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reface

Over the last decade there has been much work concemed with the act ive control of

vibrat ions of f lexible structures. The great major i ty of the work has been concerned

with feedback control of large f lexible systems at re la t ively low frequencies. The

general topic of act ive vibrat ion control in this context has been admirably treated in

recent texts by Professors Meirovitch (1990) and Inman (1989) . The subject matter of

their texts has largely been devoted to modem control systems with an emphasis on

mult i-channel feedback control . More recently, as a result of a perceived need in the

noise control community, advances have emerged in the act ive control of vibrat ions a t

audio f requencies and for s teady sta te excitat ion. These adv ances have been largely due

to the recent development of fast digi ta l s ignal processors together with fast mult i-

channel analogue- to-digita l and digita l- to-analogue converters . In addit ion, there have

been signif icant advances in the developm ent and use of control t ransducers which have

enabled the real is t ic implementat ion of many active vibrat ion control techniques. The

overal l a im o f the book is to sum marise these new advances in act ive vibration control

with an emphasis on the fundamental sc ientif ic pr inciples that form the basis of these

techniques.

In writing this book we have chosen to provide both a basic understanding of the

subject and a research reference text. The book is thus aimed towards graduate students,

researchers and engineers who have some knowledge of the theory of vibrat ions,

mechanics and control. The book is written as a companion to the text by Nelson and

Elliott (1992) which covers, in detail, the related area of active control of sound. In the

interest of brevity, much of the material which is common to both fields has not been

included in this text and references to Nelson and Elliott (1992) are provided where

pertinent. However, when the material is essential to the understanding and continuity of

the text, it is included in this book. In a similar way to the companion text, the book

attempts to combine in a unified manner, material from mechanical vibrations, acoustics,

signal processing, mechanics and control theory. Key new areas discussed in the text are

the use of feedforward control, the modelling and use of distributed strain actuators and

sensors, the control of waves in structures, the theory and implementation of active

isolation of vibration and the active control of structurally radiated sound. Throughout

the text considerable effort is directed tow ards highlighting and clarifying the dual nature

of the w av e and m od e descriptions of the vibrations of structures. I t is dem onstrated

that either form of description has its advantages, depending upon the type of

application, understanding of the control problem and designing of the controller.



x PREFACE

The book begins with a br ief review of mechanical vibrat ions and wave propagation

in structures. This material is well known, but it is necessary to review and introduce the

material within the context of the subject matter of the book in order to develop a solid

foundation of understanding. Effort is directed towards describing basic vibrations and

wave propagation in order to understand the more advanced topics or iented towards

active control that are described in later chapters. Ch apter 1 is intended as an introduction

to vibrations of lumped parameter systems and also includes a brief description of finite

element analysis. I t thus provides the basic equations for modelling the control of lightly

damped structures with only a small number of modes. Chapter 2 summarises basic

material describing longitudinal and flexural wave propagation in long slender beams and

cylinders. These equations are then applied to modelling the motion of finite beams,

plates and cylinders and a discussion is presented of the interpretation of the response in

terms of e i ther waves or modes. The equations descr ibing the response of the above

systems to var ious input force and m omen t configurations are developed and provide the

basis of modelling the control of vibrations of such structures.

Chapters 3 and 4 introduce some o f the control concepts used in the book. In Chapter

3 feedb ack control is discussed, ini t ial ly for a s ingle-input s ingle-output (SISO) system

using a transfer function approach. The general isa t ion to mult iple- input , mult iple-

output (MIMO) feedback control systems is then descr ibed using sta te var iables. The

sta te var iable formulation is a useful way of descr ibing feedback controllers , and in

par t icular i t suggests a way of descr ibing the independent control o f the syste m s

modes. Chapter 4 introduces feedforward control , again beginning with the SISO case

analysed in the f requency domain. Adaptive digi ta l f i l ters are widely used for

feedforward control and these are introduced in this chapter , and the general isa t ions

required for mult i-channel (MIMO) operat ion are descr ibed in some deta i l . The

adaptivity of the feedfo rwa rd controller ensures that i t is not ope n loo p , and a br ief

analysis is presented that shows how adaptive feedforward controllers can be

represented as equivalent feedba ck systems.

In Chapter 5 we descr ibe mater ia l re la ted to the use of actuators and sensors for

active control of vibrat ion. The chapter begins with a summary of recent work in

modell ing the use of distr ibuted, piezoelectr ic s tra in actuators and sensors in var ious

configurat ions. The use of point sensors configured in arrays in order to provide

est imates of modal and wavenumber response is a lso discussed. Decomposit ion of

wave f ie lds into individual wave components is outl ined. The chapter f inishes with a

br ief descr ipt ion of advanced actuators such as those based on shape memory a l loy

which can be used for semi-active , adaptive , or s teady sta te control of system

parameters . Chapter 6 descr ibes the act ive control of vibrat ion in var ious distr ibuted

mechan ical systems. Ini tia l ly the act ive control of the mechanical respon se is analysed

in terms of the structural modes of the complete system. The act ive suppression of the

resonant response of these modes is then i l lustra ted using both feedforward and

feedback control methods. Alternatively, the motion of a system can be analysed in

terms of the structural waves which propagate within i t , and the act ive control of

structural waves is described in the second half of this chapter. Particular attention is

paid to the active control of flexural waves since they are dispersive and have near-field

components, and both effects introduce their own complicat ions into the act ive control

problem.

Chapter 7 deals specifically with the active isolation of vibrations. The first topic to



PREF CE x

be addressed is the isolation of a periodic source of vibrations from a resonant

receiving structure . This problem is widely encountered in engineer ing practice and

occurs whenev er a rota ting o r reciprocating mac hine is mounted on a f lexible structure .

This problem is then general ised to include the isola t ion of t ransient machinery

vibrat ions and a lso to deal with the case where one wishes to provide isola t ion for a

system a sensi tive instrument for example ) f rom externally generated vibrat ions. The

considerable promise shown by the applicat ion of act ive techniques to these problems

is clearly demonstrated.

The f inal two chapters of the book deal with the new f ie ld of Active Structural

Acoustic Control ASA C) in which structurally radia ted sound is directly controlled by

active structural inputs. In Chapter 8 the concepts of ASAC are first outlined and then

the mechanisms o f sound radiat ion f rom v ibrat ing structures are discussed. A review is

presented of the applicat ion of ASAC to pla te s tructures excited by var ious c lasses of

disturbances and controlled using var ious transducer configurat ions. The b ehaviou r and

control of the system in the wavenumber domain is discussed. An example of an

ASAC configuration and experimental results are described to illustrate the practical

implementat ion of the approach. The use of mult i-channel feedback control to

implement ASAC is then outl ined. Chapter 9 discusses the applicat ion of ASAC to

cylindrical systems. The acoustic radiation and coupling with vibrating cylinders is

br ief ly reviewed. The applicat ion of ASAC to the control of sound radiated by

cylinders, interior cavity noise and vibrational power flow in fluid-filled cylinders is

summarised. The chapter f inishes with the descr ipt ion of the use of ASAC to control

interior noise in an aircraft fuselage in order to illustrate a practical application of the

technique.

There are many well established topics in act ive control of vibrat ion that have not

been covered in this book and only those most appropria te to vibrat ion control in the

audio f requency range have been dealt with in depth. Although the theory of feedback

control and its application to various structural systems are discussed briefly in the text,

the reader is referred to the texts of Meirovitch 1990) and Inman 1989) for a more

detai led descr ipt ion of this mater ia l . The main purpose of the mater ia l on feedback

control included in this text has been to illustrate where it has been used recently at the

higher audio frequencies and for the control of structurally radiated sound. It was also

thought necessary to introduce this material in order to relate it to the newer field of

feedforward control. Similarly, the control of vibrations in large flexible structures at

very low frequencies has not been cove red in this text . Throughout the book, num erous

references to other books, research publications and text are provided. This list is not

intended to be comp rehensive but is intended to provide the reader with the information

and guidance to find more detail on a particular subject.

I t is our hope that in this book w e have descr ibed new mater ia l which wil l lead to the

application, and stimulate research in the field, of active vibration control at audio

frequencies. I t is our view that the active control of vibrations shows much potential

for solving many difficult noise and vibration problems. If the book provides the basis

for guiding the reader towards using these new solutions then we believe i t wil l have

largely achieved its aim.

In writ ing this book the authors have had the benef i t of many valuable suggestions

and cr i t ic isms from a large number of colleagues. In par t icular the authors would l ike

to acknowledge the help given by Professor Ricardo Burdisso, Dr Gary Gibbs, Dr



xii PREFACE

Cathy Guigou Dr Bertrand Brevart Dr Tao Song and Julien Maillard of Virginia

Polytechnic Institute and State University. Professor Robert C lark of Duke University

Professor Jim Jones of P urdue University Professor Peter Wan g of National Pingtung

Polytechnic Institute Dr Andy von Flotow of Hood Technology Corporation Dr Kam

Ng of the Office of Naval Research Dr Dean Thomas of the ISVR University of

Southampton and Drs Rich Silcox and Harold Lester of NASA Langley Research

Center also provided very useful input. The authors are also indebted to NAS A Langley

Research Center the U.S. Office of Naval Research and the U.K. Department of Trade

and Industry for financially supporting much of the research which form s the basis of

this book. W e are also grateful to the Office of Naval R esearch for providing support

for the preparation of the text. Many reviewers contributed their valuable time to

reading and providing suggestions for improvement of the material and to Dr E.

Anderson Dr A. Baz Dr M. Brennan Professor J. Cuschieri Dr C. Hansen Professor

M. Heckl Professor D. Inman M. Johnson Professor C. Knight Professor C. Liang

Dr B. Ma ce Dr R. Pinnington Professor W. Saunders Dr J. Scheuren Dr S. Snyder

Pro fesso r S. Somm erfeldt and Dr T. Sutton the authors wish to offer their thanks.

Finally the authors are indebted to their families for unlimited tolerance during the

difficult parts of the writing process and to Daw n Williams Crystal Carter Ma ureen

Strickland Susan Hellon Karl Estes and Cathy Gorman for their excellent typing and

graphics skills.

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