Norton M.P., Karczub D.G. Fundamentals of Noise and Vibration Analysis for Engineers

Fundamentals of Noise and Vibration Analysis for Engineers

Noise and vibration affects all kinds of engineering structures, and is fast becoming an integral part

of engineering courses at universities and colleges around the world. In this second edition, Michael

Nortons classic text has been extensively updated to take into account recent developments in the

eld. Much of the new material has been provided by Denis Karczub, who joins Michael as second

author for this edition.

This book treats both noise and vibration in a single volume, with particular emphasis on wave

mode duality and interactions between sound waves and solid structures. There are numerous case

studies, test cases and examples for students to work through. The book is primarily intended as a

text book for senior level undergraduate and graduate courses, but is also a valuable reference for

practitioners and researchers in the eld of noise and vibration.

Fundamentals of Noiseand Vibration Analysis

for EngineersSecond edition

M. P. NortonSchool of Mechanical Engineering, University of Western Australia

and

D. G. KarczubS.V.T. Engineering Consultants, Perth, Western Australia

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

Information on this title: www.cambridge.org/9780521495616

First edition Cambridge University Press 1989

Second edition M. P. Norton and D. G. Karczub 2003

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written

permission of Cambridge University Press.

First edition published 1989

Reprinted 1994

Second edition published 2003

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-49561-6 hardback

ISBN 978-0-521-49913-2 paperback

Transferred to digital printing 2007

Toour parents,the rst authors wife Erica,and his young daughters Caitlin and Sarah

Contents

Preface page xvAcknowledgements xviiIntroductory comments xviii

1 Mechanical vibrations: a review of some fundamentals 1

1.1 Introduction 11.2 Introductory wave motion concepts an elastic continuum viewpoint 31.3 Introductory multiple, discrete, massspringdamper oscillator concepts

a macroscopic viewpoint 81.4 Introductory concepts on natural frequencies, modes of vibration, forced

vibrations and resonance 101.5 The dynamics of a single oscillator a convenient model 12

1.5.1 Undamped free vibrations 121.5.2 Energy concepts 151.5.3 Free vibrations with viscous damping 161.5.4 Forced vibrations: some general comments 211.5.5 Forced vibrations with harmonic excitation 221.5.6 Equivalent viscous-damping concepts damping in real systems 301.5.7 Forced vibrations with periodic excitation 321.5.8 Forced vibrations with transient excitation 33

1.6 Forced vibrations with random excitation 371.6.1 Probability functions 381.6.2 Correlation functions 391.6.3 Spectral density functions 411.6.4 Inputoutput relationships for linear systems 421.6.5 The special case of broadband excitation of a single oscillator 501.6.6 A note on frequency response functions and transfer functions 52

1.7 Energy and power ow relationships 52

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1.8 Multiple oscillators a review of some general procedures 561.8.1 A simple two-degree-of-freedom system 561.8.2 A simple three-degree-of-freedom system 591.8.3 Forced vibrations of multiple oscillators 60

1.9 Continuous systems a review of wave-types in strings, bars and plates 641.9.1 The vibrating string 641.9.2 Quasi-longitudinal vibrations of rods and bars 721.9.3 Transmission and reection of quasi-longitudinal waves 771.9.4 Transverse bending vibrations of beams 791.9.5 A general discussion on wave-types in structures 841.9.6 Mode summation procedures 851.9.7 The response of continuous systems to random loads 911.9.8 Bending waves in plates 94

1.10 Relationships for the analysis of dynamic stress in beams 961.10.1 Dynamic stress response for exural vibration of a thin beam 961.10.2 Far-eld relationships between dynamic stress and structural

vibration levels 1001.10.3 Generalised relationships for the prediction of maximum

dynamic stress 1021.10.4 Properties of the non-dimensional correlation ratio 1031.10.5 Estimates of dynamic stress based on static stress and

displacement 1041.10.6 Mean-square estimates for single-mode vibration 1051.10.7 Relationships for a base-excited cantilever with tip mass 106

1.11 Relationships for the analysis of dynamic strain in plates 1081.11.1 Dynamic strain response for exural vibration of a constrained

rectangular plate 1091.11.2 Far-eld relationships between dynamic stress and structural

vibration levels 1121.11.3 Generalised relationships for the prediction of maximum

dynamic stress 1131.12 Relationships for the analysis of dynamic strain in cylindrical shells 113

1.12.1 Dynamic response of cylindrical shells 1141.12.2 Propagating and evanescent wave components 1171.12.3 Dynamic strain concentration factors 1191.12.4 Correlations between dynamic strain and velocity spatial

maxima 119References 122Nomenclature 123

ix Contents

2 Sound waves: a review of some fundamentals 128

2.1 Introduction 1282.2 The homogeneous acoustic wave equation a classical analysis 131

2.2.1 Conservation of mass 1342.2.2 Conservation of momentum 1362.2.3 The thermodynamic equation of state 1392.2.4 The linearised acoustic wave equation 1402.2.5 The acoustic velocity potential 1412.2.6 The propagation of plane sound waves 1432.2.7 Sound intensity, energy density and sound power 144

2.3 Fundamental acoustic source models 1462.3.1 Monopoles simple spherical sound waves 1472.3.2 Dipoles 1512.3.3 Monopoles near a rigid, reecting, ground plane 1552.3.4 Sound radiation from a vibrating piston mounted in a rigid bafe 1572.3.5 Quadrupoles lateral and longitudinal 1622.3.6 Cylindrical line sound sources 164

2.4 The inhomogeneous acoustic wave equation aerodynamic sound 1652.4.1 The inhomogeneous wave equation 1672.4.2 Lighthills acoustic analogy 1742.4.3 The effects of the presence of solid bodies in the ow 1772.4.4 The PowellHowe theory of vortex sound 180

2.5 Flow duct acoustics 183References 187Nomenclature 188

3 Interactions between sound waves and solid structures 193

3.1 Introduction 1933.2 Fundamentals of uidstructure interactions 1943.3 Sound radiation from an innite plate wave/boundary matching

concepts 1973.4 Introductory radiation ratio concepts 2033.5 Sound radiation from free bending waves in nite plate-type structures 2073.6 Sound radiation from regions in proximity to discontinuities point and

line force excitations 216

x Contents

3.7 Radiation ratios of nite structural elements 2213.8 Some specic engineering-type applications of the reciprocity principle 2273.9 Sound transmission through panels and partitions 230

3.9.1 Sound transmission through single panels 2323.9.2 Sound transmission through double-leaf panels 241

3.10 The effects of uid loading on vibrating structures 2443.11 Impact noise 247

References 249Nomenclature 250

4 Noise and vibration measurement and control procedures 254

4.1 Introduction 2544.2 Noise and vibration measurement units levels, decibels and spectra 256

4.2.1 Objective noise measurement scales 2564.2.2 Subjective noise measurement scales 2574.2.3 Vibration measurement scales 2594.2.4 Addition and subtraction of decibels 2614.2.5 Frequency analysis bandwidths 263

4.3 Noise and vibration measurement instrumentation 2674.3.1 Noise measurement instrumentation 2674.3.2 Vibration measurement instrumentation 270

4.4 Relationships for the measurement of free-eld sound propagation 2734.5 The directional characteristics of sound sources 2784.6 Sound power models constant power and constant volume sources 2794.7 The measurement of sound power 282

4.7.1 Free-eld techniques 2824.7.2 Reverberant-eld techniques 2834.7.3 Semi-reverberant-eld techniques 2874.7.4 Sound intensity techniques 290

4.8 Some general comments on industrial noise and vibration control 2944.8.1 Basic sources of industrial noise and vibration 2944.8.2 Basic industrial noise and vibration control methods 2954.8.3 The economic factor 299

4.9 Sound transmission from one room to another 3014.10 Acoustic enclosures 3044.11 Acoustic barriers 3084.12 Sound-absorbing materials 3134.13 Vibration control procedures 320

xi Contents

4.13.1 Low frequency vibration isolation single-degree-of-freedomsystems 322

4.13.2 Low frequency vibration isolation multiple-degree-of-freedomsystems 325

4.13.3 Vibration isolation in the audio-frequency range 3274.13.4 Vibration isolation materials 3304.13.5 Dynamic absorption 3324.13.6 Damping materials 334References 335Nomenclature 336

5 The analysis of noise and vibration signals 342

5.1 Introduction 3425.2 Deterministic and random signals 3445.3 Fundamental signal analysis techniques 347

5.3.1 Signal magnitude analysis 3475.3.2 Time domain analysis 3515.3.3 Frequency domain analysis 3525.3.4 Dual signal analysis 355

5.4 Analogue signal analysis 3655.5 Digital signal analysis 3665.6 Statistical errors associated with signal analysis 370

5.6.1 Random and bias errors 3705.6.2 Aliasing 3725.6.3 Windowing 374

5.7 Measurement noise errors associated with signal analysis 377References 380Nomenclature 380

6 Statistical energy analysis of noise and vibration 383

6.1 Introduction 3836.2 The basic concepts of statistical energy analysis 3846.3 Energy ow relationships 387

6.3.1 Basic energy ow concepts 3886.3.2 Some general comments 3896.3.3 The two subsystem model 391

xii Contents

6.3.4 In-situ estimation procedures 3936.3.5 Multiple subsystems 395

6.4 Modal densities 3976.4.1 Modal densities of structural elements 3976.4.2 Modal densities of acoustic volumes 4006.4.3 Modal density measurement techniques 401

6.5 Internal loss factors 4076.5.1 Loss factors of structural elements 4086.5.2 Acoustic radiation loss factors 4106.5.3 Internal loss factor measurement techniques 412

6.6 Coupling loss factors 4176.6.1 Structurestructure coupling loss factors 4176.6.2 Structureacoustic volume coupling loss factors 4196.6.3 Acoustic volumeacoustic volume coupling loss factors 4206.6.4 Coupling loss factor measurement techniques 421

6.7 Examples of the application of S.E.A. to coupled systems 4236.7.1 A beamplateroom volume coupled system 4246.7.2 Two rooms coupled by a partition 427

6.8 Non-conservative coupling coupling damping 4306.9 The estimation of sound radiation from coupled structures using total

loss factor concepts 4316.10 Relationships between dynamic stress and strain and structural vibration

levels 433References 435Nomenclature 437

7 Pipe ow noise and vibration: a case study 441

7.1 Introduction 4417.2 General description of the effects of ow disturbances on pipeline noise

and vibration 4437.3 The sound eld inside a cylindrical shell 4467.4 Response of a cylindrical shell to internal ow 451

7.4.1 General formalism of the vibrational response and soundradiation 451

7.4.2 Natural frequencies of cylindrical shells 4547.4.3 The internal wall pressure eld 4557.4.4 The joint acceptance function 4587.4.5 Radiation ratios 460

xiii Contents

7.5 Coincidence vibrational response and sound radiation due to higherorder acoustic modes 461

7.6 Other pipe ow noise sources 4677.7 Prediction of vibrational response and sound radiation characteristics 4717.8 Some general design guidelines 4777.9 A vibration damper for the reduction of pipe ow noise and vibration 479


8 Noise and vibration as a diagnostic tool 488

8.1 Introduction 4888.2 Some general comments on noise and vibration as a diagnostic tool 4898.3 Review of available signal analysis techniques 493

8.3.1 Conventional magnitude and time domain analysis techniques 4948.3.2 Conventional frequency domain analysis techniques 5018.3.3 Cepstrum analysis techniques 5038.3.4 Sound intensity analysis techniques 5048.3.5 Other advanced signal analysis techniques 5078.3.6 New techniques in condition monitoring 511

8.4 Source identication and fault detection from noise and vibrationsignals 5138.4.1 Gears 5148.4.2 Rotors and shafts 5168.4.3 Bearings 5188.4.4 Fans and blowers 5238.4.5 Furnaces and burners 5258.4.6 Punch presses 5278.4.7 Pumps 5288.4.8 Electrical equipment 5308.4.9 Source ranking in complex machinery 5328.4.10 Structural components 5368.4.11 Vibration severity guides 539

8.5 Some specic test cases 5418.5.1 Cabin noise source identication on a loadhauldump vehicle 5418.5.2 Noise and vibration source identication on a large induction

motor 5478.5.3 Identication of rolling-contact bearing damage 5508.5.4 Flow-induced noise and vibration associated with a gas pipeline 554

xiv Contents

8.5.5 Flow-induced noise and vibration associated with a racingsloop (yacht) 557

8.6 Performance monitoring 5578.7 Integrated condition monitoring design concepts 559


Problems 566Appendix 1: Relevant engineering noise and vibration control journals 599Appendix 2: Typical sound transmission loss values and sound absorption

coefcients for some common building materials 600Appendix 3: Units and conversion factors 603Appendix 4: Physical properties of some common substances 605

Answers to problems 607

Index 621

Preface

The study of noise and vibration and the interactions between the two is now fastbecoming an integral part of mechanical engineering courses at various universities andinstitutes of technology around the world. There are many undergraduate text booksavailable on the subject of mechanical vibrations and there are also a relatively largenumber of books available on applied noise control. There are also several text booksavailable on fundamental acoustics and its physical principles. The books onmechanicalvibrations are inevitably only concerned with the details of vibration theory and donot cover the relationships between noise and vibration. The books on applied noisecontrol are primarily designed for the practitioner and not for the engineering student.The books on fundamental acoustics generally concentrate on physical acoustics ratherthan on engineering noise and vibration and are therefore not particularly well suitedto the needs of engineers. There are also several excellent specialist texts availableon structural vibrations, noise radiation and the interactions between the two. Thesetexts do not, however, cover the overall area of engineering noise and vibration, andare generally aimed at the postgraduate research student or the practitioner. There arealso a few specialist reference handbooks available on shock and vibration and noisecontrol these books are also aimed at the practitioner rather than the engineeringstudent.

The main purpose of this second edition is to attempt to provide the engineeringstudent with an updated unied approach to the fundamentals of engineering noise andvibration analysis and control. Thus, the main feature of the book is the bringing ofnoise and vibration together within a single volume instead of treating each topic inisolation. Also, particular emphasis is placed on the interactions between sound wavesand solid structures, this being an important aspect of engineering noise and vibration.The book is primarily designed for undergraduate students who are in the latter stagesof their engineering course. It is also well suited to the postgraduate student who is inthe initial stages of a research project on engineering noise and vibration and to thepractitioner, both of whom might wish to obtain an overview and/or a revision of thefundamentals of the subject.

This book is divided into eight chapters. Each of these chapters is summarised in theintroductory comments. Because of the wide scope of the contents, each chapter has

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xvi Preface

its own nomenclature list and its own detailed reference list. A selection of problemsrelating to each chapter is also provided at the end of the book together with solutions.Each of the chapters has evolved from lecture material presented by the rst author to(i) undergraduate mechanical engineering students at the University of WesternAustralia, (ii) postgraduate mechanical engineering students at the University ofWestern Australia, and (iii) practising engineers in industry in the form of short special-ist courses. The complete text can be presented in approximately seventy-two lectures,each of about forty-ve minutes duration. Suggestions for subdividing the text intodifferent units are presented in the introductory comments.

The authors hope that this book will be of some use to those who choose to purchaseit, and will be pleased and grateful to hear from readers who identify some of theerrors and/or misprints that will undoubtedly be present in the text. Suggestions formodications and/or additions to the text will also be gratefully received.

M. P. Norton and D. G. Karczub

Acknowledgements

This bookwould not have eventuated had it not been for several people who have playedan important role at various stages in our careers to date. Whilst these people have, inthe main, not had any direct input into the preparation of this book, their contributionsto the formulation of our thoughts and ideas over the years have been invaluable to saythe least.

Acknowledgements are due to several of our colleagues and the rst authorspostgraduate students at the University of Western Australia. These include GrahamForrester, Paul Keswick, Melinda Hodkiewicz, Pan Jie, Simon Drew and Gert Hoe-fakker.

Last, but not least, special acknowledgements are due to our families: our parentsfor encouraging us to pursue an academic career; and the rst authors wife Erica, forenduring the very long hours that we had to work during the gestation period of thissecond edition, and his young daughters, Caitlin and Sarah.

xvii

Introductory comments

A signicant amount of applied technology pertaining to noise and vibration analysisand control has emerged over the last thirty years or so. It would be an impossibletask to attempt to cover all this material in a text book aimed at providing the readerwith a fundamental basis for noise and vibration analysis. This book is therefore onlyconcerned with some of the more important fundamental considerations required for asystematic approach to engineering noise and vibration analysis and control, the mainemphasis being the industrial environment. Thus, this book is specically concernedwith the fundamentals of noise and vibration analysis for mechanical engineers, struc-tural engineers, mining engineers, production engineers, maintenance engineers, etc. Itembodies eight self-contained chapters, each of which is summarised here.

The rst chapter, on mechanical vibrations, is a review of some fundamentals. Thispart of the book assumes no previous knowledge of vibration theory. A large part ofwhat is presented in this chapter is covered very well in existing text books. The maindifference is the emphasis on the wavemode duality, and the reader is encouraged tothink in termsof bothwaves andmodesof vibration.As such, the introductory commentsrelate to both lumped parameter models and continuous system models. The sectionson the dynamics of a single oscillator, forced vibrations with random excitation andmultiple oscillator are presented using the traditional mechanical vibrations approach.The section on continuous systems utilises both the traditional mechanical vibrationsapproach and the wave impedance approach. It is in this section that the wavemodeduality rst becomes apparent. The wave impedance approach is particularly usefulfor identifying energy ow characteristics in structural components and for estimatingenergy transmission and reection at boundaries. A unique treatment of dynamic stressand strain has been included due to the importance of considering dynamic stress ina vibrating structure given the risk of fatigue failure. The treatment provided usestravelling wave concepts to provide a consistent theoretical framework for analysis ofdynamic stress in beams, plates and cylindrical shells. The contents of chapter 1 arebest suited to a second year or a third year course unit (based on a total course lengthof four years) on mechanical vibrations.

The second chapter, on sound waves, is a review of some fundamentals of physicalacoustics. Like the rst chapter, this chapter assumes no previous working knowledge

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xix Introductory comments

of acoustics. Sections are included on a classical analysis of the homogeneous waveequation, fundamental sound source models and the inhomogeneous wave equationassociated with aerodynamic sound, with particular attention being given to Lighthillsacoustic analogy and the PowellHowe theory of vortex sound. The distinction betweenthe homogeneous and the inhomogeneous acoustic wave equations is continually em-phasised. The chapter also includes a discussion on how reecting surfaces can affectthe sound power characteristics of sound sources (this important practical point isoften overlooked), and the use of one-dimensional acoustics to analyse sound trans-mission through a duct with mean ow (with applications including mufer/exhaustsystem design, air conditioning ducts, and pulsation control for reciprocating compres-sor installations) based on the use of acoustic impedance and travelling wave conceptsdeveloped earlier in the chapter. The contents of chapter 2 are best suited to a third yearor a fourth year course unit on fundamental acoustics.

The third chapter complements chapters 1 and 2, and is about the interactions betweensoundwaves and solid structures. It is very important for engineers to come to gripswiththis chapter, and it is the most important fundamental chapter in the book. Wavemodeduality concepts are utilised regularly in this chapter. The chapter includes discussionson the fundamentals of uidstructure interactions, radiation ratio concepts, soundtransmission through panels, the effects of uid loading, and impact noise processes.The contents of chapter 3 are best suited to a third year or a fourth year course unit. Theoptimum procedure would be to combine chapters 2 and 3 into a single course unit.

The fourth chapter is a fairly basic chapter on noise and vibration measurements andcontrol procedures. A large part of the contents of chapter 4 is readily available in thenoise and vibration control handbook literature with three exceptions: rstly, constantpower, constant volume and constant pressure sound source concepts are discussed inrelation to the effects of rigid, reecting boundaries on the sound power characteristicsof these sound sources; secondly, the economic issues in noise and vibration control arediscussed; and, thirdly, sound intensity techniques for sound power measurement andnoise source identication are introduced. The contents of chapter 4 are best suited to afourth year course unit on engineering noise and vibration control. By the very natureof the wide range of noise and vibration control procedures, several topics have had tobe omitted from the chapter. Some of these topics include outdoor sound propagation,community noise, air conditioning noise, psychological effects, etc.

The fth chapter is about the analysis of noise and vibration signals. It includesdiscussions on deterministic and random signals, signal analysis techniques, analogueand digital signal analysis procedures, random and bias errors, aliasing, windowing,and measurement noise errors. The contents of chapter 5 are best suited to a fourthyear unit on engineering noise and vibration noise control, and are best combined withchapters 4 and 8 for the purposes of a course unit.

The sixth and seventh chapters involve specialist topics which are more suitedto postgraduate courses. Chapter 6 is about the usage of statistical energy analysis

xx Introductory comments

procedures for noise and vibration analysis. This includes energy ow relationships,modal densities, internal loss factors, coupling loss factors, non-conservative coupling,the estimation of sound radiation fromcoupled structures, and relationships between dy-namic stress and strain and structural vibration levels. Chapter 7 is about ow-inducednoise and vibrations in pipelines. This includes the sound eld inside a cylindricalshell, the response of a cylindrical shell to internal ow, coincidence, and other pipeow noise sources. These two chapters can be included either as optional course unitsin the nal year of an undergraduate course, or as additional reading material for thecourse unit based on chapters, 4, 5 and 8.

The eighth chapter is a largely qualitative description of noise and vibration as adiagnostic tool (i.e. source identication and fault detection). Magnitude and timedomain signal analysis techniques, frequency domain signal analysis techniques, cep-strum analysis techniques, sound intensity analysis techniques, and other advancedsignal analysis techniques are described here. The chapter also includes ve specicpractical test cases; discussions on new techniques used in condition monitoring suchas expert systems and performance monitoring; and a review of design concepts fora plant-wide condition monitoring system integrating performance monitoring, safetymonitoring, and on-line and off-line condition monitoring. The contents of chapter 8are best suited to a fourth year unit on engineering noise and vibration noise control,and are best combined with chapters 4 and 5 for the purposes of a course unit.

Based upon the preceding comments, the following subdivision of the text is recom-mended for the purposes of constructing course units.(1) 2nd year unit mechanical vibration (14 hrs)

chapter 1 (sections 1.11.8)(2) 3rd year unit waves in structures and uids (14 hrs)

chapter 1 (section 1.9), chapter 2 (sections 2.1, 2.2)(3) 3rd or 4th year unit structuresound interactions (18 hrs)

chapter 2 (sections 2.3, 2.4), chapter 3(4) 4th year unit engineering noise control (18 hrs)

chapters 4, 5, 8(5) optional specialist units statistical energy analysis and pipe ow noise

and/or additional reading (8 hrs) chapters 6, 7.

Chapters 2 and 3 should be a prerequisite for the engineering noise control unit.

1 Mechanical vibrations: a reviewof some fundamentals

1.1 Introduction

Noise and vibration are often treated separately in the study of dynamics, and it is some-times forgotten that the two are inter-related i.e. they simply relate to the transfer ofmolecular motional energy in different media (generally uids and solids respectively).It is the intention of this book to bring noise and vibration together within a singlevolume instead of treating each topic in isolation. Central to this is the concept ofwavemode duality; it is generally convenient for engineers to think of noise in termsof waves and to think of vibration in terms of modes. A fundamental understanding ofnoise, vibration and interactions between the two therefore requires one to be able tothink in terms of waves and also in terms of modes of vibration.

This chapter reviews the fundamentals of vibrating mechanical systems with refer-ence to both wave and mode concepts since the dynamics of mechanical vibrationscan be studied in terms of either. Vibration deals (as does noise) with the oscillatorybehaviour of bodies. For this oscillatorymotion to exist, a bodymust possess inertia andelasticity. Inertia permits an element within the body to transfer momentum to adjacentelements and is related to density. Elasticity is the property that exerts a force on a dis-placed element, tending to return it to its equilibrium position. (Noise therefore relatesto oscillatory motion in uids whilst vibration relates to oscillatory motion in solids.)

Oscillating systems can be treated as being either linear or non-linear. For a linearsystem, there is a direct relationship between cause and effect and the principle ofsuperposition holds i.e., if the force input doubles, the output response doubles. Therelationship between cause and effect is no longer proportional for a non-linear system.Here, the system properties depend upon the dependent variables, e.g. the stiffness ofa non-linear structure depends upon its displacement.

In this book, only linear oscillating systemswhich are described by linear differentialequations will be considered. Linear system analysis adequately explains the behaviourof oscillatory systems provided that the amplitudes of the oscillations are very smallrelative to the systems physical dimensions. In each case, the system (possessing inertiaand elasticity) is initially or continuously excited in the presence of external forces

1

2 1 Mechanical vibrations

which tend to return it to its undisturbed position. Noise levels of up to about 140 dB(25m from a jet aircraft at take off) are produced by linear pressure uctuations.Mostengineering and industrial type noise sources (which are generally less than 140 dB)and the associated mechanical vibrations can therefore be assumed to behave in a linearmanner. Some typical examples are the noise and vibration characteristics of industrialmachinery, noise and vibration generated from high speed gas ows in pipelines, andnoise and vibration in motor vehicles.

The vibrations of linear systems fall into two categories free and forced. Freevibrations occur when a system vibrates in the absence of any externally applied forces(i.e. the externally applied force is removed and the system vibrates under the actionof internal forces). A nite system undergoing free vibrations will vibrate in one ormore of a series of specic patterns: for instance, consider the elementary case of astretched string which is struck at a chosen point. Each of these specic vibrationpatterns is called a mode shape and it vibrates at a constant frequency, which is calleda natural frequency. These natural frequencies are properties of the nite system itselfand are related to its mass and stiffness (inertia and elasticity). It is interesting to notethat if a system were innite it would be able to vibrate freely at any frequency (thispoint is relevant to the propagation of sound waves). Forced vibrations, on the otherhand, take place under the excitation of external forces. These excitation forces may beclassied as being (i) harmonic, (ii) periodic, (iii) non-periodic (pulse or transient), or(iv) stochastic (random). Forced vibrations occur at the excitation frequencies, and it isimportant to note that these frequencies are arbitrary and therefore independent of thenatural frequencies of the system. The phenomenon of resonance is encountered whena natural frequency of the system coincides with one of the exciting frequencies. Theconcepts of natural frequencies, modes of vibration, forced vibrations and resonancewill be dealt with later on in this chapter, both from an elastic continuum viewpointand from a macroscopic viewpoint.

The concept of damping is also very important in the study of noise and vibration.Energy within a system is dissipated by friction, heat losses and other resistances,and any damped free vibration will therefore diminish with time. Steady-state forcedvibrations can be maintained at a specic vibrational amplitude because the requiredenergy is supplied by someexternal excitation force.At resonance, it is only the dampingwithin a system which limits vibrational amplitudes. Both solids and uids possessdamping, and the response of a practical system (for example, a built-up plate orshell structure) to a sound eld is dependent upon both structural damping and acousticradiation damping. The concepts of structural dampingwill be introduced in this chapterand discussed in more detail in chapter 6 together with acoustic radiation damping.

A macroscopic (modal) analysis of the dynamics of any nite system requires anunderstanding of the concept of degrees of freedom. The degrees of freedomof a systemare dened as theminimumnumber of independent co-ordinates required to describe itsmotion completely.An independent particle in spacewill have three degrees of freedom,

3 1.2 Wave motion concepts

a nite rigid body will have six degrees of freedom (three position components andthree angles specifying its orientation), and a continuous elastic body will have aninnite number of degrees of freedom (three for each point in the body). There is alsoa one to one relationship between the number of degrees of freedom and the naturalfrequencies (or modes of vibration) of a system a system with p degrees of freedomwill have p natural frequencies and p modes of vibration. Plates, shell and acousticvolumes, for instance, have many thousands of degrees of freedom (and thereforenatural frequencies/modes of vibration) within the audible frequency range. As far asmechanical vibrations of structures (shafts, machine tools, etc.) are concerned, certainparts of the structures can often be assumed to be rigid, and the system can therefore bereduced to one which is dynamically equivalent to one with a nite number of degreesof freedom. Many mechanical vibration problems can thus be reduced to systems withone or two degrees of freedom.

An engineering description of the time response of vibrating systems can be obtainedby solving linear differential equations based upon mathematical models of variousequivalent systems. When a nite-number-of-degrees-of-freedom model is used, thesystem is referred to as a lumped-parameter system. Here, the real system is approx-imated by a series of rigid masses, springs and dampers. When an innite-number-of-degrees-of-freedom model is used, the system is referred to as a continuous or adistributed-parameter system. The differential equation governing the motion of thestructure is still the same as for the lumped-parameter system except that the mass,damping and stiffness distributions are now continuous and a wave-type solution to theequations can therefore be obtained. This wavemode duality which is central to thestudy of noise and vibration will be discussed in some detail at the end of this chapter.

1.2 Introductory wave motion concepts an elasticcontinuum viewpoint

A wave motion can be described as a phenomenon by which a particle is disturbedsuch that it collides with adjacent particles and imparts momentum to them. Aftercollision, the particles oscillate about their equilibrium positions without advancing inany particular direction, i.e. there is no nett transport of the particles in the medium. Thedisturbance, however, propagates through themedium at a speed which is characteristicof the medium, the kinematics of the disturbance, and any external body forces onthe medium. Wave motion can be described by using either molecular or particulatemodels. The molecular model is complex and cumbersome, and the particulate modelis the preference for noise and vibration analysis. A particle is a volume elementwhich is large enough to contain millions of molecules such that it is considered tobe a continuous medium, yet small enough such that its thermodynamic and acousticvariables are constant. Solids can store energy in shear and compression, hence several


types ofwaves are possible, i.e. compressional (longitudinal)waves, exural (transverseor bending) waves, shear waves and torsional waves. Fluids, on the other hand, can onlystore energy in compression. Wave motion is simply a balance between potential andkinetic energies, with the potential energy being stored in different forms for differentwave-types. Compressional waves store potential energy in longitudinal strain, andexural waves store it in bending strain.

Some elementary examples of wave motion are the propagation of sound in theatmosphere due to a source such as blast noise from a quarry, bending motions in ametal plate (such as a machine cover) which is mechanically excited, and ripples ina moving stream of water due to a pebble being thrown into it. In the case of thesound radiation associated with the blasting process at the quarry, the waves that aregenerated would travel both upwind and downwind. Likewise, the ripples in the streamwould also travel upstream and downstream. In both these examples the disturbancespropagate away from the source without being reected. For the case of the nite metalplate, a series of standing waves would be established because of wave reection at theboundaries. In each of the three examples there is, however, no nett transport of massparticles in the medium.

It is important to note at this stage that it is mathematically convenient to modelthe more general time-varying wave motions that are encountered in real life in termsof summations of numerous single frequency (harmonic) waves. The discussions inthis book will therefore relate to such models. The properties of the main types ofwave motions encountered in uids and solids are now summarised. Firstly, there aretwo different velocities associated with each type of harmonic wave motion. They are:(i) the velocity at which the disturbance propagates through the medium (this velocityis characteristic of the properties of the medium, the kinematics of the disturbance,and any external body forces on the medium), and (ii) the velocity of the oscillatingmass particles in the medium (this particle velocity is a measure of the amplitude ofthe disturbance which produces the oscillation, and relates to the vibration or soundpressure level that is measured). These two types of velocities which are associatedwith harmonic waves are illustrated in Figure 1.1 for the case of compressional andexural wavemotions on an arbitrary free surface. For the compressional (longitudinal)wave, there are alternate regions of expansion and compression of the mass particles,and the particle and wave velocities are in the same direction. The propagation of soundwaves in air and longitudinal waves in bars is typical of such waves. For the exural(transverse or bending) wave, the particle velocity is perpendicular to the direction ofwave propagation. The bending motion of strings, beams, plates and shells is typicalof this type of wave motion. It will be shown later on (in chapter 3) that bending wavesare the only type of structural waves that contribute directly to noise radiation andtransmission through structures (e.g. aircraft fuselages). The main reason for this is thatthe particle velocity (and structural displacement) is perpendicular to the direction ofwave propagation, as illustrated in Figure 1.1(b). This produces an effective disturbanceof the adjacent uid particles and results in an effective exchange of energy between


Fig. 1.1. Illustration of wave and particle velocities.

the structure and the uid. It will also be shown in chapter 3 that the bending wavevelocity varies with frequency whereas other types of wave velocities (compressional,torsional, etc.) do not.

Any wave motion can be represented as a function of time, of space or of both.Time variations in a harmonic wave motion can be represented by the radian (circular)frequency . This parameter represents the phase change per unit increase of time, and

= 2/T, (1.1)where T is the temporal period of the wave motion. This relationship is illustrated inFigure 1.2. The phase of a wave (at a given point in time) is simply the time shift relativeto its initial position. Spatial variations in such a wave motion are represented by thephase change per unit increase of distance. This parameter is called the wavenumber,k, where

k = /c, (1.2)


Fig. 1.2. Time variations for a simple wave motion.

Fig. 1.3. Spatial variations for a simple wave motion.

and c is the wave velocity (the velocity at which the disturbance propagates through themedium). This wave velocity is also sometimes called the phase velocity of the wave it is the ratio of the phase change per unit increase of time to the phase change per unitincrease of distance. Now, the spatial period of a harmonic wave motion is describedby its wavelegth, , such that

k = 2/. (1.3)This relationship is illustrated in Figure 1.3, and the analogy between radian frequency,, and wavenumber, k, can be observed.

If the wave velocity, c, of an arbitrary time-varying wave motion (a summationof numerous harmonic waves) is constant for a given medium, then the relationship


Fig. 1.4. Linear and non-linear dispersion relationships.

between and k is linear and therefore non-dispersive i.e. the spatial form of the wavedoes not change with time. On the other hand, if the wave velocity, c, is not constant(i.e. it varies with frequency), the spatial form of the wave changes with time and istherefore dispersive. It is a relatively straightforward exercise to show that a singlefrequency wave is non-dispersive but that a combination of several waves of differentfrequencies is dispersive if they each propagate at different wave velocities. Dispersionrelationships are very important in discussing the interactions between different typesof wave motions (e.g. interactions between sound waves and structural waves). Whena wave is non-dispersive, the wave velocity, c, is constant and therefore /k (thegradient of equation 1.2) is also constant. When a wave is dispersive, both the wavevelocity, c, and the gradient of the corresponding dispersion relationships are variables.This is illustrated in Figure 1.4. The gradient of the dispersion relationship is termedthe group velocity,

cg = /k, (1.4)and it quanties the speed at which energy is transported by the dispersive wave. It is thevelocity at which an amplitude function which is impressed upon a carrier wave packet(a time-varying wave motion which can be represented as a summation of numerousharmonic waves) travels, and it is of great physical importance. Plane sound wavesand compressional waves in solids are typical examples of non-dispersive waves, andexural waves in solids are typical examples of dispersive waves. If the dispersion rela-tionship of any two types of wave motions intersect, they then have the same frequency,wavenumber, wavelength andwavespeed. This condition (termed coincidence) allowsfor very efcient interactions between the two wave-types, and it will be discussed insome detail in chapters 3 and 7.


1.3 Introductory multiple, discrete, massspringdamper oscillatorconcepts a macroscopic viewpoint

When considering the mechanical vibrations of machine elements and structures onegenerally utilises either the lumped or the distributed parameter approach to study thenormal modes of vibration of the system. Engineers are often only concerned withthe estimation of the rst few natural frequencies of a large variety of structures, andthe macroscopic approach with multiple, discrete, massspringdamper oscillators istherefore more appropriate (as opposed to the wave approach). When modelling thevibrational characteristics of a structure via the macroscopic approach, the elementsthat constitute the model include a mass, a spring, a damper and an excitation. Theelementary, one-degree-of-freedom, lumped-parameter oscillator model is illustratedin Figure 1.5.

The excitation force provides the system with energy which is subsequently storedby the mass and the spring, and dissipated in the damper. The mass, m, is modelledas a rigid body and it gains or loses kinetic energy. The spring (with a stiffness ks) isassumed to have a negligiblemass, and it possesses elasticity.A spring force existswhenthere is a relative displacement between its ends, and the work done in compressing orextending the spring is converted into potential energy i.e. the strain energy is storedin the spring. The spring stiffness, ks, has units of force per unit deection. The damper(with a viscous-damping coefcient cv) has neither mass nor stiffness, and a dampingforce will be produced when there is relative motion between its ends. The damper isnon-conservative because it dissipates energy. Various types of damping models areavailable, and viscous damping (i.e. the damping force is proportional to velocity) isthemost commonly usedmodel. The viscous-damping coefcient, cv, has units of forceper unit velocity. Other damping models include coulomb (or dry-friction) damping,

Fig. 1.5. One-degree-of-freedom, lumped-parameter oscillator.

9 1.3 Massspringdamper concepts

Fig. 1.6. A simplied, multiple, discrete massspringdamper model of a human body standing ona vibrating platform.

hysteretic damping, and velocity-squared damping. Fluid dynamic drag on bodies, forexample, approximates to velocity-squared damping (the exact value of the exponentdepends on several other variables).

The idealised elements that make up the one-degree-of-freedom system form anelementary macroscopic model of a vibrating system. In general, the models are some-what more complex and involve multiple, discrete, massspringdamper oscillators.In addition, the masses of the various spring components often have to be accountedfor (for instance, a coil spring possesses both mass and stiffness). The low frequencyvibration characteristics of a large number of continuous systems can be approximatedby a nite number of lumped parameters. The human body can be approximated as alinear, lumped-parameter system for the analysis of low frequency (


model, and the subsequent effects of external shock and vibration can therefore beanalysed.

The concepts of multiple, discrete, massspringdamper models can be extended toanalyse the vibrations of continuous systems (i.e. systems with an innite number ofdegrees of freedom, natural frequencies, and modes of vibration) at higher frequenciesby re-modelling the structure in terms of continuous or distributed elements.Mathemat-ically, the problem is usually rst set up in terms of the wave equation and subsequentlygeneralised as an eigenvalue problem in terms of modal mass, stiffness and damping.The total response is thus a summation of the modal responses over the frequency rangeof interest.

It should be noted that the generally accepted convention in most of the literature isthe symbol c for both the wave (phase) velocity and the viscous-damping coefcient,and the symbol k for both the wavenumber and the spring stiffness. To avoid thisconicting use of symbols, the symbol c will denote the wave (phase) velocity, thesymbol cv, the viscous-damping coefcient, the symbol k the wavenumber, and thesymbol ks, the spring stiffness.

1.4 Introductory concepts on natural frequencies, modes of vibration,forced vibrations and resonance

Natural frequencies, modes of vibration, forced vibrations and resonance can be de-scribed both from an elastic continuum and a macroscopic viewpoint. The existenceof natural frequencies and modes of vibration relates to the fact that all real physicalsystems are bounded in space. A mode of vibration (and the natural frequency associ-ated with it) on a taut, xed string can be interpreted as being composed of two wavesof equal amplitude and wavelength travelling in opposite directions between the twobounded ends. Alternatively, it can be interpreted as being a standing wave, i.e. thestring oscillates with a spatially varying amplitude within the connes of a specicstationary waveform. The rst interpretation of a mode of vibration relates to the wavemodel, and the second to the macroscopic model. Both describe the same physicalmotion and are mathematically equivalent this will be illustrated in section 1.9.

The concepts discussed above can be illustrated by means of a simple example. Letus consider a piece of string which is stretched and clamped at its ends, as illustratedin Figure 1.7(a). The string is plucked at some arbitrary point and allowed to vibratefreely. At the instant that the string is plucked, a travelling wave is generated in eachdirection (i.e. towards each clamped end of the string). It is important to recognisethat, at this instant, the shape of the travelling wave is not that of a mode of vibration(Figure 1.7b) since a standing wave pattern has yet to be established. The travellingwaves move along the string until they meet the clamped ends, at which point they arereected. After these initial reections (one from each clamped end) there is a further

11 1.4 Natural frequencies and resonance

Fig. 1.7. Schematic illustration of travelling and standing waves for a stretched string.

short time interval during which time the total motion along the stretched string isthe resultant effect of the incident waves and the reected waves which have yet toreach the starting point. During this time, the standing wave pattern has still not yetbeen established. Once the reected waves meet, a situation arises where there is acombination of waves of equal amplitudes travelling in opposite directions. This givesrise to a stationary vibration with a spatially dependent amplitude. The standing wavepattern (mode of vibration) is thus established and the wave propagation process keepson repeating itself. Depending on how the string is excited, different modes of vibrationwill be excited the fundamental mode ismost easily observed and this is assumed to bethe case for the purposes of this example. The frequency of the resulting standing waveis a natural frequency of the string, and its shape is a mode of vibration, as illustratedin Figure 1.7(c) each point on the string vibrates transversely in simple harmonicmotion, with the exception of the nodal points which are at rest. The points of zero and


maximum amplitude for the standing wave are xed in space and the relative phase ofthe displacements at various points along the string takes on values of 0 or i.e. thereis continuity of phase between the incident and reected waves. Because the string isin free vibration and it possesses damping (all real physical systems possess dampingto a smaller or greater extent), its amplitude of vibration will decay with time. Thestanding wave pattern (or mode shape) will subsequently also decay with time. This isillustrated in Figure 1.7(d). If the string were continuously excited at this frequency bysome external harmonic force, it would resonate continuously at this mode shape and itsamplitude would be restricted only by the amount of damping in the string resonanceoccurs when some external forced vibration coincides with a natural frequency.

The travelling wave concepts illustrated in Figure 1.7(b) are not considered in themacroscopic viewpoint, but the standing wave concepts illustrated in Figures 1.7(c) and(d) are i.e. the macroscopic viewpoint relates directly to the various standing wavepatterns that are generated due to the physical constraints on the oscillating system.Lumped-parameter models can subsequently be used to study the various natural fre-quencies, forced vibrations and resonances, and the vibrations can be analysed in termsof normal modes. Alternatively, the vibrations can be studied from the wave motionviewpoint. When considering this viewpoint, it should be recognised that the physicalconstraints upon a real system produce four types of wave motion diffraction, re-ection, refraction and scattering. Reection is the wave phenomenon which results inthe production of natural frequencies and is therefore of great practical signicance.Because of the phenomenon of reection, nite (or bounded) structures can only vi-brate freely at specic natural frequencies. Innite (unbounded) structures on the otherhand, can vibrate freely at any frequency. This point is particularly important whenconsidering the interactions of sound elds (waves in an unbounded uid medium) andbounded structures.

1.5 The dynamics of a single oscillator a convenient model

In this section, oscillatorymotion ismathematically described for the simplest of cases a macroscopic, single-degree-of-freedom, massspring oscillator. The single oscillatoris a classical problem, and it is covered in great detail in a variety of texts onmechanicalvibrations. Some of the more important results relating to free and forced vibrations ofthe single oscillator will be presented in this section, and most of these results will beused repeatedly throughout this text.

1.5.1 Undamped free vibrations

It can be intuitively recognised that the motion of a rigid mass, m, on the end of amassless springwith a stiffness ks, as illustrated in Figure 1.8(a), will be of an oscillatorynature.

13 1.5 Single oscillators

Fig. 1.8. Free-body diagrams for undamped free vibrations of a single oscillator.

The simple oscillatory system has one degree of freedom and its motion can bedescribed by a single co-ordinate, x . The spring is stretched by an amount static due tothe mass, m, and this stretched position is dened as the equilibrium position for thesystem. All dynamic motion is subsequently about this equilibrium position. A simpleforce balance during equilibrium shows that

mg = ksstatic, (1.5)where g is the gravitational acceleration constant. If the mass is now displaced below itsequilibrium position (as illustrated in Figure 1.8b) and released, its equation of motioncan be obtained from Newtons second law. Hence,

mx = mg ksx ksstatic, (1.6)and therefore

mx + ksx = 0. (1.7)Note that x = 2x/t2 and x = x/t , etc.If a constant, n = (ks/m)1/2, is now dened, thenx + 2nx = 0. (1.8)Equation (1.8) is an important homogeneous, second-order, linear, differential equationwith a solution of the form x(t) = A sint + B cost . Differentiation and substitutioninto equation (1.8) readily show that this is a solution if = n . The complete generalsolution is thus

x(t) = A sinnt + B cosnt, (1.9)where A and B are arbitrary constants (evaluated from the initial conditions) and n isthe radian (circular) frequency at which the system oscillates. It is, in fact, the naturalfrequency of the massspring system.


If the mass has an initial displacement x0 and an initial velocity v0 at time t = 0,then equation (1.9) becomes

x(t) = (v0/n) sinnt + x0 cosnt. (1.10)Now, if x0 = X sin and v0/n = X cos , such that X is the amplitude of the

motion and is the initial phase angle, then

X = {x20 + (v0/n)2}1/2, (1.11)tan = (nx0)/v0, (1.12)and

x(t) = X sin(nt + ). (1.13)Equation (1.13) illustrates that the motion of the massspring system is harmonic, i.e.the cycle of the motion is repeated in time t = T such that nT = 2 . Thus,T = 2/n = 2 (m/ks)1/2, (1.14)and the natural frequency of vibration, fn , is

fn = 12

(ksm

)1/2= 1

2

(g

static

)1/2. (1.15)

Equation (1.15) illustrates how the undamped natural frequency of a massspringoscillator can be obtained simply from its static deection. The equation is widelyused in practice to estimate the fundamental vertical natural frequency during vibrationisolation calculations for various types of machines mounted on springs.

Equation (1.8) can also be solved by using complex algebra, and it is instructive todemonstrate this at this point. Complex algebrawill be used later on in this book both fornoise and vibration analyses. The solution to equation (1.8), using complex algebra, is

x(t) = A ein t + B ein t . (1.16)It should be noted here that complex quantities are presented in bold type in this book.The complex constants A and B are complex conjugates and can be obtained from theinitial conditions (initial displacement x0 and initial velocity v0 at time t = 0). It caneasily be shown that

A = {x0 i(v0/n)}/2, (1.17a)and

B = {x0 + i(v0/n)}/2. (1.17b)Substitution of A and B into equation (1.16) yields

x(t) = (v0/n) sinnt + x0 cosnt = x(t), (1.18)


which is in fact equation (1.10). Both the initial conditions are real, therefore thesolution also has to be real. When using complex algebra one therefore only needs toconcern oneself with the real part of the complex solution.

It should be noted that if z = x + iy is a complex number, then x = (z + z)/2 =Re(z) where Re stands for the real part of the complex quantity z, and represents thecomplex conjugate.

1.5.2 Energy concepts

The equation of motion (equation 1.8) can also be obtained from energy concepts.Energy is conserved for the paticular case of free undamped vibrations since there areno excitation or damping forces present. This energy therefore is the sum of the kineticenergy of the mass and the potential energy of the spring. If damping were introduced,an energy dissipation functionwould have to be included. For free undamped vibrations,

T + U = constant, (1.19a)and

d(T + U )/dt = 0, (1.19b)where T is the kinetic energy and U is the potential energy.

The kinetic energy of the mass is established by the amount of work done on themass in moving it over a specied distance. Hence,

T = x0

mdv

dtdx =

v0

mdx

dtdv =

v0

mv dv,

and thus

T = 12mv2 = 12mx2. (1.20)The potential energy of the spring is associated with its stiffness i.e.

U = x0

ksx dx = 12ksx2. (1.21)

By substituting equations (1.20) and (1.21) into equation (1.19), and dening n =(ks/m)1/2 as before, the equation of motion (equation 1.8) can be obtained.

When the mass is at its maximum displacement, xmax, it is instantaneously at restand therefore has no kinetic energy. Since there is conservation of energy, the totalenergy is therefore now equal to the maximum potential energy. Alternatively, whenx = 0 (i.e. themass passes through its equilibrium position), the system has no potentialenergy. The kinetic energy is a maximum at this point and the mass has maximumvelocity. Thus,

Tmax = Umax. (1.22)


Later on in this book, the spatial and time averages of a variety of different signals,rather than the instantaneous values, will be considered. For instance, one might beconcerned with the average vibration on a machine cover. Here, one would have to takevibration measurements at numerous locations and over some specied time interval tosubsequently obtain an averaged vibration level. Average values are particularly usefulwhen using energy concepts to solve noise and vibration problems. For an arbitrarysignal, x(t), the mean-square value over a time period, T , is

x2 = 1T

T0

x2(t) dt, (1.23)

and the root-mean-square value, xrms, is

xrms = x21/2. (1.24)If the signal, x(t), is harmonic (i.e. x(t) = X sin(t + )), then

xrms = x21/2 = X2. (1.25)

Equation (1.25) relates to a time-averaged signal at a single point in space and its phase, , has been averaged out. The spatial average is subsequently obtained by an arithmeticaverage of a number of point measurements. It is represented in this book by . Aspace and time-averaged signal is thus represented as .

Following on from the above discussion, it can be shown that the average kinetic andpotential energies are equal. The time-averaged kinetic energy of a vibrating mass is

T = mv2

2= mV

2

4, (1.26)

where v is its velocity and V is the velocity amplitude. (Note that v2 = V 2/2.) Now,the time-averaged potential energy of the spring is

U = ksx2

2= ks

(V 2/2n

)4

= mV2

4, (1.27)

since n = (ks/m)1/2, and x2 = v2/2n . Thus,T = U , (1.28)and the total energy of the undamped system is mV 2/2, or mv2.

1.5.3 Free vibrations with viscous damping

All real systems exhibit damping energy is lost and the vibration decays with timewhen the excitation is removed. The exact description of the damping force associatedwith energy dissipation is difcult it could be a function of displacement, velocity,stress or some other factors. In general, damping can be modelled by incorporating


Fig. 1.9. Schematic illustration of an exponentially damped, sinusoidal motion.

an arbitrary function, a(t), which decreases with time and a constant, , which isrelated to the amount of damping into the equation of motion for a massspring system(equation 1.13). Thus,

x(t) = a(t)X sin(nt + ). (1.29)

The viscous-damping model, which is proportional to the rst power of velocity, iscommonly used in engineering to model the vibrational characteristics of real systems.With viscous damping, it can be shown that the function a(t) is an exponential suchthat

x(t) = XT et sin(nt + ), (1.30)

and the resulting motion lies between two exponentials a(t) = XT et , as illustratedin Figure 1.9. XT is the amplitude of the transient, damped, oscillatory motion. Theviscous-damping force is represented by

Fv = cv x, (1.31)

where cv is the viscous-damping coefcient. Symbolically, it is designated by a dashpot(Figure 1.5). For free vibrations with damping, the equation of motion now becomes(from Newtons second law)

mx + cv x + ksx = 0. (1.32)This is a homogeneous second-order differential equation with a solution of the formx = A est , where A and s are constants. Substitution of this solution into equation (1.32)yields

ms2 + cvs + ks = 0. (1.33)


This is the characteristic equation of the system and it has two roots:

s1,2 = 12m

{cv (c2v 4mks)1/2}. (1.34)Hence,

x(t) = B1 es1t + B2 es2t , (1.35)where B1 and B2 are arbitrary constants which are evaluated from the initial conditions.The following terms are now dened:

(i)2n = ks/m; (ii) cv/m = 2n; (iii) = cv/(4mks)1/2,where n is the natural frequency (as dened previously), and is the ratio of theviscous-damping coefcient to a critical viscous-damping coefcient. The criticalviscous-damping coefcient is the value of cv which reduces the radical to zero inequation (1.34), i.e. cvc = (4mks)1/2, and therefore = cv/cvc. Equations (1.32)(1.34)can now be re-expressed as

x + 2n x + 2nx = 0, (1.36)s2 + 2ns + 2n = 0, (1.37)s1,2 = n n( 2 1)1/2. (1.38)Three cases of interest arise. They are (i) > 1, (ii) < 1, and (iii) = 1.

(i) > 1. Here, the roots s1,2 are real, distinct and negative since ( 2 1)1/2 < ,and the motion is overdamped. The general solution (equation 1.35) becomes

x(t) = B1 e{+( 21)1/2}n t + B2 e{( 21)1/2}n t , (1.39)and the overdamped motion is not oscillatory, irrespective of the initial conditions.Because the roots are negative, the motion diminishes with increasing time and isaperiodic, as illustrated in Figure 1.10. It is useful to note that, for initial conditions x0

Fig. 1.10. Aperiodic, overdamped, viscous-damped motion ( > 1.0).


and v0, the constants B1 and B2 are

B1 = x0n{ + (2 1)1/2} + v0

2n( 2 1)1/2 ,

and

B2 = x0n{ (2 1)1/2} v0

2n( 2 1)1/2 .

(ii) > 1. Here, the roots are complex conjugates, the motion is underdamped andthe general solution (equation 1.35) becomes

x(t) = en t{B1 ei(1 2)1/2n t + B2 ei(1 2)1/2n t}= XT en t sin{(1 2)1/2nt + }. (1.40)

The underdamped motion is oscillatory (cf. equation 1.30) with a diminishing ampli-tude, and the radian frequency of the damped oscillation is

d = n(1 2)1/2 = n. (1.41)

The underdamped oscillatory motion (commonly referred to as damped oscillatorymotion) is illustrated in Figure 1.11. The amplitude, XT, of the motion and the initialphase angle can be obtained from the initial displacement, x0, of the mass and itsinitial velocity, v0, and they are, respectively,

XT = {(x0d)2 + (v0 + nx0)2}1/2

d, (1.42a)

and

= tan1 x0dv0 + nx0 . (1.42b)

Fig. 1.11. Oscillatory, underdamped, viscous-damped motion ( < 1.0).


(iii) = 1. Here, both roots are equal to n , and the system is described as beingcritically damped. Physically, it represents a transition between the oscillatory and theaperiodic damped motions. The general solution (equation 1.35) becomes

x(t) = (B1 + B2) en t . (1.43)Because of the repeated roots, an additional term of the form ten t is required to retainthe necessary number of arbitrary constants to satisfy both the initial conditions. Thus,the general solution becomes

x(t) = (B3 + B4t) en t , (1.44)where B3 and B4 are constants which can be evaluated from the initial conditions. Forinitial conditions x0 and v0, the constants are

B3 = x0,and

B4 = v0 + nx0.Critically damped motion is the limit of aperiodic motion and the motion returns torest in the shortest possible time without oscillation. This is illustrated in Figure 1.12.The property of critical damping of forcing the system to return to rest in the shortestpossible time is a useful one, and it has many practical applications. For instance, themoving parts of many electrical instruments are critically damped.

Some useful general observations can now be made about damped free vibrations.They are:(i) x(t) oscillates only if the system is underdamped ( < 1);(ii) d is always less than n;

Fig. 1.12. Aperiodic, critically damped, viscous-damped motion ( = 1).


(iii) the motion x(t) will eventually decay regardless of the initial conditions;(iv) the frequency d and the rate of the exponential decay in amplitude are properties

of the system and are therefore independent of the initial conditions;(v) for < 1, the amplitude of the damped oscillator is XT et , where = n .The parameter is related to the decay time (or time constant) of the damped

oscillator the time that is required for the amplitude to decrease to 1/e of its initialvalue. The decay time is

= 1/ = 1/n. (1.45)If < n themotion is underdamped and oscillatory; if > n themotion is aperiodic;and if = n , the motion is critically damped and aperiodic. The case when < n(i.e. < 1) is generally of most interest in noise and vibration analysis.

The equation for underdamped oscillatory motion (equation 1.40) can also be ex-pressed as a complex number. It is the imaginary part of the complex solution

x(t) = XT et eidt , (1.46)where XT = XT ei . The imaginary part of the solution is used here because equa-tion (1.40) is a sine function. If it were a cosine function the real part of the complexsolution would have been used. Equation (1.46) can be rewritten as

x(t) = XT ei(d+i)t = XT eidt , (1.47)where d = d + i is the complex damped radian frequency. The complex dampedradian frequency thus contains information about both the damped natural frequencyof the system and its decay time.

The equation of motion for damped free vibrations (equation 1.32) can also beobtained from energy concepts by incorporating an energy dissipation function into theenergy balance equation. Hence,

d(T + U )/dt = , (1.48)where is power (the negative sign indicates that power is being removed from thesystem). Power is force velocity, and the power dissipated from a systemwith viscousdamping is

= Fv x = cv x2. (1.49)

1.5.4 Forced vibrations: some general comments

So far, only the free vibrations of systems have been discussed.A linear systemvibratingunder the continuous application of an input excitation is now considered. This isillustrated schematically in Figure 1.13. In general, there can be many input excitationsand output responses, together with feedback between some of the inputs and outputs.Some of these problems will be discussed in chapter 5.


Fig. 1.13. A single inputoutput linear system.

It is useful at this stage to consider the different types of input excitations and outputresponses that can be encountered in practice. The input or output of a vibration systemis generally either a force of some kind, or a displacement, or a velocity, or an accelera-tion. The time histories of the input and output signals can be classied as being eitherdeterministic or random. Deterministic signals can be expressed by explicit mathemat-ical relationships, whereas random signals have to be described in terms of probabilitystatements and statistical averages. Typical examples of deterministic signals are thosefrom electrical motors, rotating machinery and pumps. In these examples, a few spe-cic frequencies generally dominate the signal. Some typical random signals includeacoustical pressures generated by turbulence, high speed gas ows in pipeline systems,and the response of a motor vehicle travelling over a rough road surface. Here, thefrequency content of the signals is dependent upon statistical parameters. Figure 1.14is a handy ow-chart which illustrates the different types of input and output signals(temperatures, pressures, forces, displacements, velocities, accelerations, etc.) that canbe encountered in practice. Therefore, the chart is not limited to only noise and vibra-tion problems. It is worth reminding the reader at this stage that, in addition to all thesevarious types of input excitation and output response functions, a systems responseitself can, in principle, be either linear or non-linear. As mentioned in the introduction,only linear systems will be considered in this book.

1.5.5 Forced vibrations with harmonic excitation

Now consider a viscous-damped, springmass system excited by a harmonic (sinu-soidal) force, F(t) = F sint , as illustrated in Figure 1.15. As mentioned in the previ-ous sub-section, both the input and output to a system can be one of a range of functions(force, displacement, pressure, etc.). In this sub-section, an input force and an outputdisplacement shall be considered initially. The differential equation of motion can bereadily obtained by applying Newtons second law to the body. It is

mx + cv x + ksx = F sint. (1.50)This is a second-order, linear, differential equation with constant coefcients. Thegeneral solution is the sum of the complementary function (F sint = 0) and theparticular integral. The complementary function is just the damped, free, oscillator.This part of the general solution decays with time, leaving only the particular solutionto the particular integral. This part of the general solution (the particular solution)is a steady-state, harmonic, oscillation at the forced excitation frequency. The output


Fig. 1.14. Flow-chart illustrating the different types of input and output signals.

Fig. 1.15. Free-body diagram for forced vibrations with harmonic excitation.


displacement response, x(t), lags the input force excitation, F(t), by a phase angle, ,which varies between 0 and 180 such that

x(t) = X sin(t ). (1.51)It should be noted here that the symbols XT and relate to the transient part of thegeneral solution (equation 1.40), whereas X and relate to the steady-state part. Thegeneral solution (total response) is thus the sum of equations (1.40) and (1.51).

Phasors, Laplace transforms and complex algebra can all be used to study the be-haviour of an output, steady-state, response for a given input excitation. The complexalgebra method will be adopted in this book. This technique requires both the inputforce and the output displacement to be represented as complex numbers. Since theforcing function is a sine term, the imaginary part will be used if it were a cosine, thereal part would have been used. Thus,

F sint = Im[F eit ], (1.52a)and

X sin(t ) = Im[X eit ], (1.52b)where F is the complex amplitude of F(t) and X is the complex amplitude of x(t), i.e.

F = F ei0 = F, (1.53a)and

X = X ei. (1.53b)The output displacement is thus

x(t) = X sin(t ) = Im[X ei(t)] = Im[X eit ]. (1.54)The complex displacement,X, contains information about both the amplitude and phaseof the signal. By replacing x(t) byX eit and F sint byF eit in the equation ofmotion(equation 1.50), with the clear understanding that nally only the imaginary part of thesolution is relevant, one gets

m2X eit + icvX eit + ksX eit = F eit . (1.55)Several important comments can be made in relation to equation (1.55). They are:(i) the displacement lags the excitation force by a phase angle , which varies be-

tween 0 and 180;(ii) the spring force is opposite in direction to the displacement;(iii) the damping force lags the displacement by 90 and is opposite in direction to the

velocity;(iv) the inertia force is in phase with the displacement and opposite in direction to the

acceleration.


Solving for X yields

X = F{ks m2 + icv} . (1.56)

The output displacement amplitude, X , is obtained by multiplying equation (1.56) byits complex conjugate. Hence,

X = F{(ks m2)2 + (cv)2}1/2 . (1.57)

The phase angle, , is obtained by replacing X by Xei and F by F ei0 = F inequation (1.56) and equating the imaginary parts of the solution to zero. Hence,

= tan1 cvks m2 . (1.58)

Equations (1.57) and (1.58) represent the steady-state solution. They can be non-dimensionalised by dening X0 = F/ks as the zero frequency (D.C.) deection ofthe springmassdamper system under the action of a steady force, F . In addition,n = (ks/m)1/2; = cv/cvc; cvc = 2mn as before. With these substitutions,X

X0= 1

[{1 (/n)2}2 + {2/n}2]1/2 , (1.59)

and

= tan1 2/n1 (/n)2 . (1.60)

Equations (1.59) and (1.60) are plotted in Figures 1.16(a) and (b), respectively.The main observation is that the damping ratio, , has a signicant inuence on theamplitude and phase angle in regions where n . The magnication factor (i.e. theamplitude displacement ratio), X/X0, can be greater than or less than unity dependingon the damping ratio, , and the frequency ratio, /n . The phase angle, , is simplya time shift (t = /) of the output displacement, x(t), relative to the force excitation,F(t). It varies from 0 to 180 and is a function of both and /n . It is useful to notethat, when = n, = 90. This condition is generally referred to as phase resonance.

The general solution for the motion of the massspringdamper system is, as men-tioned earlier, the sum of the complementary function (transient solution, i.e. equa-tion 1.40) and the particular integral (steady-state solution, i.e. equation 1.51). It istherefore

x(t) = XT en t sin(dt + ) + X sin(t ). (1.61)The transient part of the solution always decays with time and one is generally onlyconcerned with the steady-state part of the solution. There are some exceptions tothis rule, and a typical example involves the initial response of rotating machineryduring start-up. Here, one is concerned about the initial transient response before thesteady-state condition is attained.


Fig. 1.16. (a) Magnication factor for a one-degree-of-freedom, massspringdamper system;(b) phase angle for a one-degree-of-freedom, massspringdamper system.

It can be shown that the steady-state amplitude, X , is a maximum when

n= (1 2 2)1/2. (1.62)

The maximum value of X is

X r = X02 (1 2)1/2 , (1.63)

and the corresponding phase angle at X = X r is

= tan1 (1 22)1/2

. (1.64)

This condition is called amplitude resonance. In general, it is different from phaseresonance ( = 90). If > 1/2, the maximum value of X would occur at = 0;i.e. it would be due to the zero frequency deection of the massspringdamper. Thisis illustrated in Figure 1.16(a).


Fig. 1.17. Half-power bandwidth and half-power points for a linear oscillator.

For most practical situations, however, is small (


Table 1.1. Different types of frequencyresponse functions.

Displacement/force ReceptanceForce/displacement Dynamic stiffnessVelocity/force MobilityForce/velocity ImpedanceAcceleration/force InertanceForce/acceleration Apparent mass

The Q factor is also related to the decay time (see equation 1.45) such that

Q = n2

. (1.70)

So far in this sub-section, solutions have been sought for the output steady-statedisplacement, X . The complex ratio of the output displacement to the input force,X/F, (i.e. equation 1.56) is a frequency response function and it is commonly referredto as a receptance. There are a range of different forceresponse relationships thatare of general engineering interest. The more commonly used ones are presented inTable 1.1. In many applications in noise and vibration, the impedance (force/velocity;F/V), and the mobility (velocity/force; V/F), are often of interest. Expressions similarto equation (1.56) can be readily obtained by solving the equation of motion. It is arelatively straightforward exercise to show that the mechanical impedance, F/V, of themassspringdamper system in Figure 1.15 is

FV

= Zm = cv + i(m ks/). (1.71)

The real part of the impedance is called the mechanical resistance, and the imaginarypart is called the mechanical reactance (m is the mass or the mechanical inertanceterm, and/ks is the mechanical compliance term). If the mechanical resistance term isdominant, the systems response is damping controlled; if themechanical inertance termis dominant, the systems response is mass controlled; if the mechanical complianceterm is dominant, the systems response is stiffness controlled. In noise and vibrationcontrol it is often important to identify which of the three (mass, stiffness or damping)dominates.

Frequency response functions such as impedance and mobility are important toolsand will be used throughout this book. For a known input, the knowledge of the fre-quency response function of a system allows for the estimation of the output response.In complex systems, impedance and mobility concepts are very useful for analysingvibrational energy and power ow. They are used extensively in the dynamic analy-sis of structures and can be applied to either lumped-parameter, oscillator models orwave-motion models.


The instantaneous power developed by a force F(t) = F sint producing a displace-ment x(t) = X sin(t ) on the system in Figure 1.15 is

= F(t)dxdt

= XF sint cos(t )= 12XF{sin + sin(2t + )}. (1.72)

The rst term in the brackets is a constant and it represents the steady ow of work perunit time. The second term represents the uctuating component of power. It averagesto zero over any time interval which is a multiple of the period. The time-averagedpower is thus

= 12XF sin. (1.73)Now, from equation (1.71),

Zm = |Zm| ={c2v + (m ks/)2

}1/2, (1.74)

and

sin = cvZm

. (1.75)

Thus, the time-averaged power is

= XFcv2Zm

= V Fcv2

V

F= 12V 2cv, (1.76)

where V = |V|.The time-averaged power can also be obtained by using complex numbers, and it is

instructive to obtain it this way at this stage. It is

= 1T

T0

(t) dt. (1.77)

Hence,

= 2

2/0

Re[F eit ] Re[V eit ] = 12Re[FV]. (1.78)

It should be noted that the real parts of force and velocity are used here. The reason forthis is explained in section 1.7.

can now be represented in terms of the mechanical impedance, Zm, where = 12 |F|2 Re

[Z1m

] = 12 |V|2 Re[Zm] = 12V 2cv. (1.79)Equations (1.76) and (1.79) represent the time-averagedpower delivered to the oscillatorby the force. During steady-state oscillations this has to equal the power dissipated bythe damper. The maximum power delivered (and dissipated) occurs when sin = 1,i.e. at resonance.


1.5.6 Equivalent viscous-damping concepts damping in real systems

Damping exists in all real systems and very rarely is it viscous viscous damping onlyexists when the velocity between two lubricated surfaces is sufciently low such thatlaminar ow conditions exist. Many different types of damping can exist in practice,and the most commonly encountered include structural (hysteretic) damping, coulomb(dry-friction) damping, and velocity-squared (aerodynamic drag) damping. Becausemost mechanical systems are essentially lightly damped (i.e. the effect of damping isinsignicant except near a resonance), it is possible to obtain approximate models ofnon-viscous damping in terms of equivalent viscous dampers. This subsequently allowsfor the continued usage of the simple vibration models, based upon viscous damping,developed in the last sub-section.

In proceeding to develop the concept of an equivalent viscous damper, one rst needsto evaluate the energy dissipated per cycle by the damping force. The criteria for equiva-lence between the actual dampingmechanism and viscous damping are (i) equal energydissipation per cycle of vibration, and (ii) similar harmonic relative displacements.

For viscous damping, the energy dissipated per cycle by the damping force is

Ud = T0

cv xdx

dtdt = cvX22

2/0

cos2(t ) dt = cvX2. (1.80)

The equivalent viscous damping can subsequently be determined from the equation

Ud = cveqX2, (1.81)

where Ud has to be evaluated for the particular type of damping force.The principles of equivalent viscous damping can best be illustrated by means of an

example. The form of damping that is most relevant to engineering noise and vibrationcontrol is structural damping, and this will now be considered. When structural ma-terials such as steel or aluminium are cyclically stressed, energy is dissipated withinthe material. A hysteresis loop is formed, hence the commonly used term hystereticdamping. Experimental observations clearly show that the energy dissipated per cycleof stress is proportional to the square of the strain amplitude. The constant of propor-tionality is generally only valid over specic ranges of frequency and temperature i.e. there will be different constants of proportionality over different frequency andtemperature ranges. Hence, for a given frequency and temperature range,

Ud = X2, (1.82)

where X is the displacement amplitude. This can now be equated to equation (1.81) toobtain the equivalent viscous-damping coefcient, and thus

cveq =

. (1.83)


The complex differential equation of motion for a one-degree-of-freedom system istherefore

m2X eit + i

X eit + ksX eit = F eit . (1.84)Equation (1.84) can be re-written as

m2X + ks(1 + i)X = F, (1.85)where

= ks

(1.86)

is the structural loss factor and

ks = ks(1 + i) (1.87)is the complex stiffness.

The structural loss factor, , is an important parameter which is extensively used instructural dynamics. It will be discussed in some detail in chapter 6. The analysis insub-section 1.5.5 for the magnication factor and the phase angle can now be repeated,and it can be readily shown that for structural damping

X

X0= 1

[{1 (/n)2}2 + 2]1/2 , (1.88)

and

= tan1 1 (/n)2 . (1.89)

For a viscous-damped system, X/X0 = 1/2 at resonance. Hence

= 2 = 1Q

, (1.90)

i.e. the structural loss factor is twice the viscous damping ratio and inversely propor-tional to the quality factor.

The two other most commonly encountered forms of non-viscous damping arecoulomb (dry-friction) and velocity-squared damping. The analyses for obtaining theequivalent viscous-damping coefcients are available in most fundamental texts onme-chanical vibrations (see reference list at the end of this chapter) and therefore only theresults will be presented here. The equivalent viscous-damping coefcient for coulombdamping is

cveq = 4FNX

, (1.91)

where is the coefcient of friction and FN is the normal force. The equivalent viscous-damping coefcient for velocity-squared (aerodynamic) damping is

cveq = 83

CFX, (1.92)


where CF is a constant which is related to the drag coefcient, CD, the exposed sur-face area, A, of the body, and the density, , of the uid in which it is immersed(i.e. CF = CDA/2). It should be noted that both of these types of damping are non-linear, i.e. they are functions of the amplitude of the vibration.

1.5.7 Forced vibrations with periodic excitation

Harmonically related periodic signals are often encountered in forces in machinery,and the vibration models developed in the previous sub-sections therefore need to begeneralised. Periodic signals are deterministic and can thus be expressed by explicitmathematical relationships i.e. they can be developed into a Fourier series.

A function, F(t), is periodic if F(t) = F(t + T ) where T = 2/. The Fourierseries expansion of F(t) is

F(t) = a02

+

n=1,2(an cos nt + bn sin nt), (1.93)

where

a0 = 2T

T0

F(t) dt, (1.94a)

an = 2T

T0

F(t) cos nt dt, (1.94b)

and

bn = 2T

T0

F(t) sin nt dt. (1.94c)

Thus, for a periodic force, F(t), applied to aone-degree-of-freedomsystem, the equationof motion is

mx + cv x + ksx = a02

+

n=1,2(an cos nt + bn sin nt). (1.95)

The coefcients a0, an , and bn are the Fourier coefcients, and the periodic force F(t) isnow expressed as a Fourier series. The steady-state response to each harmonic compo-nent is thus calculated separately and the total response obtained by linear superposition.It is

x(t) = a02ks

+

n=1,2

(an/ks)

[{1 n2(/n)2}2 + {2n/n}2]1/2 cos(nt n)

+

n=1,2

(bn/ks)

[{1 n2(/n)2}2 + {2n/n}2]1/2 sin(nt n). (1.96)


The phase angle, n , is given by

n = tan1 2n/n1 n2(/n)2 . (1.97)

The rst term in equation (1.96) is a static term and the termswithin the summation signsare the contributions of the various harmonically related terms. Each individual term issimilar to equation (1.59) and its corresponding phase is similar to equation (1.60).

1.5.8 Forced vibrations with transient excitation

When the forcing function, F(t), is non-periodic it cannot be represented by a Fourierseries, and other forms of solution have to be utilised. Several techniques are availablefor the solution of the equations of motion, including Fourier transforms, Laplacetransforms, and the convolution integral. The convolution integral procedure willbe adopted here initially and it will subsequently lead to the usage of the Fouriertransform.

An arbitrary, non-periodic, forcing function, F(t), can be approximated by a seriesof pulses of short duration, , as illustrated in Figure 1.18. The convolution integralprocedure for the estimation of the output response involves the linear summation of theproduct of each of the pulses in the input signal with a suitable pulse response functionwhich is associated with the system.

To understand the behaviour of this pulse response function, the concept of animpulse response needs to be introduced. Consider a rectangular pulse of unit area,as illustrated in Figure 1.19. A unit impulse is obtained by letting the time duration,T , of the pulse approach zero whilst maintaining the unit area. In the limit, a unitimpulse, (t), of innite height and zero width is produced. The unit impulse concept

Fig. 1.18. The approximation of an arbitrary function, F(t), by a series of pulses of shortduration, .


Fig. 1.19. A rectangular pulse of unit area.

Fig. 1.20. Unit impulse at t = 0 and t = .

is an important one in noise and vibration studies and, as will be seen later on in thisbook, it has signicant practical applications in noise and vibration.

A unit impulse has two main properties. They are:

(i) (t) = 0 for t = 0, and (ii)

(t) dt = 1. (1.98)

If the unit impulse occur

Norton M.P., Karczub D.G. Fundamentals of Noise and Vibration Analysis for Engineers

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