Control of surge in centrifugal compressors by active magnetic bearings : Theory and implementation

Advances in Industrial Control

For further volumes:

www.springer.com/series/1412

Se Young Yoon Zongli Lin Paul E. Allaire

Control of Surge

in Centrifugal

Compressors by

Active Magnetic

Bearings

Theory and Implementation

Se Young Yoon

Charles L. Brown Dpt of El. & Comp. Eng.

University of Virginia

Charlottesville, USA

Zongli Lin

Charles L. Brown Dpt of El. & Comp. Eng.

University of Virginia

Charlottesville, USA

Paul E. Allaire

Dept. of Mechanical & Aerospace Engin.

University of Virginia

Charlottesville, USA

ISSN 1430-9491 ISSN 2193-1577 (electronic)

Advances in Industrial Control

ISBN 978-1-4471-4239-3 ISBN 978-1-4471-4240-9 (eBook)

DOI 10.1007/978-1-4471-4240-9

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To Our Families

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technol￾ogy transfer in control engineering. The rapid development of control technology

has an impact on all areas of the control discipline. New theory, new controllers,

actuators, sensors, new industrial processes, computer methods, new applications,

new philosophies, ... , new challenges. Much of this development work resides in

industrial reports, feasibility study papers and the reports of advanced collaborative

projects. The series offers an opportunity for researchers to present an extended ex￾position of such new work in all aspects of industrial control for wider and rapid

dissemination.

Monographs in the Advances in Industrial Control series can be considered to

range in type from the “art of the possible”, a “proof of principle” type and then

a “state of the art” type, where the latter often reports on control as it exists in

today’s industry. For example, some “art of the possible” monographs explore a new

theoretical development and demonstrate how it might find application in the control

field. A good example of this type of monograph is Process Control by J. Bao and

P.L. Lee (ISBN 978-1-84628-892-0, 2007). Other monographs examine the present

“state of the art” of control and its technology as found in current industrial practice

and look at how better control might enhance efficiency and minimise pollution.

Recent exemplars of this category are Advanced Control and Supervision of Mineral

Processing Plants by D. Sbárbaro, R. del Villar (ISBN 978-1-84996-105-9, 2010)

or the monograph Hydraulic Servo-systems by M. Jelali and A. Kroll (ISBN 978-1-

85233-692-9, 2002).

However, this present, comprehensive Advances in Industrial Control mono￾graph Surge Control of Active-Magnet-Bearing-Suspended Centrifugal Compres￾sors: Theory and Implementation by Se Young Yoon, Zongli Lin and Paul E. Allaire

is an example of the “proof of concept” monograph. It is an excellent addition to the

series since its content has broad but complementary contributions from a new tech￾nology, from advanced control and from an advanced controller demonstration and

assessment using an industrial-standard experimental rig.

The phenomenon of surge and stall in compressor technology is long standing

and when the widespread industrial use of compressors is considered, a successful

vii

viii Series Editors’ Foreword

control strategy that optimally maximises performance and eliminates compressor

downtime would be of significant economic benefit to industry. This particular con￾trol problem has received exposure in the Advances in Industrial Control series pre￾viously through the published monograph Compressor Surge and Rotating Stall by

J.T. Gravdahl and O. Egeland (ISBN 978-1-85233-067-5, 1999), a monograph that

is often cited in the literature of the compressor control field. Some related material

can be found in another monograph in the series, namely, Dynamic Modelling of Gas

Turbines edited by G.G. Kulikov and H.A. Thompson (ISBN 978-1-85233-784-1,

2004).

However, this monograph by Se Young Yoon, Zongli Lin and Paul E. Allaire is

distinctive in that it investigates the particular technology of active-magnet-bearing￾suspended centrifugal compressors and assesses the authors’ own original advanced

control strategies. The assessment takes place using “an industrial-size centrifugal

compressor test rig... designed, built, and commissioned by the Rotating Machinery

and Controls Laboratory (ROMAC) at the University of Virginia” (USA). A descrip￾tion of this experimental set-up can be found in Chap. 4 of the monograph.

Access to and use of this industrial-sized test rig is just one of the distinctive

features of the research reported in the monograph. Another feature is the compre￾hensiveness of the contents since the authors have taken special care to address the

requirements of two readerships, one being readers from the control field, and the

second being a more general engineering readership.

The industrial and academic control community will be interested in the outcome

of the linear-quadratic-Gaussian (LQG) and H∞ advanced control trials performed

using the experimental rig. This group of readers will also find the technical knowl￾edge extracted in terms of models and parameters needed for computer simulation

tests before the instrumented control trials of value. However, to ensure that the

industrial and academic control community can fully comprehend the fundamen￾tals of compressor technology there are invaluable and detailed presentations on the

problem of surge and stall (Chap. 1), rotor dynamics (Chap. 2), magnetic bearings

(Chap. 3) and on the experimental rig and its associated instrumentation (Chap. 4).

In addition, to facilitate and ensure a full appreciation of the advanced control

developments presented in Chaps. 7 and 8 by a more general readership from the

mechanical, manufacturing, mechatronics, rotating machinery and other engineer￾ing disciplines, the authors have included an introductory chapter on control systems

theory (Chap. 6). Even readers from the control community might find this chapter

useful as a “refresher course” before reading the chapters covering the advanced

LQG and H∞ control strategies.

The original contributions made by the authors in describing the various aspects

of the technology, in devising and testing the advanced control strategies and the

careful and thorough construction of this monograph make it a very welcome ad￾dition to the Advances in Industrial Control series and to the wider literature of

compressor technology.

M.J. Grimble

M.A. Johnson

Industrial Control Centre

Glasgow, Scotland, UK

Preface

Compressors are essential machines for a large number of modern manufacturing

processes. Like the hearts pumping life to the production lines, compressors are

vital to the operation of key industrial sectors, such as the petrochemical and the

mining industries, which rely on compressors for critical tasks, ranging from tem￾perature control to gas transportation and mixing. As a result, there have been con￾tinual efforts by the academic and industrial communities to improve the reliability

and performance of such turbomachinery as new technologies become available.

Active magnetic bearing (AMB) is one such enhancing technology that has been

gaining strong momentum in recent years. Among other benefits, the low main￾tenance requirements and small parasitic energy losses have made these bearings

highly desirable for high performance compressors, particularly those designed to

operate in harsh or inaccessible environments. Additionally, with their ability to ac￾tively change the rotor-dynamic characteristics of the compressor by controlling the

bearing parameters in real time, the AMBs can provide a smoother and more reliable

operation of the compressor over a wider range of operating conditions.

Stability is a critical factor that limits the performance of compressors. The max￾imum mass flow output of a compression system is capped by choke, which is gen￾erally not a destabilizing phenomenon, and it is caused by the compressed medium

reaching sonic conditions. At the opposite end, the minimum mass flow is limited

by the compressor instabilities known as stall and surge. Stall is a localized phe￾nomenon that can be observed in some compression systems, and it is sometimes

accompanied by a sudden drop in the average compressor output flow. On the other

hand, surge is a system-wide instability that is characterized by large amplitude

oscillations in the output pressure and mass flow. These oscillations can cause ex￾tensive damage to the compressor casing and internal components due to high vi￾brational loads. They can even lead to a catastrophic mechanical failure of the com￾pressor if they are not addressed properly. A conservative way of dealing with surge

is to avoid it, by operating far away from the instability. A more efficient way is to

implement an active method to stabilize surge and stall, so that the stable operating

region of the compression system is extended, resulting in both higher productivity

and safer operation.

ix

x Preface

Unfortunately, a majority of current compressors operate conservatively to avoid

surge. In other words, many compressors trade the peak performance at the maxi￾mum pressure rise for the stability at the higher mass flow rates. The focus in surge

avoidance is on guaranteeing the mechanical integrity of the machines and the safety

of the work place by keeping a precautionary margin between the operating output

flows and the known surge points. Additionally, a reset mechanism is built in the

system that quickly releases the built-up pressure in the compressor if surge is de￾tected by the different safety triggers. An active surge controller, on the other hand,

stabilizes the compressor flow during the initiation of surge, effectively extending

the operational range of the compressor with no loss in performance. The implemen￾tation of a control mechanism is much rarer in industrial applications than the surge

avoidance strategies for several reasons. The main reason is that the modifications

to compressors in the field required for the installation of a surge control mechanism

are very often complicated and involve very specialized equipments. More impor￾tantly, there has not been an univocal experimental demonstration of the potential

benefits that an effective surge controller could offer to an actual industrial-size

compressor.

Recently, promising results have been presented in the literature on an active

surge control scheme that modulates the impeller position to stabilize the flow in an

AMB supported single stage centrifugal compressor. With the AMB acting as a high

bandwidth actuator to regulate the displacement of the impeller, the compressor flow

states can be restored to the equilibrium operating point during the early stages of the

surge instability, when the amplitude of the limit cycle is relatively small. The main

advantage of this active surge control scheme is that it can be easily implemented in

existing AMB suspended compressors, generally with a simple modification in the

control software.

The purpose of this book is to present the fundamentals on the integration of the

AMBs for the suspension of the rotor in compressors, and how this relatively new

bearing technology can be employed to actively control and potentially eliminate

the compressor surge. The material presented here is intended to serve as a com￾prehensive reference in the areas of compressor surge control and AMB application

in turbomachinery. For readers who are unfamiliar with compressors, rotor dynam￾ics and magnetic bearings, brief introductions to these topics are presented in the

earlier chapters of this book. A brief discussion on compressors and compressor in￾stabilities is presented in Chap. 1, where the literature on the surge modeling and

control is also reviewed. Chapter 2 contains a review of the basic theories and tools

in the study of rotor dynamics. Chapter 3 presents a brief discussion on the operat￾ing principles of the AMBs and a summary of the potential benefits that come from

the implementation of this bearing technology in compressors. Both Chaps. 2 and 3

are intended to be a self-contained reference for control engineers.

In order to develop the theory in a physical context, and to provide experimental

validation of the theory developed throughout this book, an industrial-sized AMB

suspended compressor system was designed, constructed and commissioned for the

study of surge control. A thorough description of this compressor test rig is pre￾sented in Chap. 4. This description includes the integration of the AMBs to the

Preface xi

compressor for rotor support and for surge control. The derivation of the dynamic

models for both the AMB/rotor system and the compression system flow, along with

their experimental validations, are presented in Chaps. 5 and 7. The experimental

identification of the system dynamics included in these chapters will demonstrate

that the assumptions made in the derivation of the mathematical models are sound.

These models will serve as the basis on which the AMB levitation controller and the

active surge controller are designed, in Chaps. 7 and 8, respectively.

In the design of the AMB levitation controller, performance and robustness spec￾ifications that are desirable for AMB suspended compressors are included in the

discussion. In the design of the surge controller, the performance degradation of the

surge controller due to dynamic limitations in the AMB system will be studied. For

both controllers, the theoretical derivation is accompanied by the experimental data

to show their effectiveness in industrial-size compressors.

Finally, it is important to note that this book is not intended to be reference mate￾rial for general design and operation of compressors. There exists an extensive list of

excellent references on the topics of compressor design and flow modeling. Instead,

this book is intended to serve as a guide for the application of the AMB technology

in turbomachinery, and to demonstrate the advantages that this rotor support sys￾tem can provide in the stabilization of the compressor surge for a particular group

of single stage centrifugal compressors. Since active magnetic bearings play a cen￾tral role in the surge control method to be presented in the book, their theory and

applications are extensively discussed. The stabilization of the compressor surge is

mainly discussed from a control theory perspective.

This book builds on years of work invested by many engineers and scientists

from the Rotating Machinery and Controls (ROMAC) Laboratory at the University

of Virginia. The authors would like to acknowledge those who participated in the

different stages of the research presented here. The derivation of the theoretical

concept for the surge control strategy presented here, as well as the design and

the initial preparation of the experimental setup, was executed in the early stages

of this project by the team led by Professor Eric Maslen and Dr. Dorsa Sanadgol.

The experience in industrial compressors brought by Kin Tien Lim and the advice

of Professor Chris Goyne in experimental fluid dynamic testing came to be of great

value during the construction and commissioning of the compressor test rig. Finally,

the authors would also like to express their appreciation for the generous donations

made by Kobe Steel Ltd., Kobe, Japan, and the constant support and funding by the

ROMAC Laboratory and its industrial partners around the world.

Se Young Yoon

Zongli Lin

Paul E. Allaire

Charlottesville, Virginia, USA

Contents

1 Introduction ................................ 1

1.1 Compressors and Compressor Systems . . . . . . . . . . . . . . . 1

1.2 Active Magnetic Bearings in Compressors ............. 4

1.3 Compressor Instability ........................ 5

1.3.1 Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Compressor Surge Modeling . . . . . . . . . . . . . . . . . . . . 10

1.5 Surge Avoidance and Suppression . . . . . . . . . . . . . . . . . . 11

1.5.1 Surge Avoidance . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.2 Surge Suppression and Control . . . . . . . . . . . . . . . 13

1.6 Objectives of This Book . . . . . . . . . . . . . . . . . . . . . . . 15

2 Introduction to Rotor Dynamics . . . . . . . . . . . . . . . . . . . . . 17

2.1 Föppl/Jeffcott Single Mass Rotor . . . . . . . . . . . . . . . . . . 18

2.1.1 Undamped Free Vibration . . . . . . . . . . . . . . . . . . 20

2.1.2 Damped Free Vibration . . . . . . . . . . . . . . . . . . . 22

2.1.3 Forced Steady State Response . . . . . . . . . . . . . . . . 23

2.2 Rotor Gyroscopic Effects . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Rigid Circular Rotor on Flexible Undamped Bearings . . . 28

2.2.2 Model of Rigid Circular Rotor with Gyroscopic

Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.3 Undamped Natural Frequencies of the Cylindrical Mode . . 31

2.2.4 Undamped Natural Frequencies of the Conical Mode . . . 32

2.3 Instability due to Aerodynamic Cross Coupling . . . . . . . . . . . 36

2.3.1 Aerodynamic Cross Coupling in Turbines . . . . . . . . . 36

2.3.2 Aerodynamic Cross Coupling in Compressors . . . . . . . 37

2.4 Rotor-Dynamic Specifications for Compressors . . . . . . . . . . 38

2.4.1 Lateral Vibration Analysis . . . . . . . . . . . . . . . . . . 39

2.4.2 Rotor Stability Analysis . . . . . . . . . . . . . . . . . . . 43

2.5 Rotor Finite Element Modeling . . . . . . . . . . . . . . . . . . . 47

2.5.1 Discretizing Rotor into Finite Elements . . . . . . . . . . . 47

xiii

xiv Contents

2.5.2 Approximating Element Displacement Functions

and Nodal Displacement . . . . . . . . . . . . . . . . . . . 48

2.5.3 Formulating Equations of Motion for Each Element . . . . 50

2.5.4 Element Mass and Gyroscopic Matrices . . . . . . . . . . 51

2.5.5 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . 52

2.5.6 Element Damping Matrix . . . . . . . . . . . . . . . . . . 53

2.5.7 Adding Lumped Mass, Stiffness and Damping

Components . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5.8 Assembling the Global Mass, Gyroscopic, Stiffness,

Damping Matrices, and Force Terms . . . . . . . . . . . . 54

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Fundamentals of Magnetic Bearings . . . . . . . . . . . . . . . . . . 57

3.1 Electromagnetic Field and Flux . . . . . . . . . . . . . . . . . . . 57

3.1.1 Field Generated by Current in Straight Wires . . . . . . . . 58

3.1.2 Field Generated by Current in a Solenoid . . . . . . . . . . 59

3.2 Magnetic Permeability . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Single Sided Magnetic Bearing Actuator . . . . . . . . . . . . . . 60

3.4 Double-Sided Magnetic Bearing Actuators . . . . . . . . . . . . . 64

3.5 Linearized Force Equation . . . . . . . . . . . . . . . . . . . . . . 65

3.6 Coil Inductance and Slew Rate . . . . . . . . . . . . . . . . . . . 66

3.7 AMB Load Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.8 Magnetic Bearing Design for Applications . . . . . . . . . . . . . 68

3.9 Amplifiers and Displacement Sensors . . . . . . . . . . . . . . . . 71

3.10 Losses in Magnetic Bearings . . . . . . . . . . . . . . . . . . . . 71

3.10.1 Flux Leakage and Fringing . . . . . . . . . . . . . . . . . 72

3.10.2 Eddy Current Losses . . . . . . . . . . . . . . . . . . . . . 72

3.10.3 Hysteresis Losses . . . . . . . . . . . . . . . . . . . . . . 73

3.11 Auxiliary Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.12 PID Control of AMB . . . . . . . . . . . . . . . . . . . . . . . . 75

3.12.1 Decentralized PID Control . . . . . . . . . . . . . . . . . 77

3.12.2 Tilt and Translate Control . . . . . . . . . . . . . . . . . . 77

3.12.3 Unbalance and Synchronous Vibration Compensation . . . 79

3.12.4 Shortcomings of the PID Controller . . . . . . . . . . . . . 80

3.13 Modern Control of AMB Systems . . . . . . . . . . . . . . . . . . 81

3.13.1 LQR and LQG Control . . . . . . . . . . . . . . . . . . . 81

3.13.2 H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.13.3 μ-Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.13.4 Combined H∞/μ-Synthesis Control . . . . . . . . . . . . 85

3.13.5 Self-Tuning, Neural Network and Adaptive Controls . . . . 86

3.14 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Design of AMB Supported Centrifugal Compressor . . . . . . . . . . 89

4.1 Compression System . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 High Speed Motor . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Compressor Instrumentation . . . . . . . . . . . . . . . . . . . . . 95

Contents xv

4.4 Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5 Active Magnetic Bearings . . . . . . . . . . . . . . . . . . . . . . 101

4.5.1 Radial AMB . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5.2 Thrust AMB . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.5.3 AMB Control . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6 Auxiliary Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.7 Experimental Surge Characterization . . . . . . . . . . . . . . . . 116

4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Derivation of the Surge Dynamic Equations . . . . . . . . . . . . . . 125

5.1 Greitzer Compression System Model . . . . . . . . . . . . . . . . 125

5.2 Variation of the Impeller Tip Clearance . . . . . . . . . . . . . . . 130

5.2.1 Simulation and Experimental Results . . . . . . . . . . . . 132

5.3 Compression System with Piping Dynamics . . . . . . . . . . . . 136

5.3.1 Fluid Transmission Line Model . . . . . . . . . . . . . . . 136

5.3.2 Piping Acoustics at Compressor Exhaust . . . . . . . . . . 138

5.3.3 Piping Acoustics at Plenum Output . . . . . . . . . . . . . 140

5.3.4 Modal Approximation of Pipeline Acoustics . . . . . . . . 143

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Introduction to Control Theory . . . . . . . . . . . . . . . . . . . . . 149

6.1 Classical Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.1.1 Objectives of a Control System . . . . . . . . . . . . . . . 150

6.1.2 Power of Feedback Control . . . . . . . . . . . . . . . . . 151

6.1.3 Input–Output Stability . . . . . . . . . . . . . . . . . . . . 155

6.1.4 PID Control of a Rigid Rotor on AMBs . . . . . . . . . . . 155

6.1.5 Transient Response . . . . . . . . . . . . . . . . . . . . . 160

6.1.6 Steady-State Response . . . . . . . . . . . . . . . . . . . . 165

6.2 Modern Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.2.1 State Space Representations . . . . . . . . . . . . . . . . . 168

6.2.2 Solution to the State Space Equation . . . . . . . . . . . . 170

6.2.3 Stability of Systems with State Space Representation . . . 172

6.2.4 Controllability . . . . . . . . . . . . . . . . . . . . . . . . 174

6.2.5 Observability . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2.6 Optimization-Based Control Designs . . . . . . . . . . . . 178

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7 Control Design for Rotor Suspension . . . . . . . . . . . . . . . . . . 185

7.1 Specifications for Machines with AMBs . . . . . . . . . . . . . . 186

7.1.1 Vibration Level . . . . . . . . . . . . . . . . . . . . . . . 187

7.1.2 Stability Margin . . . . . . . . . . . . . . . . . . . . . . . 188

7.2 Modeling of the AMB Suspension System . . . . . . . . . . . . . 190

7.2.1 Rotor Lateral Dynamics . . . . . . . . . . . . . . . . . . . 190

7.2.2 Rotor Axial Dynamics . . . . . . . . . . . . . . . . . . . . 196

7.2.3 AMB Actuator . . . . . . . . . . . . . . . . . . . . . . . . 196

7.2.4 Power Amplifiers, Sensor Electronics and Time Delays . . 197

xvi Contents

7.2.5 Losses Due to Eddy Current in the Thrust AMB . . . . . . 198

7.3 Control of Rotor Lateral Dynamics . . . . . . . . . . . . . . . . . 199

7.3.1 Linear Quadratic Gaussian (LQG) Controller . . . . . . . . 199

7.3.2 Design of LQG Controller for Lateral Rotor Suspension . . 201

7.3.3 Experimental Testing . . . . . . . . . . . . . . . . . . . . 209

7.4 Control of Rotor Axial Dynamics . . . . . . . . . . . . . . . . . . 213

7.4.1 Design of H∞ Controller . . . . . . . . . . . . . . . . . . 213

7.4.2 Design of H∞ Controller for Axial Rotor Support . . . . . 214

7.4.3 Experimental Testing . . . . . . . . . . . . . . . . . . . . 218

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8 Control of Compressor Surge . . . . . . . . . . . . . . . . . . . . . . 221

8.1 Compressor Model for Controller Derivation . . . . . . . . . . . . 221

8.1.1 Linearization of the Tip Clearance Effect . . . . . . . . . . 222

8.1.2 Piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

8.1.3 Throttle Valve . . . . . . . . . . . . . . . . . . . . . . . . 227

8.1.4 Overall Assembly . . . . . . . . . . . . . . . . . . . . . . 228

8.1.5 System Model Linearization . . . . . . . . . . . . . . . . . 229

8.2 Robustness to Surge/Levitation Controllers Interaction . . . . . . . 231

8.3 Surge Controller Derivation . . . . . . . . . . . . . . . . . . . . . 234

8.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 238

8.5 Surge Controller Implementation . . . . . . . . . . . . . . . . . . 240

8.6 Experimental Testing . . . . . . . . . . . . . . . . . . . . . . . . 243

8.6.1 Surge Control Test at 10,290 rpm . . . . . . . . . . . . . . 245

8.6.2 Surge Control Test at 13,950 rpm . . . . . . . . . . . . . . 247

8.6.3 Surge Control Test at 16,290 rpm . . . . . . . . . . . . . . 252

8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Nomenclature

Rotor dynamics

Aci Vibration magnitude at Nci

AF Amplification factor

API American Petroleum Institute

c Damping constant

C Damping matrix

D Diameter

eu Unbalance eccentricity vector

E Elastic modulus of the beam

fr Frequency ratio

G Gyroscopic matrix

I Area moment of inertia

ISO International Organization of Standardization

J Moment of inertia

k Stiffness

K Stiffness matrix

L Length

m Mass

M Mass matrix

N Rotating speed (rpm)

Nci ith critical speed (rpm)

P Moment of inertia ratio Jp/Jt

qa Alford’s cross coupling stiffness coefficients

Qa Predicted total cross coupling stiffness

R Radius

SM Separation margin

T Torque

u Lateral displacement

u Lateral displacement vector

Ub Specified rotor unbalance

W Journal static weight

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