular solution over the speed 

range. The irregularities observable in the figure, 

however, show the difficulties associated with this 

approach. 

The poles of the controlled rotor system using the 

gains of Fig. 4.7 are shown in Fig. 4.8. Enlarged views 

of various sections are shown in Fig. 4.9 through 

Fig. 4.11. Due to the somewhat irregular nature of the 

control gain solutions, the behavior of the poles, especially at lower speeds, is somewhat erratic. 

For comparison, the control gains from Fig, 4.7 at 

the stability threshold speed (R = 7.2) are used and 

held constant over the entire speed range. Fig. 4.12 

shows the rotor poles for these constant gains. As seen 

in the enlarged views of Fig. 4.13 through 4.15, t.he 
58 

poles are more well behaved than those obtained using 

speed dependent gains. More importantly, however, is 

the fact that the system is stable over the same speed 

range. A close comparison of Fig. 4.8 with Fig. 4.12 

reveals that the overall behavior of the poles does not 

differ significantly. This indicates that, at least for 

this model, there is little advantage to using speed 

dependent gains. Therefore, the rest of this chapter 

will deal with systems controlled by the constant gains 

obtained at R = 7.2 
I 

i 

00000000000000 

?3u?tY?Tq-?Y?u3? 

00000000000000 

111111 

a 

al 

al 

a 

en 

E 

d 

-a 

en 

II) 

3 

2 

.rl 

a cl 

r( 

0 

L.l 

U 

E 

0 

u 

rU 

M 

d 

Fz, 
60 

0 

0 

0 

8 

0 

0 

0 

-7- 

0 

0 

I 

F 

'0 

0 

0 

'? 

1 

0 

0 

L? 

I 

0 

0 

? 
v, 

W 

0 

0 

0 

O0 

0 

0 

0 

0 

moo 0 

0 

0 

0 

0 

0 

I I I I 

0 0 

7 

7 

c'! 

7 

0 

9 

7 

0 

0 

o! 

0 

0 

a9 

61 

0 

17 

0 

0 

0 

'9 

0 

7 

0 

I 

0 

0 

@! 

I 

0 

'? 

0 

01 

2 

a al 

M 

Ll 

a 

n 

c 

w 

4 

c 

0 

d 

U 

0 

aJ 

c/J 

* 

2 

d 

a 

c3 

U 

c 

aJ 

5 

C 

.a 

a 

aJ 

a 

aJ 

aJ 

a 

c/J 

n 

5 

d 

a 

u 

a 

Ll 

w 

9) 

rl 

0 nl 

Ll 

0 

U 

3 

2 

a 

aJ 

rl 

rl 

0 

h 

U 

C 

0 

u 

Q\ 

U 

M 

-74 

ir, 
62 

00 

00 

00 

00 

0 0 

0 0 

0 0 

0 0 

0 0 

0 0 

00 

00 

V V 

OO 0 

0 

00 

00 

0% 

00 

00 

go 

00 

00 

0 00 

8 *O 

OO eo 0 

O ooo*og 0 00 

ow 

I 

0 

.r 

0 

0 

0 

9 

0 

F 

d 

1 

0 

q 

0 

I 

0 

0 

'? 
0 

u 

P 

0 

0 

89 

0 

0 

0 

0 

0 # 

0 

0 

*O 

OO 

00 

0 

0 

0 

0 

0 OO 

'0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 2 

0 

0 

0 

O0 9% 

I 1 I I I 

63 

0 

0 

,q 

I 

0 

0 

I 

$ 

0 

4 

0 

I 

0 

0 

'4 

I 

0 

0 

? 

a 

0) 

M 

k 

a 

rl 

c 

w 

u 

c 

0 

d 

U 

c) 

al 

a 

2 

d 

(d 

c3 

U 

C 

al 

w 

C 

-a 

a 

al 

-0 

al 

0) 

a 

a 

n 

5 

d 

al 

0 

(d 

k 

w 

al 

rl 

0 

PI 

k 

0 

U 

0 

d 

a 

al 

rl 

rl 

0 

Fc 

U 

u 

3 

g 

4 

4 

U 

M 

d 

tr, 
64 

n 

'5 

E 

(3 

C 

0 

0 

0 

Y 

Y 

E 

c 

7 

0 

0 

0 

0 

0 

0 

4R 4 

U 

0 

u 

I I I I I I I I I I I 1 I I 

0 

0 

'9 

0 

0 

1 

I 

0 

+: 

0 

I 

0 

0 

'9 

I 

0 

0 

as 

b 

0 

U 

2 

-a 

a 

d 

d 

0 

b 

U 

C 

0 

u 

N 

e 

M 

d 

k 

4 
65 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 0 

0 

0 

0 

9 

0 

0 

r': 

0 

0 

Q? 

0 

-r 

0 

0 

0 

-9 

0 

F 

0 

I 

0 

0 

I 

0 

0 

? 

0' 

u! 

0 
66 

vl 

W 

-I 

0 

a, 

QL 

0 

t0 

QL 

0 

QL 

I7 

0 

0 

I 

0 

I? I" 

a 

a 

eo 

f.4 

Q 

d 

G 

w 

a 

G 

0 

d 

U 

U 

a 

VJ 
0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 * O\ 8 

0 

0 

0 

0 0 

0 

0 0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I I I I I 1-1 I I A 

b/ 

c1 

.: 

I 

0 

0 

r! 

I 

0 

0 

* 

1 

0 

u! 

0 

I 

0 

0 

'9 
I 

68 

Control Performance 

The performance of the control is evaluated by 

examining the maximum control force required to achieve 

stabilization, This control force is obtained by simulating the rotor at steady state, or in a limit cycle if 

one exists, Both the linear and nonlinear rotors are 

simulated. A power spectral density of a nonlinear 

simulation reveals its frequency components, 

Fig. 4.16 shows the maximum control force required 

to stabilize the rotor over the speed range 0 5 R 5 6. 

Though the rotor is stable for speeds up to R = 7.2, the 

amplitudes of vibration become unrealistically large as 

this speed is approached. For the nonlinear systems, 

the deadband g is equal in both the end supporting bearings. Fig. 4.16 shows that, as deadband increases, the 

force required to stabilize the rotor does also. The 

slight peak in the force curves for s' = 0 and = 1 

occurs at a speed slightly less than R = 1. This is 

approximately the same as the first critical speed. 

Comparison of Fig. 4.16 with the corresponding plot of 

the previous chapter (Fig. 3.7) shows that nearly the 

same level of force is required for this model with the 

exception of that for the = 10 case, which for this 

model requires an order of magnitude greater force. 

A typical simulation of the nonlinear system with 

3 = 1 is shown in Fig. 4.17 through Fig. 4.19. At R = 2 
69 

this system exhibits mainly synchronous motion. Increasing the speed to R = 6, as shown in Fig. 4.20 through 

Fig. 4.22 changes the system dramatically. While the 

center mass exhibits mainly synchronous motion, the end 

masses exhibit complex subsynchronous motions. This 

results from the self -exciting mechanisms being located 

at both end masses, and none at the center. In all 

cases, the amplitudes at the mass where the control is 

applied are significantly lower than at the other 

masses. Fig. 4.23 shows the frequency components of the 

end mass response indicating mainly subsynchronous 

motion. The response of the rotor with g I 10 at 

R = 6.0 can be seen on Fig. 4.24 through Fig. 4.26. 

High amplitude subsynchronous motion is clearly evident, 

resulting in high required control forces to maintain 

stability. 

Summary 

In this chapter, a SIX degree-Of-freedOM model was 

presented and stabilized to the maximum possible speed. 

The control was supplied by one magnetic bearing set, 

optimally located by examining the rotor’s modal orbit 

shapes. For the model parameters chosen, this location 

was found to be the end mass of the rotor. The output 

measurement combination for the highest degree of st,aS;- 

lization combined the signals from the center mass and 
70 

the end mass where the magnetic bearing was located. 

The control gains were determined by use of a search 

method which maximized the damping of the least damped 

mode. Speed dependent gains offered little improvement 

over constant gains for this model. All three natural 

modes could be stabilized for speeds up to R = 7.2, 

which resulted in nearly a 300X increase in the threshold speed of the original rotor. The control forces 

required to achieve stabilization were determined for 

various spin speeds and bearing deadband values, and 

were found to increase with increasing speed and bearing 

deadband. Though the magnitudes of these forces grew 

relatively large as the threshold speed was approached, 

their magnitudes for the linear and low value (2 = 1) 

deadband systems were comparable to those found for the 

two degree-of-freedom model for speeds up to three times 

that of the original threshold speed. 
71 

0 

-9 

0 

* '9 

0 

cy 

9 

0 

0 

9 
72 

N 

ROTOR CENTER POSITION 

R = 2,O , Moss 1 

1.00 I # 

0.70 
73 

ROTOR CENTER POSITION 

N 

R = 2.0 , Mass 2 

1.10 I 

1 .oo 

0.90 

0.80 

a70 

0.60 

0.50 

a40 

0.30 

0.20 

0.10 

aoo 

410 

420 

430 

-0.40 

-0.50 

460 

-0.70 

480 

-0.90 

-1 .oo 

-1.10 

-1.20 -0.80 -0.40 0.00 0.40 0.80 1.20 

Y 

Fig. 4.18. Rotor Response at Mass 2 with R = 2, = 1. 
74 

ROTOR CENTER POSITION 

N 

R = 2.0 , Moss 3 

4,100 ,I 

-4.00 -2.00 0.00 2.00 

Y 

Fig. 4.19. Rotor Response at Mass 3 with R = 2, = 1. 

4.00 
75 

ROTOR CENTER POSITION 

N 

R = 6,O , Mass 1 

a90 

Q70 

a60 

1 

- 

‘3 

0.50 - 

I I I I I I I I I I 1 I I 

-1.60 -1.20 4.80 4.40 0.00 0.40 0.80 1.20 

Y 

Fig. 4.20. Rotor Response at Mass 1 with R = 6, = 1. 
76 

ROTOR CENTER POSITION 

N 

zoo 

1 a50 

1 .00 

0.50 

aoo 

450 

-1 .oo 

-1.50 

-200 

-3.00 -1 .# 1 .00 3.00 

Y 

Fig. 4.21. Rotor Response at Mass 2 with R = 6, = 1. 
77 

ROTOR CENTER POSITION 

1.20 

1 .00 

0.80 

0.60 

a40 

0.20 

N 0.00 

420 

440 

-0.60 

480 

-1 ,00 

-1.20 

-1.00 4.80 -0,60 4.40 -0.20 0.00 0.20 0,40 0.60 0.80 1.00 

Y 

Fig. 4.22. Rotor Response at Mass 3 with R = 6, = 1. 
78 

I I I I I I i I 1 I I 1 

0 

aD 

-9 

- 

0 

<o 

-9 

0 

* -9 

0 

e4 

'9 

0 

0 

0 

hl 

U 

M 

d 

cr 

w 

0 

aJ 

0 

a 

.. (II 

2 

d 

w 

0 

h 

U 

d 

(II 

C 

al 

17 

d 

a 

&I 

U 

U 

PI 

a 

rn 

$4 

al 

F4 

m 

hl 

U 

M 

d 

!& 
79 

5.00 

4#00 

3.00 

200 

1 ,00 

N 0.00 

-1,OO 

-200 

-300 

-400 

-5.00 

ROTOR CENTER POSITION 

R = 6,O , Mass 1 

I 

-9.00 -7.00 -5.00 -3.00 -1,w 1.00 3.00 5.00 7.00 9.00 

Y 

Fig. 4.24. Rotor Response at Mass 1 with R = 6,.g = 10. 
80 

ROTOR CENTER POSITION 

R = 6,O , Mass 2 

I 

N 

-7.00 

I 

I I I I I I I I I I I I 

I 

I I i 

-8.00 -6.00 -4.00 -2.00 0.00 2,00 4.00 6.00 8.00 

Y 

Fig. 4.25. Rotor Response at Mass 2 with R = 6, = 10. 
81 

800 

7,00 

boo 

500 

4.00 

3Ao 

200 

1 .oo 

N 0.00 

-1 ,oo 

-200 

-3.00 

-4.00 

-500 

-6.00 

-7m 

-a00 

ROTOR CENTER POSITION 

R = 6.0 , Mass 3 

I I I I I I I 1 i I 

-6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 

Y 

Fig. 4.26. Rotor Response at Mass 3 with R = 6, = 10. 
Using active 

CHAPTER V 

CONCLUSIONS 

ontrol, the stabi iza ion of 

self-excited rotor systems has been investigated. Two 

different rotor models have been used, and linear 

and nonlinear cases of each examined. For each case, 

the applied control has enhanced the stability characteristics of the rotor significantly] allowing it to 

operate above otherwise unstable speeds. 

For the two degree-of-freedom model, the control 

was structured to counteract self-exciting forces. In 

the absence of these destabilizing forces, the rotor was 

stabilized for any spin speed. The control forces 

required to maintain stability were found to vary nearly 

linearly with spin speed, 

affected by bearing deadband. While this control 

approach seemed somewhat simplistic, it was shown to be 

quite effective. The versatile nature of magnetic bearings could allow implementation of such a control scheme 

in a physical rotor, especially where dimensions would 

permit locating a magnetic bearing near each major selfexciting mechanism. 

and were not significantly 

82 
83 

The control approach used for the six degree-offreedom model was shown to be effective at improving the 

stability of the rotor, although the improvement was 

limited. The system was stabilized for speeds up to 

nearly 300X of the original instability threshold speed. 

The significant result was the stabilization of each of 

the rotor’s three natural modes using one magnetic bearing. The placement of the magnetic bearing was such that 

it maximized the effect that the bearing could e- ~ert on 

each of the rotor’s natural modes. The feedback signal 

for the control was structured to be a composite output 

signal, with less than full state information. To offer 

the most system improvement] it was found that this signal should contain information from at least the center 

mass location and end mass location where the bearing 

was located. The control forces required to stabilize 

the rotor were shown to increase nonlinearly with spin 

speed and deadband. However, the relative magnitudes of 

these forces were comparable to those found for the two 

degree-of-freedom model. Obviously, the part.icular 

parameters chosen to describe the model affect the 

improvement capabilities of the control. However, in 

many cases even a small improvement in the stability 

threshold speed is worthwhile goal. 

Several interesting topics relating to this study 

remain to be addressed. First, the function used to 
84 

determine the optimal control gains for the six degreeof-freedom model in this study was somewhat poorly 

suited for iterative methods. The parameter being maximized, the minimum relative damping, was a value associated with only one eigenvalue. As the eigenvalues 

"moved" under the effect of the control, the one with 

which the relative damping was associated would switch. 

This led to there being many solutions for the optimal 

gains, and erratic behavior of the eigenvalues under 

speed dependent gain solutions. One possible research 

question would be to examine alternative optimization 

criteria, possibly based on independent eigenvalue 

excursions. 

Secondly, the a c c u r a t e m o d e l i n g of many of the 

physical phenomena affecting rotor systems is presently 

an issue of uncertainty. These often highly nonlinear 

effects are generally treated by variable coefficients, 

the values of which may vary significantly from one 

effort to another. Of interest then, would be the 

sensitivity of a control application to variations in 

the modeling parameters. 

Lastly, the dynamics associated with the control 

loop should be investigated as they affect high speed 

rotor systems. Delays from signal measurement, processing, and control actuators could greatly affect the 

Performance of the control on machinery of this type. 
I LIST OF REFERENCES 

1. W. J. Rankine, '*Centrifugal Whirling of Shafts, 

Engineer, April 9, 1869. 

2. H. H. Jeffcott, nLateral Vibration of Loaded 

Shafts in the Neighborhood of a Whirling Speed - 

The Effect of Want of Balance," Philosophical 

Magazine, Vol. 37, 1919, pp. 304-314. 

3. F. F. Ehrich, ,*Identification and Avoidance of 

Instabilities and Self-Excited Vibrations in 

Rotating Machinery, ASME Paper No. 72-DE-21, New 

York, May 1972. 

4. F. F. Ehrich, "Self-Excited Vibrations," Shock 

Vibration Handbook, McGraw-Hi 11 Inc., New Ypork, 

1975. 

5, D. W. Childs, "Fractional-Frequency Rotor Motion 

Due to Nonsymmetric Clearance Effects,,, ASME 

Journal for Power, July 1982, pp. 533-541. 

6. nEffects of Bearing Deadband on Bearing Loads and 

Rotor Stability, Control Dynamics Company Final 

Report on NASA Contract No. NAS8-35050, January 

1984. 

7. F. F. Ehrich, and J. J. O'Connor, ,*Stator Whirl 

with Rotors in Bearing Clearance," ASME Journal of 

Engineering Industry, August 1967, pp. 381-390. 

8. H. Habermann, and G. L. Liard, "Practical Magnetic 

Bearings, '* IEEE Spectrum, September 1979, 

pp. 26-30. 

9. G. Schweitzer, and R. Lange, "Characteristics 

of a Magnetic Rotor Bearing for Active Vibration 

Control," Institute of Mechanical Engineers 

Conference on Vibrations in Rotating Machinery, 

Cambrige, 1976, Paper No. C239/76. 

85 
86 

10. V. Gondhalekar, and R. Holmes, "Design of 

Electromagnetic Bearing for Vibration Control of 

Flexible Transmission Shaft, NASA Conference 

Publ ication 2238, 1984, pp. 351-356. 

11. C. R. Burrows, and M. N. SahinBaya, "Control of 

Oil-Whirl by Pole Assignmentl9* Proceedings of the 

American Controls Conference, 1984, pp. 354-359. 

12. R. Stanway, and C. R. Burrows, nActive Vibration 

Control of A Flexible Rotor on Flexibly-Mounted 

Journal Bearings, Transactions of ASME, Journal 

of Dynamic Systems. Measurement, and Control, 

December 1981, pp. 383-388. 

13. G. Schweitzer, "Stabilization of Self-Excited 

Rotor Vibrations by an Active Damperlit Dynamics of 

Rotors. ed. F. I. Niordson, Springer-Verlag, New 

York, 1975, pp. 472-493. 

14. G. Schweitzer, NMagnetic Bearings for Vibration 

Control, I( NASA Conference Publ ication 2409, 1985, 

pp. 317-326. 

15. G. V.. Reklaitis, A. Ravindran, and K. M. 

Ragsdell, Engineering Optimization I Methods and 

Application. John Wiley and Sons, New YorB, 1983, 

pp. 87-98.