Active Control of Vibration

Preface

Over the last decade there has been much work concemed with the active control of vibrations of flexible structures. The great majority of the work has been concerned with feedback control of large flexible systems at relatively low frequencies. The general topic of active vibration 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 multi-channel feedback control. More recently, as a result of a perceived need in the noise control community, advances have emerged in the active control of vibrations at audio frequencies and for steady state excitation. These advances have been largely due to the recent development of fast digital signal processors together with fast multichannel analogue-to-digital and digital-to-analogue converters. In addition, there have been significant advances in the development and use of control transducers which have enabled the realistic implementation of many active vibration control techniques. The overall aim of the book is to summarise these new advances in active vibration control with an emphasis on the fundamental scientific principles 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 vibrations, 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 towards highlighting and clarifying the dual nature of the 'wave' and 'mode' descriptions of the vibrations of structures. It is demonstrated 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 brief review of mechanical vibrations 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 oriented towards active control that are described in later chapters. Chapter 1 is intended as an introduction to vibrations of lumped parameter systems and also includes a brief description of finite element analysis. It 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 either waves or modes. The equations describing the response of the above systems to various input force and moment configurations are developed and provide the basis of modelling the control of vibrations of such structures. Chapters 3 and 4 introduce some of the control concepts used in the book. In Chapter 3 feedback control is discussed, initially for a single-input single-output (SISO) system using a transfer function approach. The generalisation to multiple-input, multipleoutput (MIMO) feedback control systems is then described using state variables. The state variable formulation is a useful way of describing feedback controllers, and in particular it suggests a way of describing the independent control of the system's modes. Chapter 4 introduces feedforward control, again beginning with the SISO case analysed in the frequency domain. Adaptive digital filters are widely used for feedforward control and these are introduced in this chapter, and the generalisations required for multi-channel (MIMO) operation are described in some detail. The adaptivity of the feedforward controller ensures that it is not 'open loop', and a brief analysis is presented that shows how adaptive feedforward controllers can be represented as equivalent feedback systems. In Chapter 5 we describe material related to the use of actuators and sensors for active control of vibration. The chapter begins with a summary of recent work in modelling the use of distributed, piezoelectric strain actuators and sensors in various configurations. The use of point sensors configured in arrays in order to provide estimates of modal and wavenumber response is also discussed. Decomposition of wave fields into individual wave components is outlined. The chapter finishes with a brief description of advanced actuators such as those based on shape memory alloy which can be used for semi-active, adaptive, or steady state control of system parameters. Chapter 6 describes the active control of vibration in various distributed mechanical systems. Initially the active control of the mechanical response is analysed in terms of the structural modes of the complete system. The active suppression of the resonant response of these modes is then illustrated 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 it, and the active 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 complications into the active control problem. Chapter 7 deals specifically with the active isolation of vibrations. The first topic to

PREFACE

xi

be addressed is the isolation of a periodic source of vibrations from a resonant receiving structure. This problem is widely encountered in engineering practice and occurs whenever a rotating or reciprocating machine is mounted on a flexible structure. This problem is then generalised to include the isolation of transient machinery vibrations and also to deal with the case where one wishes to provide isolation for a system (a sensitive instrument for example) from externally generated vibrations. The considerable promise shown by the application of active techniques to these problems is clearly demonstrated. The final two chapters of the book deal with the new field of Active Structural Acoustic Control (ASAC) in which structurally radiated sound is directly controlled by active structural inputs. In Chapter 8 the concepts of ASAC are first outlined and then the mechanisms of sound radiation from vibrating structures are discussed. A review is presented of the application of ASAC to plate structures excited by various classes of disturbances and controlled using various transducer configurations. The behaviour 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 implementation of the approach. The use of multi-channel feedback control to implement ASAC is then outlined. Chapter 9 discusses the application of ASAC to cylindrical systems. The acoustic radiation and coupling with vibrating cylinders is briefly reviewed. The application 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 finishes with the description 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 active control of vibration that have not been covered in this book and only those most appropriate to vibration control in the audio frequency 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 detailed description of this material. The main purpose of the material 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 covered in this text. Throughout the book, numerous references to other books, research publications and text are provided. This list is not intended to be comprehensive but is intended to provide the reader with the information and guidance to find more detail on a particular subject. It is our hope that in this book we have described new material which will lead to the application, and stimulate research in the field, of active vibration control at audio frequencies. It 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 it will have largely achieved its aim. In writing this book the authors have had the benefit of many valuable suggestions and criticisms from a large number of colleagues. In particular the authors would like 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 Clark of Duke University, Professor Jim Jones of Purdue University, Professor Peter Wang 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 NASA 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 forms the basis of this book. We are also grateful to the Office of Naval Research 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. Mace, Dr R. Pinnington, Professor W. Saunders, Dr J. Scheuren, Dr S. Snyder, Professor S. Sommerfeldt, 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 Dawn Williams, Crystal Carter, Maureen Strickland, Susan Hellon, Karl Estes and Cathy Gorman for their excellent typing and graphics skills.

Introduction to Mechanical Vibrations

1.1

Introduction

All mechanical systems composed of mass, stiffness and damping elements exhibit vibratory response when subject to time-varying disturbances. The prediction and control of these disturbances is fundamental to the design and operation of mechanical equipment. In particular, the use of secondary, active inputs to the system in order to modify the system response in a controllable way is the topic of this book. The analysis of controlled systems is founded on the same analytical approaches used to study the vibrations of elastic structures. A brief review of the main concepts of vibration analysis and the associated techniques of solution is necessary to set the foundation for the following chapters. In this chapter we begin by defining terminology and the mathematical methods for describing the linear response of vibrating systems. The equations of motion and linear behaviour of single-degree-of-freedom systems are outlined for both free and forced response. The use of the Laplace transform to solve for transient response is reviewed. The extension to multi-degree-of-freedom systems and then the use of finite element analysis are briefly introduced. These approaches are valid for lightly damped structures or elements that are small relative to the wavelength of motion. For more detail the reader is referred to the texts of Thomson (1993), Meirovitch (1967) and Inman (1994).

1.2 Terminology The following is a brief list of the main terminology and definitions used in analysing the vibratory response of mechanical systems. Mechanical system. A mechanical system is composed of distributed elements which exhibit characteristics of mass, elasticity and damping. Degrees of freedom. The number of degrees of freedom of a system is equal to the number of independent coordinate positions required to completely describe the motion of the system. System response. All mechanical systems exhibit some form of vibratory response when excited by either internal or external forces. This motion may be irregular or may repeat itself at regular intervals, in which case it is called periodic motion. Period. The period T is the time taken for one complete cycle of motion.

2

ACTIVE CONTROLOF VIBRATION

Harmonic motion is the simplest form of periodic motion whereby the actual or observed motion can be represented by oscillatory functions such as the sine and the cosine functions. Motion that can be described by a continuous sine or cosine function is called steady state. For example one may write the actual displacement w ( t ) in the form

w(t) = [A I cos(~ot + q~),

(1.2.1)

where w ( t ) and IAI are real, a~ is circular frequency in radians per second and q~ is an arbitrary phase angle in radians. Equation (1.2.1) can also be expressed as a superposition of a sine and a cosine function as w ( t ) - AR cos cot- At sin cot,

(1.2.2)

where AR and At are real numbers such that AR = I Alcos ¢,

(1.2.3a)

A, = I alsin ¢,

(1.2.3b)

where the phase angle is specified by qb = tan - ~(AJAR).

(1.2.4)

The constant I AI in equation (1.2.1) is related to the constants At and AR in equation (1.2.2) by

[al

= ( a z + a l z)1/2.

(1.2.5)

Frequency is the number of cycles per second (also called hertz) of the motion and is the reciprocal of the period. Therefore frequency is specified by

1 f= --

T

(1.2.6)

and circular frequency co (radians per second) is given by o9 = 2z~f. Amplitude is the measure of For example if the motion is corresponds to the amplitude of The mean square amplitude response. Thus, for example

(1.2.7)

the maximum response of the system during a period. specified by equation (1.2.1) then the constant I AI the motion. is defined as the time average of the square of the

= lim --1 [ r w2(t) dt. T--->~* Z J°

(1.2.8)

The root mean square (rms) amplitude is the positive square root of the mean square amplitude. For the harmonic oscillation of equation (1.2.1) the rms amplitude is independent of phase and is equal to Ial/~. Free vibrations are the motions of the system in the absence of external disturbances and as a result of some initial conditions. Forced vibrations are the motions of the system produced by external, persistently applied disturbances.

3

INTRODUCTION TO MECHANICAL VIBRATIONS

Natural frequencies are those frequencies at which response exists during free vibration. The lowest natural frequency is called the fundamental frequency. Transient motion is motion other than steady state response. If damping is present, transient response will decay with increasing time. Phasor. A phasor is a rotating vector representation of the harmonic motion of the system. The periodic motion of equations (1.2.1) and (1.2.2) can be represented in a complex form, which is more convenient for mathematical manipulations and is given by w(t) = A e j°~t, (1.2.9)

where A and w(t) are complex with the complex amplitude specified as (1.2.10)

A= AR + jAI,

The phasor representation of equation (1.2.9) is shown in Fig. 1.1. The length of the vector, I AI, is the real amplitude of motion. As the vector rotates with angular velocity to in a counterclockwise direction, its projection on the real and imaginary axis of the complex w plane varies harmonically with time t. A rotation of the vector through 360 ° corresponds to a cycle of motion. In this text the convention used is that the real component of the phasor or the complex description of the motion corresponds to the actual, observed or measured motion. Therefore the actual motion is given by w(t) = Re[A eJ°'t].

(1.2.11)

Using the relationship eJ°"=cos tot + j sin tot and substituting A = AR +jAI into the above expression yields equation (1.2.2). The phase of the motion, q~, is thus retained through the ratio of the imaginary and real components of the phasor as specified in +lm

AE J I I I I I

AR

+Re

ines phasor position at t = O)

Fig. 1.1 Phasor diagram representation of harmonic motion.

4

ACTIVE CONTROLOF VIBRATION

equation (1.2.4). Note that a negative sign has been used in equation (1.2.2) in contrast to many texts dealing with vibration. This choice ensures that the phase q~ is positive and since At is a constant to be determined by boundary conditions, the choice of negative sign does not affect the result. Since the phasor is a vector, any number of harmonic motions of the same frequency can be added vectorially. For linear motions written in complex notation, since the principle of superposition holds, this simply means separately summing the real and imaginary components of the individual motions. In this text the majority of the response equations are written using complex notation as this is the most convenient form for analysing systems where responses are superimposed (i.e. as is the case in active control simulations). The actual motion can be directly recovered by taking the real part of the complex description. Where the motion is directly described in the actual form, it is indicated in the text.

1.3 Single-degree-of-freedom (SDOF) systems Consider a mass M supported by a massless spring as shown in Fig. 1.2(a). As the displacement of the system can be completely specified at all times by a single variable w, the system is said to possess a single degree of freedom (SDOF). By appropriately w

T (a)

M

(b)

M

(d)

M

K

//

/

/ / / /

/

/ //'

1" (c)

M

K
1, both terms in equation (1.5.5) will be real and this implies a steadily decaying response with no oscillation. This is termed an overdamped system. When = 1, the system is said to be critically damped. This value of C represents the smallest possible damping required to prevent oscillatory motion and ensures that the system returns to its rest position in the shortest time as shown in Fig. 1.3. When ~ < 1 then the square root term will be real, positive and 7 in this case will be complex with a negative real part. Thus the response will oscillate at a damped natural frequency oJa = oJ~x/1 - ¢2

(1.5.6)

and decay in amplitude with increasing time. This is called light damping and it is exhibited by most structural systems which are thus described as underdamped. The observed response to the specified initial conditions defined in Section 1.4 is obtained by using the real part of equation (1.5.3) and the initial conditions to solve for the constants A ! and AR as described in Section 1.4. The actual displacement is then given by

w(t) = e -~,.¢t[w(0) cos COdt+ vi,(0) + ¢co,w(0)sin (Oat] .

(1.5.7)

(.,0d

This equation can also be written in simple harmonic form as -~o,¢t W(t) = IAle cos(~oat+ q~),

(1.5.8)

where the phase angle q~ is now given by q~= t a n _ l ( w ( 0 ) + ~ w ( 0 ) ) -

w(O)~oa

(1.5.9)

and the real amplitude by

I AI = {[w(0)]2 + [w(0)+ ~c0~w(0)]2/~03} 1/2.

(1.5.10)

8

ACTIVE CONTROL OF VIBRATION

Wo

light damping

~t)

~' 0 which is

M~

w(x) =

"~,e-jkrx - e

).

(2.6.3)

4Elk~ Note that in equation (2.6.3) the displacement field is again composed of a travelling wave component and a near-field component. The actual displacement is given by the real part of equation (2.6.3). The power flow in the positive x direction in the infinite beam can be evaluated using a similar procedure as outlined above and for moment excitation is given by (Gibbs and Fuller, 1992a) Fin = ~ . 16Elky

(2.6.4)

The total input power to the infinite beam system will be twice HB.

2.7

Free flexural motion of finite thin beams

In many control problems it is necessary to derive an expression for the response of finite beams to a disturbance. In order that the resulting motion be determined, it is first necessary to apply the appropriate boundary conditions in order to derive the free response. Commonly encountered simple boundary conditions are as follows. (a) Simply supported end. In this case the end of the beam is free to rotate but is constrained to have zero displacement and moment. The boundary condition is thus specified by

w(x) = 0

and

~2w(x) Ox2

= 0.

(2.7.1)

at the constraint location. (b) Clamped end. For a clamped boundary condition, both the displacement and rotation are constrained to zero. The boundary condition is thus specified by

w(x) = 0

and

bw(x)

= 0.

(2.7.2)

bx (c) Free end. For a free end, both the shear force and the internal bending moment must disappear. Thus the boundary condition is

~2w(x) ~x 2

~3w(x) =0

and

~x 3

= 0.

(2.7.3)

INTRODUCTION TO WAVES IN STRUCTURES

37

(d) General termination impedance. In many cases the beam is terminated by a known impedance. To completely specify the termination impedance both a bending termination impedance, Z'x and a transverse shear impedance Z) need to be specified. The appropriate boundary condition is then applied by matching the beam internal bending impedance to the termination bending impedance. Thus Ztx =

Mx(x) O(x)

(2.7.4)

and matching beam transverse shear impedance gives Z)= Ty(x)

(2.7.5)

w(x)

evaluated at the termination location. Simultaneously applying the above boundary conditions to an assumed wave field allows determination of both the travelling wave and near-field amplitudes. Let us apply the above approach to the case of a finite beam of length L, simply supported at each end. For the simply supported boundary conditions the flexural near fields can be shown to vanish. The beam response can be then written as the superposition of two travelling waves in the positive and negative directions with unknown coefficients A and B:

w(x, t)= A eJ~t-Jkl~ + B eJ°~t+Jklx.

(2.7.6)

Applying the boundary condition of (only) zero displacement at the constraints (since we do not have near-field components) we find at x = 0 that w(x) l x=O= O,

(2.7.7)

A = -B

(2.7.8)

w(x)[ x__L= O,

(2.7.9)

A e-JkIL + B eJ~;~L= 0.

(2.7.10)

which results in

and also at x = L, we can put

which therefore results in

Substituting equation (2.7.8) into equation (2.7.10) results in the displacement field being specified by

w(x, t)= - 2 A j sin klx e j~.

(2.7.11)

Applying the boundary condition of zero displacement at x = L to equation (2.7.11) allows the derivation of the system characteristic equation which is expressed in the form sin kiL = 0,

(2.7.12)

which therefore implies that the eigenvalues are given by k~= ~ ,

L

n = 1,2,3 .....

(2.7.13)

38

ACTIVE CONTROL OF VIBRATION

Thus unlike the infinite beam, a finite beam can only vibrate in free motion at discrete frequencies (i.e. resonate) such that the free flexural wavenumber equals the discrete values given by equation (2.7.13). We denote these discrete values of wavenumbers k,, n = 1,2,3, .... Figure 2.5 shows the dispersion curves of two beams of different thickness. Also shown in Fig. 2.5 are the eigenvalues given by equation (2.7.13) for a specified length, L. The resonance frequencies of the modes of the system are given by the intersection points of the eigenvalues and the dispersion curves. Figure 2.5 illustrates the duality in the wave and modal description of the vibration of finite elastic systems, since the beam motion can be thought of in terms of a standing wave (a mode) or two equal waves travelling in opposite directions in the beam. Using equation (2.7.13) and equation (2.3.10), the resonance frequencies of the simply supported beam can be calculated from n = 1,2, 3,...,

(2.7.14)

mI

where m' is mass per unit length of the beam. A more general solution of the free vibration of a finite beam with general impedances at each end would require equation (2.7.6) to be expanded to include terms describing the flexural near fields and the application of both boundary conditions of equations (2.7.4) and (2.7.5). The total normal response of the beam (i.e. without an external persistent disturbance) is given by the superposition of the individual eigensolutions, or modes, to give oo

w(x,t) = Z W~p, e

je)nt

,

(2.7.15)

n=l

where W~ is the modal amplitude, ~. is the mode shape of a simply supported beam given by ~. = sin(k.x) and co. is the resonance frequency of the nth mode. Beam (a)

..~

f,

J

2re

I ~-

Beam (b)

I--

-E

I== z //I

!

I

Frequency, co

Fig. 2.5 Relationship between eigenvalues and free wavenumbers for two differing finite beam systems.

39

I N T R O D U C T I O N TO W A V E S IN S T R U C T U R E S

Model

lo-2 V

Mode 3

Mode 1 A ~.

/ , " , ,'-2x

10-3 2

•~o

3 "

10 - 4

E ~ rn

0-5

~o-6

_ I

I

I

10

I

I

I

I

I[

I

100

I

t

I

t

I

t I

I

1000

Frequency (Hz)

Fig. 2.6

Frequency response function and mode shapes of a simply supported beam.

Figure 2.6 shows an example frequency response function (FRF) for a beam of length L = 0 . 3 8 m, bending stiffness E I = 5 . 3 2 9 N m 2 and mass per unit length, m' =0.6265 kgm -~ calculated using equation (2.8.12). Also shown are the mode shapes corresponding to the peaks, or resonance points in the FRF. The locations of the peaks are thus predicted by equation (2.7.14) while the associated mode shapes correspond to those of equation (2.7.15). An interesting characteristic of beam behaviour is that the resonance frequencies become spaced further apart with increasing mode number. This is also apparent from Fig. 2.5 and is due to the dispersive nature of flexural waves. Figure 2.6 will be further discussed in Section 2.8. Boundary conditions other than those for a simply supported beam will result in additional near-field motion located near the discontinuities. If the beam is long compared to the wavelength these near fields will decay before they reach the other constraint and thus will have little effect on the nature of the global response (i.e. they will not significantly influence the resonance frequencies and characteristic functions of the system).

2.8

Response of a finite thin beam to an arbitrary oscillating force distribution

We now study the flexural response of the finite beam to a harmonic force of arbitrary distribution f(x, t)= F(x)e j°'' where F(x) is considered positive in the upwards direction. The differential equation of motion can now be written in inhomogeneous frequency domain form as

F(x) E1

daw(x) - k~w(x)= ~ ,

dx 4

(2.8.1)

40

ACTIVE CONTROLOF VIBRATION

where w(x) describes the complex transverse displacement of the beam and e j'°' has again been omitted for brevity. If we further assume that the beam is finite and has simply supported boundary conditions then we can seek a solution of equation (2.8.1) in the form of a series composed from the eigenfunctions or free mode shapes found previously. We therefore assume that

w(x) = Z ~ sin knx,

(2.8.2)

n=l

where Wn are the unknown amplitudes of the response of the system. Substituting equation (2.8.2) into equation (2.8.1) we see that Z (k~ - ¢)Wn sin k n x

=

F(x)/EI.

(2.8.3)

n=l

In order to solve for the coefficients Wn, we use the orthogonality characteristic of the eigenfunctions (see Section 1.9) defined for the continuous beam system as if n , m,

(2.8.4a)

L o ~nWmdx = An if n = m,

(2.8.4b)

L

o

~)n~Imdx = 0

IIL

where ~Pn is the beam mode shape function and An is the mode normalisation constant 1 which is specified for the simply supported beam as An= ~. Note that equations (2.8.4a,b) are a specific form of the general orthogonality relations given in equations (1.9.14a,b) and are used when the beam is homogeneous and thus the mass and stiffness does not vary spatially. Thus to separate out individual modal contributions we multiply through both sides of equation (2.8.3) by ~Pn= sin knx and integrate over the length of the beam L. We then find an expression for the complex amplitudes Wn given by 2

I L F(x) sin k~x dx. IVn= EIL(k4-k}) o

(2.8.5)

Equation (2.8.5) demonstrates that in order to solve for the response of the system, we have conveniently expanded the forcing distribution into a series with the system free mode shapes as the basis functions; the magnitude of the modal amplitudes W, are dictated by the shape of the forcing function. Note that the choice of the mode shapes as the basis functions implies that the solution using equation (2.8.2) in conjunction with equation (2.8.5) is well-conditioned to determine the global system response and ill-conditioned to predict the near-field effects at say the drive point since the basis functions do not readily model the evanescent motions. In other words, at low frequencies the infinite sum in equation (2.8.2) can be truncated at a relatively small number of modes in order to obtain the global response of the beam but a relatively high number of modes is needed to accurately evaluate the near-field components. An alternative approach as used by Morse (1976), for example, is to use the normal modes of a free-free beam as the expansion functions. In this case the near fields are automatically included in the mode shapes.

INTRODUCTION TO WAVES IN STRUCTURES

41

For illustrative purposes we study two example forcing conditions. If the forcing function is constant over the beam, i.e. F (x) = F then 2F L W~ = EIL(k 4 - k}) Io sin k~x dx,

(2.8.6)

and when the integral is evaluated, this can be written as

W~ = - EILk,(k 4 _ k} )

~ - - ~ ]Jo

which finally reduces to W~= -

4F nTtEZ(k 4 - k})

,

n = 1,3, 5 .....

(2.8.8)

Thus the response of the beam for a uniform (or by extension a symmetric) forcing function is only in the n = 1,3,5 .... or symmetric modes. This characteristic is also intuitively obvious when one examines the spatial phase distribution of the n= 2, 4, 6 .... or antisymmetric modes. For example the phase of the n = 2 mode changes through 180 ° at the mid point of the beam and thus the integral of equation (2.8.5), which can be split into two antisymmetric terms about the mid point, is zero for this mode. It is also interesting to note that the amplitude of the response of higher order modes decreases with increasing modal order. This characteristic will be found useful when controlling structural motion with point forces which tend to couple into many modes. If it is desired to excite only a particular mode in the beam corresponding to a given integer n, then it is apparent from equation (2.8.5) that the forcing function, for the case of a simply supported beam, takes the distribution F ( x ) = F sin knx. By the orthogonality relations of equations (2.8.4a,b) it is apparent that the response in all other modes except the nth mode will be zero. These observations generally apply to all structures whose vibratory motion can be described as a series of orthogonal modes. Characteristics such as these are important in the design of distributed actuators and sensors as outlined in Chapter 5. In general we desire to control a low number of modes of vibration and wish to achieve this without exciting other modes (an effect termed control spillover by Balas, 1978). In this case we can sometimes tailor the input control function F (x) to achieve the required modal excitation in the control field. A point force excitation is also of prime interest and is specified, using the Dirac delta function, by F ( x ) = F r ( x ) . Equation (2.8.5) then becomes

W~ = - E I L ( k 4 - k} ) I c 6(x - xi)sin knx dx.

(2.8.9)

Using the 'sifting' property of the Dirac delta function discussed in Section 1.7, we can evaluate the integral in equation (2.8.9) which leads to W~=

sin knxi ~ . ElL k 4 - k} 2F

(2.8.10)

42

ACTIVE CONTROLOF VIBRATION

The total response of the beam including the harmonic time component is then given by

w(x,t) =

2F ~

ElL ~=

sin k~xi sin k~x ejO~, .

k4 - k}

(2.8.11)

The response relations described by equations (2.8.5), (2.8.8) and (2.8.10) can easily be rewritten in terms of frequency by using equations (2.3.10) and (2.7.14) manipulated to give k 4 = m'to2JEl and k}= m'to2/El where m' is mass per unit length of the beam. For example the total response of the beam due to the point force can be written as a function of frequency in the form

w(x,t) =

- 2F ~ M

=

sin knX i sin knx _io~t to2_ to2

e ,

(2.8.12)

where ton is the natural frequency of the nth mode and M is the total mass of the beam. When kn= kI, or to = ton, the excitation frequency corresponds to the resonance frequency of the nth mode of the beam, and as expected the system response approaches infinity. This singularity in the response function can be overcome by the approximate method of introducing hysteretic damping into the beam. This involves expressing the Young's elastic modulus of the beam material as E ' = E(1 +jr/,) where JT, is the total loss factor (Cremer and Heckl, 1988) when evaluating kI, the free flexural wavenumber at frequency to. Figure 2.6 shows the velocity response of a simply supported beam (with material properties specified for the previous example of Section 2.7) to an oscillating point force of unit amplitude located at xi=O.1L. The beam is assumed to have a damping represented by a value of JTt= 0.001. It should be noted that the hysteretic model of damping is only strictly valid for steady state harmonic motion. Use of the hysteretic model for an impulsively excited structure leads to a non-causal response, i.e. the response of the system apparently anticipates the excitation. The poles of the system transfer functions which correspond to the roots of the system characteristic equation are found in both the positive and negative parts of the s-plane (see Section 1.7 for a brief description of the Laplace domain described by the complex variable s). As will be discussed in Chapter 3, these characteristics of the hysteretic damping model have important implications in active control in terms of controllability and stability. In these situations it may be advantageous to use a viscous model of damping, although this is certainly prohibitive when the system has many degrees of freedom. The reader is referred to the papers by Crandall (1970) and Scanlan (1970) for more details on this aspect of the hysteretic damping model. Equation (2.8.11) again illustrates the duality in the interpretation of the system response as either a sum of modes or as a superposition of travelling waves. As the system is being forced by a harmonic steady state input, the finite system will respond at the same frequency of excitation as the forcing function and not at the discrete wavenumbers defined by the eigenfrequencies. If the excitation frequency is away from the resonance frequencies of the beam, the response will be composed of free waves with wavenumbers +kI that travel backwards and forwards, without constructive reinforcement in the beam system. However, when the excitation frequency is such that ki= kn then the free wavenumber (or wavelength) also corresponds to the natural eigenvalue of the free system and the free waves reflect from the beam terminations

INTRODUCTION TO WAVES IN STRUCTURES

43

with constructive interference and reinforce the beam motion. The net result is a large response associated with the resulting resonance. Behaviour such as this will be shown later to be critical to the performance of active vibration control systems in terms of stability, power requirements and performance. The response in beam systems can also be seen to be critically dependent upon the location and distribution of the disturbance force. It has been demonstrated that a distributed input can be shaped in order to excite selected modes. In contrast, a point force input which is described using the Dirac delta function can be seen from the definition of the Dirac delta function (Section 1.7 and equation (2.4.2a)) to have a spatial Fourier transform value of unity. Thus a point force can also be thought of as being composed of a sum of all wavenumber components with equal amplitude, a characteristic termed spectrally white in a wavenumber sense. The point force will then couple into all modes if appropriately located. Equation (2.8.11) predicts, on the other hand, that if the point force is located on a nodal line of a particular mode then that mode will not be excited at all. Likewise the position of the force relative to the mode shape strongly influences the magnitude of the resultant modal amplitude. Obviously if control of a particular mode is required then the position of application of the point force control input is important. Finally equation (2.8.10) predicts that for a particular frequency, as the modal order increases, the contribution of the higher order modes to the total response will decrease. Thus structures such as beams act as low pass filters of a disturbance excitation, when considered in terms of their modal response. This effect can often work to advantage in curtailing the aforementioned control spillover problem.

2.9

Vibration of thin plates

The previous sections have been concerned with the vibration of thin beams. It is of interest to extend the scope of our discussions to consider elastic motion in two dimensions, i.e. the vibrations of thin plates or panels. The equation of motion of a plate can be written as a two-dimensional extension of that governing the motion of beams. Following Cremer and Heckl (1988) the equation of motion can be written as

E1( ~4w+ 2 ~4w + ~4wI + ph ~2w= -p(x, y, t), ~X 4

~X2c~y2

~y41

~t 2

(2.9.1)

where h is the thickness of the plate, I is the moment of inertia per unit width and p is the applied external pressure or load. For the in-vacuo case when considering the free motion of the plate, the applied pressure is set to zero. As discussed in Section 2.2 the value of modulus of elasticity given by equation (2.2.9) should be used when dealing with plates. Hence, in this case E l = h 3 E / [ 1 2 ( 1 - v2)]. Important approximations in deriving equation (2.9.1) are similar to those made in deriving the equations for beam motion. It is assumed that the plate is thin with respect to a wavelength of motion and transverse shear as well as rotary inertia of the plate motion are ignored. This effectively limits the mathematical description to low frequencies (Cremer and Heckl, 1988).

44

ACTIVE CONTROL OF VIBRATION

Waves in two-dimensional structures can be described in a variety of coordinate systems. A structural wave travelling in the plate at an angle a to the x axis (see Fig. 2.7) can be expressed as (2.9.2)

w(x, y, t) = A eJ~t-;kxX-jky y.

Substitution of equation (2.9.2) in equation (2.9.1) and using some algebraic manipulations in conjunction with equation (2.3.10) results in k~= k2 + k2.

(2.9.3)

It is apparent that equation (2.9.3) represents a vector relationship between the three wavenumber components and thus the x and y wavenumber components are related to the free wavenumber by kx = k I cos a,

(2.9.4)

k,= klsin a. Equation (2.9.3) demonstrates an important characteristic of wavenumber. The free wave travels at angle a (illustrated in Fig. 2.7) at the speed of flexural waves in plates. The free wavenumber kI can also be vectorially decomposed into x and y trace components as dictated by equation (2.9.4). Thus in two-dimensional systems whose boundaries are parallel to the coordinate axes, resonance will occur when the trace wavenumber components kx and ky in the x and y directions simultaneously equal an eigenvalue (see Section 2.10 for an example of this) in each direction. Free waves can exist in a variety of different forms expressed in terms of different coordinate systems. Choice of the appropriate coordinate system is dependent upon the system configuration. For example, the excitation of an infinite plate by a point force is more conveniently studied in cylindrical coordinates, as discussed in the text by Junger

/

Simply supportededges

a

y~ ~r v

L.,

X Vl

Fig. 2.7

Simply supported rectangular plate coordinate system.

INTRODUCTION TO WAVES IN STRUCTURES

45

and Feit (1986). In general, however, it is more appropriate to choose the coordinate system of the equation of motion based on the alignment of the system boundaries rather than the geometric characteristics of the forcing function.

2.10

Free vibration of thin plates

In many cases the system to be examined is such that it is either not possible to solve the system differential equation or the shape of the boundary cannot be easily described in terms of a coordinate system. In such cases, one may be able to pursue the analysis using approximate methods in which the continuous system is approximated by an N degree-of-freedom system as outlined in Sections 1.8-1.12. Such techniques are also well described by Meirovitch (1967). For the moment we will restrict ourselves to the study of the free vibrations of a rectangular thin plate which is simply supported along the edges. On the basis that a simply supported plate is a two-dimensional extension of a simply supported beam it is appropriate to choose a separable solution of the transverse modal displacement of the form Wmn(X ,

y, t)

= Wren s i n k m x sin

k,y

e )°Jt,

(2.10.1)

where Wm, is modal amplitude and m and n are modal indices. Applying the boundary conditions of zero transverse displacement at the plate edges, shown in Fig. 2.7, results in expressions for the wavenumber eigenvalues in each coordinate direction that are given by

met/a, k,, = net~b, km =

m = 1,2,3 .... , n = 1,2, 3 . . . . .

(2.10.2)

Substituting the eigenvalues associated with resonance of the trace wavenumber components kx and ky in the x and y direction into equation (2.9.1) and using the relation between frequency and free wavenumber given by equation (2.3.10) we can solve for the discrete frequencies at which the system resonates in two dimensions. These are given by

+

]

(1)ran ~-'-~/ L~-~] ~ b I J' where the subscripts m, n denote the (m, n)th mode of vibration. Examples of the shapes of different modes of vibration are shown in Fig. 2.8. The fundamental (1,1) mode has an associated motion with no phase change across the plate surface. The higher order modes are characterised by nodal lines through which the relative phase of the displacement function flips 180 °. The discussion presented in Chapter 8 will demonstrate that modal order and the corresponding mode shape function have a significant effect on sound radiation and control. Another important aspect of plate response is the modal density or number of modes within a frequency bandwidth. The modal density of thin plates is large and the 'modal overlap' increases with frequency (Cremer and Heckl, 1988). Thus modal control of two-dimensional systems is a far more difficult problem than the equivalent onedimensional problem since there is a much larger number of significant degrees of freedom in a given band of frequencies (see Chapter 6).

46

ACTIVE C O N T R O L O F VIBRATION

Mode (m,n) (1, I)

(2, 1)

+

+

f

Nodal line (3, 1)

(1,3)

+ +

+ 4-

Fig. 2.8 Selected mode shapes of a simply supported rectangular plate.

2.11

Response of a thin rectangular simply supported plate to an arbitrary oscillating force distribution

Following a similar procedure to that outlined in Section 2.8, the in-vacuo response of a rectangular plate to a two-dimensional forcing function F(x, y) e l~'t c a n be written as

EI(~)aw +2 O4W + ~4wI + ph OZw = -F(x,y) e_i,,,t, ~X4 ~x2~y2 ~y4 ] ~)t2

(2.11.1)

where for the plate, I = h3/[12(1- v2)] and F(x,y) has the units of pressure. We assume, as previously, that the forced response can be written in terms of a sum of modes of the free response of the plate vibrating at the forcing frequency, i.e. oo

oo

w(x,y,t) = Z Z Wrensin kmx sin knY ej~t.

(2.11.2)

m=l n=l

On substituting the assumed response into equation (2.11.1) and using the orthogonality property of the plate mode shapes in the x and y directions as discussed in Section 2.8, we obtain an expression of the plate response amplitudes given by

Wmn-- M(to24_(-Omn)2I aOI boF(x, y) sin kmx sin kny dx dy,

(2.11.3)

I N T R O D U C T I O N TO WAVES IN STRUCTURES

47

where M( = pshab) is the total mass of the plate. As before we have found a solution by expanding the input force into components with the system mode shapes as the basis functions. If the input forcing function is a point force, f ( x , t)= F 6 ( x - x i ) 6 ( y - y i ) e j'°', located at x/, y/, then, using the 'sifting' property of the Dirac delta function, the integrand has a value only at xi, y~ and the modal amplitudes are given by Wmn =

4F sin kmxi sin k~ Yi M(o)2

.

2 -- (Dmn)

(2.11.4)

Once again the plate modal response is observed to depend strongly upon the location of the input force, the modal input impedance and the input frequency. If the input frequency, co, equals a resonance frequency, com,, the (m, n) mode response will again approach infinity, due to lack of system damping. In order to evaluate the total plate response, the double summation of equation (2.11.2) is truncated at a finite number of modes chosen to ensure a satisfactory convergence of the series. In many cases of interest the plate is excited by input moments. It is then convenient to write the inhomogeneous plate equation in moment form as (Timoshenko and Woinowsky-Kreiger, 1984) c-:-:32Mx c32M~ ~2My ~2w _j~ot +2 +~ - ph ~ = M(x, y) c , ~x 2 axSy ay 2 at 2

(2.11.5)

where in equation (2.11.5), M ( x , y) the disturbance moment distribution has the units of moment per unit length per unit area of plate. The internal moments of the two-dimensional plate element are specified by (Ugural, 1981) M x = -EI.-z-7~ + v ----'S"

[ 02w Ox

02wI Oy l

(2.11.6a)

[ b2w 82w I My = -El~-~y 2 +V ~x'--S]'

(2.11.6b)

~2 W

M~y = -EI(1 - v)

.

(2.11.6c)

Oxay

The shear forces acting on the edges aligned along the x and y axes are respectively (Ugural, 1981) x Tf =

=

ay/xy

+ ~, Oy Ox

(2.11.7a)

" +~.

(2.11.7b)

by

Substitution of these expressions into equation (2.11.5) confirms its equivalence with equation (2.9.1).

48

ACTIVE C O N T R O L OF VIBRATION

An important forcing function associated with piezoelectric distributed actuators is a line moment around an axis located at Xl. In this case the plate equation is written in moment form as

Ox2

+2

~.

OxOy

.

Oy2

.ph .

Ot2

. Md'(x

X1)_jtat c ,

(2.11.8)

where t~' (-) is the derivative of the Dirac delta function with respect to its argument and the moment amplitude M has units of moment per unit length. This form and other two-dimensional moment configurations will be useful in Chapter 5 for analysing the response of plates to excitation by piezoelectric distributed elements.

2.12

Vibration of infinite thin cylinders

In the previous sections we have studied the vibrations of planar structures. In this section we will deal with a brief study of an important curved structure, the thin-walled shell. Shell theory is often used to model common structures such as the fuselages of aircraft or the hulls of submarines. Figure 2.9 shows the cylindrical coordinate system and the notation used in the analysis for the displacement in the radial, axial and torsional directions. Various theories describing the motion of the shell with different approximations have been derived and are summarised by Leissa (1973). The most significant aspect of the vibration of curved bodies is that the motion must be considered in three axes. Thus in thin-walled shell vibration, the equations are written in terms of the in-plane (axial) motion, u, the out-of-plate (radial) motion, w and the torsional motion, v. The simplest thin-walled shell equations are the Donnell-Mushtari equations which are written for in-vacuo motion as (Leissa, 1973) O2u

+

(1 - v) O2u

c)x2 (1 +v)

2a v Ou

l Ov

aOx

a 2 c)0

2a 2 O2u

i)xi)O w

+~+f12 a2

+

002 (l-v)

+

2

~x 4

O2v

2a

i)xi)O

O2v OX2

a

cL2

Ox

1 Ow

a 2 002

+ ~x2~02

//

I

~4W

+2

v Ow

1 O2v

~-t-

2

(a O4W

(1 + v)

a 2 t)04

iJ

=0,

~

2 CL pa(1- V2)

c t2

Eh

a 2 ~0

l ~4w )

=0,

+~=

(2.12.1a)

(2.12.1b)

. (2.12.1C)

In the above equations, the stiffness factor fl is given by fl= h2/(12a 2) and the longitudinal or axial phase speed CL is given by equation (2.2.10) and is the same as for plate motion. Setting f l - 0 results in the equations reducing to those describing the motion of a curved membrane. The variable Pa is again an external forcing function or load with the units of pressure. Unlike the wave equations used to describe the motion of beams and plates, the shell equations consist of three coupled equations for each axis of motion which must be solved simultaneously. Important assumptions used in deriving the Donnell-Mushtari equations are similar to those used previously for thin beams and plates with the additional assumption that the variation in transverse shear stress in the circumferential direction is ignored (Junger and Feit, 1986). Higher order

INTRODUCTION TO WAVES IN STRUCTURES

49

X

h

i Fig. 2.9

Cylindrical coordinate system for an infinite thin cylinder.

thin shell theories can be obtained by adding correction factors to the Donnell-Mushtari shell equations as described in Leissa (1973). In order to solve for free wave motion in an infinite cylindrical shell we first assume displacement distributions for the shell wall of the form jwt-jknsx + j:r/2

u(x, O,t)= Z Z U~scos nO e n=l

,

(2.12.2a)

s=l

jwt-jknsx

v(x, O,t) = Z Z V,s sin nO e

,

(2.12.2b)

n=Os=l oo

oo

w(x, O,t)= Z Z W,s cos nO e n=0

jwt-jknsx

.

(2.12.2c)

s= 1

The above assumed distributions have appropriately chosen angular and axial functions to ensure that the circumferential variation will be a stationary mode pattern. The subscripts n and s correspond to azimuthal modal order (n = 0, 1,2 .... ) and branch order (s = 1,2, 3 .... ) respectively. The modal order n can be seen to correspond to the number of radial nodal lines, while s indicates the order of particular eigensolutions

n=l

n=O

\

t

f"-~

/ t

Fig. 2.10

I

\ I

n=2

/

Xln=3

Circumferential mode shapes of an infinite thin cylinder.

50

ACTIVE CONTROL OF VIBRATION

for a fixed n. Typical lower order circumferential mode shapes are shown in Fig. 2.10. A complete solution should use a circumferential distribution with a form e ±j"° which can result in rotating angular distributions or 'spinning modes'. Substituting the above distributions into the shell equation of motion and setting the disturbance input pressure p~ to zero results in a system of equations which can be conveniently described in matrix form for a particular mode (n, s) as

where

Ll1

L12

~

Z31

I

Uns

0

/~. /~3

Vns =

O,

L32

Wns

o

L~ ._ _~-~2

+

L13

L33

(knsa) 2 + ~1 (1 - v)n 2 ,

1

L~2 = ~(1 + v)n(k,~a),

(2.12.3)

(2.12.4a) (2.12.4b)

L13 = v(k,~a),

(2.12.4c)

L21 = L12, L22 = _f~2 + ~1 (1 - v)(k,~a) 2 + n 2

(2.12.4d) (2.12.4e)

L23 = n,

(2.12.4f)

L31 = L13,

(2.12.4g)

L32 = L23, L3 3 = _~-~2 +

1 + fl2[(k,~a)2 + n2] 2.

(2.12.4h) (2.12.4i)

In the above equations (2.12.4a-i), f~ is the non-dimensional frequency, ~ = (Da/cL and c~. is longitudinal phase speed of the shell material with effective modulus of elasticity given by equation (2.2.9). A value of ~ = 1 corresponds to the ring frequency of the cylinder when the system resonates as a ring due to longitudinal waves travelling around the shell with a wavelength equal to the circumference of the shell. Thus f~ is alternatively defined as ~ = o.)/(.t) r where (.or the ring frequency of the cylinder is given by (D r - - C L / a . Expansion of the determinant of the amplitude coefficients in equation (2.12.3) provides the system characteristic equation from which dispersion plots can be generated which relate the non-dimensional axial wavenumber, k~sa to the nondimensional frequency, ~. A typical dispersion plot for free waves in a steel cylindrical shell with a thickness ratio of h/a=O.05 and Poisson's ratio v=0.31 is shown in Fig. 2.11 for a circumferential mode number n = 1. The results of Fig. 2.11 were calculated using the Fliigge shell equations in order to be accurate at very low values of ~. This is achieved by adding small correction terms to the Donnell-Mushtari equations as outlined in Leissa (1973). Dispersion curves for different values of n have shapes similar to those of Fig. 2.11, the main difference being that the waves cut-on at different frequencies. Unlike waves in plates, cylindrical shell waves can be seen to have three forms of roots; purely real roots which correspond to propagating axial, torsional and flexural type motion (for f~> 1), a purely imaginary root which asymptotes to the same value as a plate bending near field for ~ ~>1 and complex roots which when paired together correspond to an attenuated, near-field standing wave (see Fuller, 1981). The behaviour of the waves can generally be divided into two regions,

INTRODUCTIONTO WAVES IN STRUCTURES

Re knsa

51

+

5.0

/ o/

I

2.0

/i

! 4 / / ~ / , / \

-5.0 / f

Bending near field "\.

f

f f I

~"

(b)

Fig. 2.11 Dispersion plot for waves in an infinite thin-walled shell, v=0.31, h/a=O.05, n = 1; ~, purely real solutions; - - - - - , purely imaginary solutions, - - -, complex solution (after Fuller, 1981).

either below or above the ring frequency, ~ = 1. Above the ring frequency the behaviour of the shell is similar to a flat plate and most of the energy of vibration is in bending. Below the ring frequency, the behaviour is far more complex due to the increased relative curvature of the wall to the wavelength and most of the energy is in stretching. The solution denoted (a) in Fig. 2.11 corresponds to beam motion of a long slender rod at low frequencies ( ~ ~ 1) and approaches wave motion similar to that of flexural vibration of a flat plate at high frequencies (f~-> 1). The solution denoted (c) is purely imaginary at very low frequencies and cuts-on at f~ = 0.5 to approach torsional motion at higher frequencies. Solution (b) consists of two complex solutions at low frequencies which when combined together form an attenuated standing wave (or near field). As the frequency is increased the two solutions meet in the purely imaginary plane and then diverge to become either that of the bending near field on a fiat plate or, after cut-on, approach longitudinal motion in a fiat plate. More detailed discussion of the free-wave characteristics of cylinders is given by Fuller (1981). It should be noted that for some of the wave solutions, in particular the complex branches, the Donnell-Mushtari shell equations do not behave well at extremely low frequencies ( ~ ~ 0). In this case it is better to use a higher order shell theory such as the Fliigge shell equations.

52

2.13

ACTIVE CONTROL OF VIBRATION

Free vibration of finite thin cylinders

The above analysis can be modified to study the response of finite cylinders by using a similar procedure to that used for the study of the response of finite beams described in Section 2.7. We assume that the cylinder is of length L and has 'simply supported' boundary conditions (also called 'shear diaphragm', Leissa, 1973); that is the out-ofplane displacement w, the bending moment M x, the torsional displacement v and the shear force TxI are simultaneously constrained to be zero at the cylinder ends. Note that this 'simply supported' boundary condition is more complex than that used for thin beams due to the additional components of motions used in shell analysis. Assuming a wave field in the cylinder, for one circumferential mode of vibration, the radial displacement can be expressed as w,~(0, x, t) = (W/s cos nO e -Jk,~x + W,,r cos nO e.ik,~X)e j'°t

(2.13.1)

and applying the boundary conditions at the end of the cylinder we obtain a similar result for the transverse displacement distribution to that obtained in the beam analysis. This is given by w,~(x, O, t ) = - 2 j W i cos nO sin k,~x e j~'

(2.13.2)

and the corresponding eigenvalues are written as kr~ = s : r / L ,

s = 1,2, 3 ...

(2.13.3)

Resubstituting the eigenvalues back into the system characteristic equation results in a cubic equation in the squared non-dimensional frequency, f~2 (Junger and Feit, 1986). This is given by (~"~2)3 __ A2(~-~2)2 + A1 (~'~2) __ Ao = 0,

(2.13.4)

1 2 v){(l_

(2.13.5)

where Ao=

A1 -

{[(k,~a)2 + 2

+

v2)(knsa) 4 +

+ n214},

( )

+ (3 + 2v)(k,~a) 2 } + f12 3 - v [(k~,a)2 + n213' 2

a2 = 1 + ( 3 ) 2

[(k,~a) + n 2] + flZ[(k, sa)2 +

n212.

(2.13.6)

(2.13.7)

Finding the roots of the polynomial given by equation (2.13.4) results in the values of corresponding to the non-dimensional resonance frequencies of the system. Two points are important to note. First, the procedure for obtaining the resonance frequencies is identical to that used for beams, although it is more complicated due to the form of the three individual coupled shell equations. Second, the resonance frequencies could also be obtained, as in the case of the beam, by the intersection points of the eigenvalues specified by sz~/L with the free dispersion curves as shown in Fig. 2.12. For illustration purposes the location of the second flexural resonance frequency, ~2, and the first longitudinal resonance frequency, f~, are shown graphically.

53

INTRODUCTION TO W A V E S IN STRUCTURES

Flexural ./____ __/Torsional ~ ~

2~: L

/i I

.Q

/

I

E

-/

l>

.~ x

/Longitudinal

--!

L

!

. . . . .




c 1000

o o

o

E = E

500

~

c--

N

o -80

|

-60

-40

i

-20 0 20 Phase error (deg)

40

60

80

Fig. 4.6 The convergence coefficients (a) required to give the fastest convergence time in a simulation of the filtered-x LMS algorithm with a sinuisoidal reference signal having four samples per cycle with various phase errors between the estimate of the secondary path G(q) and the true secondary path G(q). The three graphs correspond to pure delays in the secondary path G(q) of 4, 8 and 12 samples. Also shown in (b) is the convergence time resulting from the use of these optimum convergence coefficients under the same conditions.

4.5

Multichannel feedforward control

A single-channel active control system is, in principle, able to completely control the vibration in one direction, at a single point on a structure. It is often found, however, that what needs to be controlled is either the vibration in several directions, or the vibration at several points on a structure. Multiple control actuators must then be used to achieve active control and the performance of these actuators is generally sensed by multiple response (or error) sensors. We will leave aside the design and positioning of such actuators and sensors for the moment and concentrate on the feedforward control problem posed by such multi-channel systems.

103

FEEDFORWARD CONTROL

Vector of primary onse signals u

.

Vector of reference signals

Matrix of control filters

Matrix of mechanical paths

Vector f response (error) signals

Fig. 4.7 Block diagram of a multi-channel feedforward control system. The block diagram of a multi-channel feedforward control system is illustrated in Fig. 4.7, which is a generalisation of that shown in Fig. 4.3. Note that multiple excitation signals, x, have also been assumed, which may, for example, be harmonic reference signals at different frequencies, or estimates of independent random excitations. The controller, H, consists of a matrix of electronic filters which drive each of the actuators with the sum of the filtered versions of each of the excitation signals. The response of the mechanical system, G, is also assumed, in the most general case, to be fully coupled, with the output to each actuator, u, affecting every response signal, e. As in the single-channel case, feedback paths may again exist from the controller output, u, back to the excitation signals, x. It is still possible, however, to lump these feedback paths into the response of an effective controller response, without affecting the analysis of the system performance, as described by Nelson and Elliott (1992). In the following sections the behaviour of such a multi-channel feedforward controller will be considered in both the frequency domain and the sampled time domain, and adaptive algorithms for the adjustment of the controller will be discussed.

4.6 Adaptive frequency domain controllers Assume that the set of excitation signals; depicted in the general multi-channel block diagram of Fig. 4.7, consists of a set of sinusoids. These may be the harmonics of a periodic primary disturbance, for example. The spectrum of each of the elements of the response vector, e, will thus contain tonal components at these frequencies. Providing each of the mechanical paths from actuator to sensor is linear, however, and the system is in the steady state, the action of the set of filters in the controller which affect one frequency will have no effect on the response at any other frequency. The analysis of the control problem is considerably simplified in this case because the adjustment of each of the sets of filters affecting each reference frequency can be considered independently. The analysis of the performance of such a multi-channel system thus has to be performed only at a single frequency. It is convenient to adopt complex notation to denote the amplitude and phase of the various signals, and of the frequency responses of the mechanical paths at the reference frequency. We do not need to explicitly include the reference signal in the analysis, however, since by assuming it takes the form of a complex exponential at

104

ACTIVE CONTROLOF VIBRATION

the reference frequency tOo, it disappears from the analysis. The vector of complex response signals may now be expressed as (4.6.1)

e(flOo) = d (flOo) + G (jto0)u (jto0),

where u(jto0) denotes the vector of contributions to each of the actuators at the frequency too. The block diagram of the multbchannel feedforward controller with frequency domain variables is shown in Fig. 4.8. Note that the amplitude and phase of the controller at too (H in Fig. 4.7) have been absorbed into the definition of u(jto0) and it is now the real and imaginary parts of the vector u(jto0) which are the variables which can be adjusted by the controller. The general properties of several different algorithms which could be used to adjust the components of u(jto0) to minimise the sum of the squares of the error signals, e(jto0), have been discussed in Chapter 12 of Nelson and Elliott (1992). In this section we will concentrate on the minimisation of a cost function which is consistent with that used in optimal feedback control, as discussed in the previous chapter, and has particular application to active vibration control. In the remainder of this section we will drop the explicit dependence of the variables on too for notational convenience. Equation (4.6.1) can thus be expressed as e = d + Gu.

(4.6.2)

The cost function we seek to minimise by the adjustment of the real and imaginary parts of the components of u can now be defined as J = eHQe + uHR U,

(4.6.3)

in which the superscript H denotes the Hermitian (conjugate transpose) of the vectors, and Q and R are positive definite, but not necessarily diagonal, Hermitian weighting matrices (so that QH= Q, RH= R) and J is a real scalar. Note the similarity between this cost function and that minimised in optimal feedback control (equation (3.10.1)). The first term in equation (4.6.3) depends on the response of the system under control. The use of the general weighting matrix Q, however, allows particular aspects of the response to be emphasised, such as that corresponding to the sound power radiated by the mechanical system, for example, as described by Elliott and Rex (1992) and in Chapter 8. The second term in equation (4.6.3) depends on the 'effort' expended by the actuators and the weighting matrix R allows the effort of some actuators to be discriminated against more than others, for example. By suitable choice of the matrix R, the effort term could also be made proportional to the mean square excitation of a set of structural modes not detected by the error sensors (Elliott and Rex, 1992).

u

(jco o)~

G

(j~o o)

)~

e q,oo)

Fig. 4.8 Block diagram of the steady state behaviour of the multi-channel feedforward controller at the reference frequency tOo.

FEEDFORWARDCONTROL

105

Substituting equation (4.6.2) into (4.6.3), the cost function can be expressed as J = uH[GHQG + R ]u + uHGHQd + dHQGu + dnQd,

(4.6.4)

which can also be written in the standard Hermitian quadratic form (Nelson and Elliott, 1991, Section A.5) as J = uHAu + uHb + bHu + C,

(4.6.5)

where the definitions of A, b and c are obvious from equation (4.6.4), and it should be noted that the matrix A in equation (4.6.5) is not the same as the state variable system matrix used in Section 3.6. The vector of control variables which minimise the cost function in equation (4.6.3), u0, and the resulting minimum value of J(Jmin) can then be immediately identified as being u0 = A-~b and J m i n = c - bHA-lb (Nelson and Elliott, 1992, Section A.5). In this case the optimal set of actuator signals can be written as Uo = - [GHQG + R ] -1GHQd.

(4.6.6)

The complex Hessian matrix [A = GHQG + R] associated with this cost function is guaranteed to be positive definite. In particular, the assumed positive definiteness of the effort weighting R ensures this condition even if G HQG is ill-conditioned, or is rank deficient, as would be the case if there were fewer response sensors than secondary actuators, for example. The cost function is thus guaranteed to have a unique global minimum value for u = u0 as given above. One of the standard methods of adjusting the control variables, when the cost function is a quadratic function of these variables with a guaranteed global minimum, is the method of steepest descent. Some care needs to be taken in the development of this algorithm for complex variables, but it is shown in Section 12.4 of Nelson and Elliott (1992), that such an algorithm can be expressed in the standard form u(k + 1) - u(k) - a [Au(k) + b],

(4.6.7)

in which u(k) denotes the vector of control variables at the kth iteration, A and b are the terms defined by equation (4.6.5) and a is a convergence coefficient. Substituting the expressions for these terms (deduced from equation (4.6.4)) into equation (4.6.7) gives the steepest descent algorithm which minimises the cost function defined by equation (4.6.3). This adaptive algorithm can then be written, using equation (4.6.2), as u(k + 1)= [ I - a R ] u ( k ) - aGHQe(k),

(4.6.8)

where e(k) is the vector of complex response signals measured in the steady state after the application of the control variables u(k). The convergence behaviour of a gradient descent algorithm such as this is described, for example, by Widrow and Steams (1985) and Nelson and Elliott (1992). In particular, the convergence behaviour of the cost function can be described in terms of the decay of a number of independent 'modes' of convergence, leaving a residual level which is equal to the exact least squares solution, J m i n = c - b HA-~b. The decay rates of these modes of convergence are determined by the eigenvalues of the matrix [GHQG + R l, and the level to which they are initially excited depends upon the primary disturbance vector, d. Figure 4.9 shows an example of the overall convergence measured for a 32-sensor, 16-actuator control system operating at 88 Hz (Elliott et al., 1992) together with the calculated decay curves of the individual modes

106

ACTIVE CONTROL OF VIBRATION

-10

II

-20 A

1:13 "o

--~ -30 ID

._1

-40 -50 _60

1H, ! 1i I , 0 2

\

,

4

, 6

k,, 8

ll0

, 12

, 14

, -.. , 16 18

I 20

Sample number (thousands)

Fig. 4.9 The convergence of the sum of the squared outputs of the 32 error microphones predicted from equation (4.6.8), - - -, together with the convergence of each of the individual modes of the control system. of this control system. One problem with such an algorithm may be the slow convergence of control 'modes' associated with small eigenvalues of the matrix [GHQG + R ]. It should be noted, however, that the effect of the effort weighting term in the cost function is to increase the value of these small eigenvalues and so reduce this problem (Elliott et al., 1992). In addition, the effort weighting term also makes the algorithm more robust to errors in the measurement of the matrix of responses of the mechanical system at the reference signal, G, which are used in equation (4.6.8) (Boucher et al., 1991). Figure 4.10, for example, shows the measured convergence of a 32-sensor, 16actuator acoustic control system at a frequency of 88 Hz, in which the estimate of the transfer responses from each actuator to each sensor was corrupted by random errors (Elliott et al., 1992). With no effort weighting, the sum of the squared sensor outputs begins to rise after about 15 000 samples due to the presence of slow unstable modes. With a small effort weighting (dashed line), however, the control is stabilised, but achieves a lower level of control (22 dB) than that achieved with a control system with an exact model of the transfer responses (33 dB). To further reduce the problems associated with the slow 'modes' of the steepest descent algorithm, other algorithms may be used which 'rotate' the direction of the gradient algorithm. One such algorithm uses the Gauss-Newton method which, in general, can be written (Nelson and Elliott, 1992, Section 12.5) as u(k + 1) = u(k) - aA-1 [Au(k) + b].

(4.6.9)

in which A and b are again defined by equation (4.6.5). In the case being considered here, equation (4.6.8) reduces to u(k + 1) = u ( k ) - a[GHQG + R]-~ [GHQe(k) + Ru(k)].

(4.6.10)

FEEDFORWARD CONTROL

107

-5

-10 v

-~-15

-20 -25 0

,

,

10

20

,

,

I

l

30 40 50 60 Sample number (thousands)

,

70

80

Fig. 4.10 An example of the behaviour of the control system when the estimate of the transfer response (from each secondary source to each microphone) used in the update equation is corrupted by random errors with a variance which is approximately equal to the real and imaginary parts of the true response. The solid curve is with no effort weighting; the dashed curve is with a small effort weighting.

4.7

Adaptive time domain controllers

Instead of assuming that the set of reference signals (x in Fig. 4.7) are continuous-time sinusoids, as in the previous section, we now assume that they are sampled sequences. It may be that these sequences still represent sinusoids at the excitation frequencies of the primary source, but more generally they could also now represent sampled estimates of a number of random primary excitations. In this section, we will consider the adjustment of the coefficients of an array of digital FIR filters whose inputs are these K reference sequences, xk(n), and whose outputs Um(rl), drive the M secondary actuators. We denote the ith coefficient of the filter driving the mth actuator from the kth excitation signal as h mki, SO that the output of this filter can be generally represented as K

1-1

Um(n)=ZZhmkiXk(rl--i),

(4.7.1)

k=l i=0

which may be regarded as a generalisation of equation (4.4.1), such that there are now MK control filters which each have I coefficients. Again using the operator notation introduced in Section 4.4, equation (4.7.1) can be rewritten as K

Um(n) = Z Hmk(q)xk(n), k=l

(4.7.2)

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ACTIVE

CONTROL

OF VIBRATION

in which the operator Hmk(q) is defined to be I-1

Hm~(q)= Z hmkiq-i.

(4.7.3)

i=O

Again adopting the philosophy developed in the single-channel case, we assume that the overall response, including the analogue filters, data converters, actuator response and mechanical system, in the path from the mth output of the controller to the /th sampled error (response) signal, is represented by a fixed digital filter whose response is denoted a s G tin(q). The lth error sequence can thus be written as M

et(n) = dr(n) + Z Gtm(q)um(n),

(4.7.4)

m=l

where dr(n) is the lth error sequence in the absence of control. Using equation (4.7.2) this error sequence can also be written explicitly in terms of the controller response as M

K

et(n) = dr(n)+ Z Z Gtm(q)Hmkt(q)xk(n).

(4.7.5)

m=lk=l

This can in turn be expressed as M

K

et(n) = dr(n) + Z Z H,~(q)rtmk(n),

(4.7.6)

m=lk=l

where the filtered reference signals are now defined to be

rtm~(n) = Gtm(q)xk(n ).

(4.7.7)

The expression for the lth error signal can now be explicitly expressed in terms of the coefficients of the filters in the digital controller, using equation (4.7.6), as M

K l-1

el(n)- dr(n) + Z Z Z h,~i rt,,~(n- i),

(4.7.8)

m = l k = l i=O

which can be written in vector notation as

et(n) = di(n) + r~(n)h,

(4.7.9)

where rt(n) = [rlll(n ) rill(n- 1)... rlMx(n-- I+ 1)] w and •

]T.

h = [hi10 h111 .. hMKI_ 1

Defining the vectors of all the error signals, and primary signals as e ( n ) = [el(n) e2(n).., eL(n)] T, d ( n ) = [d~(n) d2(n).., dL(n)] ~,

(4.7.10)

we can now express the steady state sampled response of the multi-channel

FEEDFORWARDCONTROL

109

feedforward control problem in the matrix form (Elliott et al., 1987) as e(n) = d(n) + T (n)h,

(4.7.11)

where T ( n ) = [rl(n) r2(n) ... rL(n)] T. Returning to equation (4.7.1), we can also express this sequence feeding the mth actuator as T(n)h, Um(Fl)'- X m

(4.7.12)

where xT= [0 0... xl(n) x l ( n - 1)... 0 ... x2(r/) x2(r/- 1) ... 0], so that the vector of signals driving the actuators can be written as (Elliott and Nelson, 1988) u(n) = X(n)h,

(4.7.13)

where X ( n ) = [xl(r/) x2(r/) ... XM(r/)]T. We now define a generalised cost function (Elliott, 1993), similar to that used in optimal feedback control theory, and in the previous section, which includes both error and 'effort' terms, as

J= E[eH(n)Qe(n) + un(n)Ru(n)],

(4.7.14)

in which the superscript H denotes the Hermitian (complex conjugate transpose) and E denotes an expectation operator. The Hermitian transpose is retained here to allow for the possibility that the sampled signals may be complex, and could represent transformed variables, for example. Q is an error weighting matrix, which is Hermitian and positive definite but not necessarily diagonal, and R is an effort weighting matrix which is also Hermitian and positive definite but not necessarily diagonal. Using the equations for e(n) and u(n) above, this cost function can be expressed in the complex quadratic form J = hnAh + hUb + bnh + c,

(4.7.15)

in which A = E[TH(n)QT (n) + XH(n)RX(n)],

b=E[TH(n)Qd(n)], and

c = E[dH(n)Qd(n) ].

This equation has a unique global minimum, assuming A is positive definite, for a set of control filter coefficients given by hopt = - A - l b ,

(4.7.16)

which result in the least squares value of the cost function Jmin

-" c -

bHA - lb.

(4.7.17)

110

ACTIVE CONTROLOF VIBRATION

The vector of derivatives of the real and imaginary components of the vector of control filter coefficients, hR and hi, can be written as (Haykin, 1987; Nelson and Elliott, 1992) =

~J

g ~hR

+j

aJ

-~i

= 2[Ah + b]

(4.7.18)

which, in this case, can be written as g = 2E[TU(n)Qe(n) + XU(n)Ru(n)].

(4.7.19)

In practice only an approximation to each of the paths from secondary source to error sensor can be measured and used to generate the practically implemented filtered reference signals, the matrix of which may be denoted T (n). Using the instantaneous estimate of g, with 1" (n), to update all the control filter coefficients at every sample, yields the algorithm (Elliott, 1993): h(n + 1) = h(n) - a[qf"(n)Qe(n) + X"(n)Ru(n)].

(4.7.20)

If all the error signals are equally weighted (Q = I) and no effort term is used (R =0), this algorithm reduces to the Multiple Error LMS algorithm (Elliott and Nelson, 1985). In this simplified case, equation (4.7.20) can be written in terms of the adaptation of the individual coefficients of the controller as L

hmki(rt +

1) =

hmki(n) - a

(4.7.21)

~ ftmk(n)el(n - i), 1=1

where f~mkis the filtered reference signal obtained by passing the reference signal xk (n) through an estimate of the path from the mth actuator to the lth error sensor, (~ lm(q)" The algorithm with diagonal weighting matrices for error and effort has also been discussed by Elliott et al. (1987, 1992). The Multiple Error LMS algorithm has been used by Jenkins et al. (1993) for the feedforward control of the harmonic signals fed to four active mounts, to minimise the sum of the squared signals from eight accelerometers on the receiving structure. It has also been used by Fuller et al. (1990a) to minimise the sum of the squared harmonic outputs from two piezoelectric sensors on a beam in a study of the simultaneous control of flexural and extensional waves, and in investigations by Fuller et al. (1989a) and Thomas et al. (1990) of active control of sound transmission through panels. A convergence analysis of this algorithm can be performed in a similar manner to that generally used for the LMS algorithm (Widrow and Steams, 1985). One difference in this case is that the algorithm, if stable, is found to converge to the solution (4.7.22)

hoo= - E [ T ( n ) Q T ( n ) + X H ( n ) R X ( n ) ] - ~ E [ t H ( n ) Q d ( n ) ] ,

which is not, in general, equal to the optimal solution, hop t above, since t (n)¢ T (n). Using this expression for h=, substituting for e ( n ) = d ( n ) + T (n)h(n), and making the usual assumption that the filter weight vector is statistically independent of the reference signals, the update equation can be written as E[h(n + 1) - hoo] = [I - aE[TH(n)QT (n) + XH(n)RX(n)]]E[h(n) - ho.],

(4.7.23)

the convergence of which depends on whether the real parts of the eigenvalues of the generalised autocorrelation matrix, E[I"H(n)QT (n) + XH(n)RX(n) ], are positive. Note ,

FEEDFORWARD CONTROL

111

that the eigenvalues of ~'H(n)QT(n) are, in general, complex since l"(n) is not necessarily equal to T(n), and the real parts of these eigenvalues are also not guaranteed positive (as they would be in the normal LMS analysis) for the same reason. The effort term in this expression, XH(n)RX(n), is guaranteed to be positive definite, however (assuming the control filters are persistently excited), and thus will have positive real eigenvalues which can have the effect of stabilising an otherwise unstable system (Elliott et al., 1992).

4.8 Equivalent feedback controller interpretation When the reference signal is a sinusoid, the adaptive feedforward controller has an interesting interpretation as an equivalent fixed feedback control system. This interpretation follows from an analysis of electrical adaptive cancellers presented by Glover (1977), which was originally used to analyse the behaviour of time domain digital feedforward controllers by Elliott et al. (1987) and Darlington (1987). A similar approach has also been used to analyse both analogue and digital time domain and frequency domain feedforward controllers by Sievers and von Flotow (1992), and by Morgan and Sanford (1992), who also present an interesting explanation of the analysis in terms of the manipulation of block diagrams. In this section we present the results of such an analysis for a single-channel digital feedforward controller which employs the filtered reference LMS algorithm, as discussed in Section 4.4. The block diagram of such a controller is shown in Fig. 4.11 (a) in which the reference signal is a synchronously sampled sinusoid so that x ( n ) = cos o90n,

(4.8.1)

where o90 = 2 e r / N and N is the number of samples per period. The mechanical system being controlled is represented by G, and the adaptive feedforward controller has coefficients updated according to equation (4.4.13). By taking the single-channel case of the result derived by Elliott et al. (1987), it can be shown that the relationship between the z transform of the control filter output, U ( z ) , and that of the error signal, E ( z ) for this algorithm, is given exactly by the response of a linear time-invariant system whose z transform is H(z) = U(z) _ E(z)

aAJ

2

[

]

z cos(og0 - q~) - cos q~

1 - 2z cos o90+ z 2

(4.8.2)

"

In this expression a is the convergence coefficient of the adaptive algorithm, the estimated response of the mechanical system at the frequency of the reference signal is given by (-~(Jog0) = A e j~, and I is the number of coefficients in the control filter. The implication of this analysis is that the behaviour of the adaptive feedforward controller illustrated in Fig. 4.11 (a) is exactly the same as the fixed feedback controller shown in Fig. 4.11 (b), in which H represents the system whose z transform is given by equation (4.8.2). By way of illustration, the modulus of the frequency response of this equivalent linear feedback controller is shown in Fig. 4.12(a), for the case in which too = er/2. Notice that the response of this equivalent linear system tends to infinity at the frequency of the reference signal. As the convergence coefficient of the adaptive

112

ACTIVE CONTROLOF VIBRATION

x(n)

t "~1 hi(n)

u(n)

/

_~n)

G

._e(n)

(a)

~e(n)

u(n)

(b) Fig. 4.11 Block diagram of an adaptive feedforward controller (a) and the equivalent fixed feedback control system (b).

I

I

(a)

IGc(j )l

(b)

1

v

o~o

~o

to

Fig. 4.12 The modulus of the frequency response of (a) the equivalent fixed feedback controller (equation 4.8.2) and (b) the net closed loop response obtained from equation (4.8.3). feedforward algorithm increases, and hence its speed of response becomes more rapid, the bandwidth of the significant response in Fig. 4.12(b) is increased and a wider range of frequencies near to the reference frequency is affected by the controller. The overall frequency response of the control system, from original disturbances to residual error, is given, from the closed loop response of the block diagram in Fig. 4.11 (b), as

E(jto) 1 = • Gc(j~) = ~ O(jto) 1 - G(ja~)H(jog)

(4.8.3)

FEEDFORWARD CONTROL

113

For the case in which the response of the mechanical system corresponds only to a pure delay, of duration equal to one sample of the reference frequency, the modulus of the closed loop frequency response is shown in Fig. 4.12 (b). When the frequency of the disturbance corresponds exactly to the reference frequency, ~o0, the overall closed loop response, equation (4.8.3), is zero, because the response of the equivalent feedback controller, equation (4.8.2), tends to infinity. For a sinusoidal disturbance with a frequency which is far removed from ~o0, the residual signal is approximately equal to the disturbance, since the closed loop response is unity at these frequencies for which the equivalent feedback controller has a very small response. The closed loop response of the control system away from the reference signal may have a magnitude which becomes significantly greater than unity, however, if the mechanical system has resonances in this frequency region. In extreme cases this can lead to instability of the controller at these frequencies, as described by Morgan and Sanford (1992). The equivalent linear system approach has been used to analyse the variation of the maximum convergence coefficient with delay in the secondary path by Elliott et al. (1987) and Morgan and Sanford (1992). The effect of errors in the model of the secondary path has been investigated using the equivalent linear feedback system by Boucher et al. (1991) and Darlington (1991). Darlington has shown that the relative heights of the two peaks in the frequency response of the system, on either side of the reference frequency, depend on the phase error of the secondary path model, and suggests that this asymmetry could be exploited as a diagnostic tool to detect such phase errors. Further parallels between the behaviour of harmonic adaptive feedforward and linear feedback systems have been discussed by Sievers and von Flotow (1992), who point out that a similar technique has been used to analyse algorithms for the higher harmonic control of helicopter vibration, by Hall and Wereky (1989). Because of the requirement for an infinite gain at the reference frequency in the feedback controller, the direct feedback control method illustrated in Fig. 4.11 (b) is probably not a practical method of implementing a narrow-band active control system. Its equivalence to the adaptive feedforward controller of Fig. 4.11 (a) can, however, provide a useful way of analysing the feedforward case, and may allow the extensive array of analytical tools developed for feedback control to be brought to bear on the feedforward control problem.

5 Distributed Transducers for Active Control of Vibration

5.1

Introduction

An important element of any practical control system are the transducers used for implementation of the control. Sensors are needed for measurements which can be used to estimate important disturbance and system variables. Actuators are used to apply control signals to the system in order to change the system response in the required manner. In general sensors provide information to the controller to determine the performance of the control system or to provide control signals related to the system response. Thus sensors and actuators provide the link between the controller and the physical system to be controlled and their design and implementation is of prime importance. In general, control transducers come in three main forms; point transducers, arrays of point transducers or continuously distributed transducers. Each particular format has its advantages and disadvantages and these are outlined in this chapter. Types of actuators range from point force actuators, electrostrictive and piezoelectric distributed actuators, to those based on the materials of shape memory alloy (SMA) and magnetostrictive systems. The choice of actuator is dependent upon system requirements such as required control authority (amount of control force, moment, strain or displacement, etc.), power consumption, frequency response, and physical constraints such as size and mounting requirements, etc. Sensors also range from conventional transducers such as accelerometers, strain gauges and proximity detectors to the systems based on piezoelectric material, optical fibres and shape memory alloy as well as advanced non-contacting sensors such as laser vibrometers. Choice of the particular sensor configuration is dependent upon the system variable to the observed, and to some degree, the form of signal processing to be used. The following sections are a brief discussion of the basic theory and issues regarding the implementation of distributed actuators and sensors commonly encountered in active vibration control systems. Electrodynamic shakers can be readily modelled as point force inputs and the response of structures to these types of forcing functions have been outlined in Chapters 1 and 2. For more information on these devices the reader is referred to the text of Cremer and Heckl (1988) and the handbook by Broch (1984). This chapter is mainly concerned with the relatively new theory of distributed piezoelectric actuators and sensors. The analysis presented here is limited to a static approach where inertial effects associated with the actuator itself are ignored. However,

116

ACTIVE CONTROL OF VIBRATION

the static approach is shown to be satisfactory in many cases and more recent work on advanced theories which include material dynamic terms, actuator-structure coupling, impedance effects etc. as well as the dynamics of the associated electrical circuitry, is briefly reviewed. Some brief discussion is also given of actuators and sensors based on advanced new materials such as shape memory alloy. It should be noted that although the theory is derived in terms of piezoelectric material, it is generally applicable to all distributed strain-inducing actuators and sensors.

5.2

Piezoelectric material and definitions

The piezoelectric effect was first discovered in 1880 by Pierre and Jacques Curie who demonstrated that when a stress field was applied to certain crystalline materials, an electrical charge was produced on the material surface. It was subsequently demonstrated that the converse effect is also true; when an electric field is applied to a piezoelectric material it changes its shape and size. This effect was found to be due to the electrical dipoles of the material spontaneously aligning in the electrical field. Due to the internal stiffness of the material, piezoelectric elements were also found to generate relatively large forces when their natural expansion was constrained. This observation ultimately has led to their use as actuators in many applications. Likewise if electrodes were attached to the material then the charge generated by straining the material could be collected and measured. Thus piezoelectric materials can also be used as sensors to measure structural motion by directly attaching them to the structure. Most contemporary applications of piezoelectricity use polycrystalline ceramics instead of naturally occurring piezoelectric crystals. The ceramic materials afford a number of advantages; they are hard, dense and can be manufactured to almost any shape or size. Most importantly the electrical properties of the ceramics can be precisely oriented relative to their geometry by poling the material as described below. The relationship between the applied forces and resultant responses of piezoelectric material depend upon a number of parameters such as the piezoelectric properties of the material, its size and shape and the direction in which forces or electrical fields are applied relative to the material axis. Figure 5.1 shows an element of piezoelectric material. Three axes are used to identify directions in the piezoelectric element termed 1, 2 and 3 in respective correspondence with the x, y and z axes of rectangular coordinates. These axes are set during the poling process, which induces the piezoelectric properties of the material by applying a large d.c. voltage to the element for an extended period. The z axis is taken parallel to the direction of polarisation and this is represented in Fig. 5.1 by the vector p pointing by convention from the positive to negative poling electrode (shown in the figure) or in the negative z direction. Piezoelectric coefficients, usually written in a form with double subscripts, provide the relationship between electrical and mechanical quantities. The first subscript gives the direction of the electrical field associated with the voltage applied or the charge produced. The second subscript gives the direction of the mechanical strain of the material. Several piezoelectric constants are used and the interested reader is referred to the IEEE Standards on Piezoelectricity for a full definition of these (IEEE, 1988). Anderson (1989) has also provided a good introduction to piezoelectric materials and their associated definitions.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

117

3(z) E,ectro

+1 V

Applied field, E

I

P

,,,,,--

e

#2(y)

l(x) Fig. 5.1

Piezoelectric element and notation.

The constitutive equations for a linear piezoelectric material when the applied electric field and the generated stress are not large can be written as (Uchino, 1994) ~i

E

= Sijcr~ + dmiEm,

(5.2.1a)

D m = dmio i + ~ikEk,

(5.2.1b)

where the indices i, j = 1 , 2 , . . . , 6 and m, k = 1, 2, 3 refer to different directions within the material coordinate system. In equations (5.2.1a) and (5.2.1b) e, o, D and E are respectively the strain, stress, electrical displacement (charge per unit area) and the electrical field (volts per unit length). In addition S e, d and ~ are the elastic compliance (the inverse of elastic modulus), the piezoelectric strain constant and the permittivity of the material respectively. The piezoelectric strain constant d is defined to be the ratio of developed free strain to the applied electric field E. In particular the strain constants d33, d31 and d32 are of major interest. The subscript 33 implies that the voltage is applied or charge is collected in the 3 direction for a displacement or force in the same direction. The subscript 31 implies that the voltage is applied or charge is collected in the 3 direction while the displacement or force occurs in the 1 direction. For much of the following analyses, while values of d32 and d31 are often significantly different in real materials, it is assumed that the piezoelectric material behaves identically in the 1 and 2 directions. The d32 and d31 constants are related to d33 by a Poisson effect with a negative sign. The above relations also imply the use of electrodes to apply or collect the electrical field and for the type of motion discussed here, these are shown as a shaded region on the top and bottom of the element depicted in Fig. 5.1. The strain of the piezoelectric element in the z direction can be simplified, for onedimensional motion, in the absence of an applied stress, to the relation l~.p e3

=

d3 3V~ h a,

(5.2.2)

where V is the applied voltage (note E V/ha) and h~ is the element thickness in the z direction. For the following discussion we restrict the voltage V to relatively small values so that the piezoelectric material can be considered relatively linear. The subscript pe now denotes strain of the piezoelectric element. For the same applied field =

118

ACTIVE CONTROLOF VIBRATION

the actuator will also deflect in the x and y directions and the resultant strains are in the x direction 1 = d31 V / h a Epe

(5.2.3)

2 = d32 V/ha. Epe

(5.2.4)

and in the y direction

By convention when a field (relatively small in value compared to the poling field) is applied to the piezoelectric element in the same direction as the poling vector as shown in Fig. 5.1, the element will expand in the z direction. At the same time, due to Poisson coupling, the element will contract in the x and y directions. Thus the d33 constant is typically specified as a positive value while the d31 and d32 are negative for piezoelectric ceramics. The above simplified definitions are sufficient to perform the analyses required in this chapter. For a more detailed description of piezoelectric terminology and behaviour the reader is referred to the text of Moulson and Herbert (1990), the work by Anderson (1989) and the IEEE Standards on Piezoelectricity (IEEE, 1988).

5.3

Piezoelectric stack actuators

The following analysis of the actuation of elastic structures is based upon what is known as the static approach. The static response of an interaction between a piezoelectric element and a structure is first determined by coupling the constitutive relations of the piezoelectric element and structure with their equilibrium and compatability equations. Once the equivalent static force or moment due to the actuator is obtained it is then used as a frequency-independent amplitude for a harmonically varying input to the system. This approximate approach has been found to provide reasonable results for relatively lightweight piezoelectric elements driven well below their internal resonance frequency. Most importantly, the static approach includes the distributed forcing function effects of the piezoelectric elements which will be shown to be a very important attribute for selective control of the states of the structural system. The first configuration of piezoelectric material we consider is the stack arrangement shown in Fig. 5.2 which is working against an applied external force F and an external stiffness represented by a spring. A stack is defined to be a single or multi-layered piezoelectric element which is relatively long in the z direction. This configuration is intended to induce motion in the 3 direction by applying voltages over electrodes at the top and bottom of the element. Two configurations of the actuator are shown. In Fig. 5.2(a) the actuator is working against an external spring stiffness arranged in parallel with the actuator while in Fig. 5.2 (b) the stiffness is positioned in series. In both cases for zero voltage the extemal spring is in equilibrium and applies no stiffness force. Note that the actuator also has an internal stiffness associated with its material Young's modulus of elasticity. The objective is to find the resultant displacement of the actuator and thus the effective stiffness when a voltage is applied to the actuator. The following static analysis of the parallel configuration of Fig. 5.2(a), although simple, does illustrate the basic technique for solving for coupled response of piezoelectric-structural systems. The

DISTRIBUTEDTRANSDUCERSFORACTIVECONTROLOFVIBRATION

119

F

tW

l

:t W •:_ ":_ .:•:- .:_ .:•:_ .:- .:. ,.- ..- .":-':-':- l .:- .:- .." !i- .'.":-:.":'. La

K

Piezoelectric ~ stack actuator

External .l--e" spring

~i~j

load

///////

// (a)

(b)

Fig. 5.2 Piezoelectric stack actuator working against an external stiffness load in (a) parallel, (b) series with the load.

piezoelectric material is of area A a, length La and is assumed to have a Young's elastic modulus o f E~. (E~ is assumed to be the ' short-circuited' modulus.) The unconstrained strain (i.e. when no external resisting stiffness is present) of the actuator in the z (or 3) direction is given by

F_,pe:

(5.3.1)

d33 V/La

and thus the unconstrained displacement of the actuator is

(5.3.2)

W a = d33V.

When a stiffness resists the motion of the actuator to w as illustrated in Fig. 5.2(a) then the internal force F a that the actuator exerts in the positive z direction is related to the constrained motion of the actuator by Ea(w a - w)

F~ =

(5.3.3)

Aa. L~

Applying a force balance to the arrangement of Fig. 5.2(a) gives an expression for the external force F in terms of the actuator and external spring constants and their deflections. This is given by F = Kw -

E~(w~-w) L~

Ao,

(5.3.4)

where K is the constant of the external spring. Solving this equation for the displacement of the actuator gives d33V + F/K~ w =

,

1 + K/Ka

(5.3.5)

120

ACTIVE CONTROL OF VIBRATION

where K~ is the actuator equivalent spring constant given by K~ = E~Aa/L a. Setting the applied external force F to zero results in an expression for the deflection when a voltage is applied given by w=

d33V

(5.3.6)

1 + X/X,, and the actuator force is then given by F~ =

d33VK

.

(5.3.7)

1 + K/K~ When the actuator displacement is constrained to zero, an important quantity called the blocked force of the actuator is given by F~

.

=

(5.3.8)

L~

These relationships demonstrate some fundamental aspects of coupled piezoelectric behaviour. Increasing the equivalent stiffness of the actuator relative to the system stiffness that the actuator is working upon increases the displacement of the actuator. However, increasing the stiffness of the piezoelectric material does not have such a significant effect on the maximum force exerted; the maximum force will be exerted when the actuator is working against a very stiff material. Although the above simple analysis is for a static situation it does illustrate an important point regarding piezoelectric actuators; the best configuration of actuator will depend upon the impedance of the system to be driven. The above analysis could be extended to a dynamic formulation by including material inertial effects. In this case different conclusions will be drawn, particularly when the coupled system is being excited near resonance. The coupled displacement of the series configuration of Fig. 5.2(b) can be found by using a similar procedure as outlined above. Use of piezoelectric stack actuators in a series configuration has been reported by Scribner et al. (1993) and the reader is referred to this reference for more details of their implementation.

5.4 Piezoelectric one-dimensional asymmetric wafer actuators The other common form of arrangement of a piezoelectric actuator is the asymmetric, wafer configuration shown in Fig. 5.3. In this arrangement the actuator is bonded to the surface of the structure and when a voltage is applied across the electrodes (in the direction of polarisation) the actuator induces surface strains to the beam through the d31 and d32 mode of the piezoelectric material response. We term this configuration a wafer arrangement since the piezoelectric element is very long (in the x and y directions) compared to thickness (in the z direction) through which it is polarised. For the 1-D analysis we shall follow the early work of Bailey and Hubbard (1985), Fanson and Chen (1986) and Crawley and de Luis (1987). Important assumptions are that the beam is covered by a layer of thin piezoelectric material of thickness, h~ which is perfectly bonded to the beam and strains only in the x direction. The following

121

DISTRIBUTED T R A N S D U C E R S FOR ACTIVE C O N T R O L OF VIBRATION

Z

,._

Piezoelectric actuator ~ Beam

Fig. 5.3 Piezoelectric asymmetric wafer configuration and associated strain distribution. derivation is an approximation using a static approach; inertial effects of the piezoelectric element are ignored, which is valid if the element is thin and lightweight compared to the beam system. In connection with this approximation, work by Pan et al. (1992a) has shown little difference in behaviour of a dynamic and static model for the geometries and frequencies of interest studied here. When a voltage is applied across the unconstrained (i.e. not attached) piezoelectric element the actuator will strain by an amount e, pe in the direction 1 which is parallel with the x axis as dictated by

d31V

~'pe = ~ ,

ha

(5.4.1)

where V is the applied voltage in the direction of polarisation, ha is the actuator thickness and d31 is the piezoelectric material strain constant When a voltage is applied across the bonded piezoelectric element it will attempt to expand but will be constrained somewhat due to the stiffness of the beam. Due to the symmetric nature of the load the beam will both bend and stretch, leading to an asymmetric strain distribution as shown in Fig. 5.3, where the origin of the z axis lies on the centre of the beam. The method of analysis outlined here follows previous work by Gibbs and Fuller (1992a). We assume that the strain distribution is linear as a result of Kirchoff's hypothesis of laminate plate theory (Jones, 1975) and thus can be written e(z) = Cz + e0,

(5.4.2)

where C is the slope and e0 is the z intercept. Equation (5.4.2) can be decomposed into the sum of an antisymmetric distribution Cz (i.e. flexural component) about the centre of the beam and a uniform strain distribution e0 (i.e. longitudinal component) as shown in Fig. 5.4. Using the strain distribution of Fig. 5.3 and Hooke's law, the stress distribution within the beam is given by Ob(Z) = Eo(Cz + Co),

where

E b

(5.4.3)

is Young's elastic modulus of the beam material. The stress distribution

122

ACTIVE CONTROL OF VIBRATION

I ,~X

m m Z

x

-I- -

Flexural

/

~X

Longitudinal

Fig. 5.4

Decomposition of asymmetric strain distribution.

within the piezoelectric a c t u a t o r ape(Z) is a function of the unconstrained piezoelectric actuator strain, the Young's elastic modulus of the actuator material Epe and the strain distribution shown in Fig. 5.3. This stress distribution can be written as a p e ( Z ) -- E p e ( C z "Jr"E 0 -- Epe)"

(5.4.4)

Applying moment equilibrium about the centre of the beam produces the relation

I'-hb ao(z)z dz + Ihb'+' (Tpe(Z)Z dz = 0,

(5.4.5)

where ho is the half-thickness of the beam. Next we apply the condition of force equilibrium in the x direction which shows that hb

I-hb ao(z)z dz + I hohb+h°%e(Z) dz = 0.

(5.4.6)

After integration, equations (5.4.5) and (5.4.6) can be solved for the unknowns C and e0, which, after some algebra, are given by

eo = K Lepe,

(5.4.7)

where the material-geometric constant is specified by

KL =

Epeha( 8 Eb h ~ -I-Epeh]) 2 4 16E2h 4 + EoEpe(32h3bha + 24h2h2a + 8hoh 3) + Epeha

(5.4.8)

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

123

and the slope is given by (5.4.9)

C---U~_,pe.

In this expression the material-geometric constant is specified by

Kf =

12E~pehbha(2hb + hu) 2 2 + 8hbh 3)+Epeha 2 4 16E2h~ + EbEpe(32h3h,~ + 24hbha

(5.4.10)

The induced moment distribution, mx, in the beam beneath the actuator is given by

mx(x) = Et, IKfepe •

(5.4.11)

The uniform strain component across the beam cross-section is 6 , ( X ) - " 6,0 -" KLr, pe,

(5.4.12)

where ~pe is related to the applied voltage by equation (5.4.1). Thus the response of the beam to the asymmetric actuator consists of a moment distribution mx(x), specified by equation (5.4.11) and a longitudinal strain distribution e(x), specified by equation (5.4.12). Both of these fields will exist at every point beneath the infinite piezoelectric element. We now examine the excitation of an infinite thin beam by an asymmetric piezoelectric wafer element of finite extent. In order to induce motion, a harmonically oscillating voltage v(t) = V e j°~' is applied to the electrodes of the piezoelectric element. For the following derivation we make further assumptions. We assume that the piezoelectric element is very long and thin and hence end effects (such as where the stress field vanishes) are ignored. This assumption is supported by the work of Liang and Rogers (1989) and Anderson (1989) who showed that the actuator strain field for a distributed actuator is unaffected by a free edge beyond approximately four actuator thicknesses distance from the boundary. Therefore for actuators that are large with respect to their thickness, the strain distribution of Fig. 5.3 can be taken to exist in the actuator-beam system. The other important assumptions are that the actuator is perfectly bonded and inertial effects of the actuator material are ignored. First we solve for the flexural response of the beam associated with the induced moment field. The Bernoulli-Euler equation of motion of the thin beam has been derived in Chapter 2. Written in moment form and including the actuator induced moments the beam-actuator equation of motion is given by 02[Mx(x) - mx(x)] Ox2

- o)2pSw = O,

(5.4.13)

where M~ is the internal beam bending moment and mx(X) is the actuator-induced bending moment, while p and S denote the density and cross sectional area of the beam. Following the approach of Crawley and de Luis (1987), equation (5.4.11) can be modified for a finite patch of L~ and then substituted into equation (5.4.13). For a finite length element, equation (5.4.11) is written as

m~(x)

= Corpe[H(x

) -

H ( x - L~) ],

(5.4.14)

where H(.) is the unit Heaviside step function defined as

H(x) = I i' x > O, [ 0, x < 0 .

(5.4.15)

124

ACTIVE CONTROL OF VIBRATION

Equation (5.4.14) implies that the induced moment only exists at every point under the location of the finite actuator. Substituting the moment distribution into equation (5.4.13), taking the second partial derivative with respect to x, and moving the actuator terms to the fight hand side results in ~2Mx(x) - t o 2 D S w = Col~pe[6'(x ) - 6 ' ( x OX2

L~)]

(5.4.16)

where Co = EDIKI and 6' (.) represents the derivative of the Dirac delta function with respect to its argument. Substituting the relation for bending moment, M~(x) =-EIO2w/Ox 2, it can be shown that equation (5.4.16) can be written in similar form to that of equation (2.3.8). Equation (5.4.16) demonstrates the classic result, pointed out by Fanson and Chen (1986) and Crawley and de Luis (1987), that the induced bending of the actuator can be represented as an external load consisting of a pair of line moments of opposite sign located at the actuator edges. As shown previously, the magnitude of these line moments is proportional to the applied voltage. We next consider the longitudinal motion of the beam. The equation of motion in this case (see equation (2.2.1)), written in terms of displacement and including a forcing term, is given by

to2pu

d2u +

d,x 2

E

de(x) =

dx

,

(5.4.17)

where u is in-plane displacement and e(x) is the applied strain distribution due to the actuator. Equation (5.4.12) can be modified for a finite length of patch La by assuming excitation strain only under the element such that ~,(X) = K L r p e [ n ( x ) -

H ( x - La) ].

(5.4.18)

Equation (5.4.18) is then substituted into equation (5.4.17) which produces the inhomogeneous longitudinal equation of motion given by 1

dZu -t-

2 tO U = KLgpe[6(X) -- d ( X --

La)],

(5.4.19)

where c~ = E/p. For the longitudinal motion, the finite piezoelectric element can be seen to be equivalent to an external load of two equal and opposite line forces acting in the x direction, longitudinally at the element edges and along the central axis of the beam. Simultaneously controlling both flexural and extensional motion in beams has been shown to be an important problem if vibration in beams are to be controlled effectively by active means (Fuller et al., 1990b). In order to efficiently control these types of vibration it is important to excite these wave forms to a varying relative degree. In order to demonstrate this we analyse an infinite thin beam excited by a pair of independently driven, but symmetrically located wafer actuators as shown in Fig. 5.5. The flexural equation of motion, equation (5.4.16), is solved using the procedure outlined in Chapter 2 for each individual element (with complex excitation voltages V~

125

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

Z

a

?

i

_~~_ Beam

Actuator, V1

I I I

I I

~X

N\\\\\\\'q ,

I

Actuator, V2 Fig. 5.5

Infinite thin beam excited by two co-located piezoelectric wafer elements.

and V2). The total motion w,(x) of the beam for the two actuators is found by superposition to be (Gibbs and Fuller, 1992a)

wt(x, t) = d31KI(V1 - V2) [(1 - e ~TL°)e -~Tx- (1 - e -jk~L°)e -jkzx] c"J~", 4k~h~

(5.4.20)

where 09 is the frequency of oscillation of the voltages and kI is the flexural wavenumber. The power flow associated with this motion is derived using the method outlined in Chapter 2 and is given by (Gibbs and Fuller, 1992a) 1-IB , -

8hike

IV1 - V2i [ 1 - cos kfta].

(5.4.21)

The longitudinal equation of motion, equation (5.4.19), is solved as demonstrated in Chapter 2 (see Section 2.2) and the solution is (Gibbs and Fuller, 1992a)

ut(x, t) = jKtd31(V1 + V2) (1 - e -jk'~L°)e j°~'-jk'x 2kLha ,

(5.4.22)

where kL is the longitudinal wavenumber. The power flow associated with this wave motion as outlined in Chapter 2 is (Gibbs and Fuller, 1992a) 1-I~ =

(KL )2d231t°ES 4kLh ]

12 IV1 + V2 [1 - cos kLLa].

(5.4.23)

As an illustrative example, an aluminium beam of thickness 2h b equal to 3.175 mm and width 7.62 cm is considered. The piezoelectric actuators are assumed to be of G1195 ceramic material with properties given in Table 5.1. The excitation frequency is 800 Hz. For the first result the voltage amplitude input to both actuators is fixed at V~- V: = 400 V p.-p. and the length is fixed at 3.81 cm. Figure 5.6 presents the flexural and longitudinal power flow (dB relative to 10 -~2 W) plotted as a function of the relative phase between V~ and V2. When the actuators are perfectly in phase, only longitudinal waves are generated and conversely when the actuators are 180 ° out of phase only flexural waves are generated. Variation in the relative magnitudes of flexural and longitudinal wave power flow can be achieved by choosing the phase between 0 and 180 °.

126

ACTIVE CONTROL OF VIBRATION Table 5.1

Typical piezoelectric ceramic actuator properties (PZT, G1195).

Vpe = 0.30 ha = 0.1905

gpe = 6.3 x 101° N/m 2 P pe =

7650 kg/m 3

mm

d31 =

d32 =

d36 =

0

-

166 x 10 -12 m/V

120

~

100

A

~ ' x Bending wave

80

133 "0 v

Longitudinal

% % % %

_o

60

o a.

40-

200

0

I

I

I

I

I

I

I

I

20

40

60

80

100

120

140

160

180

Relative phase, V 1-V 2 (deg)

Fig. 5.6 Power flow as a function of actuator relative phase, f = 800 Hz, La= 3.81 cm (after Gibbs and Fuller, 1992a). In the second illustrative result, the phase of V~ is chosen to be 90 ° in advance of V2. The power flow for both longitudinal and flexural waves as a function of actuator length is presented in Fig. 5.7. It can be seen that as the actuator length approaches zero, the power flow for both wave types, as expected, also approaches zero. When the actuator length is equal to 9.6 cm (corresponding to half a flexural wavelength at 800 Hz), the flexural wave power flow is at a maximum, since, as shown previously, an individual piezoelectric actuator effectively acts as two line moment sources in antiphase that are externally applied to the beam at the ends of the element. The waves generated from each end in this case, are thus perfectly in phase. Conversely when the actuator is 19.2 cm long, the element length corresponds to a flexural wavelength and the flexural power flow drops to zero. It is interesting to view the above wave based results in terms of the excitation of finite beams. For example, if a simply supported beam was completely covered with a piezoelectric element, then the element length would be equal to a complete wavelength corresponding to the second mode of motion of the finite beam. As discussed above, from a wave point of view it is apparent that the second mode would not be excited. Likewise from a modal point of view using the orthogonality principle outlined in Chapter 2, it is also apparent that a symmetric actuator cannot couple into an antisymmetric beam motion. The above observation is another useful illustration of the wave-mode duality exhibited by vibrations of extended systems.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

127

110

100

nn 1:} v _o

o 13.

90

80

_ i////

Longitudinal w

/I

/ 70 -I

I I

60

50

I

I

I

I

I

I

I

I

I

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Actuator length,

0.2

La (m)

Fig. 5.7 Power flow as a function of actuator length, f=800 Hz, (/)12 "-90 ° (after Gibbs and Fuller, 1992a). Finally, due to the higher wave speeds of longitudinal waves, the longitudinal power flow does not have a maximum power flow until the actuator is relatively much longer. At 800 Hz the longitudinal half-wavelength in aluminium is approximately 3.1 m. Thus, the results indicate that in order to dominantly drive longitudinal motion with this configuration, longer, extended arrays of actuators are needed.

5.5

Piezoelectric one-dimensional anti-symmetric wafer actuators

A simplification of the asymmetric wafer configuration is when two, identical wafer piezoelectric elements are located symmetrically about the beam and driven 180 ° out of phase with the same signal as shown in Fig. 5.8. Due to the 180 ° phase difference of each element we call this an anti-symmetric configuration. The analysis presented here essentially follows that of Section 5.4 except that, because the system is being excited in an anti-symmetric fashion, pure bending of the beam will occur without any excitation of longitudinal waves. By applying moment equilibrium about the centre of the beam we find that

I

-h -h b - ha

~ ~pe(Z)Z dz

+

I hb I hb + ha (9 OpeZ d z = 0, Oo(Z)Zdz + -h b hb

(5.5.1)

where superscripts ® and ® denote wafer elements shown in Fig. 5.8. Writing the stress in the piezoelectric elements and beam in terms of the Young's elastic modulus of the material, the strain slope C and the unconstrained strain of the piezoelectric elements can be deduced in the manner used above. We can then solve

128

ACTIVE CONTROL OF VIBRATION Z

l

I

S

Piezoelectrielcement(~ be/beam 01 Piezoelectrielc ement(~)

Fig. 5.8 Piezoelectric anti-symmetric wafer actuator configuration and associated strain distribution. equation (5.5.1) for the strain slope which gives c=

where the material-geometric constant is now specified by Kf=

3Epe[(hb+

ha)z- h 21

2{Epe[(hb + ha)3 - h 31 + Ebh3}

•

(5.5.2)

The moment surface density mx induced in the beam by the actuator is again given by (5.5.3)

mx(x) = EblKf epe•

It is interesting to note that the value of mx for anti-symmetric excitation is not exactly twice that given for a single anti-symmetric wafer which is specified by equation (5.4.11). This result arises because the expression for the asymmetric actuator does not include the stiffness of the bottom-located actuator. If this stiffness is included in the analysis (with applied voltage set to zero on that element) then it can be shown that the anti-symmetric actuator provides exactly twice the input moment. As an example application of the above analysis we will consider excitation of a simply supported Euler-Bemoulli beam by multiple anti-symmetric piezoelectric actuators depicted in Fig. 5.9 (Clark et al., 1991). In this case we again assume that actuators will be excited by an oscillating voltage, V e jc°t, and their inertial effects are ignored. Since each actuator is now finite, we can again assume that the induced moments of the actuator(s) are only present directly under the actuator(s) location. Following the analysis of the previous section, the equation of motion for the beam-actuator system becomes "-'-'----~2 X -- o')2~Sw "- Z C°EP i=1

'(X - x1)i _

6'(x

-

x2)],~

(5.5.4)

where 6' (') again represents the derivative of the Dirac delta function with respect to its argument.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION f!!iiiiiii]

tiii!!!iiit

-76mm-~"

129

40~mm

i 266 mm

I"

380 mm I ....

':1

I: ....

I

Simply supported beam Piezoelectric ceramic element

Fig. 5.9 Excitation of a simply supported beam by multiple anti-symmetric piezoelectric actuators. In equation (5.5.4), x~ and x2 are the location of the edges of the ith actuator for a series of N~ anti-symmetric actuators. We find a solution of equation (5.5.4) by expanding the response of the simply supported beam in terms of its basis mode shapes such that oo

w(x, t) = Z W~ sin(mrx/L) e j~'.

(5.5.5)

n--1

Using equation (5.5.3), substituting equation (5.5.5) into equation (5.5.4) and using the orthogonality property of the modes shapes outlined in Chapter 2, we find an expression for the modal amplitudes of the beam response given by

IV, = i ~1 2Coep cos "=

e

- cos

L

(to~ - toZ)LZm"

,

(5.5.6)

where the unconstrained strain of the ith actuator is given by i ~ - d31V i ej~, +j~' Epe

(5.5.7)

and where V i is the voltage amplitude and q~i is the phase applied to the ith actuator. In equation (5.5.6), m" is the mass per unit area of the beam and to, is the resonant frequency of the nth mode of vibration of the beam. Equation (5.5.6) can now be used to evaluate the modal amplitudes of a typical beam responding to an array of anti-symmetric actuators. The example results given below are for a steel beam of 380 mm length, 40 mm width and 2 mm thickness. Figure 5.10 presents modal amplitudes of the beam, plotted as acceleration amplitude, for two identical piezoelectric anti-symmetric actuators (note, here 'actuator' implies two antisymmetric elements positioned as in Fig. 5.9 and driven out of phase with the same voltage magnitude). The excitation frequency is 200 Hz which is an off-resonance excitation case for the beam system. The piezoelectric elements were of length 38.1 mm, 15.8 mm width and 0.2 mm thickness and were constructed from a ceramic material, G1195 with properties specified in Table 5.1. The actuators were each driven with a voltage of 60 V p.-p.

130

ACTIVE CONTROLOF VIBRATION 1.8

r---1 1.6

Theoretical

Experimental

1.4 E

1.2

t-

.9

1

"~ 0.8 o o

0.6

-

"o o

0.4 0.2 1

2

3

4

5

6

7

8

Modal number

Fig. S.10 Modal amplitudes of beam response, f=200 Hz, both actuators out of phase (after Clark et al., 1991). In the analytical predictions the moments along the edges of the actuators parallel with the beam edges were ignored due to the extremely high stiffness of the beam about its width. In addition, the magnitude of the moments were scaled by the ratio of the width of the actuator to the width of the beam, since the actuator did not completely cover the beam across its width. Also shown in Figs 5.10 and 5.11 are experimentally measured values of amplitude taken from the work of Clark et al. (1991). 2

r-'n

1.8 ¢kl

E tO .m

Theoretical

Experimental

1.6 1.4 1.2 1

o o "o o

0.8 0.6 0.4 0.2 0

1

2

3

4

5

6

7

8

Modal number

Fig. S.ll Modal amplitudes of beam response, f = 200 Hz, both actuators in phase (after Clark et al., 1991).

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION

131

As can be seen in Fig. 5.10, the dominant response is in the n = 2 mode and this is expected as the driving frequency is near the corresponding n = 2 resonance frequency of 124 Hz. Figure 5.11 presents the same configuration, except now the actuators are driven in phase. It is apparent that the n = 2 mode has now largely been suppressed and the dominant response is that given by the n = 3 mode. This behaviour can be understood by studying the phasing of the actuators relative to the beam normal mode shapes. For example, by considering the orthogonality property of the modes of the beam, it would be theoretically impossible for two actuators symmetrically positioned about the beam mid point as shown in Fig. 5.9 and driven in phase, to couple into the n = 2 mode which has a 180 ° phase change in response through the beam mid point. The above example is important as it outlines the fundamental basis of distributed actuators. In order to selectively control required modes, without exciting unwanted modes (the control spiUover phenomenon discussed in Chapter 3), it is necessary to create a distributed actuator configuration with the required amplitude and phase distribution. Obviously the more independent actuators that are employed, the larger the success in achieving this goal. It is also apparent from the agreement with experimental results shown in Fig. 5.10 and 5.11, that the above static theory provides a reasonable model of the excitation of one-dimensional structures by piezoelectric ceramic wafer elements and to a large degree validates the approximations made in the theory for the geometries and frequencies considered here. Such one-dimensional anti-symmetric wafer elements have also been used to excite and control bending waves in thin beams (Gibbs and Fuller, 1992b).

5.6

Piezoelectric two-dimensional anti-symmetric wafer actuators

The above discussions relate to the excitation of one-dimensional structures. We now turn our attention to the excitation of two-dimensional plates in pure bending produced by a piezoelectric patch configuration in an anti-symmetric arrangement. The analysis is a logical extension of the one-dimensional case.

Fig. 5.12

Two-dimensional piezoelectric wafer element and structure.

132

ACTIVE CONTROLOF VIBRATION

Figure 5.12 shows a piezoelectric patch element located on a flat plate and defines the coordinate system used in the analysis. Figures 5.13(a) and 5.13(b) show the assumed strain distribution resulting from two identical piezoelectric elements located synunetricaUy on the plate and driven 180° out of phase with the same signal. The analysis presented below essentially follows that of Dimitriadis et al. (1991) except that the strain slope is assumed to ~ continuous through all laminae in correspondence to the previous analyses. Due to the anti-symmetric nature of the piezoelectric wafer excitation the strain distributions in the x and y direction of the plate are given by ex = Cx z,

(5.6.1a)

~,y-- CyZ,

(5.6.1b)

where Cx and Cy a r e the slopes of the strain distribution for the x - z and y - z planes respectively. For the 2-D case we assume that the piezoelectric material has similar properties in the 1 and 2 directions such that d31 = d32. The unconstrained strain of the actuator in both the x and y direction is then given by e,pe =

d31V / h a.

(5.6.2)

Piezoelectric element C) Plate

i

I ,

I

J

/x

i

Piezoelectric element (~ (a)

Piezoelectric element (~ Plate

I ~Piezoelectric element (~

(b) Fig. 5.13 Piezoelectric anti-symmetric wafer element two-dimensional strain distribution: (a) x - z plane; (b) y - z plane.

133

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

Following the approaches of the previous sections, the stresses in the plate for the x and y direction are given respectively by =

Ep (ex + Vpey) l_v 2

(5.6.3a)

op =

Ep. (ey + Vpex), l_v 2

(5.6.3b)

off and

where Vp is Poisson's ratio of the plate material. The stresses in the top piezoelectric element are given by OPe(l)--"

Epe

[~-'x "~ "12pe~'y- (1 + 'Vpe)l?,pe]

(5.6,4a)

E~y "st"'llpel?,x - - ( 1

(5.6.4b)

1 -

and 0 pe~-"

Epe

+ 'Ppe)l?,pe],

1 - Vp2e where 'l,'pe is Poisson's ratio of the piezoelectric element. The stresses in the bottom piezoelectric element are given by pe®__

Epe

[l?,x .-1-llpel?,y "4" (1 q- llpe)E, pe ]

(5.6.5a)

1 - 'pp2e and pe®

o~

=

Epe

1 -"fft2e [ey-1" 'llpel?.x "l- (1 + "llpe)~.pe].

(5.6.5b)

Since F_,pe is the same in both axes and the plate is assumed homogenous then Ex-- Ey-- E. Therefore the strain can be written as

(5.6.6)

~, -- CxZ-- C y z - " Cz.

As previously, we apply the condition of moment equilibrium about the x and y axis by integrating over the stress distributions in the x - y and y - z planes. After much algebra we can solve for the strain distribution in both the x and y directions (which are equal) yielding the result (5.6.7)

C = Kff, pe,

where the material-geometric constant is given by Kf =

3Epe[(h o + h,~)2- h2](1 -

vp)

2Epe[(ho + ha) 3 - h31(1 - vp) + 2Eph3(1 - "Vpe) '

where the plate is of thickness 2h b and the actuator elements are of thickness ho.

(5.6.8)

134

ACTIVE CONTROLOF VIBRATION

As an example of the use of the above two-dimensional analysis we now consider harmonic excitation of a simply supported thin plate by a single anti-symmetric actuator. As previously, we assume that the piezoelectric actuator will induce internal moments in both the x and y direction which are only present under the piezoelectric patch extent. As shown above, these moments will be equal and are thus specified by

ms = my-- CoEpe[n(x- x1)- H ( x - x 2 ) ] [ H ( y - y ~ ) - H ( y - Y2)],

(5.6.9)

where (Xl, Yl) and (x2, Y2) are the coordinates of the patch comers and Co = EIK I. We also assume that the actuator is driven by a harmonically oscillating voltage and ignore inertial and end effects of the actuator. The actuator is also assumed to be perfectly bonded to the plate. Using the classical thin plate theory outlined in Chapter 2, substituting the moment distribution of equation (5.6.9) into the 2-D thin plate equation, evaluating the differential operators and moving the piezoelectric terms to the fight hand side (as in the 1-D case) we derive the inhomogeneous plate equation given by ~2w

E1 V4w + m"

c3t2

= Coepe[b'(x - Xl) - 6'(x - Xz)][H(y - Yl) - H(y - Y2)]

+ Co~.pe[H(x- x 1 ) - H(x- x2)][6' ( y - Y l ) - 6' (Y- Y2)], (5.6.10) where E1 denotes the bending stiffness of the plate and V 2 is the Laplacian operator. Equation (5.6.10) implies that the 2-D anti-symmetric piezoelectric actuator effectively applies line moments of amplitude CoEpet O the plate at the location of the actuator edges. The solution of equation (5.6.10) can be found by using the modal expansion of the plate response w(x, y) as described in Chapter 2, which is given by oo

w(x, y, t) = Z

oo

Z wmnsin kmx sin k~y e j~'t,

(5.6.11)

m=ln=l

where Wmn is the plate displacement modal amplitudes which can be calculated by substituting w(x, y) back into the equation of motion, and using the orthogonality property of the plate modes. The modal amplitudes are found to be given by

Wmn =

,,,~4C°epe,2 [ - (k~'2'+ k2n) (cOs kmxl - cOs kmx~)(cOs knyl - cOs kny2)]' m ~pt~Om,,-092) kmkn

(5.6.12)

where m" is the mass per unit area of the plate, Sp is the plate area, (.Dmnare the plate natural frequencies and k m and kn are the plate eigenvalues given in Chapter 2. Example results are presented for a steel plate whose dimensions are: width a = 0 . 3 8 m, height b = 0 . 3 0 m and thickness h = 1.588 mm. Two different configurations of piezoelectric ceramic actuators were considered, as shown in Fig. 5.14. In configuration (a), the element is long in the y direction, narrow in the x direction and symmetric about the b/2 line with x~ = 0.32 m, x2 = 0.36 m, Yl = 0.04 m and Y2 = 0.26 m. In the second case, configuration (b), the actuator was rotated such that it was long in the x direction and narrow in the y direction, i.e. x~ =0.04 m, x2=0.34, y~ =0.23 m and y2=0.27 m. Thus in configuration (b) the actuator is symmetric about the line corresponding to x = a/2.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION

135

Actuator

Simply supported plate .. .°%%%° °.Oo. %%° °°°°° °°o° ..... %%. ...°. .°-. -°.. .... ..°° ...° .°.°. ... .°°. °.. %%.........

X

X

(a)

Fig. 5.14

(b)

Piezoelectric actuator test configurations.

Figure 5.15 illustrates the plate normalized displacement distribution along the line corresponding to y = b/2 at a frequency of f = 9 5 . 5 Hz for configuration (a). The displacement distribution shows evidence of multi-modal excitation due to the excitation frequency being between the (1, 1) and (2, 1) resonance frequencies. It is also apparent from Fig. 5.15 that for this case there is a nodal line being excited close to the x = x~, actuator boundary. This behaviour is associated with the forcing function of the piezoelectric element which was shown to be a line moment along its edge. The piezoelectric element thus tends to induce a rotation at its edge rather than out-of-plane displacement. Tables 5.2 and 5.3 present modal amplitudes of plate response (normalised to the largest value in each table) for an excitation of f = 148 Hz in configurations (a) and (b) respectively. In configuration (a), the (2, 1) mode is dominant as to be expected since the excitation frequency is close to the (2, 1) resonance frequency of 149.8 Hz. Note that the response in the anti-symmetric y distributions (even n indices) is effectively zero. This result is due to the orthogonality between the symmetric actuator forcing

133 "lD v

-o m

-10

c~

E

-20

E

o ..i...,

.Q >

= E o Z

-30 -40 _50

I 0.0

! 0.2

!

I 0.4

I

I 0.6

I

I 0.8

I 1.0

Axial location, x/a

Fig. 5.15 Plate normalised displacement, f=95.5 Hz, configuration (a) (after Dimitriadis et al., 1991).

136 Table 5.2 1991).

m

ACTIVE CONTROL OF VIBRATION Plate displacement amplitudes (dB), f = 148 Hz, case (a) (after Dimitriadis et al.,

1

2

Plate displacement amplitudes (dB), f = 148 Hz, case (b) (after Dimitriadis et al.,

1 2 3 4 5 6

0.0 -345.8 -55.4 -358.5 - 84.7 -357.4

- 32.8 -348.1 -54.3 -356.1 - 81.6 -352.2

3 - 39.1 -352.5 -57.2 -357.9 - 82.6 -355.2

4 -45.7 -358.5 -62.4 -362.3 - 86.5 -348.9

- 74.2 -69.4 -67.9 - 68.2 -72.1

6

Table 5.3 1991).

2

- 670.3 -665.9 -664.8 - 665.6 -670.3

5

-42.1 0.0 -32.7 - 38.1 -46.6

1

- 64.4 -60.9 -60.8 - 62.4 -68.2

4

1 2 3 4 5

m

- 648.3 -650.0 -652.6 - 655.6 -622.8

3

5 - 54.9 -367.3 -70.8 -370.3 -94.1 -349.1

- 668.5 -663.6 -661.8 - 661.8 -665.1

6 - 343.4 -655.6 -358.8 -658.0 - 381.5 -350.6

function and the anti-symmetric response in these distributions. This behaviour is further illustrated in the results of Table 5.3 which is for configuration (b) with the actuator rotated through 90 °. In this configuration the dominant modes are the (1, 1) and (1,2) modes and the (2, 1) mode is now not excited even though it is being driven close to its resonance frequency. The above results illustrate an important feature of the behaviour of piezoelectric actuators. Since they are of a distributed nature, the shape and location of the actuator can be chosen to excite a required mode or modal distribution. This observation has important considerations in terms of reducing control spillover. In addition, experiments carried out by Clark et al. (1993) have demonstrated that the above static model provides a reasonable approximation to the dynamic excitation of thin plates by wafer type piezoelectric actuators. The above analyses have provided the equivalent forcing functions for 1-D asymmetric and anti-symmetric actuator configurations as well as a 2-D anti-symmetric configuration. The results show that the actuator can be replaced by equivalent line moments acting along the edges of the actuator whose magnitude is given by

m ( x ) = m ( y ) - Coep,,

(5.6.13)

where the m ( y ) value is used in the 2-D case. Relations for Co~E1 = K I are summarised in Table 5.4 for the three actuator configurations where E1 is bending stiffness. The previous formulations are based upon simplified static models to estimate the piezoelectric induced strains. Although experiments have shown that these models

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

137

Table 5.4 Summaryof strain constants. Excitation condtion 1-D asymmetrict 1-D anti-symmetric 2-D anti-symmetric

Geometric constant, KI = ColE1 12EbEp~hbha(2hb + ha) 2 2 2 4 16E~h4~ + EbEpe(32h3bh~ + 24hbha + 8hbh3) + Epeha

3Epe[(hb + ha)z - h 2] 2Epe[(hb + ha) 3 - ha] + 2Ebh3b 3Epe[(hb + ha)2 - h2](1 - Vp)

2Epe[(hb+ ha)3- h3](1 - v,,) + 2Eoh3(1 - 'llpe)

~f For the asymmetric configuration there will also be in-plane excitation.

provide reasonable predictions of dynamic forcing functions there are many, more accurate, extensions of the static models. Crawley and Anderson (1989) have developed detailed models for piezoceramic actuator effects including the effects of dynamics. Hagood et al. (1990) have developed models for the dynamics of piezoelectric actuators for structural control and have included the dynamic influence of the electrical power circuit through the piezoelectric effect. Since the analysis of Haygood et al. (1990) is based upon a variational approach it is also very useful for analysing complex piezoelectric-structure systems. More recently Stein e t a l . (1993) have derived expressions for a coupled piezoelectric-structure system including the impedance of the electrical network. These expressions allow derivation and design of the actuator in terms of power consumption and impedance matching. The single-layered actuator analysis has been extended to multi-layered actuators by Cudney et al. (1990) while Jia and Rogers (1989a) have developed models for embedded distributed actuators in structural systems using classical laminate plate theory. Numerical models have been developed by a number of workers. Ha et al. (1992) have presented a finite element formulation for modelling the dynamic as well as static response of laminated composites containing distributed piezoceramics subjected to both mechanical and electrical loading. Work has also been carried out for actuators on curved surfaces such as cylinders. Dimitriadis and Fuller (1992) developed expressions for a piezoelectric actuator in 2-D cylindrical coordinates. Banks et al. (1995) have developed a model for actuation of cylinders by finite 2-D patches. Tzou and Gadre (1989) investigated the axisymmetric excitation of multi-layered cylinders by embedded piezoelectric layers. Lester and Lefebvre (1991) analytically and experimentally studied harmonic excitation of a finite composite cylinder by piezoceramic patches. The above brief review indicates that there has been much work carried out in the study of piezoelectric actuation. The analyses developed in Sections 5.3-5.6 are valid for all forms of distributed induced strain actuators, i.e. those forms of actuators that apply distributed tractions to a structure due to their internal expansion or contractions. The use of piezoelectric actuators in sound radiation control from structures will be illustrated in Chapter 8. Other forms of advanced actuator are based on magnetostrictive materials which are similar to piezoelectric materials except that they are activated by a magnetic rather than an electrical field. They show much potential for their use in active vibration control due to their extremely high strain rates (Goodfriend and Shoop, 1991; Hiller et al., 1989).

138

5.7

ACTIVE CONTROL OF VIBRATION

Piezoelectric distributed sensors

The previous sections have dealt with the use of piezoelectric elements as control actuators. We now turn to a related problem; the use of distributed piezoelectric elements as sensors in active control. Much previous work has concentrated on the use of discrete point sensors in controlling vibrations in distributed structures. The reader is referred to Beranek (1988) and Broch (1984) for information on conventional point transducers. Point sensors are usually employed in arrays of transducers whose outputs are processed to obtain some estimate of a required variable or state of the system to be controlled (see Chapter 3 and the next section for a discussion on some of these techniques). The basis of this approach is that in order for the control to be effective without 'observation spillover', then the controller has to be designed to observe only those motions which are required to be reduced. The main disadvantage of this approach is due to the signal processing requirements necessary to process the transducer outputs and thus obtain estimates of the required variables. Distributed piezoelectric sensors show potential to overcome this disadvantage in that they can be shaped so as to act as spatial filters which only observe certain motions. As the piezoelectric sensors are continuous, this spatial filtering is achieved by what is effectively a continuous analogue integration of the measured variable over the sensor surface and thus does not require any signal processing. The main disadvantage of these types of sensor are that they are fixed in shape and are thus fixed in terms of the characteristic which they observe, unlike the point array sensor which can be reconfigured through the use of different signal processing techniques. In addition the output of the sensor is sensitive to the accuracy of the shape of the sensor as well as its positioning on the structure. Figure 5.16 shows a one-dimensional thin beam covered with a thin layer of piezoelectric material. Additional assumptions to those made in the previous sections are that the piezoelectric material is mechanically isotropic, and that the sensor has constant properties along its length and is also thin compared to its length. We also assume that the sensor has no effect on the motion of the beam, i.e. the stiffness and inertial force of the sensor are very small compared to the beam.

Z

l

/ Piezoelectsensor ric

L ~ : : J . ' : : : ~ ' . ~

~' Beam

Fig. 5.16 Piezoelectric distributed sensor-beam distribution.

configuration and

associated

strain

DISTRIBUTEDTRANSDUCERSFOR ACTIVECONTROLOF VIBRATION

139

The strain in the x direction at the surface of the sensor can be written as ~2w

e(x, t)= ~

~X 2

(hb + hs)

(5.7 1)

'

where hb is half thickness of the beam and hs is the thickness of the piezoelectric sensor. Due to the reciprocity of the piezoelectric effect, deformation of the sensor will produce a charge across the sensor electrodes. As discussed by Lee and Moon (1990) the sensor output is a function of the effective electrode width F(x,y) and the polarisation of the piezoelectric material P(x, y). Following Collins (1990) we define an arbitrary sensor shape as +b/2F(x, y)P(x, y) dy, f(x) = [J-b/2

(5.7.2)

where +b define the transverse limits of the sensor. In general P will be +1 for piezoelectric material poled through its thickness and F(x, y) denotes the spatial pattern of the electrodes, i.e. F(x, y) will be either 1 or 0 at coordinate x, y depending upon whether that point is covered by the electrode or not. For predicting the electrical output of piezoelectric sensors it is more convenient to use the piezoelectric stress constant e~q. The stress constant e~q is directly related to the strain constant d~q introduced in Section 5.2 by the Young's elastic modulus and the Poisson's ratio of the piezoelectric material as discussed in Auld (1990). The stress constant has the units of coulombs per square metre in the metric system. The total charge q(t) generated by the piezoelectric sensor when it is deformed can then be calculated by integrating the local beam surface strain multiplied by the piezoelectric material stress constant e31 and weighted with the sensor shape to obtain the one-dimensional sensor relationship given by q ( t ) = - JL ° ~e3d°(x) Ox2 (hb + h,)dx,

(5.7.3)

where L s is the total length of the sensor. The strain variation through the sensor is assumed negligible (due to its thinness). Lee and Moon (1990) have also derived an expression for the charge generated by a two-dimensional distributed sensor which is given by

[

~2W

~2W

~2W ]

q( t) = - j[ s f (x, y) e31 ~~X 2 + e32 ~y2 + 2e36 ~x ~y (hb + hs) dx dy,

(5.7.4)

where S is the area covered by the sensor and e36 is the stress constant in the shear direction. When a sensor is used with no skew angle then e36 = 0 (Lee and Moon, 1990). As an example of a one-dimensional sensor we consider a simply supported beam model. We desire to develop a modal sensor, that is a distributed piezoelectric sensor whose output is only related to the motion of one particular mode of the beam. As described in Chapter 2 the motion of a simply supported beam can be written as ¢,o

Ajwt

w(x, t) = ,2~ W~q~n(x)e ,

n=l

(5.7.5)

140

ACTIVE CONTROL OF VIBRATION

where Wn are the modal amplitudes and ~pn are the modes shape functions given by ~p, = sin k,x,

k n =

ner/L.

(5.7.6)

Substituting equation (5.7.5) into the one-dimensional sensor equation (5.7.3) yields an expression for the charge output given by oo

q(t) = Z W,Bn,

(5.7.7)

n=l

where

B~ = -(hb + h~)e311~"f(x) ~2p~ dx. ~x 2

(5.7.8)

As is well known, the modes of a self-adjoint system such as the one-dimensional simply supported beam are orthogonal (see Chapter 2). We can thus take advantage of this property to design a sensor shape in order to observe only the required modes. From equation (5.7.8), it is apparent that if we choose f ( x ) to be proportional to the second derivative of a mode shape then q(t) will be proportional only to the amplitude of the corresponding mode, i.e. we have constructed a modal sensor. In this case the sensor charge output will be given by

q. = - (hb + h~)e3~K.A.L~W.,

(5.7.9)

where K~ is a constant related to the sensor gain and A. is the mode normalisation constant specified by

1 L, A~ = ~ I0 ~p~dx.

(5.7.10)

Figures 5.17(a) and (b) show two sensor shapes designed to observe the first and second mode of a simply supported beam respectively. Note that for the first mode, the sensor weighting (i.e. width) is largest where the strain of the first mode is largest. For the second mode, the polarisation factor P is + 1 in the left half and - 1 in the fight half of the beam in accordance with the response of the second mode which flips through 180 ° in phase over the beam mid point. Other important work in shaped distributed sensors has been performed by Collins et al. (1991) who designed shaped piezoelectric sensors based on the sinc function in order to respond with unity gain to all structural vibrations below a cut-off frequency o)~ and not to observe motions above w ~. In the investigations of Collins the piezoelectric material used was the polymer polyvinylidene fluoride (PVDF). As discussed by Collins and von Flotow (1991), PVDF material makes an excellent sensor. It is lightweight and flexible and thus causes little change to the system response. It has a high piezoelectric charge constant and can readily be shaped into complicated forms either by cutting or etching the electrodes. Changing the polarity of the sensor is achieved by simply flipping the sensor material over in the required areas so that the polarity direction is reversed relative to the system coordinate axis. Another important aspect of the implementation of piezoelectric sensors is the development of the necessary high input impedance electronics (from the sensor side) to measure the

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

141

(a)

(b) Fig. 5.17 Distributed sensor shapes for a one-dimensional simply supported beam modal observer: (a) n = 1 mode; (b) n = 2 mode.

charge over the sensor without causing 'leakage'. Circuit arrangements to provide high input impedance voltage amplification etc. are discussed by Collins (1990) and Clark et al. (1992a). As mentioned previously, care must be taken in accurately shaping and positioning the distributed sensors. Errors in the charge output due to these aspects have been investigated by Gu et al. (1994). Lee and Moon (1990) and Burke and Hubbard (1991) have also considered the design of two-dimensional modal sensors. In this case, it is more difficult to shape the sensor in order to obtain the necessary sensor weighting since variation in the vibration profile occurs in both the x and y directions. In order to solve this problem, efforts have been directed towards locally varying the piezoelectric material properties or thickness in order to obtain the necessary integrated rejection of the structural motions that are not required to be sensed. Chapter 8 will also illustrate the use of two-dimensional piezoelectric sensors in the control of sound radiation from structures. Typical properties of PVDF can also be found in Table 8.3.

5.8

Modal estimation with arrays of point sensors

In many situations it is advantageous to use arrays of point sensors and to electrically combine their outputs in order to construct with point sampling an output that is equivalent to a distributed sensor. Here a point sensor is defined as one whose size is small compared to the wavelength of structural motion. As the method of combining the individual sensor outputs is usually electronically based, it is possible to construct a sensor that is adaptive by changing the configuration of the electronics. A point sensor

142

ACTIVE CONTROLOF VIBRATION

can of course be seen to be a particular case of a distributed sensor, with its spatial weighting factor represented by a Dirac delta function. It can then be shown that the single point sensor observes all modally weighted motions equally at that point. By combining the outputs of individual sensors, as shown in the simple example of Fig. 5.18 we can construct an array of sensors whose combined shape function has reduced modal observability. We first discuss an array of point sensors which can be used to estimate modal amplitudes of known systems. Let us assume that the system is defined by a series of response characteristics such as mode shapes, ~Pm,-Thus the out-of-plane motion of the system could be described by Ajo)t

w(x,y,t)= Z Z AmJpmn(x'y) c ' m=ln=l

(5.8.1)

where Amn are modal amplitudes. The above relationship could easily be written for other variables such as velocity, acceleration, strain etc. We desire to measure or estimate a mode or state of the system. In order to do this we can sample the structural response at J positions and represent their values as a vector ws. Assuming that the structural response is dominated by J modes, the vector w, is then related to the system modal amplitudes and known mode shape functions by

I

wi

°

~P 1ll

~

~P~N 1

A11 ,

°,,

(5.8.2)

LWs where the elements of Ws comprise the measured complex displacements and M + N = J. The e j~' time factor has been omitted for convenience. By solving the above system of equations, we can obtain the modal amplitudes as a = W-~w ~,

(5.8.3)

where a [All A12 ... A12 ... AMN] T, W s = [W] W s2 ... WsJ] T and W is a matrix of the modal contributions at the sample points. Thus if we position an array of J sensors on a structure, we can process the output of the sensors using equation (5.8.3) in order to obtain information related to individual modes. Several points are important. The individual =

Sensor output

Summer

Gain

Point sensor Structure

Fig. 5.18

Array of point sensors configured as a general distributed observer.

DISTRIBUTED TRANSDUCERSFOR ACTIVECONTROLOF VIBRATION

143

sensors need to provide estimates of absolute amplitudes and phase at all points simultaneously. This necessitates the use of a multi-channel data acquisition system and the processing of the information in the frequency domain in order to resolve the relative phases. If the system is being driven at a steady frequency then a reference and a roving sensor can be used with the phases measured relative to the reference sensor. The above approach also requires an accurate knowledge of the system mode shapes or characteristic functions and this often is not readily available. Finally, spatial aliasing will occur if the individual modes are not sampled with sufficient spatial resolution. In general, two sampling points are needed in a wavelength of a motion of the structure to be observed. If modes of significant amplitude occur at wavelengths shorter than the Nyquist wavelength (equal to twice the spacing of the sensors) then the information from these modes will fold back around the Nyquist value and corrupt the estimates of the magnitudes of lower order modes in a manner analogous to discrete time-frequency processing of signals. The ad hoc approach is to increase the number of sample points until modal amplitude is observed to roll-off at some set value. Errors in the modal estimation associated with noise and positioning of the sensors are described by Clark et al. (1993). The technique is, of course, aided by the fact that, as discussed in Chapter 2, structures act as low pass filters to broadband disturbances in terms of modal response. The method can also be made more robust by using more measurement points than required modal amplitudes. In other words, a system of overdetermined simultaneous equations is formed. In this case it is appropriate to use a pseudo inverse or least mean squares technique to solve for the modal amplitudes as a = [WTW]-~WT W~.

(5.8.4)

The above method can be applied to systems described in terms of waves as well as modal response functions. Fuller et al. (1990c) have used the procedure to estimate complex amplitudes of travelling and near-field waves in vibrating thin beams. Arrays of sensors are also used in state feedback control of systems to estimate the states of the system. Such an estimate is usually in the form of a Kalman filter and this technique is described in Chapter 3.

5.9

Wavenumber estimation with arrays of point sensors

In many cases it is advantageous to estimate directly (or sense) the wavenumber components of a structural motion. As discussed in Chapter 8, sound radiation is directly related to structural wavenumber components having a supersonic phase speed. Thus a sensor that provides wavenumber information from structural measurements may in principle be used to infer properties of the sound radiation. If a onedimensional structure has a motion described by

w(x, t) = w(x) e j°~

(5.9.1)

we can apply a one-dimensional spatial Fourier transform given by

W(kx) = I = w ( x ) ~ + dx,

(5.9.2)

where W(k,) are the spectral wavenumber amplitudes at a particular wavenumber value

144

ACTIVE CONTROL O F VIBRATION

kx. In effect, since kx = 2z~/2, we have decomposed F(x) into its Fourier wavenumber components of different wavelengths when equation (5.9.2) is applied to a spatial response for a particular frequency as discussed in Section 2.4. In practice it is impossible to obtain a continuous estimate of F(x); hence we can reduce equation (5.9.2) to a discrete Fourier transform (DFT) for a finite record length (again analogous to time-frequency manipulations) which can be written as I

g'(kx) = x~ w(xi) eft~i Ax,

(5.9.3)

i-1

where if' are the spectral estimates and I discrete samples are taken over equal spacings of width Ax (Maillard and Fuller, 1994). Equation (5.9.3) thus enables an estimate of the amplitude of a particular wavenumber component to be made for particular frequencies. It is also possible to evaluate the discrete wavenumber transform using fast Fourier transform (FFT) algorithms as demonstrated by Williams and Maynard (1982) and Wahl and Bolton (1992). Note that the usual sampling requirements in terms of finite record length and bandwidth apply as discussed by Maillard and Fuller (1994) and Nelson and Elliott (1992). In order to build a wavenumber sensor, we can measure the motion at a required number of points and apply the relationship given by equation (5.9.3). Approaches of this type as well as finite difference techniques for separating wave components have been pursued by Pines and von Flotow (1990a). Different wavenumber components can then be directly used as sensor information. Note that this method applies at a particular frequency and is thus a frequency domain method, and that the equations can readily be extended to two-dimensional systems. Often it is required to estimate the wavenumber components in the time domain. Maillard and Fuller (1994) have developed a system which estimates time domain, structural wavenumber information from an array of accelerometers whose outputs are passed through a bank of digital FIR filters and then summed to provide an estimate of the wavenumber components of the structural response.

5.10 Wave vector filtering with arrays of point sensors In many active control problems the vibrational field to be attenuated consists of multiple waves. For example the vibrational field can consist of waves travelling in opposite directions (i.e. a standing wave) due to scattering from a discontinuity located at the end of the structural element. As discussed in Chapter 6, an effective control strategy in this case is to minimise the power flow travelling towards the discontinuity. In order to implement such a control system is necessary to use a sensor which provides error information proportional to the positive travelling wave component. On the other hand, if one wished to make the beam discontinuity act like a perfect absorber then the sensor would provide error information proportional to the reflected wave. The following analysis outlines a procedure based upon an approach developed by Elliott (1981) which filters out positive and negative wave components from a standing wave field in a non-dispersive medium (see Chapter 2).

DISTRIBUTED

TRANSDUCERS

FOR ACTIVE CONTROL

OF VIBRATION

145

Let us assume that we can measure the displacement field of a beam vibrating in longitudinal motion at two points. If the wavelength of motion is long compared to the spacing between the sensors, Ax, then the total displacement ut is given by ut(x, t) = [ul(x, t) + u2(x, 0 ] / 2 ,

(5.10.1)

where the subscripts 1 and 2 refer to the sensor positions. If the beam is carrying both positive and negative travelling waves then the total displacement field is also given by ut(x, t) = ui(x, t) + Ur(X, t)

(5.10.2)

2 U i + 2/,/r = U 1 q- U2,

(5.10.3)

and thus where the arguments of the displacement u have been eliminated for the sake of simplicity. For longitudinal wave motion, as discussed in Chapter 2, the in-plane force f ( x , t) is given by

0u (5.10.4)

f(x,t) = -SE ~,

i)x

with S and E denoting area and modulus of elasticity of beam respectively. Using the finite difference method as outlined by Elliott (1981) this force can be estimated from the two sensor positions by evaluating the expression f, .~ SE u 2 - u_______L1"

(5.10.5)

Ax We also know that for a single longitudinal wave the internal force is related to the particle displacement by (5.10.6)

f = + pLcLjtOu

for positive and negative propagation respectively, where P L is mass per unit length and CLis longitudinal wave speed given by c~ =~-ET-~L. Hence the total force in the beam in terms of the assumed wave field is ft = p l c L j o o ( u i - Ur).

(5.10.7)

This total force should equal that estimated from the finite difference expression given by equation (5.10.5), i.e. /62 -- U 1

pr.CLflO(Ui - Ur) -- - S E ~ . Ax

(5.10.8)

Manipulating this equation we see that 2U~ -- 2Ur =

2SCL

(Ul -- U2).

(5.10.9)

joAx

If we now sum equations (5.10.3) and (5.10.9) we can solve for the incident or positive travelling wave field such that 4u~ = (Ul + u2) +

2Sct. floAx

(Ul - u2).

(5.10.1 O)

146

ACTIVE CONTROLOF VIBRATION

If we assume that both sensors are of equal sensitivity and have a gain G then 1 u = -- V G

(5.10.11)

and we can re-write equation (5.10.10) in terms of V + which is the sum of the voltage outputs of sensors 1 and 2 and V- which is the difference between sensors 1 and 2, i.e. 4Gu/= V÷ +

2Scz.

V-.

(5.10.12)

jcoAx Similarly, if we take the difference between equations (5.10.3) and (5.10.9) we obtain an estimate of the negative travelling wave component that can be written as

4Gur = V +

2SCL V-. j~oAx

(5.10.13)

Figure 5.19 shows a block diagram of a circuit designed to perform the above functions in order to produce signals proportional to u~ and Ur. The outputs of both sensors are fed into the circuit via buffer amplifiers, one of which has a variable gain so that the apparent sensitivities of both sensors may be equalised. The signals are then summed and differenced and then passed through high pass filters in order to remove spurious low frequency noise which could be passed through the integrator and thus overload the circuit. In order to implement the above expression the signal of the bottom channel of the circuit in Fig. 5.19 is passed through an integrator with gain 2ScL/Axjw where S is the cross-sectional area of the beam, cL is phase speed and Ax is the sensor spacing. The above method can be used to resolve wave fields in various non-dispersive media, by calculating different gains for the integrator depending upon the wave type and material properties (for example, Elliott, 1981, discusses separation of acoustic plane waves). An important aspect of the technique is that it can be implemented so as to provide time domain information of the wave components which is compatible with the time domain implementation of the adaptive LMS algorithm (see Chapter 4). Practical implementation aspects of the above method and a discussion on the accuracy can be found in the work of Elliott (1981).

-..t Sensor 1 ~

G

4Ku i

Sensor 2

4Ku r 2Sc L Axj~

Fig. 5.19 Circuit diagram for an analogue wave vector filter in a one-dimensional, nondispersive medium (after Elliott, 1981).

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

147

Many structural systems carry dispersive waves (such as beams and plates in flexure) and in this case it is impracticable to use the above analogue method. Gibbs et al. (1993) have developed and demonstrated a method for separating out positive and negative travelling waves in a thin beam excited by broad band noise. The procedure is very similar to the above method except that the differenced output of two piezoelectric sensors designed to observe flexural strain is passed through an IIR filter designed to give the correct transfer functions which are causal over a band of frequencies. Similar to the above approach, the method of Gibbs et al. (1993) provides time domain signals which are proportional to both positive and negative travelling flexural waves. The interested reader is referred to the work of Gibbs et al. for more details on implementation of wave vector filtering in dispersive media. The method could also be extended to two-dimensional systems. In this case the vector component of the resolved wave field will point in the same direction as the finite difference array. Pines and von Flotow (1990a) discuss in some detail the theory and difficulties associated with wave separation from sampled information in a dispersive medium. Chapter 6 will present more discussion of the use and performance of various configurations of sensor arrays designed to separate out travelling flexural waves and flexural near fields in the active control of vibration in long beams.

5.11 Shape memory alloy actuators and sensors The previous sections have been concemed with actuators that apply an oscillating control input to the system usually at the same frequency (or frequencies) as the disturbance. These forms of control inputs have been previously defined in Chapter 3 as 'fully active' actuators. We now consider the use of an actuator which applies relatively steady-state or slowly time-varying control inputs which tend to change the system response by altering the system characteristics itself. As these types of inputs are not vibratory in nature and do not add energy to the dynamic system we call these systems 'semi-active' or 'adaptive' as defined in Chapter 3. The following discussion will centre on the use of filaments or wires of shape memory alloy (SMA) embedded in composite panels. However, the description could be applied to any actuator capable of inducing static strain that is attached to, or embedded in, a structure. The shape memory effect can be described very basically as follows: a material in the low-temperature martensitic condition, when plastically deformed and with the external stresses removed, will regain its original shape when heated. The material phenomenon that 'memorises' its original shape is the result of the reverse transformation of the deformed martensitic phase to the higher temperature austenite phase (Jackson et al., 1972). The solid-solid phase transformation also yields other useful characteristics such as the ability to reversibly and reliably control the material properties such as Young's elastic modulus and the yield strength. Many materials are known to exhibit the shape memory effect. They include the copper alloy systems of Cu-Zn, Cu-Zn-A1, C u - Z n - G a , C u - Z n - S n , C u - Z n - S i , Cu-A1-Ni, C u - A u - Z n , Cu-Sn, and alloys of Au-Cd, Ni-A1, Fe-Pt and others (Jackson et al., 1972). The most common of the shape memory alloys or transformation metals is a nickel-titanium alloy known as Nitinol. Nitinol is the SMA of choice for use as an embedded distributed actuator because of its unusually high resistivity which allows for resistive heating through the passage of an electrical current.

148

ACTIVE CONTROLOF VIBRATION

Nickel-titanium alloys (Ni-Ti) of proper composition exhibit unique mechanical 'memory' or restoration force characteristics and the ability to provide reversible changes of the material properties. The material can be plastically deformed in its lowtemperature martensite phase and then restored to the original configuration or shape by heating it above the characteristic transition temperature. This behaviour is limited to Ni-Ti alloys having near-equiatomic composition. Plastic strains of typically 6-8% may be completely recovered by heating the material in order to transform it to its austenite phase. Figures 5.20 and 5.21 show typical non-linear mechanical properties of Nitinol SMA as a function of temperature (Jackson et al., 1972). It can be seen that the Young's elastic modulus of Nitinol can increase by three to four times and the 0.2% yield strength increases from about 83 MPa to about 620 MPa; the recovery stress, which is the stress caused by the restoration tendency when the edges of the Nitinol are fully restrained when activated, changes as a function of temperature and initial strain. The class of the material referred to as SMA hybrid composite is simply a composite material that contains SMA fibres (or films) in such a way that the material can be stiffened or controlled by the addition of heat to the SMA. One of the many possible configurations of the SMA hybrid composite material is one in which the SMA fibres are plastically elongated before embedding and constrained from contracting to their 'normal' of 'memorised' length upon curing the composite material with high temperature. The plastically deformed fibres are therefore an integral part of the composite material and the structure. When the fibres are heated, generally by passing a current through the fibre shape memory alloy material, they 'try' to contract to their 'normal' or 'memorised' length and therefore generate a predictable in-plane force and

80 70 A

co

10% 7% 6% 5%

60

o x

-~

Q.

50 4%

oO t_

~"

40

3%

30

-.-

2%

•- -

1%

o

~ n-.

20

Initial strain

10 I.

0 ~J.....__._.a...._...__t......._..__t.~_--.-.------

60

100

140

180

220

260

300

340

380

Temperature (°F)

Fig. 5.20 Recovery stress versus temperature for Nitinol shape memory alloy. Heat rate equals 40 ° per minute (after Jackson et al., 1972).

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION 13

149

130

-

l

oo

70 50 ~'-

3

,--

x

1

x •~ 10

(D O

30

(a) Cooling

¢'~

I

vco

v

Z3

"o

o

11

Mf 1

~O

13

M

I

Md

S

m 130 -

i

_~ 110

-

90

-

70

-

50

-

30

-

10

O0

-20

j

s ~

60

140

~(Hb)atlng 220

300

Temperature (°F)

Fig. 5.21 Young's elastic modulus of Nitinol shape memory alloy versus temperature: r-l, modulus; ©, 0.2% yield stress; (a) cooling; (b) heating (after Jackson et al., 1972).

moment which can be used to provide active control of the dynamic and static response of the composite. This technique is referred to as active strain energy tuning (ASET) (Rogers et al., 1989a). SMA fibres can also be embedded into a composite material without prestrain. When the embedded SMA fibres are electrically activated, the Young's elastic modulus of the SMA fibres increases three to four times, as illustrated in Fig. 5.21, the overall stiffness of the composite structure will be increased, and therefore the response of the composite structure will be modified. In this case, significant numbers of SMA fibres relative to the structural volume is needed to achieve a usable tuning effect. This technique is referred to as active property tuning (APT) (Rogers et al., 1989a). To provide adaptive control of the response of the SMA hybrid composites, one can either 'tune' the mechanical properties of the embedded actuator fibres (APT) or impart large internal restoring stresses throughout the structure (ASET). Tuning the mechanical properties means that the SMA fibres or films are embedded without being plastically deformed, i.e. there is no shape memory effect, and it may avoid introducing large internal loads and stress concentrations in the structures. Active strain energy tuning is accomplished with SMA fibres (or actuators) that are embedded with a certain

150

ACTWE CONTROLOF VIBRATION

amount of initial strain. Once the fibres are actuated, the overall stiffness of the structure changes and an internal force and restoring stress, is created because of the shape memory effect. This allows control of various material properties of the structure. Liang et al. (1991) have investigated analytically the structural acoustic behaviour of adaptive shape memory alloy reinforced laminate plates in terms of their structural and acoustic response characteristics. Liang et al. use a Rayleigh-Ritz approach to find the normal mode response of a simply supported rectangular plate with embedded SMA fibres by including the recovery stress due to activation of the SMA fibres directly into the governing laminate equations (the ASET principle). The acoustic field radiated is then coupled to the plate motion using a procedure similar to that outlined in Chapter 8. The analysis is reported in full detail in Liang et al. (1991). Here we discuss an illustrative example of the use of adaptive SMA fibres. For the results discussed, a composite plate of dimensions 1.1 m by 0.8 m by 8 mm was studied. The stacking sequence of the composite plate layup was [0/-45/45/90 ° ] resulting in a quasi-isotropic plate (Jones, 1975). The plate was assumed to be made of equal thickness graphite/epoxy and Nitinol/epoxy laminae. Nitinol/epoxy obeys the rule of mixture and detailed information about the constitutive relations of SMA hybrid composites is discussed in detail in the work by Jia and Rogers (1989b) and material properties are available in the work of Rogers et al. (1989b). The Nitinol/ epoxy laminae are considered to be the top and bottom laminate, which has a 40% Nitinol fibre volume fraction which yields a SMA fibre volume fraction of 10% for the entire plate. It is assumed that the recovery stress of the embedded SMA fibres upon activation is 280 MPa. The recovery stress is related to the initial strain, the recovery stress-strain relation of the SMA fibre, the curing process of the composite plate which could cause stress relaxation, and the boundary conditions of the plate (Rogers and Barker, 1990). The plate is assumed to have a damping coefficient of 0.01. Table 5.5 contains the calculated natural frequencies of the first ten modes of the plate when inactivated and activated (i.e. with heating of the SMA fibres). The results show that, for example, the first natural frequency has been increased by approximately 70% upon actuation of the embedded SMA fibres. Typical changes in mode shapes for the third to the sixth mode of the free plate vibration are given in Fig. 5.22. The result of both Table 5.5 and Fig. 5.22 indicate significant changes in the plate natural frequencies and associated mode shapes upon activation of the embedded SMA fibres. As discussed in Chapter 8, sound radiation from a harmonically vibrating plate is strongly influenced by the value of the plate resonance frequencies (relative to the drive frequency) and plate mode shapes. Figure 5.23 shows sound intensity radiation Table 5.5 Changeof the first ten natural frequencies (after Liang et al., 1991) Natural frequencies (Hz) Mode

1

Inactive 41.3 Active 71.5

2

3

4

5

6

7

8

9

10

82.8 114.8~ 144.4 166.9 224.0 233.7 245.5 2 9 0 . 7 317.9 129.7 146.6 203.4 239.5 246.9 296.5 3 2 2 . 4 355.4 403.4

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION

151

Activated

Inactivated

,

.

3rd

3rd

4th

4th

.

.

,

.

,/c 5ih

5th

'6th

'

"

6th

Fig. 5.22 Change in mode shapes for an SMA composite, simply supported plate (after Liang et al., 1991).

0 i

4.5

/

2.25

xq~,,-

0

i

2.25

I

I

4.5

Intensity (W/m2x 106)

Fig. 5.23 Radiation directivity pattern of an SMA composite plate, f=220 Hz: inactivated; - - - , activated (after Liang et al., 1991). directivity patterns for the same SMA composite panel located in a rigid baffle and excited on one side by a sound wave incident at 45 ° with an amplitude of 1 Pa. The frequency of the wave is 220 Hz and the far-field directivity patterns are in the y = b/2 mid plane of the plate (see Fig. 8.10(b) for the radiation coordinate system). The results show a significant change in level and shape of the radiation pattern when the SMA is activated. This is due to two mechanisms. Firstly, as the plate resonance frequencies are changed, its response amplitude is modified (see Chapter 2), leading to

152

ACTIVE CONTROLOF VIBRATION

an overall change in the radiation level. Secondly, the coupling between the plate motion and the sound radiation is altered due to the change in activated mode shapes (see Chapter 8). This results in a drastically different sound radiation pattern. The work discussed in this section illustrates the use of SMA actuators that apply relatively steady state loads to the system or change the overall system properties. Fluids can also be used as a form of actuation in the sense that they are employed to change the system properties similarly to the SMA implementation termed APT and discussed above. Electrorheological (ER) fluids are suspensions of highly polarised fine particles dispersed in an insulating oil (Stangroom, 1983). When an electric field is applied to the ER fluid the particles form chains which lead to changes in viscosity of the medium of several orders of magnitude, as well as alteration of elasticity. The ER fluid can be thus embedded in a structure, for example, and used to tune its overall mechanical properties such as damping and stiffness by electrical actuation (Gandhi and Thompson, 1989). It is worthwhile to note that SMA can also be used as a 'fully active' actuator to control vibrations in dynamic systems. Baz et al. (1990) have analytically and experimentally demonstrated the use of SMA fibres to control transient motion in a cantilevered beam. The results show that the use of an SMA-based active system significantly increased the damping of the system. In general SMA is limited to control of very low frequencies due to its large thermal time constant and provision must be made for quickly dissipating the thermal output of the SMA fibres in order to quicken the cooling phase. In the work of Baz et al. (1990) the fibres were located outside of the material of the beam to facilitate cooling. Shape memory alloy can also be used as a distributed sensor. Work reported by Fuller et al. (1989b, 1992) demonstrated that an embedded SMA fibre when used in a Wheatstone bridge configuration can give accurate estimations of oscillating strains in a cantilevered beam. As SMA material can be manufactured to be super-elastic, very large strains can be measured before the failure of the sensor. This technique has been extended by Baz et al. (1993) who used arrays of different lengths of SMA fibres embedded in a cantilevered beam in conjunction with a matrix technique (similar to that of Section 5.8) to provide estimates of modal amplitudes of response. The technique shows much promise where large strains and hostile environments are encountered. The use of multiple embedded actuators and sensors in structures is part of a rapidly growing field termed adaptive, intelligent or smart structures. For more information on this field the reader is referred to the review paper by Wada et al. (1990).

6 Active Control of Vibration in Structures

6.1

Introduction

In this chapter we review a number of different approaches to the active control of mechanical vibration in structures. These approaches are distinguished from those discussed in Chapters 3 and 4 in that the structure is now assumed to be governed by a partial differential equation rather than an ordinary differential equation. In other words, the structure is assumed to be distributed rather than having 'lumped' springs, masses and dampers. There are a number of ways of describing the motion of such a structure, each of which is consistent with the governing partial differential equation. One way of expressing the velocity distribution over an entire structure, for example, is in terms of the sum of the contributions from a number of structural modes. Another approach is to describe the motion in terms of the amplitudes of a number of different types of mechanical w a v e s in the structure. Both of these representations and their relation to each other were discussed in Chapter 2. The most 'efficient' description, which requires the fewest parameters to describe the motion of the structure, will depend very much on the geometry of the structure, its boundary conditions and the frequency of excitation. These two descriptions of the motion of a distributed structure lead to two rather different approaches to active control. The first concentrates on controlling the m o d e s of a structure. By actively reducing the amplitudes of these structural modes the spaceaverage mean square velocity over the whole structure is reduced, and the control can be said to be 'global'. It should be emphasised that reducing the total vibrational energy of a distributed structural system, for example, does not guarantee that the s o u n d radiated by the structure will be correspondingly reduced (due to the nature of the structure-acoustic coupling). The active reduction of sound radiation from a structure will be discussed in more detail in Chapters 8 and 9, and for now we restrict ourselves to considering only the reduction of vibration p e r se. This is still an important area of practical application, however; for example in order to improve the positioning or pointing accuracy of an antenna or robot arm, or to reduce fatigue in highly-stressed structures. Whereas the control of structural modes tends to imply the global control of vibration throughout the structure, the active control of structural w a v e s is normally used when the flow of vibrational energy from one part of a structure to another is important. This would be true, for example, where there was a concentrated source of

154

ACTIVE CONTROLOF VIBRATION

vibration, and a particularly sensitive component was located at another point on the structure, connected via a relatively long structural component in which only a reasonable number of structural waves can transmit power. In the active control of structural waves we thus tend to be concerned with the suppression of vibration transmission rather than global control of the entire structure. The active control of vibrations transmitted through vibration isolation mounts could be said to also fall into this classification, but because of their practical importance they are described separately, in Chapter 7. It should be noted that suppressing vibration transmission into one part of the structure may well increase the vibrational energy in another part of the structure, and will generally not achieve global control. In this chapter we will first consider the modal approach to global active vibration control using both feedforward and feedback techniques. The feedforward approach, in which we assume a single-frequency excitation, is reasonably straightforward to describe analytically and it allows model problems to be solved which illustrate the ultimate performance limits of any active control system. Feedback control is a more practical strategy when knowledge of the primary disturbance is limited. The active control of structural waves is similarly treated, except that in this case feedforward control can be applied to a wider class of disturbances. This is because wave control implies that something is known about the direction of structural power flow, so that it is sometimes possible to position a sensory array between the primary and secondary sources of excitation which can detect the incident structural wave, whatever its waveform (see Nelson and Elliott, 1992, Chapters 5 and 6, for a discussion of the equivalent acoustical problem). We particularly concentrate on the active control of flexural waves, because of their importance in practice. A brief discussion of feedback control of flexural waves is also included, in which it is shown that the most successful strategy, of displacement feedback, is equivalent to a conventional method of wave control using a linear spring.

6.2

Feedforward control of finite structures

We will assume that the structure to be controlled is excited by a primary excitation at a single frequency and that all altemating quantities are proportional to e j~'t. The excitation is described by a force distribution f(x, y, to) acting over the structure, which for convenience is assumed to be a plate extending in the two directions x and y. The distribution of transverse displacement over the plate, w(x, y, to) is expressed, as in Sections 2.10 and 2.11, as the sum of the contributions from a finite number of natural modes of the structure, N, so that N

w(x, y, a~) - Z A~(a~)~Pn(X,y),

(6.2.1)

n=0

in which An(to) is the complex amplitude of the nth natural mode at frequency to. ~Pn(x, y) is the spatial distribution of the nth natural mode shape, which is orthogonal with respect to any other mode shape, Is ~'n(x,Y)~'m(X,Y) dx dy

0

if n

:#

m,

(6.2.2)

155

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

and is assumed normalised such that 1

I ~n2(x'y) dx dy S s

1,

if n = m,

(6.2.3)

where S is the area of the plate. In principle an infinite number of modes must be used in the summation of equation (6.2.1), but in practice the distribution of displacement can be described to any arbitrary accuracy by a finite number of modes. For excitation frequencies in the range of the natural frequency of the first few structural modes the number of modes required for an accurate description is typically fairly small, indeed close to the natural frequency of one of these lower order modes in a lightly damped structure it is often only necessary to account for the contribution of one 'dominant' mode. The amplitudes of each of the structural modes in equation (6.2.1) can be expressed as

An(a)) = An(a)) Isf(X, y, (.O)t~n(X,y) dx dy.

(6.2.4)

The second-order resonance term, with damping, can be written A'(a)) =

1

M(w 2 - 092 + j2~nOgnO))

,

(6.2.5)

where M is the total mass of the plate, w, is the natural frequency of the nth mode and ~, the damping ratio associated with this mode. The total time-averaged vibrational kinetic energy of the plate when driven by a pure tone force distribution can be written as Mo) 2 g

Ek(w) = ~

4S

J w(x, y, w)w* (x, y, w) dx dy. s

(6.2.6)

Using the expansion in terms of the normal modes given by equation (6.2.1), and noting that the mode amplitudes are not dependent on the position on the plate, the total kinetic energy can be written as Ek(~o) =

Mw 2 4 Zn Zm A"(o)A*m(m) -S1 I s

t~n(X' Y)lPm(X'

y) dx dy.

(6.2.7)

The orthonormality of the natural modes described by equations (6.2.2) and (6.2.3), can now be used, so that the total kinetic energy can be written as M~o2 N Ek(m) =

(6.2.8)

[, 4

n--0

i.e., proportional to the sum of the modulus squared mode amplitudes. The total vibrational potential (or strain) energy of the plate can also be written as

Ep((,o)=

4

s

(I2w2 ~~x 2

+ ~~y2

)

dxdy,

(6.2.9)

which can itself be expressed in terms of the amplitudes of the natural modes. For

156

ACTIVE cONTROLOF VIBRATION

many common boundary conditions, however, including free-free, simply supported and clamped-clamped, the kinetic and potential energies are equal (Morse and Ingard, 1968). We thus concentrate on the problem of the feedforward active control of the kinetic energy in a harmonically excited plate. The analysis closely follows the analogous problem of minimising the total acoustic potential energy in an enclosure, considered in detail in Chapter 10 of Nelson and Elliott (1992). The modal expansion of the transverse displacement at some point on the plate (equation (6.2.1)) is first expressed as a vector inner product

w(x, y, o))= aT(to)~(x, y),

(6.2.10)

where aT(to) = [Al(to) A2(to)...AN(to)], gtT(x, Y) = [~Pl(X,Y) ~P2(X,y)... ~PN(x,Y)].

(6.2.11)

The total vibrational kinetic energy of the plate given by equation (6.2.8), can then be written as Ek(co) =

Mto 2

aH(o))a(to).

(6.2.12)

4 The vector of structural mode amplitudes is itself the superposition of components due to the primary source ap(to) and those due to an array of M secondary force distributions, whose complex amplitudes are contained in the vector f s, coupled to the structural mode amplitudes via a matrix of modal coupling coefficients B, so that in general a = ap + Bf~.

(6.2.13)

The total vibrational kinetic energy can thus be written in the standard Hermitian quadratic form as Me92 Ek-

4

H

H

H

H

(fsHBHBfs+ fs B ap + ap Bfs + apap).

(6.2.14)

E k is thus a quadratic function of each component in fs which is guaranteed to have a minimum value, since the matrix BHB is positive definite, for a value of f~ given by (Nelson and Elliott, 1992, Appendix)

f~o - - [BHB ] -~BHap

(6.2.15)

with a corresponding minimum value of E k given by Ek(m~n)-

Mto 2

a n [ l - B[BHB]-~BH]ap.

(6.2.16)

4 To illustrate these general principles, several authors have taken specific examples of structure and primary and secondary force distribution. Curtis (1988), for example, performed numerical simulations of the minimisation of the total vibrational strain energy in an undamped beam of length L. Some typical results from these simulations are presented in Fig. 6.1.

157

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

/ t"-

g

t-

09 0

.¢ ""

I

0

I

I

I

100

200 300 Frequency (Hz)

400

500

Fig. 6.1 Vibrational strain energy as a function of pure tone excitation frequency for a beam excited by a primary point force at Xp -0.3L ( ~ ) and after the energy has been minimised by the action of a secondary point force at Xs- 0.1L (- - -) (after Curtis, 1988). The formulation for global control presented above was generalised somewhat by Post (1990) and Post and Silcox (1990), who considered the minimisation of the mean square transverse displacement over some part of the length of a beam, say from x = xl to x:. The cost function being minimised is thus of the form

j =

l dx.

Xl

(6.2.17)

This results in a similar quadratic optimisation problem to that considered above, which can be solved at a number of excitation frequencies to give the minimum cost function. The control strategy is still, however, entirely feedforward and is illustrated Primary frequency generator (o~)

J Hqco) J ~ "-I

wI P,I,

Fig. 6.2

xp

xs

Primary force

Secondary force

Block diagram of the feedforward control system on a beam (after Post, 1990).

158

ACTIVE CONTROLOF VIBRATION

in Fig. 6.2, in which H ( j t o ) is the complex number which describes the amplitude and phase of the secondary source relative to that of the primary source. As another example of global control, Fig. 6.3 shows the results computed by Post (1990) and Post and S ilcox (1990) for a simply supported beam with a damping factor ~, of 1%, in which a cost function equal to the total kinetic energy on the beam was minimised. The physical parameters in this simulation were normalised so that the first mode had a normalised natural frequency of to = 10. From equation (6.2.6) it is clear that the total kinetic energy in this case is proportional to the mean square transverse displacement of the beam, averaged over its length. Significant reductions in cost function can be achieved near the natural frequencies of the beam, and almost no reductions can be achieved at the frequencies in between these resonances. The distributions of transverse displacement on the beam with and without active control are shown in Fig. 6.4 for excitation frequencies corresponding to the second normalised natural frequency ( t o - 40) and half way between the first and second normalised natural frequencies (to = 25). As an example of the minimisation of a local cost function, Fig. 6.5 shows the cost function given by the mean square displacement averaged only over the final quarter of the beam length, with and without active control. This cost function is even more significantly reduced than the global cost function for excitation frequencies close to the natural frequencies of the beam. The distributions of transverse displacement on the beam before and after local control are shown in Fig. 6.6 for the same on-resonance and off-resonance excitation frequencies as were used in Fig. 6.4. The residual displacement distribution for the on-resonance excitation frequency, Fig. 6.6(a) is similar to that obtained by global control, Fig. 6.4(a). The effect of local control offresonance, however, is that although some reductions have been obtained in the final quarter of the beam, significant increases in the displacement are created elsewhere on the beam. -10

-20

. uncontrolled

-30

"0 v

3

------ c o n t r o l l e d

-40 -50

-60 -70

8°o J

I

2'o

I

I

do

I

do

I

J

Normalised frequency, to Fig. 6.3 The global cost function, proportional to the total kinetic energy on the beam, due to the primary force alone, at xp= L/6, (uncontrolled) and (controlled) with the effects of an optimally adjusted secondary force at Xs = 3L/4 (after Post, 1990).

0.028 0.024

159

a)

0.02 0.016 0.012 0.008 0.004 00

0.2

0.4

0.6

0.8

1

x/L 0.016

(

0.014 0.012 0.001 0.0008 0.0006 0.0004 0.0002

f

k3(/ --con,ro,,e

oo

0.2

0.4

0.6

0.8

t ffL

Fig. 6.4 Transverse displacement distributions on the beam after minimisation of the global cost function at (a) an on-resonance excitation frequency (~ =40) and (b) an off-resonance excitation frequency (co = 25) (after Post, 1990). -20 -30

uncontrolled controlled

-40 -50 1:13 -.o v -60 ~" -70 -80 -90 -100 0

20

40

60

80

100

Normalised frequency, o9

Fig. 6.5 The local cost function, proportional to the kinetic energy from x = 3L/6 to x = L on the beam, due to a primary force at Xp = L/6 alone (uncontrolled) and controlled with the effects of an optimally adjusted secondary force at Xs = 3L/4 (after Post, 1990).

ACTIVECONTROLOFVIBRATION

160 0.028[

~

0.024f

/

o.o~ol/ * °°'°I /

k

~" 0"014I /

\

\ \

o.oo.I/ o oo;v

/

/

/

/

~

\

(a)

\

\ \ / --unc?n][oile<J\ oo \

0.2

0.4

0.6 x/L

0.008 (b)

0.007

uncontrolled

0.006 0.005 "~" 0.004 0.003 0.002 0.001 0

0.2

0.4

0.6

t

F/////..,!

t

Xp

xs x/L

Fig. 6.6 Transverse displacement distributions on the beam after minimisation of the local cost function (proportional to the kinetic energy over the shaded region on the x axis) when excited (a) on-resonance (to = 100) and (b) off-resonance (to = 250) (after Post, 1990). Experimental approximations to the space-averaged mean square displacement minimised in the numerical simulations described above could, for example, be obtained from the sum of the squared outputs of a number of accelerometers on the beam. This is analogous to minimising the sum of the squared outputs from a number of microphones to obtain an approximation to the total acoustic potential energy. The positioning of such discrete error sensors, and the errors involved in this approximation to the true space-averaged cost function, are described in Section 10.11 of Nelson and Elliott (1992). These errors are minimised if the sensors are positioned so that they are able to observe each of the structural modes which are significantly excited. An alternative approach to the practical measurement of the residual vibrations is to use spatially distributed sensors which, for example, are preferentially sensitive to certain structural modes. Gu et al. (1994) discuss some of the practical problems of implementing such distributed sensors on two-dimensional structures, and show that such sensors

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

161

must be very accurately shaped if they are not to be sensitive to structural modes other than that for which they were designed. Nevertheless, Gu et al. (1994) experimentally observed significantly better attenuation of the (3, 1) structural mode on a plate using two specially shaped distributed sensors than when using two point accelerometers. Broadband excitation of structures is also an important problem. Vipperman et al. (1993) and Burdisso et al. (1993c) have analytically and experimentally studied the SISO feedforward control of transverse displacement at a point on a simply supported beam. In their work they investigated the use of IIR fixed filters as plant models (Vipperman et al., 1993) and studied the effects of non-causality due to delays through the control hardware, as also discussed in Section 6.13 of Nelson and Elliott. Their results demonstrate that as the control system becomes increasingly non-causal, the amount of achievable attenuation across the band is reduced. However, this can be somewhat compensated for if the number of coefficients in the adaptive filter is increased (Burdisso et al., 1993c).

6.3

Feedback control of finite structures

The various approaches to the feedback control of distributed structures have been described by Meirovich (1990), for example, and were also briefly discussed in Chapter 3. There has been a great deal of theoretical work in this area, with experimental investigations being rather thin on the ground. For this reason we concentrate on two examples of practical implementations, reported by Hodges (1989) and Rubenstein et al. (1991). The block diagram of an experimental arrangement used by Hodges (1989) is shown in Fig. 6.7. The objective of the experiment was to independently control the two lowest

F Cantileverbeam Accelerometers | Primary Secondary_l ~..~l~l, Force transducer voltageVp(t) shaker I~ "~lh' IFI Primary I 1]1 slaker Sec°ndary i~ Dual I voltage Vs( t ) i~I channelI , Current \\\\\\\\\\\" analyserII ~xamplifier IChargelf(t) I amp. I ~2(t) I Charge Reconstruction filter amp. I~ 1 t) ] Charge ] amp. I Micro-, I A ,,I DAC processor ADC , TMS 320C25+ host PC 4

I I

I I I

N I

I I I

Fig. 6.7 Block diagram of the modal feedback experiment on a cantilever beam described by Hodges (1989).

162

ACTIVE CONTROLOF VIBRATION

natural modes of a cantilever beam, which was 0.63 m long, 50 mm wide and 5 mm thick. A primary force was provided by a shaker which was connected to the beam via a force transducer, and driven by random noise, low pass filtered so as to have a bandwidth of 100 Hz. The secondary force acted at the same position on the beam, on which were also mounted two accelerometers, one close to the force inputs, and one close to the tip. The force inputs were arranged to be close to the node of the third structural mode on the beam, so that this mode was not significantly excited by either the primary or secondary force input. The outputs from the charge amplifiers driven by the two accelerometers were fed to a pair of analogue-to-digital converters. No anti-aliasing filters were necessary because the input force was band limited. The absence of antialiasing filters reduced the delay in the feedback loop, which was shown to be detrimental to the performance in Section 3.5. The sampled accelerometer signals were passed to a Texas Instruments TMS 320C25 processor mounted on a DSP (Digital Signal Processing) card in a host personal computer. The sample rate of the system was 50 kHz, which was chosen to minimise the processing time, and again reduce the loop delay. The processor implemented a pair of digital integrators (Hodges et al., 1990) and 55

"-" 45 "7t ~ -~ 35

uncontrolled

(a)

o

"" 25

-

t__

m 15 "o r-

5 -5

-15 -25

I

0

100

50 Frequency (Hz)

55 A

45

uncontrolled ........ controlled

"7o') -~ 35

~

~

k

(b)

o

"

25

rn 15 "o

5

-'~

-5

t~

-15 -25

I

0

50

100

Frequency (Hz) Fig. 6.8 Transfer inertances measured on the cantilever beam with and without (a) feedback of the velocity associated with the first structural mode; (b) feedback of the velocity associated with the second structural mode (after Hodges, 1989).

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

163

a modal analyser, which gave two outputs, proportional to the velocities of the lowest two structural modes on the beam. These signals were weighted by two modal velocity gains and fed back to the secondary shaker via a digital-to-analogue converter, reconstruction filter and current amplifier. Figure 6.8 shows the transfer inertance (acceleration per unit force) measured between the primary force and the tip acceleration for two different modal velocity feedback gain settings. Figure 6.8(a) shows the effect of feeding back only the velocity of the lowest order structural mode. It can be seen that the response of this mode at its natural frequency has been reduced by more than 20 dB because of the additional damping due to the velocity feedback (the damping ratio of this mode was increased from 0.005 to 0.158). The response due to the second mode, however, has hardly been affected by this feedback strategy. In contrast, Fig. 6.8(b) shows the effect of driving the secondary force with a signal proportional to the velocity of the second structural mode alone. In this case the amplitude of the second modal response has now been reduced by more than 15 dB at its natural frequency, but the response due to the first mode is largely unaffected. In fact, the natural frequency of the second mode has been slightly increased by this feedback strategy, which is probably the result of a small decrease in the modal mass due to delays in the feedback loop, as predicted by equation (3.5.6) in Section 3.5. In the investigation of Rubenstein et al. (1991), state feedback control was used to optimally control one structural mode on a 3 mm thick steel plate of dimensions 0.6 m x 0.5 m. The experimental arrangement is indicated schematically in Fig. 6.9. Twelve accelerometers were used on the panel whose outputs were passed through a 'modal filter' which used the assumed mode shapes of the plate to compute the modal accelerations of the first two structural modes. These were then used to update a Kalman filter (Section 3.9), which optimally estimated the modal velocities and

7 Panel • in frame -q

Primary shaker generating disturbance

-1

Twelve accelerometers

v

;construction filter riving secondary shaker

A

-kc

C

" -~

I~,

Kalman filter -:

Modal-filter -~

A D : C ~

4

Digital-toanalogue converters

Fig. 6.9 (1991).

LQR feedback gains

L_

Y

J

Analogue-to-digitalconverters

State estimator

Block diagram of LQR state feedback experiments described by Rubenstein et al.

164

ACTIVE CONTROLOF VIBRATION

accelerations of the first two structural modes, which, together with the states of the reconstruction filter and the disturbance model, were the eight state variables used to describe the system. The use of a disturbance model to augment the vector of state variables is described by Johnson (1976), and had been previously used in the active control of a torsional system, for example, by Burdess and Metcalfe (1985). These estimates of the state variables are then fed back to an electromagnetic shaker acting as the secondary source, via a set of eight feedback gains. The complete feedback system was implemented digitally on a transputer-based system operating at a sample rate of 3.3 kHz. A second-order reconstruction filter was used before driving the shaker, but again no anti-aliasing filters were used before sampling the inputs from the accelerometers. The LQR feedback gains were chosen to minimise a quadratic cost function of the sort described in Section 3.10, but which included only the amplitude of the first structural mode of the plate. These gains depend upon the assumed form of the disturbance. The disturbance was assumed to be either a pure tone at 60 Hz, close to the natural frequency of the first structural mode at about 50 Hz, or narrow-band random noise. In both cases the disturbance was modelled as white noise passed through a second-order resonant system whose damping was chosen so that its output was either harmonic or narrow-band random. It should be noted that the centre frequency of the disturbance is assumed known, so that the output of the disturbance model can accurately model the physical disturbance. In the case of a pure tone disturbance the disturbance observer will have no damping and will become an oscillator whose frequency is assumed to be known exactly, and whose states are steady state sinusoids. Under these conditions the controller designed using LQR feedback methods becomes 10 5

(a)

0

~-5 !

-100

0.1

0.2

!

0.3 0.4 Time (secs)

0'.5

O'.6

0.7

4 t,"-

o

.m

2 (b)

0 0

1/2, as observed. The case in which the discontinuity on the beam has both linear and rotational inertia has been analysed by Cremer and Heckl (1988). They show that this combination of discontinuities can completely suppress a propagating flexural wave at a single critical (a) Acceleration feedback t°O m if}

1.0

E l'-- . E

O 13..

(b) Velocity feedback

L_

0.0

i

0

J

~

1.0

c °O ~ if}

t

i

i

~

,

Normalised gain,/t

O t'l

,

0.0

~

100

i

0

i

~

i

~

~

Normalised gain, v

,

J

100

(c) Displacement feedback 1.0 c°O ~ 0~

i,tl:i

°~ l~::

11)

O

O

a..

I

0.0

I

0

I

I

I

I

Normalised gain, e

I

I

100

Fig. 6.21 The effect on the power transmission coefficient of a beam of varying degrees of: (a) acceleration feedback control; (b) velocity feedback control; (c) displacement feedback control.

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

181

frequency. Clearly both mechanisms by which energy can be transmitted by flexural waves in the beam are affected by such an arrangement. If the controller is designed to feed back the velocity of the beam so that F~ = -g~w(0),

(6.8.7)

then by a similar line of reasoning to that used above, the resultant propagating transmitted wave is given by (j -- S ~ -

4-EI/93 ,41.

(6.8.8)

This may be expressed as

jr-4 ] Ar = - j r - (4 + v) A~, where v=~og,,/Elk}=g,,k:/copa. feedback controller is thus

at=

(6.8.9)

The power transmission coefficient for such a [at[ 2 .2 =

[A, [

v 2 +16

v2 + (4 + 1,,)2

'

(6.8.10)

which is plotted in Fig. 6.21 (b). The power transmission coefficient falls more rapidly with feedback gain in this case, reaching a value of a r=0.4 for v = 8 , but then increases for higher values of feedback gain. As the feedback gain becomes very large, the beam is effectively pinned, or simply supported, at the point of application of the secondary force. This boundary condition again reflects the power in the flexural motion associated with the transverse motion, but cannot affect that associated with the rotational motion, so that again the power transmission coefficient tends to the value of 1/2 as the feedback gain becomes very large. Finally, we will consider what turns out to be the most interesting case of feedback control; when the secondary force is proportional to the transverse displacement of the beam, so that

F~ = - gdw(O).

(6.8.11)

Following the same steps as above, we obtain the equation for the transmitted propagating wave which in this case is given by

]

(1 +J)ga + 4EIk~ A~,

(6.8.12)

which can be written, by analogy with the result of Mead (1982) for a spring attached to the beam, as AT =

j ( 4 - e)

(6.8.13)

e + j ( 4 - e) in which e= ga/Elk}= gak:/w2pA. The corresponding value of power transmission

182

ACTIVE CONTROL OF VIBRATION

coefficient becomes ( 4 - e) 2 ar = (4 - E) 2 - E2

(6.8.14)

which is plotted in Fig. 6.21 (c). It can be seen that as gd and hence e, become large, which again corresponds to the case of a pinned or simply supported discontinuity on the beam, then again a r tends to the value of 1/2. The most interesting aspect of Fig. 6.20(c), however, is that aT has a minimum which, from equation (6.8.14), is seen to be given by ar=0

if

e=4.

(6.8.15)

The value of the ideal feedback gain corresponding to e = 4 is thus g d = 4 Elk},

(6.8.16)

so for a given frequency of excitation there is a single value of displacement feedback gain which completely suppresses flexural waves. In conventional terms, this result could also be achieved by a linear spring, having an appropriate spring constant, acting between the beam and an inertial reference position. Unfortunately because of the frequency dependence of kI in equation (6.8.16), this critical value of feedback gain will change if the excitation frequency is changed, and broadband control cannot be achieved with a single gain setting. If the gain of the ideal feedforward controller, equation (6.7.9), is considered in the limit as the distance between sensor and actuator, l l, tends to zero, we obtain lim Go =

l,---)0

F~ = 4Elkf

W(0)

O.)2 '

(6.8.17)

which thus results in F~ = - (4Elk})w(O),

(6.8.18)

exactly as obtained for the displacement feedback controller in equation (6.8.16). It is interesting to note that to suppress flexural waves on a beam, the ideal feedback controller should have a finite gain, whereas the ideal feedback controller for suppressing acoustic plane waves propagating in a duct should have an infinite gain (Nelson and Elliott, 1992, Section 7.8). This difference is due to the near-field contribution to the response in the case of flexural waves on a beam. The simple feedback controller is not as useful as it first appears, however, since the optimal feedback gain, given in equation (6.6.16), has a response which is real, but proportional t o 0.) 3/2 (since klo~ 0.)1/2). Since the frequency response of the feedback controller is real, it must have an impulse response which is symmetric about the time origin. Because the frequency response increases according to to 3/2, the impulse response is not confined to the origin of time and so must be non-causal. The ideal controller is thus not realisable for broadband excitation, as discussed by von Flotow (1988) and Miller et al. (1990). Alternative feedback strategies are also possible. For example, Miller et al. (1990) and von Flotow (1988a) describe the suppression of the reflected flexural wave at the free end of a beam by applying a moment at the end proportional to the tip velocity.

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

183

Similar wave absorbing terminations for the free end of a beam using feedforward control have been described by Scheuren (1990) and Pines and von Flotow (1990b). Miller et al. (1990) also discuss the strategy of maximising the power absorption of a set of secondary sources, an idea investigated experimentally on an infinite beam by Redman-White et al. (1987). Such a control strategy should only be exercised with some caution, however, since it was shown for the analogous acoustical case by Elliott et al. (1991), that the power output of the primary source can be significantly increased by maximising the power absorption of the secondary sources in a finite system with reflections from the ends. This can result in dramatic increases in the stored energy in the system, a result which has been confirmed in the vibrational case by Brennan et al. (1993). Von Flotow (1988a) points out that the near-field contributions in flexural wave propagation can also provide the coupling necessary to increase the power output of the primary, and concludes that 'it is not yet clear whether this type of "energy vacuuming" will be useful in application'. Although power absorption does not appear to be a practical control strategy in finite systems, the use of feedback control for wave suppression does appear promising. It should be emphasised that this feedback control strategy is different from that of damping structural modes, as described in Section 6.3. Whereas the practicalities of modal control have been widely investigated, the implementation problems involved in wave control have only recently been considered, and are still under investigation.

7 Active Isolation of Vibrations

7.1

Introduction

This chapter will deal with the application of active techniques to problems in vibration isolation. Broadly speaking, there are two classes of such problem; instances where we wish to isolate a vibrating body (such as a machine of some kind) from a 'receiving structure' (such as a car body, ship hull, aircraft fuselage or building) and instances where we wish to isolate a body (such as sensitive equipment or a railway car) from vibrations imposed by another source (such as ground vibrations or railway track unevenness). In both classes of problem, the source of the vibrations may be either deterministic (i.e. having a perfectly predictable waveform) or random (i.e. having a waveform that is not perfectly predictable). However, it is generally true to say that most problems of the first kind, which involve isolating machinery vibrations from a receiving structure, have a deterministic source of vibrations. This is the case whenever the source of vibrations is a rotating or reciprocating machine. In these cases we can adopt what is essentially a feedforward control approach to the problem. Thus one only needs to know (or be able to detect) the frequency of the vibration source and the necessary control forces can be synthesised using the adaptive feedforward techniques described in detail in Chapter 4. It is also true to say that the second class of problem, which involves isolating a vibrating body from external sources of vibrations, is mostly dealt with using feedback techniques of the type described in Chapter 3. For example, one of the most commonly addressed problems of this type is the design of active vehicle suspension systems. Thus the body to be isolated is the passenger cabin of the vehicle and the source of vibrations is the variable height of the road surface, the latter being a random process. Although isolation problems associated with a deterministic excitation are mostly treated using feedforward techniques and those associated with a random process are mostly treated using feedback, feedforward control can be applied to problems with random disturbances and feedback can be applied to problems with deterministic disturbances. Many of these possibilities will be considered here. The intention of this chapter is to give an introductory survey of these types of vibration isolation problem. First, the isolation of periodic vibrations from a flexible receiving structure will be described, since this is one of the most commonly encountered problems faced by the vibration control engineer. Basic guidelines for the positioning and requirements of secondary actuators will be given and some of the performance limits of such systems will be discussed. Examples will be described of

186

ACTIVE CONTROL OF VIBRATION

the practical application of the technique to automobiles and helicopters. Some discussion will also be presented of such vibration isolation problems when the vibration source is not deterministic and feedback control techniques must be used. Finally, an introduction will be given to the application of feedback control for the active isolation of bodies from random vibrations. Much of the work in this area has been undertaken within the framework of linear quadratic Gaussian (LQG) optimal control theory and an introductory discussion of these techniques will be presented.

7.2

Isolation of periodic vibrations of an SDOF system

First we will follow Nelson et al. (1987a) and consider the isolation of an SDOF system from a flexible substructure. The vibration of SDOF systems has been considered in Chapter 1. A similar analysis to that given below has been presented by von Flotow (1988b). The problem studied represents the simplest possible model of a mounted machine whose vibrations we wish to isolate from a receiving structure. It will also be assumed that the single degree of freedom system is harmonically excited, such that in practice, an adaptive feedforward control system could be used. This in turn assumes that the primary excitation forces in the machine are deterministic (perfectly predictable) which is of course the case for a large class of problems arising in practice; the isolation of machines such as engines, pumps and compressors from flexible structures such as the hulls of ships and submarines or the bodies of automobiles. As illustrated in Fig. 7.1, a secondary force can be applied at three possible locations associated with the mass-spring-damper system. Here we have assumed that the primary complex excitation force fe (representing the unsteady forces generated by the machine) is applied to the mass M of the system. Thus an obvious approach to the problem is that illustrated in Fig. 7.1 (a) where the secondary force is also applied to the mass and that to achieve zero response of the system we require simply that f~ = -fp.

(7.2.1)

In practice therefore, this implies that the secondary force would have to be applied to the body of the vibrating machine via an inertial exciter of some kind (e.g. an electrodynamic exciter which is provided with a mass against which it can react). A second approach to the problem is that illustrated in Fig. 7.1 (b). Here the secondary force is applied directly to the receiving structure with the objective of reducing the response of the receiving structure to zero. Again the secondary force would have to be applied by using some form of inertial exciter. It is important to calculate the magnitude of the secondary force required relative to that of the primary force applied to the system. Since we are dealing with periodic vibrations it will suffice to work at a single frequency. First assume that the receiver can be characterised by a complex input receptance R (jto), such that its complex displacement wn is related to the applied force f by WR= R(jto)f. (Each of these terms is a complex number which varies with to; see Chapter 1.) The force applied to the receiver is the sum of the secondary force and the forces applied via the spring and viscous damper. Thus we can write WR = R(jco)[fs + K ( w s - WR) + j t o C ( w s - wR)],

(7.2.2)

ACTIVE ISOLATION OF VIBRATIONS

187

to

M

M

//////////

(a)

[•C

1"////////

(b)

M

fJ///'ffff~/'/f~//fff (c) Fig. 7.1 Active isolation of a harmonically excited SDOF system from a receiving structure. Three arrangements are shown for the application of a secondary force (a) directly to the mass of the system, (b) directly to the receiver and (c) directly to the receiver with reaction against the mass of the system. where Ws has been used to denote the complex displacement of the mass M, the subscript s implying the displacement of the 'source'. This equation implies that for wR = 0 we require f , = - (K + je)C)ws

(7.2.3)

and the secondary force required is that necessary to cancel the sum of the stiffness and damping forces applied to a rigid foundation by a complex displacement Ws of the mass. Note that this result holds irrespective of the receptance R(jco) of the receiver. We can deduce fs in terms of fp by examining the equation of motion of the mass given by -ogZMws + j~oC(ws - wR) + K ( w s - wR)= fp

(7.2.4)

and it therefore follows that when wR = 0, we have ( - ~oZM + jogC + K)ws = f p.

(7.2.5)

188

ACTIVE CONTROLOF VIBRATION

Substitution of this result into equation (7.2.3) therefore shows that - ( K + floC) fs =

-o92M + flo C + K

(7.2.6)

fp.

This result, which relates the required secondary force to the primary force, can be expressed non-dimensionally in terms of the natural frequency co,,='qK/M and damping ratio ~ = C / 2 M r o , , of the SDOF system (these parameters have been previously discussed in Chapter 1). First note that

-[(K/M) + (floC/M)] (7.2.7)

f~ = -09 2 + (floC/M) + ( K / M ) fp

and therefore +

f,

=

2j o¢co.)

-co 2 + 2jog~tOn + tO,,2

(7.2.8)

fp.

This expression can also be written in terms of non-dimensional frequency f~= co/~o, such that 1 +2j~

f, =

~'-

1 - 2j~O

(7.2.9)

fp.

The frequency dependence of this relationship is illustrated in Fig. 7.2 which shows a plot of the ratio of the modulus of the secondary force to that of the primary source for a damping ratio ~ = 0.01. This plot emphasises that, as one would expect, the secondary force required to cancel the vibration of the receiver is much larger than the primary force at the natural frequency of the mounted system. Thus, in the region of this natural frequency, primary force fluctuations are effectively amplified by the 100

10

(b) (a)

I

fso I

0.1

0.01

I

0

,

,O=l

i

t

J

2

3

4

Non-dimensional frequency, ,(2

Fig. 7.2 The magnitude of the secondary force required relative to that of the primary force for the three arrangements shown in Fig. 7.1: (a) cancellation of the force applied to the mass; (b) cancellation at the receiver; (c) cancellation at the receiver with reaction against the mass.

ACTIVE ISOLATION OF VIBRATIONS

189

response of the system and result in large forces transmitted to the receiver by the spring and damper. Naturally, in conventional passive isolation systems, this is the situation that one seeks to avoid by designing the mounted natural frequency of the machine to be well below the lowest excitation frequency generated by, for example, the rotational speed of the machine. A further approach to the problem, which as we shall show below, may have some distinct advantages, is that illustrated in Fig. 7.1 (c). Here the secondary force is applied in parallel with the spring and damper such that it acts on the receiver with a reaction against the source. Such a situation would arise if, for example, an electrodynamic exciter were used with the body of the exciter rigidly fixed to the source and the excitation applied to the receiver; such an arrangement generates equal and opposite forces applied to both source and receiver. Under these circumstances, equation (7.2.3) for the secondary force required still holds, but equation (7.2.4) describing the motion of the mass becomes - c o 2 M w s + jcoC(ws-

wR) + K(ws + wR) = f p - fs"

(7.2.10)

Again setting wR = 0 and using equation (7.2.3) for f, shows that WS --

-fP

(7.2.11)

o) 2M

This shows that the displacement of the mass is exactly as if it were freely suspended; the dynamic displacement of the mass is determined only by its inertia. This results because the secondary force cancels the contribution of the stiffness and damping forces which act on the mass, in addition to those that act on the receiver. The net result is that the dynamic displacement of the mass is as if it were 'floating' in free space. Combining equations (7.2.3) and (7.2.11) then shows that

fs =

(K + jcoC) ooZM fp.

(7.2.12)

This in turn can be written non-dimensionally as fs =

1 +2j~ ~2

fp"

(7.2.13)

The modulus of the ratio fs and fp is plotted in Fig. 7.2. The clear difference between the magnitude of the secondary force required in this case and that required when cancelling at the receiver is that in the region of the mounted natural frequency I L l ' - I L l . Thus the required secondary force is much less than that necessary to cancel the receiver motion at the natural frequency of the system. As far as the practical application of active control is concemed, the use of an actuator in parallel with a passive isolation stage could have distinct advantages. In a given application, if an actuator can be found that provides a secondary force of the order of the primary force, then it may be possible to use a much higher mounted natural frequency associated with the passive isolation stage than would otherwise be possible. This in turn has advantages with regard to the stability of the mounted machine; 'soft' passive isolators designed to give a low mounted natural frequency cause problems with, for

190

ACTIVE CONTROL OF VIBRATION

example, the alignment of shafts with other machines. Such stability requirements are usually met by increasing the mass of the machine with a supplementary 'inertia base'; this is very often an undesirable increase in mass.

7.3

Vibration isolation from a flexible receiver; the effects of secondary force location

One of the inadequacies of the simple models presented in the last section is that the secondary force applied to the receiver is assumed to be coincident with the point of application of the force applied via the spring and damper. In real active isolation systems, there will inevitably be some mismatch between the point (or area) of application of the primary force (via a passive isolator for example) and the point (or area) of application of the secondary force. As a first step towards evaluating the effect of this, here we examine the problem of a single point secondary force fs separated by a distance d from a point primary force fp, both forces being applied transverse to a thin infinite plate (Fig. 7.3). We calculate the secondary force necessary to ensure that the total vibrational power input to the plate is minimised. This analysis shows that the separation distance between the two forces determines a high frequency limit above which active control cannot be globally effective. Here we follow the analysis presented by Jenkins (1989) and Jenkins et al. (1993). First note that the complex transfer mobility relating the transverse velocity of an infinite thin plate to the transverse applied force at a frequency to can be written as o)

(7.3.1)

Y(r) = 8Dk} [H(°Z)(klr)- H(°Z)(-jklr)]

which is the result given by Cremer and Heckl (1988), where H(02)denotes the Hankel function of the second kind. Here k/is the wavenumber and D is the plate bending

OO

OO

OO

Fig. 7.3 A point primary force and a point secondary force applied to an infinite thin elastic plate.

ACTIVE ISOLATIONOF VIBRATIONS

191

stiffness given by Eh3/12(1 - v:), where E is Young's modulus, h is the plate thickness and v is Poisson's ratio. The wavenumber kI of flexural waves propagating in the plate is given by k}= ~ : p h / D , where p is the plate density as derived in Chapter 2. The power input to the plate by a harmonic point force f can be written as II = ~1 Re{f*ja~wR},

(7.3.2)

where joowR is the complex transverse velocity of the plate at the position of application of the force. The total power output of the force pair can be written as the sum of the power outputs of the primary and secondary forces. Each force will produce a velocity at its own point of application (determined by the input mobility of the plate at that point) and at the point of application of the other force. Thus we can write for the total power input 1

H = 7 R e { f * [ f p Y ( O ) + f s Y ( d ) l + f * [fsY(0)+fpY(d)l},

(7.3.3)

where Y(0) is the input mobility of the plate and Y(d) is the transfer mobility which determines the complex velocity produced at a radial distance d from a point force. Of course equation (7.3.3) assumes that the structure is linear such that the velocity fields produced by the two forces can simply be superposed. Equation (7.3.3) can be written as

ri = Ifsl 2'7 Re{ Y(0) } + 71Re{f*fsY(d) +fs*fpY(d)} + Ifpl 2'~ Re{ Y(0)}. (7.3.4) Since Re{~f,Y(d) } = ½ rearranged to show that

ri =

1

fsr(a) +

(a)l, the second term in this equation can be

rlfsl 2 Re{Y(0)} +fs* R e l Y ( d ) } f p + f * Re{Y(d)}fs + ILl 2 Re{Y(0)}l,

(7.3.5)

which is a quadratic function of the complex force fs having the form (7.3.6)

rI = A lf s l2 + f * b + b *f s + c.

The

parameters

in

this

equation

are

1

A=sRe{Y(0)},

1

b=sRe{Y(d)}fp

and

c = =1l L I 2Re{Y(0)}. This expression for the total power output has a well-defined minimum value II 0 = c - I b l 2 / A that is associated with an optimal secondary force given by f,o = - b / A . (See Nelson et al., 1987b, or Nelson and Elliott, 1992, for a full description.) Since it follows from equation (7.3.1) that Re{Y(0)} =oo/8Bk 3 and Re{ Y(d)} = Y(O)Jo(kfl) where J0 is a zeroth-order Bessel function of the first kind (Abramowitz and Stegun, 1972), then the expressions for the minimum power output and optimal secondary force input become 1-I0

(7.3.7a)

- 1 - j2o(kld ),

lip

Lo - - = -Jo(kld ).

.~

(7.3.7b)

The results have been expressed non-dimensionally where lip = ILI2,o/leOk 2 is the power input to the plate by the primary force in the absence of the secondary force. These results are depicted graphically in Fig. 7.4. This shows that appreciable

192

ACTIVE CONTROLOF VIBRATION I I I I I

fso

d=3Z/8

,,l

2~

d-- 3~,/8

Z

2~,

10 log 10 //o" (dB)

0

Radial distance, d

Fig. 7.4 (a) The optimal secondary force input required to minimise the total power output of the force pair illustrated in Fig. 7.3. (b) The maximum possible reduction in power output as a function of the distance d relative to the flexural wavelength in the plate at the frequency of excitation. reductions in total power output can only be achieved if the secondary force is applied at a distance well within 3;t/8 from the primary force, where 2 is the flexural wavelength in the receiving structure at the frequency of interest. This result follows since the first zero of the Bessel function depicted in Fig. 7.4 occurs at kid ~ 3 Jr~4. As an illustration of the practical implications of this result, Table 7.1 shows the primary/secondary force separation distances necessary to achieve given levels of power input reduction when two point forces are applied to an (infinite) steel plate of 2 mm thickness. This shows, for example that 10 dB reduction in power output would only be achieved for frequencies less than 100 Hz if the two forces are separated by about 30 mm. Thus the misalignment between the points of application of the primary and secondary forces imposes quite severe limitations on the frequency range of applicability of active techniques. Inevitably success in this type of problem is restricted to low frequencies.

193

ACTIVE ISOLATION OF VIBRATIONS

Table 7.1 Primary/secondary force separations for a specified input power reduction. Primary/secondary force separation (mm) for a reduction in power flow of Frequency (Hz) 50 100 200 500 1000

2 = 2zt/k I (mm)

10 dB

5 dB

0 dB

625 442 313 197 139

45 32 22 14 10

84 59 42 26 19

239 169 119 75 53

7.4 Active isolation of periodic vibrations using multiple secondary force inputs It is also inevitable in practical installations that rotating or reciprocating machinery will be mounted off flexible receiving structures via a multiplicity of mounting points. Machinery of this type, depending upon the application, is usually mounted via typically four, six or eight passive isolators. The analysis presented in Section 7.2 gives some indication of the benefits and disadvantages of applying secondary control force inputs at various locations in such an installation. Here we introduce the analytical framework appropriate to the analysis of active isolation systems where multiple secondary forces are used in order to apply control to the system. The general problem is illustrated in Fig. 7.5. The typical mounted machine constitutes an MDOF system, potentially having three translational and three rotational modes of motion. The vibration of MDOF systems has been discussed in Chapter 1. Thus, for example, as illustrated in Fig. 7.6, control could be applied 'at the source' by using six appropriately oriented inertial actuators in order to synthesise three translational and three rotational dynamic inputs. These would directly apply the requisite secondary forces and moments in order to exactly cancel the primary forces and moments generated by the machine itself. A similar problem in principle exists if control is to be applied at the position of the mounts; as illustrated in Fig. 7.5(b), at any given mounting point the machine is capable of applying primary excitation to the receiving structure via all six forms of input. One approach to this problem is that suggested by Ross et al. (1988) who advocate the use of 'an intermediate structure' in a two-stage isolator, all degrees of freedom of the intermediate structure being controlled by secondary inertial inputs (Fig. 7.7a). Another approach is that suggested by Smith and Chaplin (1983) who used passive isolators, which were very compliant to sheafing motions, in parallel with electrodynamic actuators at each mounting point, which acted to control only vertical motion. A further approach, which was used with some success by Jenkins et al. (1991), employs a pneumatic isolator which can in principle only transmit a normal stress to the receiving structure. The pneumatic isolator is in turn controlled by only a single translational actuator which is capable of controlling the pressure fluctuations

194

ACTIVE CONTROL OF VIBRATION

I Vertical ~.~

(a)

Twisting

i

I

i

Machine /

,

fo fo

f

f

of°

f o r ° f o f ° - -

"

Raft

Z Longitudinal

Pitching ,

(b)

& />---"l

Lateral

|

i Fig. 7.5 (a) The six modes of motion that can be excited by a vibrating machine. (b) The same types of motion are possible in an isolator that is compliant in all three directions.

Actuators

Fig. 7.6 Active control of primary excitation by acting on the machine to cancel excitation in six degrees of freedom. The use of inertial actuators is assumed.

ACTIVE ISOLATIONOF VIBRATIONS

195

within the isolator and through this controlling the transmitted normal stress. This is illustrated in Fig. 7.7(b). Finally, Fig. 7.7(c) shows the arrangement presented by Staple (1989) for use on helicopter vibration control systems. Irrespective of the mode of application and number of secondary force inputs, systems designed to control periodic machinery vibrations most conveniently employ a feedforward control approach as discussed in detail in Chapter 4. It is interesting to note however, that early approaches to this type of problem (see Calcaterra and Schubert, 1968) used a feedback approach with narrow-band filters in the control loop, the filters being tuned to the excitation frequency and its harmonics. Sievers and von Flotow (1990) have discussed the relationships between the various approaches to controlling periodic vibrations. However, in using a feedforward approach, one is generally faced with adjusting the magnitude and phase of a multiplicity of secondary force inputs in order to minimise some appropriate quadratic cost function. One such cost function is the total vibrational energy (kinetic or strain) in the receiving structure. Inevitably such a function must be approximated by undertaking measurements at discrete locations distributed over the receiver (a car body or helicopter passenger cabin, for example). The analysis of such systems is straightforward, again assuming linear behaviour of the structural response and the applicability of the superposition principle. At a given frequency, the displacement of the receiving structure can be characterised by the complex vector w of order N whose elements are the complex displacement at N points on the receiving structure. Clearly as N becomes larger one obtains a better approximation to the global response of the structure as long as the sensors are appropriately distributed. The displacement at each point can be considered to consist of a superposition of the displacements produced by the primary excitation and those produced by the secondary actuators. One can thus write W = Wp "+"

Rf,,

(7.4.1)

where Wp is the vector of complex displacements produced by the primary excitation and the product Rf, defines the complex vector of displacements produced by M secondary force inputs. Thus R is the N x M complex receptance matrix characterising the response at the N points considered due to the application of the M secondary forces, these in turn being characterised by the complex vector Is. If we now choose the cost function for minimisation to be the sum of the squared displacements at N positions defined by N

J-

Iw~ - w w,

(7.4.2)

n=l

then substitution of equation (7.4.1) shows that J may be written as J = fHRHRf~ + fHRHWp+ wpHRf~+ WpHWp.

(7.4.3)

This function can be written as the Hermitian quadratic form J - f~Af~ + fnb + bnf~ + c,

(7.4.4)

where the matrix A - R HR, the complex vector b= RHWp and the scalar constant c - WHWpis the sum of the squared displacements due to the primary excitation alone. As described by Nelson et al. (1987b) and dealt with in Chapter 4, a function of this

196

ACTWE CONTROLOF VIBRATION

type has a unique global minimum Jo associated with an optimal vector of secondary force inputs f~o. These are defined by J0 = c - bHA-lb,

f,o = - A - l b

(7.4.5a,b)

The unique minimum in the function is assured provided that the matrix A is positive definite. This is ensured in this case since f~Af, defines the sum of the squared displacements due to the secondary force inputs alone. This must therefore be greater

Vibrating I-1

assiveisol.ator (a)

Secondary actuator

,P' .eceiver Electrodynamic secondary force actuator

Intermediate plate Pneumatic mount

K

J

Source raft (b)

• Passive isolation

Receiver

Fig. 7.7 (a) The active mounting configuration suggested by Ross et al. (1988) for control of transmission in multiple degrees of freedom. (b) The active mounting configuration used by Jenkins et al. (1991) to enable control to be applied in only the vertical direction at the machinery mounting point. (c) The mounting arrangement described by Staple (1989) for the application of control to the interface between the fuselage of a helicopter and the raft supporting the engine and gearbox. Note that the elastomer (shown hatched) provides a stiffness giving passive isolation between raft and fuselage. The additional oscillatory input provided by the hydraulic actuator acts in parallel with this stiffness to provide isolation in the manner illustrated in Fig. 7.1 (c).

ACTIVE ISOLATIONOF VIBRATIONS

~

~

197

Raft

attachment

Primary loadpath (c)

Elastomer Fuselage~ attachment " ~

I

I_oscillatory Additional input

Actuator

Fig. 7.7 Continued. than zero for all non-zero f r Note that equation (7.4.4) is the vector equivalent of the scalar quadratic function of a complex variable given by equation (7.3.6). The objective of a feedforward controller in this case is therefore to adjust the complex secondary force inputs in order to minimise this function. As dealt with in Chapter 4, the case with which this can be achieved will to a large extent be determined by the conditioning of the matrix A. In addition to this, the cost function defined above can be made more sophisticated and include terms which also penalise the 'effort' (fHf s for example) used in achieving the minimisation. The inclusion of such terms may help in making the problem better conditioned. Bound up with the conditioning problem is the number and location of secondary force inputs used and the number of sensors used on the receiving structure in order to adequately represent its response. As a first step towards evaluating the influence of these factors on the success of control, a finite element analysis of active control applied to an MDOF system will be described in the next section. An introduction to finite element analysis of distributed elastic systems can be found in Chapter 1.

7.5

Finite element analysis of an active system for the isolation of periodic vibrations

Jenkins et al. (1993, see also Jenkins, 1989) have undertaken a detailed investigation of a specific active isolation system designed to be representative of a typical

198

ACTIVE CONTROLOF VIBRATION

machinery isolation problem. The problem analysed is one of a rigid thick plate mounted via four passive isolators onto a flexible receiving structure in the form of a clamped thin plate. Laboratory experiments were undertaken on the system depicted in Figs 7.8(a) and (b). Note that the passive isolators (in the form of foam rubber cylinders) are supplemented by electrodynamically generated secondary forces which effectively act in parallel with the passive isolators (as illustrated schematically in Fig. 7.1 (c)). The equivalent finite element model is illustrated in Fig. 7.9. In the real system the plate and raft have dimensions of 0.99 m x0.66 m x0.002 m and 0.22 m x 0.22 m x 0.01 m respectively. The foam rubber isolators are of outer radius 0.04 m and inner radius 0.025 m and each have a vertical stiffness of approximately 14.4 x 103 N m -~. The fundamental vertical mass/spring resonance of the system occurs at approximately 14 Hz. The thin receiver plate was constructed from mild steel whilst the raft was aluminium.

Primaryforcegeneratingunit (coil and magnet) Force signal / Acceleration ~ - ~~ s~ ~i ~g. ~n a l ~._~ SecondaryforceacutaO tS ras) g

Clamping

'

It

ii

I

Receiving structure Foam"cylinder"isolator Sensorinputs(accelerometers) to the controlsystem "Measuring" accelerometer (a)

Sensorinputs / ...~ Clamping

Primaryforce actuator

I~")

Q

Plate Secondaryforce actuator

(b) Fig. 7.8 The experimental active isolation system described by Jenkins view; (b) top view.

et al.

(1993): (a) side

ACTIVE ISOLATION OF VIBRATIONS

Central primarye x c ~

Spring/damperelements

199

Off-centralprimaryexcitation ,,

,,


,

m_...~ Wavenumber,kx 8

Fig. 8.5 A typical wavenumber spectrum of a vibrating plate and the identification of radiating wavenumber components (after Fahy, 1985). delta function at the origin. This reduces the condition of equation (8.3.17) to k ~ +kx. The peak in the spectrum is given by kx = m~r/a. However, as we have seen in equation (8.3.18), only those values of I kxl k, the acoustic wavenumber, will contribute to acoustic power radiation (for the one-dimensional case). This range of wavenumbers is shown cross-hatched, for example, in Fig. 8.5. Physically speaking, if one associates with the wavenumber kx, a phase speed of propagation Cx, such that kx = cO/Cx then the condition that ] kx[ >-k amounts to requiring that ] c~ ] ~ 1, the effect of the mutual radiation term decreases since each of the modes radiates independently and with approximately equal efficiency. 101 100

°11,11

10-1 _ >:,

o

¢-.

1

0.2

-

°11,31

(D

i_~

10 -3 -

,-.

10-4 _

o

1 0-5 tr'

°31,31 °21,21

10_ 6 10-7 _ 10-8 10-1

. . . .

I

100

.

.

.

.

.

.

,,I

101

i

|

|

,

,

,

,,

103

Non-dimensional frequency, ka

Fig. 8.7 The radiation efficiencies of selected structural modes on a rectangular plate. The self-radiation efficiency of the (1, 1), (1,2) and (3, 1) modes are designated al~.~, crzl.z~, cr3~.31 and mutual radiation efficiency of the (1, 1) and the (3, 1) modes is designated 0~.3~.

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATESYSTEMS

239

It is possible to define a set of velocity distributions on the plate which do radiate sound independently at any one frequency. Consider the eigenvalue/eigenvector decomposition of the matrix M in equation (8.4.6): M = pT~p,

(8.4.15)

where, since M is symmetric and positive definite, P is a real unitary matrix of eigenvectors and f2 is a diagonal matrix of positive real eigenvalues. The power radiated by the plate, equation (8.4.6), can now be written as II = ";vHPTf2P-;v= bHf~b,

(8.4.16)

where b = Pw is the set of structural mode amplitudes transformed by the eigenvectors of the radiation resistance matrix. Since fl is diagonal, equation (8.4.16) can be written as N

I-[ = ~

(8.4.17)

f ~ . l b . I 2,

n=0

so that the velocity distributions corresponding to this transformed set of velocity distributions radiate sound independently. The existence of this set of velocity distributions has been described by Borgiotti (1990) and Cunefare and Koopman (1991), and is implicit in the work of Baumann et al. (1991). They have been termed radiation modes by Elliott and Johnson (1993), who have plotted out their radiation efficiencies, proportional to the f2,'s, for the plate described above, as shown in Fig. 8.8. At low frequencies, ka ~ 1, the first-order (1) radiation mode is much more efficient than those labelled (2) and (3), and generally the variation of their radiation efficiency with frequency is simpler than that for the radiation efficiency of the structural modes (Fig. 8.7). The velocity distributions corresponding to these theoretical radiation modes are weak functions of the excitation frequencies. At low frequencies, ka ~ 1, however, the velocity distributions are almost independent of excitation frequency, and for the plate considered here are plotted in Fig. 8.9 (Elliott and Johnson, 1993). 101 10 0 I~

10-1

i

I

i

I

i

i

i

i

i

i

i

I

i

i

i

i

i i

i

Radiatiom n o d e ~

10"2

~

0.3

.E_O

10 .4

.(1:1 "0

10-5

rr

10..6

(3) (4)

10 .7 10 .8 10-1

i

10 °

I

I

I

I

I

I II

I

101

i

i J J i ii102

Non-dimensional frequency, ka

Fig. 8.8 The radiation efficiencies of the first six radiation modes of the rectangular plate.

240

ACTIVE CONTROL OF VIBRATION

velocity, ~,

(1)

(2)

(3)

(4)

(5)

(6) Fig. 8.9 The velocity distributions corresponding to the first six radiation modes of the rectangular plate at an excitation frequency corresponding to k a = O. 1. The most efficiently radiating velocity distribution (1) clearly corresponds to the net volume displacement of the plate, as expected. The advantage of the radiation mode approach is that, by using Fig. 8.8, it allows one to quantify the extent to which other velocity distributions are also significantly radiating at any one excitation frequency. It also suggests an efficient method of sensing the velocity distributions of the plate which are most important in radiating sound, as will be discussed in Section 8.8.

8.5

General analysis of Active Structural Acoustic Control (ASAC) for plate systems

In this section we outline the basic analysis of ASAC applied to plate systems. The system studied consists of a simply supported rectangular plate positioned in an infinite baffle as shown in Fig. 8.10. Two different disturbances to the system are considered as

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS

Incident pressure,Pi

241

/,,

Iii

// ~ t e d ~

pressure Pr

Y

,J v!

b2f - I - Uniforml_~ pressure

Point liD'-force

I I I I

b

y

Piezoelectricpatch

I I I I I

-

--~Ye

i

a2

x2

(c)

Fig. 8.10 Coordinate system and arrangement of a baffled rectangular plate with inputs (a) on the incident wave side, (b) on the radiated wave side and (c) configurations of the input forces. examples. Firstly, an acoustic plane wave incident on the plate at an oblique angle is taken as the noise input shown in Fig. 8.10(a). Secondly, a forcing function over a small localised area on the plate is introduced that is considered to be representative of a structural disturbance (in the limit this forcing function will be a point force) shown in Fig. 8.10(c). Both of these disturbances will excite the plate into motion resulting in radiation on the transmitted half plane of the radiated field shown in Fig. 8.10(b). As discussed in Chapter 1 and Section 8.2 the form of the disturbance has an important influence on the resultant plate motion and thus the sound field to which it is coupled. When the disturbance is a low frequency plane wave, the input phase distribution is relatively constant over the plate surface and the result is that the plate response is dominated by lower order modes that are often excited well above their resonance

242

ACTIVE CONTROLOF VIBRATION

frequencies. This behaviour leads to an overall vibration pattern with fewer nodal lines and a high radiation efficiency. For a localised structural input, the plate responds with a much richer modal distribution and higher order modes tend to dominate near their resonance frequencies. In this case at higher frequencies (however, still below the plate critical frequency) the overall radiation efficiency is lower. Thus, as demonstrated by McGary (1988), airborne disturbance inputs to structural systems will tend to radiate/ transmit higher levels of sound than structural inputs for the same frequency and total magnitude of load. In the following analysis a general procedure is outlined. The inputs are assumed to be at single frequencies and all systems (structural, acoustic and electrical) are assumed linear so that superposition of response holds. The type, number and location of the control transducers is assumed known. In this section the fluid loading is assumed small (e.g. as in air) hence the plate response can be determined using the inv a c u o equations described in Chapter 2. The following steps are taken in the general ASAC analysis: (1) An expression is derived for the response of the plate to the input or primary disturbance(s). (2) An expression is derived for the coupled radiation to the far-field from the plate due to the disturbance. (3) An expression is derived for the response of the plate to the multiple structural control or secondary inputs. (4) An expression is derived for the far-field radiation from the plate due to the structural control inputs. (5) The total far-field pressure response is found by superimposing the disturbance and control fields. (6) A quadratic cost function is formed that is based on the required observed radiated pressure field variables.t (7) Quadratic optimisation theory is used to find the optimal control inputs that minimise the cost function as outlined in Section 4.6. (The input disturbance(s) is assumed constant and known.) (8) The optimal control inputs are substituted into the relations for the total field response (i.e. far-field pressure, plate out-of-plane displacement etc.) in order to evaluate the control performance. (Note that the reduction in the cost function also provides a measure of the overall control performance in terms of the observed error variables.) Before we carry out the above steps, we first derive the basic system response equations. Note that as outlined by Pan et al. (1992b) the method can also be formulated by using transfer matrices written in terms of input and radiation impedances. In addition, as described in Section 8.8, design techniques for optimally shaping and locating the control actuators and sensors are available. Figure 8.10 shows the arrangement and coordinate system of the baffled, simply supported, rectangular thin plate excited by a harmonic disturbance pressure acting over an area of the plate. To calculate the radiated sound field, a description is required of

t One could also minimise a pressure related variable such as supersonic wavenumbercomponents.

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

243

the plate complex vibration distribution. For the simply supported thin plate, the displacement distribution is given by equation (2.11.2) in modal form as oo

oo

W(X,y, t) "- Z Z wmn sin kmx sin kny e i~t,

(8.5.1)

m=ln=l

where the eigenvalues are given by km

met =

~ ,

a

kn =

nJr b

.

(8.5.2a,b)

The plate modal amplitudes for various forms and distributions of input forcing functions have been calculated by Wang and Fuller (1991) and are given by Wren "-

Pro. ph(ooZm~-co 2)

,

(8.5.3)

where 09 are the natural frequencies, p the plate density and h is the plate thickness. The modal force, Pmn, due to the input disturbance depends upon an exact description of the external load. Gu and Fuller (1993) have studied active control of sound radiation from a plate in the presence of a heavy fluid. However, for the present discussion we limit the analysis to a light fluid loading such as air and thus radiation loading effects are ignored. Modal forces for various input disturbances acting upon an in-vacuo simply supported plate have been derived by Wang and Fuller (1991) as follows. (1) Uniformly distributed pressure. For a uniformly distributed pressure with amplitude, Q, located between coordinates a~, a2 and b~, b2, as shown in Fig. 8.10(c), the modal force is given by mn

Pmd =

4Q

m nx~ 2

(cos kma 1 -

cos

kma2)(cos knbl -

cos

knb2),

(8.5.4)

where the superscript d will hereafter denote the input disturbance. (2) Obliquely incident plane wave. An obliquely incident plane wave as shown in Fig. 8.10(a) can be described by pi(x, y, t) = Pie j°~t-jksin °i

cos q,,-jksin 0, sin ~i.

(8.5.5)

Again assuming that the fluid loading is light we can calculate the total input pressure at the plate surface from the plate blocked pressure. That is, we assume total reflection of incident waves and thus the total input pressure at the plate surface is twice the incident pressure. Using such an approach the modal force for an oblique incident plane wave has been calculated first by Roussos (1985) and later by Wang and Fuller (1991) and is given by Pare.= 8Pilmln where the coupling constants are given by

-j [m = 2

sgn(sin Oi cos q~i)

if (met)2 = [sin 0icos ~ b i ( o ) a / c ) ] 2,

(8.5.6)

244

ACTIVE CONTROL OF VIBRATION

or

mn{ 1 - ( - 1 ) m exp[-j sin Oi cos ~i(toa/c)} I n --

(mn) 2 - [sin Oi cos ~i(wa/c)] 2 if (mn) 2~ [sin Oi cos q~i(toa/c)] 2

(8.5.7a,b)

and -J sgn(sin Oi sin (])i)

rn- 2

if (n~:)2 = [sin Oi sin q~i(~ob/c)] 2, or

L-

nn{ 1 - (-1)n exp[-j sin 0 i sin qbi(wb/c)] } (net) 2- [sin Oi sin ~ i ( t o b / c ) ]

2

if (nn)2~ [sin Oi sin ~i(tob/c)] 2.

(8.5.Sa,b)

(3) Piezoelectric actuator. As discussed in Chapter 5 a rectangular piezoelectric actuator configured for pure bending is considered to consist of two wafer elements located symmetrically on each side of the plate and driven 180 ° out of phase in the d31 mode. The modal force for such a piezoelectric actuator as shown in Fig. 8.10(c) is given by Wang and Fuller (1991) as P C m = 4C°l~Pe mn~2

(kZm+

k2n)(cos k m x 1 - c o s k m x z ) ( C O S

knYl -

cos knY2)

(8.5.9)

where x~, x2, and Yl, Y2 are the coordinates of the actuator edges. The parameter Co is a constant that is function of the piezoelectric actuator/plate properties and geometries specified by C o - E b I K I, where K I is given by equation (5.6.8). The unconstrained strain, epe, of the piezoelectric element is defined by d31V l?,pe -- ~ ,

(8.5.10)

ha where d31 is the piezoelectric transverse strain constant and h a is the piezoelectric element thickness. Most importantly Epe and thus Pm~ is seen to be linearly related to V, the input complex voltage to the actuator. (4) Point force. The modal force associated with excitation by a harmonic point force of amplitude F located at (Xr, Yr) as shown in Fig. 8.10(c) is pc _

4F

ab

sin k mXf sin k,y I,

(8.5.11 )

where the superscript c denotes control force. The sound radiation caused by the vibration due to the above inputs is related to the plate velocity distribution. As outlined in Sections 8.2 and 8.3 the radiated pressure can

ACTIVI~ STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTtIM$

245

be evaluated by using the Rayleigh integral or spatially Fourier transformed variables in conjunction with the method of stationary phase. As demonstrated by Junger and Feit (1986), both mathods yield identical solutions. Roussos (1985) has derived e×pressions for the radiation from a rectangular plate using the Rayleigh integral. The radiated field shown in Fig. 8.10 (b) is given by eo

e@

p(R, 0, q~)- K Z Z

(8.5.12)

Wm, lml, e m,

m-'-I n=-I

where the radiation constant K is defined as

1[

K - =topoab exp ]to t . . . .

2r~R

c

]}

(a cos ~ + b sin ~) . 2c

(8.5.13)

The coupling constants I,, and I, for the radiated field ere identical to equations (8.5.7a,b) and (8.5.8a,b) except that the coordinate angles (0i, 'hi) am replaced by (0, ,)) which define the coordinate of the observation point in the radiated fieldand the sign in the argument of the exponential function is changed to minus. We now have all the necessary components in hand to derive the response of the total radiated field,i.e.the disturbance field (also called primary) plus the control field (also called secondary). For a feedforward control arrangement the disturbance source and the control actuators act simultaneously and am assumed Imrfectly coherent. By superposition the totalcomplex pressure can be written as p, = pp + p,,

(8.5.14)

where the subscripts t, p and s refer to total, disturbance and control pressure respectively. Using the previous relations the total pressure in the far field can be written for the following configurations. Case I. Plane wave disturbance = point control forces u~ p, = P~B + ~ F;Cj,

(8.5.15)

/-=1

where N~ is the total number of control forces, while P~ and Fj are the complex amplitudes of the disturbance wave and control forces respectively. Case I1. Localised structural disturbance = point control forces #,

p, = O.B + Z r;cj,

(s.5.16)

/=-1

where Q is the amplitude of the disturbance pressure acting over the small area specified previously. Case III. Plane wave disturbance -- piezoelectric actuators N, P,

= PIB + ~ VjCj, /=-1

where Vj is the complex control voltage applied to the jth actuator.

(8.5.17)

246

ACTIVE CONTROLOF VIBRATION

Case IV. Localised structural disturbance - piezoelectric actuators

s pt= QB + Z V~Cj, j=l

(8.5.18)

where Q is the amplitude of the disturbance pressure. In the above equations the transfer functions are defined as: plane wave disturbance, B - K

-~

ImI~,

(8.5.19a)

Imln,

(8.5.19b)

localised structural disturbance, B =K

= = y

where the appropriate form of WPmnis calculated using equation (8.5.3) with either equation (8.5.6) or equation (8.5.4) respectively; point control force, =

= -~j

Iml.,

(8.5.20a)

=

__ - ~ j

ImIn'

(8.5.20b)

piezoelectric control actuator, Cj=K

where the appropriate form of Wm~njis calculated using equation (8.5.3) with equation (8.5.11) and equation (8.5.9) respectively. In the above transfer functions, B and C are normalised with respect to their appropriate forcing function amplitudes in order to put the relationships for total response in a form which can more readily be manipulated in terms of control amplitudes. Thus in effect, B, for example, will represent a complex transfer function between the specified input pressure of an obliquely incident wave and its corresponding radiation pressure at some observation angle in the far field. Control of the disturbance field can be achieved by appropriately choosing the vector of point force amplitudes f~= [F~ F~ F~ ...]T or piezoelectric voltages v~= [V~ V~ V~... ]T in order to minimise a chosen cost function. Choice of the cost function is defined by the form of control performance required. The ideal cost function for globalt control of sound radiation is the total acoustic power radiated by the plate since this is the variable that we seek to reduce. In this case the cost function is defined as the integral of intensity flowing through a hemisphere surrounding the plate which is given by J =

1 f

Ip, 12dS

2p0c0 ' s t Here global means through an extended area or volume.

(8.5.21)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

247

and which can be described in polar coordinates as J =

2p0c0 ' 0 J0

IP,

sin 0 dO dq~.

(8.5.22)

If directional control is required then equation (8.5.22) can be modified to only attenuate power over a required directional sector and is thus written J =

1

fo2f¢2 ]p 12R2 sin 0 dO d~b, 2 p o c o ' o, ~,, '

(8.5.23)

where 01, 02, q~l, q~2 define the limits of the sector. Usually it is not possible to design a sensor such that a cost function of the form of equation (8.5.21) can be measured, and in practice microphones are often used as point error sensors. In this case the cost function is written in discrete form as Ne

1

J=

~ [pI(R~, 0 i, ~)i)12,

(8.5.24)

2poco i= 1 where N e is the number of error sensors and (Ri, Oi, ~)i) is the location of the ith error sensor microphone. Equation (8.5.24) can be written in a more accurate discrete form as 1

J =

N~

~ Ipll = A S i,

(8.5.25)

2p0c0 i= 1 where A S i is the projected area associated with the jth sensor on the hemispherical surface. In practice, however equation (8.5.24) provides a cost function which is a reasonable estimate proportional to the total radiated power as long as a suitable geometry of error sensors are chosen. Choice of number and location of error sensors is an important topic of research and will be briefly discussed later. In general, for global control, the number of point error sensors required is equal to the number of modes contributing to the overall sound power radiation. The error sensors, by analogy with time sampling theory (see, for example, Nelson and Elliott, 1992, Ch. 2), should be spaced in sectors of the radiated field defined by regions of 180 ° phase change through nodal lines in the radiation field associated with that mode. For example, Fig. 8.11 shows a hypothetical low frequency, rectangular plate, radiation pattern which is a combination of radiation from the (1, 1) and (2, 1) modes. Positioning a single error sensor at location (a) will only lead to control of the (1, 1) mode since this angle corresponds to a node in the (2, 1) radiation pattern. Positioning two sensors at locations (b) will not lead to global control since the (1, 1) and (2, 1) modes will cancel at the error sensors and reinforce in the other sector. Locations (c) shows a situation where two sensors are positioned in two sectors which are 180 ° out of phase for the (2, 1) mode. When used with two actuators, the only way the controller can attenuate the error signals is by completely attenuating each mode, thus leading to global control. Another important observation regarding the choice of point sensors is that generally the number of sensors should be equal to or greater than the number of control inputs used. If the number of error sensors equals the number of actuators then total

248

ACTIVE CONTROL OF VIBRATION

~mode

1

mode 2 + (a)

0 ensor

(b)

e error sensor

+ (c)

Fig. 8.11 Influence of error sensor position on modal radiation control. attenuation is theoretically achieved at each error sensor since the system is perfectly determined. This situation generally leads to poor attenuations at other positions especially if the number of degrees of freedom in the system exceeds the number of error sensors. If the number of control actuators used is greater than the number of error sensors then the system is underdetermined. Note, however, that Elliott and Rex (1992) have presented a methodology to ensure well-conditioned relations for underdetermined systems by introducing a control effort term into the cost function. It should again be stressed here that in ASAC, although the control action is applied directly to the structure, the cost function is derived from the far-field radiated pressure (or far-field radiated pressure-related variables). Thus, inherent in the definition of the cost function, is the natural structural acoustic coupling that relates the plate vibration to the radiated sound. This arrangement should be contrasted to the more obvious

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

249

approach in which plate vibration is directly observed and minimised. Naturally, completely reducing overall plate vibration will lead to a reduction in sound radiation. However, as will be demonstrated, this latter approach generally requires many more channels of control (i.e. number of sensors and actuators) and a much more subtle and efficient control paradigm can be implemented when far-field pressure is used as an error variable. One is thus directly observing the field variable to be controlled, which in this case is radiated sound. The following analysis for derivation of the vector of optimal control inputs is for the ideal case of minimising total radiated power and thus equation (8.5.21) is used to define the cost function. Although the following derivation is formulated for the above Case IV, it can be written in exactly the same form for the other three cases with appropriate substitution of variables. When the expression for total pressure Pt from equation (8.5.18) is substituted into the cost function definition, equation (8.5.21), it can be demonstrated that the cost function is a scalar which is quadratic in the vector of complex control voltages v, or control force amplitudes, f,. The reader is referred to the Appendix of Nelson and Elliott (1992), as well as Chapter 4 of this text, for a proof and discussion on the nature and minimum value of the quadratic cost function. It can be shown that this cost function will possess a unique minimum which will define the optimal control voltages. The minimisation procedure is based upon the setting the gradient of the cost function J with respect to the control vector v, to zero in order to find the stationary point of the quadratic form. The total complex pressure can be expressed in vector form as p, = hTq + cTv ~,

(8.5.26)

where q=[Q~Q2Q3 ...]v is the vector of complex input disturbances and h = [H~ H2 H3 ...IV is the vector of aforementioned transfer functions associated with those disturbances. For cases in which there is a single input disturbance these vectors reduce to scalars such that q=Q

(8.5.27)

h = H~.

(8.5.28)

and

In equation (8.5.26) the control transfer function vector is defined by

c = [Ci C2... Cu,]"r,

(8.5.29)

while the input control voltages or control forces can also be written in vector form as

v,=

T.

(8.5.30)

Note that a similar vector expression as equations (8.5.26) and (8.5.30) could be used for f,, the vector of complex control force amplitudes. As outlined by Nelson and Elliott (1992) it is convenient to write the cost function using matrices in a Hermitian quadratic form. For a single input disturbance the squared modulus of the total complex pressure can be written IP, [2= p,p,= v~Cv, + v~x + XHV,+

QH,H*~Q*.

(8.5.31)

250

ACTIVE

CONTROL

OF VIBRATION

In equation (8.5.31) the Hermitian matrix C is defined by C = c*c T

(8.5.32)

x = QHlC.

(8.5.33)

and the vector x is given by

The cost function as defined by equation (8.5.22) which is written in terms of total radiated acoustic power can now be specified as J = v HAv, + Vnb + bHv s +

(8.5.34)

C,

where the individual terms are now given by AN, ×U,

=

1 [2~ [,q2 [c.cT]R 2 sin 0 dO dq~ 2poco ' o Jo

1 I2, [,#2 bN'×l= 2p0c0,0 J0

c=

1012

r/o rjo,

(8.5.35)

QHlcR2 sin 0 dO dq~

(8.5.36)

sin 0 d0 d~.

(8.5.37)

2poco '

As discussed in Nelson and Elliott (1992), all the combined terms of equation (8.5.34) are scalars and as long as A is positive definite then there is a unique minimum value of J. A typical element of matrix A is defined by A~=

~. kli~ mnjI kliI mnj1" sin 0 d 0 d q~ RoCo "

k = l I=1 m--1 n--1

i = 1 , N , ; j = 1,N,.

(8.5.38)

A typical element of vector b is oo

oo

oo

1 [2=Ij/2 K1K* Z ~ Z Z QP~tlQP,~,'PktalS£nsR2sin OdO dq), s= 1,Ns.

pOCo '

k = l !=1 m = l n = l

(8.5.39) Finally the single entry of h for a single input disturbance is defined as n 1-

1 f 2=[zr/2K1K~ Z 0 dO poCo ' k=l

Z

Z

P* IP lP* D 2 sin0d0dq~ QpkllQmnl*kll*mnl*~

(8.5.40)

1=1 m = l n = l

where

w;%

QS,,j = - - , V~

wL1

Q~l = ~ . Q

(8.5.41a,b)

Note in the above equations that the quadruple infinite sums in (k, l) and (m, n) result from the multiplication of two double modal series. As discussed in Section 8.4 the cross terms are important and the additional modal indices (k, l) are employed. The

ACTIVE STRUCTURALACOUSTICCONTROL. I PLATE SYSTEMS

251

optimal solution of control strengths to minimise the cost function of equation (8.5.34) can be found using the result derived in the Appendix of Nelson and Elliott (1992) and Chapter 4 which shows that V~o= - A - l b -

(8.5.42)

Equation (8.5.42) thus defines the vector of optimal control voltages V,o for the case of piezoelectric actuators. On obtaining V,o, the minimised far-field radiated pressure can then be calculated using equation (8.5.26). The minimum of the cost function can also be calculated from Jmin = c - bHA-lb.

(8.5.43)

Equation (8.5.43) can be used to calculate the attenuation in total radiated power obtained when the control is invoked as long as enough error sensors are used to provide a reasonable estimate. Note that if there are more control actuators than error sensors then matrix A will be singular and the procedure developed by Elliott and Rex (1992) for underdetermined systems should be used. The reader is also referred to the Appendix of Nelson and Elliott (1992) for discussions on this and other aspects of finding the minimum of quadratic functions. The previous analysis enabling the derivation of the optimal control voltages can readily be applied to the different cases I, II and III outlined above with use of the appropriate variables and using similar methodology. In the next two sections, results from example applications using this analysis will be discussed.

8.6

Active control of sound transmission through a rectangular plate using point force actuators

In this section we study the active control of sound transmission through a rectangular, baffled, simply supported plate using point force actuators. Systems similar to these have been previously investigated by Fuller (1990) and Wang and Fuller (1991). All disturbance frequencies are well below the coincidence frequency of the plate which is fc_~6300 Hz. Table 8.1 presents the specifications of the steel plate, while the corresponding natural frequencies for the simply supported boundary conditions, computed using equation (2.10.3), are given in Table 8.2. The acoustic medium is assumed to be air with P0 = 1.21 kgm -3 and Co = 343 ms -~. To calculate the plate response and the corresponding radiated field it is necessary to truncate the infinite summations. In the following examples, truncating the indices k, l, m and n at a value of five (i.e. 25 modes are considered in the doubly infinite sums) was found to provide close to 0.01% error in the radiated pressure amplitude at the highest frequency considered. This choice of truncation can be seen from Table 8.2 to effectively limit the input disturbance frequency such that f < 1750 Hz. In addition, it Table 8.1

E = 207 x 1 0 9 N m pp = 7870 kg m - 3

Plate specifications. v = 0.292 h = 2 mm

a = 0.38 m b = 0.30 m

252

ACTIVI~CONTROLOF VIBRATION Table 8.2 Naturalfrequencies of the plate (Hz).

m

1

2

3

4

5

1 2 3 4 5

87.71 188.74 357.13 592.88 895.98

249.81 350.85 519.23 754.98 1058.08

519.98 621.02 789.40 1025.15 1328.25

898.22 999.25 1167.64 1403.39 1706.48

1384.53 1485.56 1653.95 1889.69 2192.79

was necessary to calculate the integrals of equations (8.5.35) to (S.5.37) and this was carried out numerically using Simpson's rule. In order to use the above equations for this case, the input control terms are replaced with those for point control forces. The plate response results presented in Sections 8.6 and 8.7 consist of the distribution of plate vibrational amplitude plotted along the ) , - b / 2 horizontal plate mid plane (see Fig. 8.10). The results, presented in decibels (dB), were normalised to the largest amplitude obtained in each figure. Radiation directivity patterns are also presented along the y-b/2 axis at a distance of R - 2 m. Although this observation point is relatively close to the plate, far-field radiation equations were used for simplicity, and thus the results also reflect the behaviour at large distances from the plate. For convenience, angular positions with a negative sign of 0 in the figures correspond to the coordinate ~ - ~ positions. For the following example the input disturbance was assumed to be a plane wave with amplitude P~-1 N m =2 incident at angles of 0~ - 45 ° , ~ - 0 e. Figure 8.12 presents the radiation directivity patterns with and without control when the excitation frequency was set to 186 Hz. Note that negative values of sound pressure level correspond to pressure magnitudes less than the reference pressure of 2 x 10 =s N m =2. From Table 8.2 it is apparent that this frequency is near the resonance frequency of the (2, 1) mode, and correspondingly, the uncontrolled radiation field appears to have the distorted version of the radiation pattern associated with a (2, 1) mode. An examination of the modal contributions confirms that the radiation field is mostly due to the (1, 1), (2, 1) and (3, 1) modes. The slight offset of the node in the radiation field is due to the presence of a monopole type radiator such as the (1, 1) mode. Applying one control force, as shown in the schematic diagram at the top of Fig. 8.12 as a black dot, leads to little reduction in radiated power. Positions of the control forces are shown to scale in the schematic figures. However, it appears that the (1, 1) radiation contribution has been controlled as the residual radiation pattern has now the characteristic dipole shape associated with the (2, 1) mode and this is confirmed by examining the modal contributions. The single, centrally located control force is unable to couple into the (2, 1) mode of the plate as shown in equation (2.11.4). Using two forces leads to a large reduction in radiated power as now the (2, 1) mode is controlled and residual field is now largely due to the (3, 1) mode contribution. On using three point force actuators positioned as shown in the schematics of Fig. 8.12, control is achievable over the (1, 1), (2, 1) and (3, 1) modes, and power reductions of the order of 67 dB are predicted.

ACTIVI~STRUCTURALACOUSTICCONTROL,I PLATI~$YSTISMS Unoontrollti¢l

I Foroti

a

llllllllll=

i1= ,

Forolill

253

3 Forollit

=III=I=I=IIIIUlIIII=I

I

48,7 (dB)

80,8 (¢IB)

I

I

i

01=_48" 0l =- O*

Power rttduotlon

o,e (aa)

5O ....... -46°

0--'0°

""':::,

I

/

/ 0

F

=50

7/

=100

.............

....... ":1:::~ ~

............ I ............. 48'1 .:=;:'"

,.................. ~

(

Illl $

/

",...,' . . . .

~

I

.::;:"......

........................ i)N /

F........"

",~:~2 ... \

[ %1

L( ( ;-; !....]i......... li '2........... (

-50

=50 =I00 -80 Sound prnmum Iovel (d8)

0

0

60

Fig, 8,12 Radiationdirectivity for different numbers of point for, o actuators, f=- 186 Hz,

Curves of the power transmission loss versus frequency for the same situations as above are plotted in Fi$, 8,13, Her~ power transmission loss is defined as (Wang, 1991) Transmission loss-~ 10 log(lli/l=I~),

(8,6,1)

where the incident power is given for an obliquely incident plane wave by the relation

hi--- Ipil~ ab cos 0~

(8.6.2)

2pete

and the radiated power is given by Jo

~,, Ip(r, 0, O)[~ r ~ sin 0 dO dO, o 2pete

(8.6.3)

whore p ( r , 0, 0) is evaluated in the far field usin8 equation (8,$,26) for the cases with and without control. Note that negative values of tr~amission lo~a are a numodoal artifact of the computer calculations; in practice transmission loss cannot be loss than zero, For the primary or uncontrolled case, the transmissiort loss cu~e ha~ a number of dips a~sociatod with the plate modal resontm,os, When one ,ontrol fo~o is u~od, it ,an be soon from Fig, 8,13 that the fall in transmission loss at the (1,1) re~onanco frequency has been eliminated as discussed above, Similarly, use of two control forces eliminates the dips at the (1, 1) and (2, 1) resonance frequencies, The figure shows that attenuation greater than 50 dB can M theoretically obtained over a frequency range of

254

ACTIVE

Uncontrolled

CONTROL

OF VIBRATION

1 Force

2 Forces

3 Forces

.................................

0i==45°~i 0° i 200

I

~'~"1

|

150

~,,,',,,

A

rn "o

o (0

100

m

tO .B

50

.B

.'~..

E

,,.

...........~~:...~ ~

\ :~"'.....

,.,,..~..-..-..-:....-.,

e--

-50

0

I

I

I

I

I

I

I

1O0

200

300

400

500

600

700

800

Frequency (Hz)

Fig. 8.13 Plate transmission loss for different numbers of point force actuators. 0 to 450 Hz with three control forces. Whether this is achieved in practice for broadband, random disturbances depends upon a number of issues such as causality, filter size, etc. as described in Chapter 4. However, these frequency domain results do define the ultimately achievable performance with the limited number of actuators used in the locations specified.

8.7

Active control of structurally radiated sound using multiple piezoelectric actuators; interpretation of behaviour in terms of the spatial wavenumber spectrum

The previous section briefly discussed results of using point force actuators as control inputs. However, there are disadvantages to using point force actuators such as their size and the need for a back reaction support. In this section we discuss a few representative examples of control of structurally radiated noise (i.e. a structural rather than an airborne input) with arrays of independently controlled piezoelectric actuators. The piezoelectric actuators were assumed to be manufactured from ceramic material with typical properties given in Table 5.1 (corresponding to G1195 material). As discussed in Chapter 5 the actuators were configured to produce pure bending in the plate. The system used for the analysis is exactly the same as that presented in the previous section except that in this case the input disturbance is assumed to act over a very small area of 40 x 40 mm approximating a localised structural input. For this case the input

255

ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS

disturbance amplitude was set to Q = 7.9 x 103 N m -~ giving an input force of 12.65 N. In the following figures the prescribed disturbance source and actuator locations and size are shown in schematics at the top of each figure, drawn to scale looking into the plate surface from the radiated field. The black rectangle represents the size and position of the disturbance source while the clear rectangles represent the size and position of the two-dimensional piezoelectric actuators. At this stage no attempt is made to optimally configure the control actuators; their selection is made on an ad hoc basis, linked with a knowledge of the acoustically significant plate modes and their response shapes. For the first case the disturbance frequency is set to 85 Hz. An examination of Table 8.2 reveals that this frequency is close to the resonance frequency of the (1, 1) mode.t Hence this case corresponds to an 'on-resonance' excitation. Figure 8.14 presents the normalised vibration amplitude distribution with and without control for four different configurations of piezoelectric actuators. In Fig. 8.14 the solid line depicts the displacement distribution of the plate only under the influence of the disturbance, and as expected is close to a (1, 1) mode shape. When the various configurations of control (1)

(2)

(3)

(4)

I

I

I

I

rn I1) "O

,m=

-50

..../.--:':~'"~.L.. -'~.-...

E

.:----, :,,

tl:l ¢...

•~

-~00 -

,.,.~,-~... -~

.dI3 >

/

t;:',, '(" #

"

.,

:

',.k/. "-

" ~1

"-,,

&, "':~, . . . . i':_..-:...... - .

"13

I

-150

i"

I,

\

|".f

',

E 0

z

-200

I

I

I

I

I

I

0

0.2

0.4

0.6

0.8

1

x/a

.............

Uncontrolled 1 Piezo

.......................... 2 P i e z o s 3 Piezos .......

4 Piezos

Fig. 8.14 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 85 Hz. 1 Since no damping is included in the model, the disturbance frequency is not set exactly to the resonance frequency.

256

ACTIVE CONTROLOF VIBRATION

actuators are applied, the vibration amplitudes are significantly reduced and the (1, 1) mode is well controlled. However, increasing the number of actuators does not lead to a significant reduction in vibration. (Note the we are using an acoustic cost function in this example, in contrast to the configurations in Chapter 6.) This effect will be discussed below. Figure 8.15 presents the radiation directivities corresponding to the cases shown in Fig. 8.14. Also presented is the total reduction in radiated acoustic power calculated from equation (8.2.7) for each case. As expected the uncontrolled field is uniform and corresponds to the monopole like radiation of the (1, 1) mode. When one control actuator is applied, reductions in radiated power of the order of 60 dB are obtained. Use of two control actuators brings a further 10 dB reduction. Further increasing the number of actuators has little effect on the radiated power until for case (4) when actuators are located off the y = b/2 line (see Fig. 8.10) and a further 30 dB of power reduction is obtained. From these cases, it is apparent that good sound control is achievable with a single actuator and increasing the number of control channels has no significant practical advantage. In general this observation is true for systems on or near resonance of an efficiently radiating structural mode such as the (1, 1) mode considered here. Figure 8.16 presents the wavenumber transform of the plate vibration calculated (1)

(2)

(3)

~5°1 ,,

(4)

I po.0°

i,,.,~° ....

................... "

!t i i :

i I. i/!

o ~ ii £,

-ooo5

i i

.~ :. :~!

i i

'i

!'" !

:

.z i

. :: "

~i

i

i ii'ii

il

i

~..

!,~i

"~

!/i

i i

ii

!

-0.015 ~ 1 / ii ! :'

V

-001:

:-

~,:

i

"

:i

~!

ii

:.

.".,

"~ ---

l

I 0.1

""'.

"

"--

""

- -

-0.02 / 0

~--

..

"'~

Mode 1

~

Mode 2 ..... Mode 3

t 0.2

i 0.3

i 0.4

t 0.5

i 0.6

i 0.7

i 0.8

t 0.9

1

Time (sec)

Fig. 8.30 Velocities of the three structural modes of the beam modelled by Baumann et al. (1991) to demonstrate state feedback control of (a) sound radiation and (b) structural vibration. which results in a feedback law of the form u(t) = KV~bx(t).

(8.11.15)

In the results presented by Baumann et al. (1991), the weighting on the effort term in equation (8.11.14), a z, was adjusted to make the total energy used by the controller

276

ACTIVI~ CONTROL OF VIBRATION 70

~

I

I

I

I

00

=

80

=

40

=

3° /

.10 0

~,

....

i 50

.....

i 100

....

:7

i 160

-,.,, .......

J 200

,

260

Frequenoy (Hz) Fig. 8.31 Totalradiatedpower without statefeedback (====) and with statefeedback (= ==) two control forces placed at x - 0.1 m, y -- 0.1 m and x - 0,25 m, y - 0.125 m with a - 10=°. Results presented from Thomas and Nelson (1993) for the predicted reduction in sound power radiated from an aluminium plate excited by a turbulent boundary layer, The plate was simply supported, measured 0.5 x 0.25 m and was 1 mm thick. A structural damping factor of 0.01 was assumed.

equal to that used in the simulation above. Figure 8.30(b) shows the velocities of the three modes modelled in the computer simulation when implementing feedback control of vibration (equation (8.11.15)). In this case, all three structural modes have been controlled to approximately the same degree, with the result that the third mode, in particular, rings for considerably longer than in Fig. 8.30(a). Baumann et al. (1991) state that the total acoustic energy radiated when using feedback control to suppress vibrations (Fig. 8.30(b)) was 38% greater in these simulations than when feedback control was used explicitlyto suppress radiation (Fig. 8.30(a)). A similar formulation has also been used to analyse the feedback control of sound radiation from a structure excited by random disturbances (Baumann et al., 1992). Thomas and Nelson (1993) have also used Baumann's theory to examine the feasibility of providing active control of the sound power radiated from a simply supported flexible plate excited by a turbulent boundary layer. An example of the results they derived is shown in Fig. 8.31 which demonstrates the reduction of sound power radiated that can be in principle achieved with optimal feedback control for an aluminium panel typical of those used in aircraft fuselage construction. The results presented are for a specificchoice of 'effortweighting' a in the cost function used; the reductions produced were found to be crucially dependent on this choice and thus the control gains used, More detailsare presented by Thomas and Nelson (1995).

9 Active Structural Acoustic Control. II Cylinder systems

9.1

Introduction

The previous chapter considered Active Structural Acoustic Control (ASAC) applied to sound radiation and transmission of two-dimensional plate systems. We now turn our attention to the application of ASAC to sound transmission through, and radiation from, cylindrical structures. Such systems are representative of many applications. The control of sound transmission into aircraft is important, and in the low frequency region the fuselage can be adequately modelled as a cylinder (Koval, 1976; Fuller, 1986b; Bullmore et al., 1990). Piping systems are common in many applications and often carry unwanted vibrations (White and Sawley, 1972) and radiate noise (Holmer and Heymann, 1980). Submarine hulls can also be approximated as cylinders and active control shows promise for the reduction of low frequency structurally radiated sound (Clark and Fuller, 1993). In the following sections we will consider ASAC applied to various cylindrical structural systems which can be considered to form the basis of the above applications. Before we do this, it is necessary to extend the previously described cylinder equations of Sections 2.12 and 2.14 to allow for coupled interior and radiated acoustic fields, various forcing functions and the effects of finiteness of the cylinder.

9.2

Coupled cylinder acoustic fields

Consider the infinite cylinder discussed in Chapter 2 and shown in Fig. 9.1. If the cylinder is harmonically vibrating and is totally immersed in a compressible fluid then the interior and exterior acoustic fields will be excited into motion by the radial vibration of the cylinder wall (ignoring the effects of viscosity). The interior acoustic field can be modelled by a double modal series as (Morse and Ingard, 1968) oo

pi(r, O, x, t)

oo

ZZ = n=l

pins cos nOJ,(k~ir) e

jsx,

(9.2.1)

s=l

where k~~is the internal radial wavenumber which is related to the axial wavenumber k~s by the vector relationship k~~= +~/(ki) 2 - kZ,s,

(9.2.2)

278

ACTIVE CONTROL OF VIBRATION

i

~

, ,,jln=o

0=1 I

!

~~"

n=2

Fig. 9.1 Coordinate system and modal shapes for an infinite thin cylinder. where k ~ is the wavenumber of the interior acoustic field. Note that J,(-) denotes a Bessel function of the first kind as described by Abramowitz and Stegun (1972). The radiated field only experiences outward radiation and can be written as (Junger and Feit, 1986)

ZZ pO cos nOH,(k~ r) e oo

pr(r, O, x, t) =

oo

to

,

(9.2.3)

n=0s=l

where H , ( k , r ) is the Hankel function of the first kind (Abramowitz and Stegun, 1972) and a similar relation to equation (9.2.2) is obtained for k r° in terms of outside field properties and wavenumbers. In equations (9.2.1) and (9.2.3) the index n corresponds to the circumferential modal order (n = 0, 1,2, 3 . . . . ) and is related to the number of diagonal nodal lines across the mode shape as shown in Fig. 2.10. The subscript s indicates a branch or wave solution (s = 1,2, 3, ...) for a particular n and does not relate to any particular characteristic of the mode shape except that increasing s implies increasing modal complexity. The superscripts o and i correspond to the 'outside' and 'inside' acoustic fields respectively. Application of the boundary condition of continuity of displacement at the shell wall in the three media results in results in P / a n d Pn° being written in terms of W,, as (Fuller and Fahy, 1982) P/=

~-P~ ri

t

W,~

(9.2.4a)

ri

ksJn(ksa) and

[ 20] P__z

rO

l

rO

k, Jn(k~ a)

(9.2.4b)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

279

The free equations of motion for the shell (i.e. without external disturbances) are subsequently modified by the fluid loading due to radiation such that all the matrix terms specified previously by equation (2.12.4) are similar except the L33 term which is now written as (Fuller and Fahy, 1982) L33 = -~'-'2 2 +

1 + fl2[(k,sa)2 + n2] 2 - FL,

where the fluid loading term FL is given by (Fuller, 1986a) FL = ~"22

Jn ( k ;ia ) ¢

ri

ri

i_\Psl Jn(ksa)ksa

n

i o,l

(9.2.5)

ro]

Hn(k s a) t

FO

,

(9.2.6)

ro

~ Ps I Hn(ks a)ks a

The system of equations represented by the modified shell equations (2.12.4) including the fluid loading term of equations (9.2.5) and (9.2.6) thus represents coupled equations of motion for the vibration of the shell and fluid media. In equation (9.2.6), P s represents the shell density, while P li and pOy are the densities of the 'inside' and 'outside' acoustic media respectively and (') denotes differentiation with respect to the argument. The symbol f~ indicates non-dimensional frequency such that ~ = wa/cL. By setting p} and/or p~ to zero the equations for the free response of the cylinder can be reduced to any of the three in-vacuo cases, with only a contained fluid, with only outward radiation or totally in-vacuo. Thus under the action of a disturbance, on solving the system of equations for Wns, the response of the interior and radiated acoustic fields can also be obtained, as described in the next section.

9.3

Response of an infinite cylinder to a harmonic forcing function

In practical problems structural systems are excited by various forms of forcing functions whether they are disturbance or control inputs. In the following analyses we will consider two forms of forcing function which are representative of those found in a number of practical situations; a point force which is applied to the structure or a monopole source which is located in the fluid medium.

Point force input The response of a fluid-filled cylinder to line and point forces has been derived by Fuller (1983). As discussed in Section 2.14, a point force input f = F e j°~' applied at x = 0, 0 = 0 can be expressed in cylindrical coordinates as

p(x, t) = 1 ~o enF cos nO 6(x) e_riot, 2Jr =

(9.3.1)

where en = 1 if n = 0 and e , = 2 if n~0. Following the analysis of Section 2.14 the response of the system is described by

1 i5~ Wn(kn) e -j(k,.)(x/a) d(kna), w(x)= ~o = 2Jra

(9.3.2)

280

ACTIVE CONTROLOF VIBRATION

where the spectral wavenumber displacement is

Wn(kn)= ( pshe)2 ~2F II ] 33

(9.3.3)

and •33 =

(LllL22- L,2L21)/ILI,

(9.3.4)

where the time variation e i~'' has been suppressed. In equation (9.3.4), for a fluid-loaded shell, the L33 term of the determinant of matrix L is given by equation (9.2.6), and the result of setting ILl to zero also provides the axial wavenumbers, k,sa, for the fluid-loaded system. Equation (9.3.3) in conjunction with equations (9.2.4a,b) enables evaluation of the interior and exterior acoustic fields. Spectral quantities can also be seen to only have one subscript n since the contributions from all waves for a particular n are included in the integral of the inverse Fourier transform.

Interior monopole source A simple form of acoustic excitation inside piping systems investigated by Fuller (1986a) is a point source of sound located at rp. The free-field pressure resulting from this arrangement is given by

p(r,O,z) = Po eJkir'/r',

(9.3.5)

where r' is the distance to the observation point from the monopole location (expressed in cylindrical coordinates) and P0 is the monopole source strength in units of force per unit length. Skelton (1982) and James (1982) have derived the expression for the interior pressure field associated with the monopole positioned on the interior of an elastic cylinder which can be written in spectral form in cylindrical coordinates as

{ri]

i P(r, n, k ri) = pi~oZaWn(kn)Jn(krir)[kriaJ'n(kria)] -1

Jn(k Fp) + 2poe,, J',,(kr'a) ' ria)Y,,(k rir)] - [J,,(krir)y',,(kria)- J,,(k

for r > rp, (9.3.6)

[ Jn(krir) ]_ + 2poe~ J'n(kr'a)

_ , Jn(kria)Yn(krirp)]

[Jn(krirp)Y'n(kria)

for r < rp, (9.3.7)

where the monopole is located at r = rp, 0 = 0, and x = 0. The factor e n = 1 if n = 0 and e, = 2 if n > 0 and radial wavenumber k ri is given by equation (9.2.2) with suppressed subscript s. Equations (9.3.6) and (9.3.7) were obtained by assuming that the interior field consisted of a direct term from the monopole source and a scattered term associated with the forced vibration of the cylinder wall. The unknown amplitude of the scattered term was obtained by applying the Euler boundary condition at the wall as

ACTIVESTRUCTURALACOUSTICCONTROL.II CYLINDERSYSTEMS

281

described by James (1982) and Fuller (1984). Note that this implies, at large values of the distance of x, the interior pressure field will be dominated by the term associated with the vibrations of the cylinder, Wn(k~), since the direct terms will decay as 1/r'. Substituting the forcing functions into the equations of motion of the cylinder provides a system of equations which can be expressed in matrix form as

L2! L22 L23 Vn(kn) = 0 , Lal L32 L33 Wn(kn) T31

(9.3.8)

where the elements of L are given by equations (2.12.4a-i) and include the fluid loading effects of FL, given by equation (9.2.6), in term L33. The source term T31 is given by (Fuller, 1984)

Jn( rp

T31(kn, n)= 2enPo J'n(k ia)k ia

(9.3.9)

p,~oZh '

where (') implies differentiation with respect to the argument. The response of the system to the interior monopole can be again found by solving for Wn(k~) and applying the inverse Fourier transform. This gives the radial displacement relationship

(

~-~2 )n~0

W(rp, x, O) = Po ~pshaco2 =

cos

too [kri]_j(kna)(x/a)J..2n(rp.....~) e d

J'n(k ia)kria

na

(9.3.10) By applying inverse transforms to equation (9.3.6) and (9.3.7), similar relations for the interior pressure response can be obtained. The radiated field can be found directly from Wns using equations (9.2.3) and (9.2.4b).

Exterior monopole source James (1982) has used a procedure similar to that described above to find the pressure and response of a cylinder excited by an exterior monopole. The spectral form of the monopole, given by equation (9.3.5), in the presence of the cylinder can be written

as P(n k r°) ,

ro rp) H'n(kr°a)kr°a

2 2

2poenHn(k =

-

ro

pf(.O Wn(kn)Hn(k a) +

H'n(kr°a)kr°a

,

(9.3.11)

where in equation (9.3.11) the pressure is evaluated at the shell surface, r = a. Using a similar procedure to that employed in the interior monopole problem, the radial displacement response of the shell to the exterior monopole is found to be given by

w(x/a, O) = Po[

~,-~2 ]~0 f~-oo[ Hn(kr°rp)] 133 e -j(kna)(x/a)d(k,a) pszcha(.o2 = e n cos nO H,n(kroa)kroa (9.3.12)

282

ACTIVE CONTROL OF VIBRATION

and the interior pressure field is given by i

pi(r/a, x/a, O) = Po pserhaw2 PlW

2a

Z en cos nO

n=O

Hn(k rp) H'n(kr°a)kr°a

[Jn(krir)]-i(k.a)(x/a) x ..... 133 e d(kna). J',(kria)kria

(9.3.13)

The integrals obtained by applying the inverse Fourier transform could be solved by contour integration (including branch cuts if necessary) and the method of residues as described in Churchill et al. (1974). Such approaches have been used in Chapter 2. However, it is often more convenient to evaluate the integral numerically by directly integrating along the kna axis as described by Fuller (1986a). Care must be taken to choose sufficiently large values of +kna to ensure that the integral converges satisfactorily. In addition, a small amount of damping is usually added to the shell system to move the poles (associated with the eigenvalues) off the real axis thus avoiding the problem of numerical instability. As discussed in Chapter 2 this form of hysteretic damping is valid for harmonic motion. A more detailed discussion of this technique has been presented by Fuller (1986a).

9.4

Active control of cylinder interior acoustic fields using point forces

The control of sound transmission through cylindrical structures to the contained interior space is an important problem in many applications. For example, much work has been carried out on controlling aircraft interior acoustic fields using arrays of active acoustic sources located in the interior space (Elliott et al., 1989; Lester and Fuller, 1990; Silcox et al., 1990). In this section we study an alternative technique based on ASAC whereby active forcesare applied to the structure and are optimised to minimise the interior acoustic field. The analysis essentially follows work by Jones and Fuller (1990). Figure 9.2 shows an axial view of the system used for this study which is similar to the interior noise model developed by Fuller (1986b) where the aircraft fuselage is modelled by an infinitely long shell with coupled interior and exterior acoustic fields. The rationale behind using the infinite shell model is that mid to large size aircraft have appreciable structural and acoustical damping. This has the effect of damping axial waves, with the consequence that the response behaviour is dominated by a wave rather than a modal behaviour in the axial direction. Likewise inhomogeneities such as ribs and stringers are assumed 'smeared' while other asymmetries such as the floor etc. are ignored. The acoustic source due to an exterior propulsion unit such as a propeller is modelled as a monopole which radiates towards the fuselage, drives it into motion, which in turn, forces the interior acoustic field to respond. The control inputs are modelled by normal point forces arranged around the cylinder circumference in the source plane as shown in Fig. 9.2. Excitation and control is assumed to occur for harmonic oscillations which is representative of the dominant noise inside the cabin of propeller aircraft.

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

283

Oi Po

0=0

a

Fig. 9.2 Coordinate system and input locations for active control of cylinder interior noise (x axis is into the page). The total complex interior acoustic field is by superposition, the sum of the contributions due to the exterior monopole disturbance (also called the primary source), pp, and the control force (also called the secondary sources) inputs, Ps, respectively and thus

p,(r/a,x/a,O)=pp+ps.

(9.4.1)

The interior field due to the monopole disturbance is specified by expressing equation (9.3.13) as a function of angular position such that

pp(rla, xlo. o) poAo(rla, xla. O..ola). =

(9.4.2)

where P0 is the monopole source strength (specified), and the internal and external media are assumed identical. The time variation e j~'' has been suppressed for conciseness. The transfer function, A0, in equation (9.4.2), is given by oo

A0 = Z ~PPcos(n0- nOo),

(9.4.3)

n=O where

/pP-- [En~a2][~sf][h]-I

H~(krr°) Jn(krr) 133 e H'n(kra)kra J'n(kra)kra

d(kna). (9.4.4)

In equations (9.4.2)-(9.4.4) r 0 is the radial location of the external source, 00 is the angular location of the source and r is the coordinate of the internal radial observation point.

284

ACTIVE CONTROL OF VIBRATION

The interior pressure response due to N, control forces located at x/a = O, 0 = 0 i is given by N,

ps(r/a, x/a, O)= Z FiBi(r/a' x/a, O, 0~),

(9.4.5)

i=1

where F i are complex force amplitudes (unknown) and the transfer function B~ is specified by oo

ni = Z ~Pn(r/a' x/a) cos (nO- nOi),

(9.4.6)

n=0

where

I?~ ~s=

Jn(krr) ] 133e j(kna)(x/a)d(k,,a).

J'n(kra)kra

era JLPsJ

(9.4.7)

Using the approach outlined in Chapter 4 we define a quadratic cost function given by

J(Fi) = --1 ~s ~r Ip,(r/a, x/a, S

o) I

dS,

(9.4.8)

which is the square of the modulus of interior pressure integrated across an axial plane of the interior acoustic field. For the following results we will simplify equation (9.4.8) by noting that for a rigid cylinder, all acoustic modes will have a pressure maximum at the wall. Thus, we can reduce equation (9.4.8) to a line integral around the circumference at the axial location of interest, and approximate the cost function as

i yg 0 Ip,(1, x/a, O) dO.

J(F3 --- --1

(9.4.9)

This simplified cost function results in substantial savings in computer time and has demonstrated very good control characteristics. The purpose of the analysis is to solve equation (9.4.9) such that the cost function J(Fi) is a minimum. The procedure for obtaining the optimal solution is similar to that outlined in Chapter 4 and is described briefly in this section. For further details the reader is referred to Jones and Fuller (1990). To obtain the optimal solution, the cost function is best expressed in matrix form by substituting equation (9.4.1) into equation (9.4.9) and performing the required integration using the orthogonality characteristics of the circumferential modes on the interval (0, 2zl). The resulting cost function is a real scalar quantity expressed in terms of unknown control amplitudes described by the vector f , = [F1 F2 F3 ... ]T, with all other variables specified or assumed known. To solve for the vector of control amplitudes which minimises J(fs), the cost function is differentiated with respect to f, and equated to zero. The resulting optimal solution f~o is given by, in matrix form (see the Appendix of Nelson and Elliott, 1992, and Chapter 4)

fso = -A-~b,

(9.4.10)

where b = xp0. For vector x of size [N, x 1 ] a typical element in the jth row is oo

= n~0 __ 2E

s

p:~

o,)],

(9.4.11)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. 11CYLINDER SYSTEMS

285

where 0] is the angle of the jth control force and 0 p is the angular location of the noise source. For the Hermitian matrix A of size [N~ x Ns] a typical element in the ith row and jth column is oo

A~j = = 2 ~ ( ~ ' ~ ) * c o s [ n ( 0 ~ - 0~)l,

(9.4.12)

where the factor e = 2 if n = 0 or e = 1 if n > 0. For the following illustrative results the cylindrical shell was assumed to be of 0.254 m radius, 1.63 mm thickness and constructed from aluminium with properties given in Table 9.1. The non-dimensional excitation frequency was set at ~ - 0.193. This value of ~ would correspond to, for example, 166 Hz in an aluminium cylinder of 2 m diameter. The objective was to minimise the interior acoustic field in the source plane, thus the integral of equation (9.4.9) was evaluated at x / a - O . Note that, as discussed in Section 2.8, a hysteretic damping q, as specified in Table 9.1, was introduced into the shell and fluid media to condition the inverse Fourier integrals for numerical evaluation and model the damping of a realistic aircraft fuselage. As shown in Fig. 9.3(a) the monopole is offset slightly from 0 = 0 ° by an angle of 0 d = - 8 ° to introduce some asymmetry into the model, corresponding to more practical situations. Figure 9.3(b) presents the interior sound pressure level distribution due to the disturbance alone, evaluated in the source plane. It can be seen that the interior pressure field is dominated by a slightly rotated cos 20 mode, with some small contribution from the other circumferential modes. In order to interpret the results it is convenient to describe the shell and interior pressure field response as an azimuthal series such that p or w = Z An cos nO + B n sin nO.

(9.4.13)

n=0

Figure 9.4 presents the shell response decomposed into modal amplitudes]" I An l a n d I Bn I for the conditions leading to the results shown in Fig. 9.3 (b). For the disturbance alone, the shell response is dominated by many modes such as A2, A3, A4, A5 and n 3 (note that the subscript refers to circumferential modal order). Figures 9.3(b) and 9.4 reveal an important characteristic of the coupled sound field. Out of all the strongly responding shell modes only the n = 2 motion is well coupled to the interior acoustic field (Fuller, 1985b; Thomas, 1992). Thus in order to control the interior acoustic field for this system, it is only necessary to control the n = 2 structural motion. Table 9.1 Medium

Aluminium Air

Youngs modulus E(N m -2) 71

×

m

10 9

Material properties

Poisson's ratio

Density p(kg m -3)

Free wave speed cL, cr (m s-')

Damping ratio

v

0.33 --

2700 1.21

5432 343

0.2 0.001

t In effect a wavenumber transform has been applied to the response in the angular direction.

rls, rlt

286

ACTIVE CONTROL OF VIBRATION

!

Pointcontrol \ ~ forces

~

I '

/

~~ Acoustic

Jsource

I i

(a)

(b)

(c)

(d) 50

70 90 SPL (dB)

110

Fig. 9.3 Interior sound pressure level distributions, x/a = 0, ~ = 0.193: (a) test configuration; (b) uncontrolled; (c) controlled with one force; (d) controlled with two forces (after Jones and Fuller, 1990). Figure 9.3(c) presents the controlled field when one control force located at 01 = 180 ° is employed. It is apparent that reductions o f the order of 10 dB have been obtained. However, the residual field now appears to have the shape of the sin 20 mode, suggesting control spillover has occurred. For the next test we apply a second control force at 0~= 45 ° and the total field given in Fig. 9.3(d) now shows reductions of the order of 50 dB with no discernible residual mode shape. In effect the use of two control forces has resulted in a distributed control input which has reduced coupling into the sin 2 0 mode, thus limiting the control spillover. The corresponding shell modal amplitudes for the case of two simultaneous control forces are also plotted in Fig. 9.4. The results show that the A2 and B2 shell amplitudes have been reduced leading to global interior noise reduction. Figure 9.4 also demonstrates significant control spillover into the higher order structural modes. As these modes are not well coupled to the interior acoustic field, the control spillover is constrained to the structure and does not lead to performance degradation in terms of

ACTIVE STRUCTURALACOUSTICCONTROL.II CYLINDERSYSTEMS Noise source o Control source A Noise and control source o

~50[

[] A O

I

100

x

E E v

287

,< -

-

50

t-

0

O ¢::

0

0 0

2

4

6

8

10

12

14

16

Circumferential mode number, n

C

150

E

~100 .~

g

50 "0 n"

0

2

4

6

8

10

12

14

16

Circumferential mode number, n

Fig. 9.4 Circumferential modal amplitudes of shell radial displacement response, x/a=O, = 0.193 (after Jones and Fuller, 1990). controlling the interior acoustic field. These results again illustrate the basic concept behind ASAC; it is only necessary to control or modify those structural motions which are associated with significant sound radiation. The results of Fig. 9.4 are also analogous to those presented in Chapter 8 which were discussed with regard to their wavenumber content. As can be seen from Figs 9.3 and 9.4, only the low angular wavenumbers (i.e. low values of n) are well coupled to the interior field, while the high wavenumbers are not well coupled and are thus not as important in terms of interior radiation. If the increase of structural response is unacceptable then shaped actuators could be used as in the work described by Dimitriadis et al. (1991), or a cost function could be used which includes a weighted contribution from the shell response. Both of these techniques can be employed in order to keep the structural vibration bounded to acceptable levels.

9.5

Active control of vibration and acoustic transmission in fluid-filled piping systems

Piping systems are common in many industrial situations, and they often can transmit unwanted vibrational energy to points away from the excitation source. Vibration transmission and energy distributions in fluid filled elastic cylindrical shells have been investigated in detail by Fuller and Fahy (1982). Due to the mixed media of the transmission path (i.e. the structural and fluid path) the passive control of vibrations in fluid filled pipes can be difficult. For example, the work of Fuller (1983) demonstrates that the internal fluid can cause a significant flanking of the energy of predominantly structural waves around a radial line constraint applied to the cylinder wall.

288

ACTIVE CONTROL OF VIBRATION

In this section we study active control of wave propagation in a fluid-filled elastic infinite shell. The work discussed is a summary of previous investigations reported by BrEvart and Fuller (1993). The configuration of the system to be analysed is shown in Fig. 9.5. The most convenient location to apply the control forces is directly to the shell wall, even though the object may be to minimise either structural power flow, fluid power flow or the sum of both. For the following analysis we restrict ourselves to axisymmetric (n = 0) and beam type (n = 1) wave motion and choose to minimise the radial displacement of the cylinder wall at up to two axial locations as shown in Fig. 9.5. The rationale again is that it is easiest to observe the structural motion; direct control of the interior field would require obtrusive arrays of control sources and sensors in the acoustic field which may impede flow of the internal fluid. Using the coordinate system of Fig. 9.5 where the incident wave has complex amplitude Wff,at x = 0, the complex disturbance displacement can be written as

Wp(x) = W~ cos n 0 e -jk~x,

(9.5.1)

where k,p, is the axial wavenumber of the chosen disturbance wave type. As discussed by Fuller and Fahy (1982), k~ is strongly dependent upon frequency and type of the wave in the shell-fluid system. The time variation e j°'' has again been suppressed, since we perform the analysis for a single frequency of motion. The control or secondary forces are considered to be axisymmetric (n = 0) or beam type (n = 1) radial line forces (corresponding to circumferential distribution of the disturbance) applied to the pipe wall as in Fig. 9.5. The azimuthal distributions of these line forces are given in Fig. 9.1, and can be written as

p,(x, O)= F, cos nO 6 ( x - x~),

n = 0, 1,

(9.5.2)

where x,i is is the axial point of application. The radial response of the shell fluid system to these line forces has been derived previously in Section 9.3 and is given by Ns

i

w~(.,x, n) = i=~1 2:rpsc2h/a _F~ cos nO I:** 133 e -jk.(x-x',) dkna,

for n = 0, 1,

(9.5.3)

where N, is the total number of control forces. In the following analysis we restrict ourselves to a maximum of two control forces, N, = 2. Input disturbance wave ~ W ° hs

Control force(s)

Error sensor(s) 7

Fluid

I I x=O

Fig. 9.5 System arrangement and coordinate system for active control of vibrations in fluidfilled cylinders.

289

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

The total displacement field in the shell system is, by superposition, the sum of the disturbance and control fields. Thus the complex radial displacements at the two error 1 Xe 2 for the case of two control forces is given by locations, x = Xe,

d WPs+ E f s,

W'=

(9.5.4)

where the vectors d, f s and w' and the matrix E are specified by -jknsxe T d = [e_J~sX~ e , 2] •

COS

1

•

dkna

133 e

f?~133 e -j~o~x~ x~, dkna

i_= 133 e

133 e

nO

E =

1

(9.5.5) 1

2

dkna

'

2Jrpsc2h/a

-

fs = [F~ w ' = [w'(x~)

(9.5.6)

F 2,T

s],

w'(x~)] T.

- y k . ( x e - x s)

~ ~

dkna

'

(9.5.7) (9.5.8)

Note that the subscript s is retained on the disturbance amplitude WPs and in the terms of vector d since the disturbance is that corresponding to a particular wave (n, s). On the other hand, the subscript s is suppressed in the integrals of matrix E as discussed in Section 9.3. For the following control strategy we choose to minimise the cost function defined by ue

J= Z I~Ztlwt(xie)12dO,

(9.5.9)

j=l

which is proportional to the out-of-plane vibrational energy per unit length of the shell at the error sensor locations. The cost function of equation (9.5.9) can be written in matrix form as J(fs) = g[fsH Afs + f~ b + bHfs+ c],

(9.5.10)

where the superscript H denotes the Hermitian transpose operator and e = 2 for n = 0 or e = 1 for n > 0. The vector b is given by b = [EHd]WPs.

(9.5.11)

A = EHE,

(9.5.12)

The Hermitian matrix A is given by

while the constant c is specified by the expression c = dHd]

WPs12.

(9.5.13)

The optimal control force vector that minimises the cost function has been previously derived in Chapter 4 and is given by

fso =

-A-lb.

(9.5.14)

In order to evaluate the performance of the active control, we now need to derive expressions for the total power flow in the shell-fluid system based upon previous work

290

ACTWECONTROLOF VIBRATION

of Fuller and Fahy (1982). We express the shell variables for a particular circumferential mode n as a series of complex wave solutions given by

u(x, O, t) =

Z

jwt-jknsx +jzt/2

U,, cos nO e

,

(9.5.15)

s=l jwt -jknsx

v(x, O, t) = Z Vr~ sin nO e

,

(9.5.16)

s=l oo

jcot-jk.sx w(x, O, t) = Z W,~ cos nO e

(9.5.17)

s=l

and the corresponding, coupled interior complex pressure as oo

p(x, r, O, t) =

Z

j~ot-jk.sx P,~ cos nO J,,(k~r) e .

(9.5.18)

s=l

Power flows in the axial direction in the fluid field, H I, and in the shell wall, H~, have already been derived in several previous references (e.g. Fuller and Fahy, 1982) and are given by oo

YL'

3,--x3

oo

i

1-If=2CL~,zpfeZZWnse

-jknsxT ~g

~ f

Wni[e -jk"ix] Fnsi,

(9.5.19)

s=l i - 1

where the fluid power factor Ffi is given by

1[ 1 ][

Ffi = -'~

kraJ,n(kra)

,ia

kraj,(Uia)

r ,

o J.(k~r)J n (k~r)r dr

(9.5.20)

and oo

i-is

oo

3 -jknsxT *r-jknix-,* f = Y('psCL~"~E Z Z wns e Wni[e J ansi, s=l i=l

(9.5.21)

where the shell power factor Sfnsiis given by

sfi = [(h/a)3/12][(k,~a)2(k.~a) * + vn2(k.ia) * + R~(k,~a)(knia)* + nRts(knia)*]

+[(h/a)/2]t(k,~a)R,,,R*. + vnRt~R,,* + vR,,*] + [ ( h / a ) / 4 ] ( 1 - v)[nRasR~ + kn~aRtsR~].

(9.5.22)

In equation (9.5.22), R~, and Rts are the ratios of axial to radial and axial to torsional amplitudes respectively, obtained by resubstituting the derived axial wavenumber k,, back into the shell equations of motion, stated in equations (2.12.3) and (2.12.4) including fluid loading if necessary, as discussed in Section 9.2. In equations (9.5.19) and (9.5.21), the radial displacement amplitude is found from the relation for the total response of the system given by equation (9.5.4) with the optimal control force value.

ACTIVE STRUCTURALACOUSTIC CONTROL. II CYLINDERSYSTEMS

291

The performance of the discontinuity caused by the active control system is then evaluated by use of the power transmission coefficient T~ defined as T~ =

Total transmitted power flow

(9.5.23)

Total incident power flow ( r I f + ]'-[ ~transmitted ~'SIX > X e (rIf.+.

(9.5.24)

l'-I ~incident xxslx

<xc

The power transmission loss is then defined as TL = 10 log~0(T~).

(9.5.25)

The above performance criteria assumes that the source strength is independent of the reflected wave which is reasonable for very long piping systems. For the following results we consider a steel shell of thickness h/a = 0.05 filled with water. Results are given for two cases of control; when one control force is applied and the radial shell displacement is minimised at one location, and when two control forces spaced a distance AXc/a=O.1 are used to minimise the radial displacement at two different axial locations closely spaced at a distance Axe/a = 0.05 apart. In the second case the two control forces have been spaced a distance of about 10 times smaller than the minimum wavelength at the highest frequency of interest (f~= 3.0). In this situation, the control inputs effectively introduce a line moment component into the shell system as well as a radial force. It was also desired to keep the control hardware reasonably compact and thus the spacing between the control forces and the error point was chosen as three radii. The inverse integrals of equation (9.5.6) were evaluated using the residue theorem as outlined in Section 2.14. This enabled the total system response to be broken down into contributions from each wave (n, s). The power flow relations of equations (9.5.19)-(9.5.22) could then be used. The infinite sums were truncated at eight roots which was found to provide sufficient accuracy in the solution. Material properties for the system are given in Table 9.2. Note that a hysteretic damping r/s = 0.02 has been added to the shell material, making E, the Young's modulus of elasticity, complex and equal to E' = E(1 - j r / s ) as discussed in Chapter 2. For the first case we consider an axisymmetric (n = 0) disturbance and control forces. Figure 9.6 presents the transmission loss when the incident wave is the branch denoted s = 2 in the notation of the work by Fuller and Fahy (1982), which behaves dominantly like a structural wave at low frequencies. For the case of one control force, even though the radial displacement is minimised at the error point, the active input can be seen to be ineffective in controlling total power flow in the system. The reason for this behaviour is Table 9.2 Materials

Steel Water

Material properties

Young's modulus E(N m -2)

Poisson's ratio

Density p (kg m -a)

Free wave speed cL, c: (m s -1)

Damping ratio

v

19.2 x 101° --

0.3 --

7800 1000

5200 1500

0.02 0.0

rls, fly

292

ACTIVE

CONTROL

OF VIBRATION

10 ~" "o v _J

o

5 |

°.,.

0 -5

tO

._~ -10 E

!."....... -. "

e--

o

:o,

~ -15 ;

"200~

0.5

1

1.i 5

2.L

215

Non-dimensional frequency, ~ Fig. 9.6 Transmission loss of the active control system shown in Fig. 9.5, s = 2 wave, n = 0: ~, one control force; - - - , two control forces (after Br6vart and Fuller, 1993). that the fluid field leads to flanking of energy around the discontinuity provided by the control input. Using two control forces leads to attenuation of between 10 to 17 dB at low frequencies except around the frequency ~ =0.85 where the shell fluid systems are strongly coupled through coincidence behaviour. At very low frequencies, since there are only two waves cut-on, one would expect total attenuation of the power (TL---> -oo) with two control forces. However, due to the compact arrangement of the control transducers, near-fields of cut-off waves generated by the control inputs are observed by the error sensors. As discussed in Chapter 6 this leads to a reduction in the attenuation achieved. Design of the system is thus a compromise between the compactness of the transducers (in terms of their axial spacing) and desired reduction in power. At higher frequencies, I> 1.3, the incident, s = 2, wave becomes a dominantly fluid type wave and most of the energy propagates via the fluid path and the attenuations achieved are negligible. Figure 9.7 shows the transmission loss when the incident disturbance is the s = 1 wave (Fuller and Fahy, 1982) which has predominantly fluid wave type characteristics. As discussed by Fuller and Fahy (1982), this wave in the fluid field is subsonic and consists of a pressure near field located near the shell wall. However, Fuller and Fahy (1982) also demonstrate that below the ring frequency ~ = 1, the power flow of the s = 1 branch is dominantly in the fluid field. Applying one control force, Fig. 9.7 shows the surprising result that the power flow is almost uniformly reduced across the frequency range by around 10 dB. Due to the resonances of the coupled system associated with cut-on of higher order duct modes, an increased power flow occurs at a few discrete frequencies. This aspect of the results illustrates a potential drawback of active control. In contrast to passive control, active approaches can lead to an increase in power carried by the system when they act by introducing energy. When two control forces are used, higher attenuations of the order of 20 to 50 dB are obtained. The explanation for this good control performance is associated with the nature of this particular wave; the near field closely hugging the shell wall behaves like an attached mass loading and is thus strongly affected by structural forces. We now turn our attention to beam like (n = 1) motion of the system. Figure 9.8 shows the transmission loss with the s = 1 branch incident. As described in Fuller and

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

293

10 0 13 v

-10 o -20

i

tO .m (/)

i:', Y'..

u) -30 -i .i : E " (/)

: i

i" .........."

\:',1

" -40 -50 ~ 0

~

0.5

1

1.'5

2'.5

3

N0n-dimensi0nal frequency, .O

Fig. 9.7 Transmission loss of the active control system shown in Fig. 9.5, s = 1 wave, n = 0: ~, one control force; - - - , two control forces (after Br6vart and Fuller, 1993). 20 ~" ...I

I0 tO

0 -20 -40

-60

~E -80

:'",.- -i

-100 -120~

0'5

'

'

'

'

1 1.5 2 2.5 Non-dimensional frequency, ~

3.0

Fig. 9.8 Transmission loss of the active control system shown in Fig. 9.5, s = 1 wave, n = 1" ~, one control force; - - - , two control forces (after Br6vart and Fuller, 1993). Fahy, (1982) this wave behaves like a slender beam at low frequencies where the presence of the enclosed fluid simply acts as an additional mass. The acoustic response consists of a forced near field located near the shell wall. At low frequencies, below the first acoustic wave cut-off frequency, high attenuations of the order of 10 to 60 dB are observed due to the dominantly structural motion of the system. The performance is again limited by the presence of control-generated near fields observed at the error sensors. However, above the first cut-off frequency, as higher order acoustic modes begin to propagate the control performance is severely decreased due to energy now propagating in the fluid path. Increasing the number of control forces to two leads to significantly better control performance particularly at low frequencies. At higher frequencies the attenuations are maintained at around 15 dB except near the cut-on frequencies of higher order modes. In this case it appears that the controller can reduce both the structural and fluid path simultaneously.

294

ACTIVE CONTROL OF VIBRATION

The preceding analysis demonstrates the potential for active control of total energy flow in fluid-filled piping systems. The results indicate that due to the coupled nature of the system it is possible in some situations to control the interior fluid power flow by applying structural inputs to the piping wall. The structural inputs effectively change the wall impedance seen by the fluid field making it more resistive and thus energy absorbing. This has obvious benefits in realistic applications such as ease of implementation and nonobtrusive (in the flow field) control hardware. It is apparent from the results that the controller needs to be of higher order, i.e. multiple input and output channels. This observation is undoubtedly related to the complex nature of the waves in fluid filled cylinders, in particular that there are many propagation paths and thus degrees of freedom in the system. In addition the performance would likely be improved by use of wave vector type sensors (as outlined in Section 5.10 and Chapter 6) which separate out positive travelling waves of a particular type and uses this information as error signals.

9.6

Active control of sound radiation from vibrating cylinders

The radiation of sound from cylindrical structures occurs in many important applications. Pertinent examples are the radiation of sound from submarine hulls, 'break out' of sound from circular air conditioning ducts and radiation of noise from piping systems. Radiation from cylinders excited by an internal flow has been studied previously by Holmer and Heymann (1980). Fuller (1986a) has analytically studied the related but idealised problem of sound radiation from an infinite cylinder excited by an internal monopole source. In this section, we study the active control of sound radiation from long cylindrical structures by radial forces applied to the wall of the structure. As an illustrative example we will consider an infinite cylindrical shell excited by an internal monopole source representative of, for example, a simplified model of an acoustic source due to a flow disturbance. Control is in the form of point forces applied radially to the shell wall. As most realistic sources are asymmetric we will locate the monopole source off the centreline at rp/a = 0.9 in order to induce asymmetry in the system response. As discussed by Kuhn and Morfey (1976) most realistic piping and duct systems carry higher order circumferential waves (n > 0) due to the source and duct asymmetries. The geometry and coordinate system used in the analysis is given in Fig. 9.1. Following the derivation presented in Section 9.1 the spectral response of a fluidfilled cylinder to a point force and an internal monopole respectively can be written in the wavenumber domain as

W~n(kn) =

a/33

p,c (h/a

F

(9.6.1)

and a Jn(krirp) W~(k,) = 2en133 p~ C2 L(h/a) (kria)J'n(k'~a) Po,

(9.6.2)

where rp is the location of the monopole and equations (9.6.1) and (9.6.2) hold for a particular circumferential mode n. All terms in the equations have been defined previously in Section 9.1.

295

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

By superposition the total spectral wavenumber displacement of the system consisting of the disturbance (primary) and control (secondary) forces is then given by t Wn(k,)

Us Z =

si Wn(kn) + WPn(kn),

(9.6.3)

i--1

where Ns is the number of control forces. The spectral radiated pressure is found by applying the boundary condition of continuity of radial displacement (similar to that carried out previously with an interior acoustic field). This leads to or-x2 2

ro

Pex(kn) = Pf~ CL Hn(krr) Wn(kn), a (kr°a)H'n(kr°a)

(9.6.4)

where r is the radial observation point. Applying an inverse Fourier transform to equation (9.6.4) gives the total radiated pressure in the spatial domain as p tex(r,

O, X) = i~1= ~a "4- --P~ £ a

cos n(O

COS ?'/(0-

n--o

foo

-

Oi)J_oo

02

nn(kr°r) -jk.x e dkna 2Z~ps(h/a) kr°an'n(kr°a) pf£) 133

}

-jknx Om) I?oo enpff2° 2133 .... Jn(k'-rir "p) nn(kr°F) e dk, a, erp~(h/a) (kria)J'n(kria) (kr°a)H'n(kr°a)

(9.6.5) where Oi, i = 1 to Ns and 0 m a r e the angular locations of the control force(s) and disturbance monopole respectively. Equation (9.6.5) assumes that the control forces, Fi, a r e applied in the source plane, x = 0. The disturbance source has amplitude p~. However the relationships could be easily modified to account for control forces out of the source plane. We now desire to minimise the sound field at a particular circular location in the far field. In the same manner as before, we form a cost function as an integral of the square of the total far-field pressure amplitude defined by J(fs) =

i IP'exl R dO, o 2p;c;

(9.6.6)

where P'e.~is evaluated at a particular axial location, x, and radial location, R. The cost function can again be written in matrix form as J = fsHAfs + where the control force vector fs = [F~

c

=

(

~-" IpPl2

fHb + bnfs + c,

F 2 . . . ]T

(9.6.7)

and b = p~0x. The scalar c is specified by

)

2nOm + --1 sinZnOm

e COS

n=0

E

I+°° 6nP;~'~2133

Jn(krirp)

'r_non(kr°?') -jk.a 12. _oo 2Ztps(h/a) (k ia)J'n(k~ria) (k ma)H~n(kr°a) e dkna

(9.6.8)

296

ACTIVE CONTROL OF VIBRATION

The vector x is of size N= x 1 and a typical element is given by

( [i_+:

1

m/

X~I = ~~o--~ e cos nOj cos nOm + --e sin nOj sin nO

x

x

pz~ 133 H.(kr°n) -ik.x 2~p=(h/a) (kr°a)g~(U°a) e dkna

gnpf~'~ 133 Jn(krirp) [I5 2zcp=(h/a) o: (kria)Jn(kria)

]"

Hn(kr°r) -jk.x ] e dk,,a . (kr°a)n'(kr°a)

(9.6.9)

The Hermitian matrix A is of size N, x Ns and a typical element is given by cos nOi cos n0j + --e sin nOi sin n

A/j = ~~o--~

o

j2.

2

I +00 PI ~ 133 Hn(kr°r) e jk,,x dk, a -00 4~ps(h/a) (kr°a)H'(kr°a)

x

(9.6.10)

In the above equations again e = 2 if n = 0, e = 1 if n ~ 0. The optimal force vector fso to minimize J is then given by

f=o = - A - ~ b .

(9.6.11)

The system used in generating the following example results consists of a steel shell of thickness h/a = 0.05 filled with water. Material properties are as used in Section 9.5. The monopole disturbance source is assumed to be located at rp/a = 0.9, 0 = 0 and the 90 ° 120 o ~

L3_O

~

60 ° ...,/

]o'~-~./

150 °

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0

"'~ ,,

..:'C"

"'"

~8o I'~',

..'"

.............

\

~ ,

i

!

"

" ...... /'.',

...............

.....

".',,

i,

"

",,

"~*

',

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, ,,

:

-~

".. i ' ..-.>'.~ z .......... ,., ]"-,. ', i \ ,. . . . . . . . ,,, z ', ." ""--,~ ." ". . . . '~"~:. ""'),/' 'ix...... .i:i

'.,. ....."'.. o

"'""

"

"

\'... " ~

% '.. % "....

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...................

~ ~ "'"" " "....

/ -~(.

/i

24o ° ~

% ~

............... ~

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