f-

Fundamentals of Noise and Vibration Analysis

for Engineers Second edition

M. P. Norton School of Mechanical Engineering, University of Western Australia

and

D. G. Karczub S.Y.T. Engineering Consultants, Perth, Western Australia

"""'d CAMBRIDGE ~~: UNIVERSITY PRESS

s

Contents

Preface Acknowledgements

Introductory comments

page xv xvii

xviii

1 Mechanical vibrations: a review of some fundamentals

1.1 Introduction 1.2 Introductory wave motion concepts - an elastic continuum viewpoint 3 1.3 Introductory multiple, discrete, mass-spring-damper oscillator concepts -

a macroscopic viewpoint 8 1.4 Introductory concepts on natural frequencies, modes of vibration, forced

vibrations and resonance 10 1.5 The dynamics of a single oscillator - a convenient model 12

1.5.1 Undamped free vibrations 12 1.5.2 Energy concepts 15 1.5.3 Free vibrations with viscous damping 16 1.5.4 Forced vibrations: some general comments 21 1.5.5 Forced vibrations with harmonic excitation 22 1.5.6 Equivalent viscous-damping concepts - damping in real systems 30 1.5.7 Forced vibrations with periodic excitation 32 1.5.8 Forced vibrations with transient excitation 33

1.6 Forced vibrations with random excitation 37 1.6.1 Probability functions 38 1.6.2 Correlation functions 39 1.6.3 Spectral density functions 41 1.6.4 Input-output relationships for linear systems 42 1.6.5 The special case of broadband excitation of a single oscillator 50 1.6.6 A note on frequency response functions and transfer functions 52

1.7 Energy and power flow relationships 52

vii

viii Contents - --------------------------------------------------------------

1.8 Multiple oscillators - a review of some general procedures 56

1.8.1 A simple two-degree-of-freedom system 56 1.8.2 A simple three-degree-of-freedom system 59 1.8.3 Forced vibrations of multiple oscillators 60

1.9 Continuous systems - a review of wave-types in strings, bars and plates 64

1.9.1 The vibrating string 64 1.9.2 Quasi -longitudinal vibrations of rods and bars 72 1.9.3 Transmission and reflection of quasi-longitudinal waves 77

1.9.4 Transverse bending vibrations of beams 79 1.9.5 A general discussion on wave-types in structures 84

1.9.6 Mode summation procedures 85 1.9.7 The response of continuous systems to random loads 91 1.9.8 Bending waves in plates 94

1.10 Relationships for the analysis of dynamic stress in beams 96

1.10.1 Dynamic stress response for flexural vibration of a thin beam 96 1.10.2 Far-field relationships between dynamic stress and structural

vibration levels

1.10.3 Generalised relationships for the prediction of maximum

dynamic stress 1.10.4 Properties of th~ non-dimensional correlation ratio 1.10.5 Estimates of dynamic stress based on static stress and

displacement 1.10.6 Mean-square estimates for single-mode vibration 1.10.7 Relationships for a base-excited cantilever with tip mass

1.11 Relationships for the analysis of dynamic strain in plates

1.11.1 Dynamic strain response for flexural vibration of a constrained rectangular plate

1.11.2 Far-field relationships between dynamic stress and structural vibration levels

1.11.3 Generalised relationships for the prediction of maximum

100

102

103

104

105 106

108

109

112

dynamic stress 113 1.12 Relationships for the analysis of dynamic strain in cylindrical shells 113

1.12.1 Dynamic response of cylindrical shells 114

1.12.2 Propagating and evanescent wave components 117 1.12.3 Dynamic strain concentration factors 119 1.12.4 Correlations between dynamic strain and velocity spatial

maxima 119

~~~ In Nomenclature 123

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contents ~ --------------------------------------------------------------

m

2 Sound waves: a review of some fundamentals 128

2.1 Introduction 128

2.2 The homogeneous acoustic wave equation - a classical analysis 131

2.2.1 Conservation of mass 134

2.2.2 Conservation of momentum 136

2.2.3 The thermodynamic equation of state 139

2.2.4 The linearised acoustic wave equation 140

2.2.5 The acoustic velocity potential 141

2.2.6 The propagation of plane sound waves 143

2.2.7 Sound intensity, energy density and sound power 144

2.3 Fundamental acoustic source models 146

2.3.1 Monopoles - simple spherical sound waves 147

2.3.2 Dipoles 151

2.3.3 Monopoles near a rigid, reflecting, ground plane 155

2.3.4 Sound radiation from a vibrating piston mounted in a rigid baffle 157

2.3.5 Quadrupoles -lateral and longitudinal 162

2.3.6 Cylindrical line sound sources 164

2.4 The inhomogeneous acoustic wave equation - aerodynamic sound 165

2.4.1 The inhomogeneous wave equ8tion 167

2.4.2 Lighthill's acoustic analogy 174

2.4.3 The effects of the presence of solid bodies in the flow 177

2.4.4 The Powell-Howe theory of vortex sound 180

2.5 Flow duct acoustics 183

References 187

Nomenclature 188

3 Interactions between sound waves and solid structures 193

3.1 Introduction 193

3.2 Fundamentals of fluid-structure interactions 194

3.3 Sound radiation from an infinite plate - wave/boundary matching

concepts 197 3.4 Introductory radiation ratio concepts 203

3.5 Sound radiation from free bending waves in finite plate-type structures 207

3.6 Sound radiation from regions in proximity to discontinuities - point and

line force excitations 216

x Contents

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4

3.7 Radiation ratios of finite structural elements 221 3.8 Some specific engineering-type applications of the reciprocity principle 227 3.9 Sound transmission through panels and partitions 230

3.9.1 Sound transmission through single panels 232 3.9.2 Sound transmission through double-leaf panels

3.10 The effects of fluid loading on vibrating structures 3.11 Impact noise

References

Nomenclature

Noise and vibration measurement and control procedures

241 244 247 249 250

254

4. I Introduction 254 4.2 Noise and vibration measurement units - levels, decibels and spectra 256

4.2.1 Objective noise measurement scales ·256 4.2.2 Subjective noise measurement scales 257 4.2.3 Vibration measurement scales 259 4.2.4 Addition and subtraction of decibels 261 4.2.5 Frequency analysis bandwidths 263

4.3 Noise and vibration measurement instrumentation 4.3.1 Noise measurement instrumentation

267 267

4.3.2 Vibration measurement instrumentation 270 4.4 Relationships for the measurement of free-field sound propagation 273 4.5 The directional charactelistics of sound sources 278 4.6 Sound power models - constant power and constant volume sources 279 4.7 The measurement of sound power 282

4.7.1 Free-field techniques 282 4.7.2 Reverberant-field techniques 4.7.3 Semi-reverberant-field techniques 4.7.4 Sound intensity techniques

4.8 Some general comments on industrial noise and vibration control 4.8.1 Basic sources of industrial noise and vibration 4.8.2 Basic industrial noise and vibration control methods 4.8.3 The economic factor

4.9 Sound transmission from one room to another 4.10 Acoustic enclosures 4.11 Acoustic barriers 4.12 Sound-absorbing materials 4.l3 Vibration control procedures

283 287 290 294 294 295 299 301 304 308 313 320

• . Contents ~ ------------------------------------------------------------

5

6

4.13.1 Low frequency vibration isolation - single-degree-of-freedom

systems 4.13.2 Low frequency vibration isolation - multiple-degree-of-freedom

systems 4.13.3 Vibration iso lation in the audio-frequency range

4.13.4 Vibration isolation materials 4.13.5 Dynamic absorption 4.13.6 Damping materials

References

Nomenclature

The analysis of noise and vibration signals

5.1 Introduction

5.2 Deterministic and random signals

5.3 Fundamental signal analysis techniques

5.3.1 Signal magnitude analysis

5.3.2 Time domain analysis

5.3.3 Frequency domain analysis 5.3.4 Dual signal analysis

5.4 Analogue signal analysis

5.5 Digital signal analysis

5.6 Statistical errors associated with signal analysis 5.6.1 Random and bias errors 5.6.2 Aliasing 5.6.3 Windowing

5.7 Measurement noise errors associated with signal analysis

References

Nomenclature

Statistical energy analysis of noise and vibration

6.1 Introduction 6.2 The basic concepts of statistical energy analysis 6.3 Energy flow relationships

6.3.1 Basic energy flow concepts

6.3.2 Some general comments 6.3.3 The two subsystem model

322

325

327 330 332 334

335 336

342

342

344 347

347 351

352 355

365 366

370 370 372 374

377 380

380

383

383 384

387 388

389 391

I I

11

7

xii Contents mmmmm ____________________________________________________________ __

6.3.4 In-situ estimation procedures 6.3.5 Multiple subsystems

6.4 Modal densities

6.4.1 Modal densities of structural elements 6.4.2 Modal densities of acoustic volumes

6.4.3 Modal density measurement techniques 6.5 Internal loss factors

6.5.1 Loss factors of structural elements

6.5.2 Acoustic radiation loss factors

6.5.3 Internal loss factor measurement techniques 6.6 Coupling loss factors

6.6.1 Structure-structure coupling loss factors

6.6.2 Structure-acoustic volume coupling loss factors

6.6.3 Acoustic volume-acoustic volume coupling loss factors 6.6.4 Coupling loss factor measurement techniques

6.7 Examples of the application of S.E.A. to coupled systems

6.7.1 A beam-plate-room volume coupled system 6.7.2 Two rooms coupled by a partition

6.8 Non-conservative coupling - coupling damping

6.9 The estimation of sound radiation from coupled structures using total

393 395 397

397 400

401 407 408

410 412 417

417 419 420

421 423 424

427 430

loss factor concepts 431

6.10 Relationships between dynamic stress and strain and structural vibration levels 433 References 435 Nomenclature 437

Pipe flow noise and vibration: a case study 441

7.1 Introduction 441

7.2 General descliption of the effects of flow disturbances on pipeline noise and vibration 443

7.3 The sound field inside a cylindrical shell 446 7.4 Response of a cylindrical shell to internal flow 451

7.4.1 General fonnalism of the vibrational response and sound radiation 451

7.4.2 Natural frequencies of cylindrical shells 454 7.4.3 The internal wall pressure field 455 7.4.4 The joint acceptance function 458 7.4.5 Radiation ratios 460

<

r

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xiii contents ~ -------------------------------------------------------

8

7.5 Coincidence - vibrational response and sound radiation due to higher

order acoustic modes 461 7.6 Other pipe flow noise sources 467 7.7 Prediction of vibrational response and sound radiation charactelistics 471

7.8 Some general design guidelines 477 7.9 A vibration damper for the reduction of pipe flow noise and vibration 479

References 481 Nomenclature 483

Noise and vibration as a diagnostic tool 488

8.1 Introduction 488 8.2 Some general comments on noise and vibration as a diagnostic tool 489 8.3 Review of available signal analysis techniques 493

8.3.1 Conventional magnitude and time domain analysis techniques 494 8.3.2 Conventional frequency domain analysis techniques 501 8.3.3 Cepstrum analysis techniques

8.3.4 Sound intensity analysis techniques 8.3.5 Other advanced signal analysis techniques

8.3.6 New techniques in condition monitoring 8.4 Source identification and fault detection from noise and vibration

signals

8.4.1 Gears 8.4.2 Rotors and shafts 8.4.3 Bearings 8.4.4 8.4.5

8.4.6 8.4.7 8.4.8

Fans and blowers

Furnaces and burners Punch presses Pumps Electrical equipment

503 504

507 511

513 514 516 518 523 525

527 528 530

8.4.9 Source ranking in complex machinery 532

8.4.10 Structural components 536 8.4.11 Vibration severity guides 539

8.5 Some specific test cases 541

8.5.1 Cabin noise source identification on a load-haul-dump vehicle 541

8.5.2 Noise and vibration source identification on a large induction motor 547

8.5.3 Identification of rolling-contact bearing damage 550 8.5.4 Flow-induced noise and vibration associated with a gas pipeline 554

xiv Contents ~ -----------------------------------------------------------------

8.5.5 Flow-induced noise and vibration associated with a racing sloop (yacht)

8.6 Performance monitoring

8.7 Integrated condition monitoring design concepts

References

Nomenclature

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

coefficients for some common building materials

Appendix 3: Units and conversion factors Appendix 4: Physical properties of some common substances

Answers to problems

Index

557 557 559

562 563

566 599

600 603 605

607

621

1 j

r 1

621

J

Index

absorption coefficients 283-4, 313-15 for typical materials 321, 602 see also sound-absorbing materials and techniques

accelerometers 270-3 placing of 491

acoustic barriers 308-12 insertion loss 309, 311-12 Fresnel diffraction 309-11

acoustic cut-off frequency, higher order modes 441, 443,445-6,447-50

acoustic enclosures 304-8 air-gap leakages 307-8 close fitting 304, 308 enclosure resonances 307-8 design guidelines 298-9 flanking transmission 307 insertion loss 307 large fitting 304-7 sound radiation 305

acoustic impedance see impedance acoustic modes, higher-order 445-6,447-50 acoustic radiation damping 410, 411, 537

see also internal loss factors acoustic radiation reactance 161

see also impedance acoustic radiation resistance 160-1

see also impedance acoustic source models 146-65

cylindrical line sound sources 164-5 dipoles - two monopoles in close proximity

151-5 aerodynamic sound 173,174,179 finite-plate sound radiation 214-15, 223-4

monopoles - spherical sound source 147-51 aerodynamic sound 167, 173, 174, 179 far-field/near-field 148-50 source strength 148 specific acoustic impedance of spherical waves

148-9 monopoles near a rigid, reflecting, ground plane

155-7 power doubling effects 156-7,280

quadrupoles 162-4 sound power source models

constant power 279-81 constant pressure 280-81 constant volume 279-81

vibrating pistons in a rigid baffle 157--62, 195-7 directivity factor 158-9 radiation impedance 160 radiation reactance 161 radiation resistance 160-1

see also directional characteristics of sound sources; inhomogeneous acoustic wave equation

acoustic velocity potential 141-2 dipoles 152-3 monopoles 147-8, 155-6 plane waves 143

acoustic wave equation 140 derivation 133-43 one-dimensional 143 velocity potential 142 see also homogeneous acoustic wave equation;

inhomogeneous acoustic wave equation acoustically slow/fast modes

acoustically slow (subsonic) 211, 226 acoustically fast (supersonic) 211, 226 critical frequency 199,210-13 sound radiation principles 197-200

aerodynamic sound see inhomogeneous wave equation,

air absorption 285 air springs, as vibration isolators 331 aliasing problems 372-3 amplitude resonance 26-7

see also resonance analogue filter characteristics 365--6 analogue signal analysers/analysis 365--6 anechoic chambers 282 apparent mass 28 auto-correlation functions 39-40,42,43,351-2

see also correlation auto-spectral density functions 41-2, 45, 49,352,

367-8 see also spectral density

A-weighting 258-9, 266-7

baffled piston 157-62, 195-7 bandwidth 366, 370-2, 376-7

filter 371 frequency analysis 263-7 half-power 27,51 mean-square 51,105--6 signal analysis 370-2

I

i.

622 Index ~

barriers, acoustic 308-12 bars and rods see rods and bars, quasi-longitudinal

vibrations baseband auto-spectra 501-2 beams, dynamic stress analysis see dynamic

stress/strain beams, response of continuous systems to random

loads 91-4 beams, transverse bending vibrations 79-84

bending wave velocity 81 boundary conditions 82 damping 83-4 drive-point mechanical impedance 83 equation of motion 79-80 Euler beam equation 80 group velocity 81 travelling wave solution 81, 97-8 see also dynamic stress/strain

bearing faults/defects detection 494-6, 514, 518-23, 550-1

cepstrum analysis technique 522-3, 548-50 crest factor detection 522 envelope power spectrum analysis 523 kurtosis technique 522 rolling-contact bearings 519-23, 550-1 sliding-contact bearings 519 spectral analysis technique 522 vibration severity guides/standards/limits

539-41 bending waves

definition 4-7 forced 213 in beams 81-6, 96-100 in pipes 226--7, 461 in plates 94-5,109-12,198-200,207-16 wave velocity 6--7,81, 198

bias error problems 370-2 blocked pressures 196 blower noise 523-5 boundary layer pressure fluctuation studies 447 broadband excitation of a single oscillator 50-1 burners, combustion noise 525-7 B-weighting 258-9, 266-7

cavitation 529-30 centrifugal pumps 528-30, 559 cepstrum analysis 353-6

for bearing fault detection 522-3, 548-50, 551-2 complex cepstrum 355-6, 503-4 power cepstrum 353-4, 503-4

characteristic mechanical impedance see impedance coefficients

absorption 283-4, 313, 315, 321, 602 reflecti on 313-14 transmission 232, 284, 308

coherence 362-4, 378-9, 509, 536 coherent output power 362, 379 coincidence

cylindrical shells 461-7, 473-4, 477, 554-7 definition 7 double leaf panels 243 panels 232-41 see also critical frequency

combustion noise/roar 296,525-7 compensation costs for hearing damage 300-1 complex modulus of elasticity 75, 83 complex stiffness 31 complex wavenumber 75,83 compression~l (longitudinal) waves 4-5, 72-5 condenser mIcrophones 267, 268-9 270 condition monitoring 488-9, 490, 492

online condition monitoring 560-2 safety monitoring 560 system design considerations 559-62 see also performance monitoring

constant percentage bandwidth 264 continuous monitoring see condition monitorina continuous systems 64-95 " control methods for industrial noise and vibration

295-9 conversion factors and units 603-4 convolution integral 33, 36, 46 correlation

coefficients 39-40 functions 39-41, 374 see also auto-correlation functions; cross­

correlation functions correlation ratios 103-4 coulomb (dry-friction) damping 31

see also damping coupling loss factors 387, 391, 417-23

acoustic volume-acoustic volume 420 coupled system examples 423-30 measurement techniques 421-3 structure-acoustic volume 419-20 structure-structure 417-9

crest factors 496, 522 critical frequency

infinite plate 199 finite plates 210-12, 216--21 radiation ratio 222, 225 panels and partitions 231, 232-41 see also acoustically slow/fast modes; coincidence

critical viscous-damping coefficient 20 see also damping

criticality and failure mode analYSis 488-9 cross-correlation functions 40, 355, 357

see also correlation cross-spectral density functions 49,358-9,379-80,

459,508 see also spectral density

cut-off frequencies 441, 443, 445-6, 447-50 cylindrical line sound sources 164-5 cylindrical shells, dynamic strain analysis see

dynamic stress/strain C-weighting 258-9, 266--7

damping 30-32 basic concepts 2, 8-10 and complex stiffness 31,75, 83 coulomb (dry-friction) damping 31 coupling 430-1 critical damping 20-1 damped natural frequency 19,20-1 damping materials 334-5 damping ratio 25

623 Index ~

decay time 21 effects on dynamic absorbers 333 with forced vibrations of multiple oscillators 62--4 generalised 85--6 hysteretic (structural) 30, 92 with low frequency vibration 323--4 structural components testing 537 structural loss factor 31, 53, 75, 83, 92, 408 viscoelastic damping 334-5 viscous damping

equivalent viscous damping concept 30-2 of free vibrations 16-21

see also internal loss factors, acoustic radiation damping

decibels addition and subtraction 261-3 noise/sound levels 222, 256 vibration levels 260-1

degrees-of-freedom 2-3, 56-60 deterministic and random signals 22, 344-7 diagnostics using noise and vibration analysis

488-565 see also diagnostic tools

diagnostic tools 493-513 cepstrum analysis 503--4 condition monitoring 488-9, 492 crest factor measurement 496 discrete wavelet transforms (DWTs) 512 envelope power spectrum analysis 507-8 frequency domain analysis 501-3 frequency response (transfer) functions 509 fuzzy logic 512 kurtosis 500 magnitude domain analysis 494-501 neural networks 512 peak signal measurement 494 phase-averaged time histories 496 probability density distributions of noise levels

497-8 propagation path identification 507-9 short time Fourier transfomls (STFTs) 512 sound intensity analysis 504-7 sound intensity mapping 505 sound source ranking 504, 532-6 temporal wavefolID recovery 510-11" time domain analysis 494-501 waterfall plots 50 I see also signal analysis techniques and functions

diffuse (reverberant) sound fields 283-7 digital signal analysis 366-70 dipoles see acoustic source models Dirac delta function 34 direct field 283, 286 directional characteristics of sound sources 278-9

directivity factor and directivity index 278-9 vibrating pistons in a rigid baffle 158-9

discrete Fourier transforms (DFTs) 366-9 discrete wavelet transforms (DWTs), as a diagnostic

tool 512 dispersion curves, in cylindrical shells 117-18,

450 dispersion relationships 7, 463-65 dual signal analysis 355--64

ducts see flow duct acoustics Duhamel convolution integral 89-90 D-weighting 258-9, 266-7 dynamic absorption/absorbers 332--4 dynamic load factor 90 dynamic stiffness (force/displacement) 28 dynamic stress/strain

beams 96-108 base-excited cantilever with tip mass 106-8 dynamic stress and fatigue 96 estimates based on static stress and displacement

104-5 mean-square estimates for single-mode vibration

105-6 non-dimensional correlation ratio 103--4 prediction of maximum dynamic stress from

velocity 102-3 relationships between dynamic stress and

velocity 100--4 evanescent wave effects and dynamic stress

concentration 98-100 spatial distributions of vibration and dynamic

stress 97-1 00 strain-<iisplacement relation 96-7

cylindrical shells 113-21,433-5 dynamic bending strain 114-15 dynamic su"ain concentration 119, 120 relationships between dynamic strain and

velocity 119-21 shell vibration dynamic response 114-17 spatial distributions of dynamic strain 115-16,

119 travelling wave equations 114

plates 108-13 evanescent waves 109-11, 113 dynamic bending strain 109-12 dynamic strain concentration 110-1 relationship between dynamic strain and velocity

112-13 travelling wave solution 109

in statistical energy analysis 433-5

economic factors in noise and vibration control 299-301

damage limitation 299 hearing damage compensation 300-1

eddy current probes 270 eigenfunctions 86 eigenvalues, mode summation procedures 86-7 eigenvectors 58 elasticity 1-3 electrical equipment, noise and vibration sources

530-2 enclosures see acoustic enclosures energy concepts

energy and power flow 52--6 energy flow relationships 387-97 oscillatory motion 15-16 potential and kinetic energy 15-16,52 see also statistical energy analysis (SEA)

energy density 146,285,304 energy spectral density functions 352-3 envelope power spectra analysis 507-8, 552-4

624 Index -equivalent viscous damping 30-2

see also damping errors in signal analysis 370-7

aliasing 372-3 bandwidth considerations 370-1 bias errors 370-2 measurement noise errors 377-80 normalised random errors 371-2 random errors 370-2 windowing 374-7

Euler beam equation 80-1 Euler-Bernoulli beam theory 96 evanescent waves

beams 81, 98-100 cylindrical shells 117-9 plates 109-11, 113 see also far-field; near-field

fan and blower noise 523-5 far-field

acoustics 148-50,207 vibration 98-101,109-10,112, 113,119-21 see also evanescent waves; near field

fast Fourier transforms (FFTs) 42, 352, 366 fault detection, prediction and source identification

513-41 about sources of industrial noise and vibration

294-5,513-14 bearings 494-9, 514, 518-23, 539-41, 548-50 with condition monitoring 488-9 electrical equipment 530-2 gears 514-16 hydraulic pumps 528-30 rotors 514, 516-18 shafts 516-18 structural components 536-9 vibration severity guides 539-41 see also diagnostic tools

finite Fourier transform 367 finite plate-type structures, sound radiation from free

bending waves 207-16 acoustically excited 208 analysis using Rayleigh's equation 207 edge and corner radiation 211, 214-16 flow resistance (porous materials) 315 forced vibration 212-14 mechanically excited 208 modal density 214 wavenumber diagrams 209-13

flow duct acoustics 183-7 area discontinuity 185 reactive silencers, acoustic performance 186-7 side-branch elements 185-6 outlet radiation impedance 187 transmission matrix modelling conventions 184-5

flow-induced sound 165-7,441-6,467-71 see also pipe flow noise and vibration; Strouhal

number; transmission loss fluid loading on vibrating structures 244-7 fluid-structure interactions 194-7 forced vibrations

arbitrary, non-periodic, forcing function 36 as input-output system 21-3

basic principles 2, 10-12,21-2 harmonic excitation 22-9 impulse response functions 33-6 linear systems; input-output relationships 42 4 multiple oscillators 60-4 ' 6-9 periodic excitation 32-3 random excitation 37-52 single oscillator 12-15 transient excitation 33-6

Fourier series expansion 32 Fourier transforms 41-9, 352-3, 366-70 free vibrations 2, 16-21 free-field sound propagation 273-8

line sources (cylindrical and semi-cylindrical) 275-7

plane sources 277-8 point ~ources (spherical and hemi-spherical) 274-5

frequencIes, natural see natural frequencies frequency analysis bandwidths 263-7, 370-2 frequency domain analysis 342-4, 352-5

for fault prediction 489-90, 491,501-3 frequency response functions (transfer functions) 28

47-9,50-2,358-62 ' as a diagnostic tool 509

Fresnel diffraction theory, acoustic barriers 309-10 furnaces and burners, combustion noise 525-7 fuzzy logic, as a diagnostic tool 512

gas pipeline flow induced noise and vibration 554-7 see also pipe flow noise and vibration

gas turbines bearing vibration limits 540 performance monitoring 558-9

Gaussian distributions 350 gear noise and vibration 514-6 generalised damping 85-6 generalised force 85-6 generalised mass 85-6 generalised stiffness 85-6 Green's function

aerodynamic sound 170-2 for structure interactions 194-7

grilles in ducted flows 469 group velocity 7, 81 Gumble distributions 350

Hankel function 225 hearing damage compensation costs 300-1 Heckl's relationships 454-5 Helmholtz cavity resonator 182, 316-19 homogeneous acoustic wave equation 129, 131-46

general solution 143 Iinearised acoustic wave equation 140-1

human body as a system of damped springs 9-10 hydraulic pumps see pumps, noise and vibration hysteretic damping 30, 92

see also structural loss factor; damping

impact noise 247-9 acceleration/deceleration noise 247-8, 249 radiated noise energy 248 ringing noise 247,249

impact testing 358-9

r ~ _1"_de_X _______________________ _

I impedance

acoustic 144, 148-50 acoustic radiation 159-62, 195-7,244-7 characteristic (wave) impedance

acoustic 144 mechanical 73, 78

drive point 67,83, 160,195,218,220,235 mechanical 28-9, 73, 78, 160, 195,235 moment 419

impedance tube, absorption coefficient measurement 313,314

impulse excitation spectra, structural components 537-9

impulse response functions 33-6, 47,357-8, 362

induction motor noise and vibration 547-50 bearing vibration acceleration 547-9 cepstrum analysis 548-50

industrial noise and vibration sources 294-6 industrial noise and vibration control techniques

295-9 inertance (acceleration/force) 28 inertia base, in vibration isolation 331-2 infinite plates, sound radiation wave!boundary

matching concepts 197-203 radiation efficiency concept 199 radiation ratio 199 wave!boundary matching condition 201-2 wavenumber concepts 200

inhomogeneous acoustic wave equation (aerodynamic sound) 129, 165-82

basic concepts 165-7 general solution 169-71 Green'sfunction 170-2 Helmholtz cavity resonator 182 Lighthill's acoustic analogy 165-7, 174-7,

178, 179 monopoles, dipoles, quadrupoles 167-80 Powell-Howe theory of vortex sound 167, 180-2

dissipation of sound concept 181 retardation time concept 168 solid bodies in the flow 177-80 solutions for simple sources 167-74 see also homogeneous acoustic wave equation; pipe

flow noise and vibration " input-output relationships 46-9 insertion loss 186-7,306,309,311-12 intensity (sound) see sound intensity internal loss factors and SEA 387, 407-17

acoustic radiation loss factor 409, 410-11 amplitude tracking 416-17 band-averaged internal loss factors 414-15 envelope decay measuring technique 412 half-power bandwidth measuring technique 412 measurement techniques 412-17 random noise burst reverberation decay measuring

technique 412, 414, 416 steady-state energy flow measuring technique

412-14 structural loss factors 408-10

for some common materials 410 inverse filtering 5 I 0 inverse Fourier transform 353, 367

jet noise 177-82, 468 jet nozzle noise 130-2 joint acceptance function 452, 458-60 journals on noise and vibration 599

kinetic energy 52 Kirchhoff-Helmholtz integral 194,200 kurtosis 500

for bearing fault detection 522

lag window functions 374-6 Lagrangian of a system 52 LighthiII's acoustic analogy 165-7, 174-7, 178, 179 linear systems; input-output relationships 42-9

and the convolution integral 46 linearised acoustic wave equation 140-1 logarithms, use of 222 longitudinal (compressional) waves 4,72-5 loss factors see internal loss factors and SEA loudness level (phon) 258 loudness scale (sone) 258 lumped-parameter models 12

machines noise and vibration control methods 129-31,298 vibration severity guides/standards 539-41 see also bearing faults/defects detection

magnification factor 27 magnitude analysis 494-501 mass law equation 237-8 material handling equipment, noise and vibration

control methods 298 Maxwell distributions 350 mean-square response 49, 50-1, 105-6 measurement noise errors 377-80 measurement of sound and vibration see signal

analysis techniques and functions; sound intensity; sound measurement; sound power; vibration measurement

mechanical impedance see impedance mechanical compliance 28 mechanical inel1ance 28 mechanical reactance 28 mechanical resistance 28 microphones 267-70

ceramic microphones 268 condenser microphones 267, 268-70

free-field, pressure and random incidence 268-9 dynamic microphones 267-8 see also sound measurement

mobility (velocity/force) 28, 328-30, 401 modal density 214, 387, 397-407

acoustic volumes 400 definition 214 honeycomb structures 399-400 mass and stiffness corrections 404-7 measurement techniques 401-7 plates 2 I 4, 398 point mobility technique 401-3 structural components/elements 397-400, 537 thin-walled cylindrical shells 398-9 uniform bars in longitudinal vibration 398 see also statistical energy analysis

626 Index IIIIII!IIIIIIII

modal frequency response function of a cylinder 452 mode pruticipation factor 90 mode shapes/eigenvectors 58, 70-1, 537 mode summation procedures 85-91

damping 90 Duhamel convolution integnil 89-90 eigenvalues 86-7 generalised co-ordinates of modes 85-6

modes of vibration 10-12 modelling sound sources see acoustic source models;

sound power monitoring see condition monitoring; performance

monitoring mono poles see acoustic source models multiple oscillators 56-64

effects of damping 62-4 forced vibrations 60-4 see also degrees-of-freedom

natural frequencies concept 10-12 fluid-loaded structures 246 multiple oscillators 56, 60, 61 plates 208, 213 rods and bars 77 single oscillator 14 strings 71 structural components 537 transverse beam vibration 82-3 near-field

acoustics 148-50,207 vibration 98-101, 102-3, 109-11 see also evanescent waves; far-field

neural networks, as a diagnostic tool 512 noise measurement see sound measurement noise reduction 303, 429 noise source ranking

as a diagnostic tool 504, 532-6 selectively wrapping/unwrapping parts of machines

533 surface intensity measurement techniques 534-5 surface velocity measurement techniques 533 vibration intensity measurement techniques 536

normal modes 85-90 normalised random errors 371-2

octave frequency bands 264 online condition monitoring 560-2

see also condition monitoring; performance monitoring

orthogonality 86-8 oscillators

damped 16-21 multiple 56-64 single 12-15,50-1 undamped 12-15

oscillatory motion basic concept 1-3 complex quantities 14-15 energy concepts 15-16 see also damping; degrees-of-freedom; forced

vibrations; vibration; waves/wave motion overdamped motion 20

panel absorbers see partitions and panels particle velocity 4,67, 142, 143,148 particular integral 25 partitions and panels 230-44

sound transmission frequency range aspects 231, 232-3, 237-9 single panels 232-41 double-leaf panels 241-4 between rooms 301-3

panel absorbers 317-20 typical TL valves 600-1

pascals (sound pressure level) 256 performance monitoring 488-9, 557-9

centrifugal pumps 559 gas turbines 558-9 see also condition monitoring

periodic excitation 32-3 phase-averaged linear spectra 503 phase-averaged signals 496 phase resonance see resonance phase velocity 6, 65 phon (loudness level) 258 physical properties of common substances 605-6

gases 606 liquids 606 solids 605

pipe flow noise and vibration 441-87 bends, effects of 456-7 boundary layer pressure fluctuation studies 447 cavity resonances 471 complete coincidence 462-6 external sound radiation 454, 472 general discussion 443-6 coincidence 461-7, 473-4, 477

a coincidence damper 479-81 principal wavenumber coincidences 464

cut-off frequencies 444, 448-50 design guidelines 477-9 diffusers and spoilers (splitter plates) 468 dispersion curves 450 flow spoilers 468 grille noise 469 internal acoustic modes 449 internal wall pressure field 455 jets 468 joint acceptance function 458-60 modal frequency response function of a cylinder

452 shell natural frequencies 454-5 prediction of vibration and sound radiation 471-6 radiation ratios 460-1 response of a cylindrical shell to internal flow

451-61 internal sound field 446-50 Strouhal number 444,468-71,477-8 vibration damper 479-81 valve noise 469-71 vOltex shedding 468, 470, 471, 477 wavenumber coincidence 462-5 see also cylindIjcal shells, dynamic stress/strain;

transmission loss; coincidence pipes see cylindrical shells; dynamic stress/strain;

pipe flow noise and vibration

·4

-

r '627 Index

~ ------------------------------------------------------------piston in a rigid baffle 157-62, 195-7 plane waves 7, 143-4 plateau method 240-1 plate-type structures, bending waves in 94-5, 109-12 plates, dynamic strain analysis see dynamic

stress/strain plates, sound radiation from see finite plate-type

structures, sound radiation from; infinite plates, sound

radiation wave!boundary matching concepts point mobility technique, and modal density 401-3 porous and fibrous materials, absorption by 313, 316 potential energy 15-16,52, 145-6 Powell-Howe theory of vortex sound 167, 180-2 power concepts

energy and power flow 52-6 instantaneous power 29 power dissipation 388-9 real and reactive power 54-5 structural loss factor 53 time-averaged power 29, 54 see also sound measurement; sound power;

statistical energy analysis power flow measurement 360 power injection measuring technique 422 power (sound) see sound power power spectrum see mean-square response; spectral

density functions probability density functions 38-9,348,497-9 probability distribution function 348 probability of exceedance 496, 500 propagation path identification, as a diagnostic tool

507-9 propagation of plane sound waves 143-4 pulse response functions 33-6

see also impulse response functions pumps, noise and vibration 528-30

bearing vibration limits 540 cavitation 529-30 hydraulic forces 528 recirculation 530

quadrupoles see 162-4 quality factor 27,452 quasi-longitudinal vibrations 72-9

radian frequency 5 radiated sound

estimation using total loss factor concept 431-3 see also acoustic source models

radiation impedance, resistance and reactance 159-62 radiation ratios

basic concepts 199,203-6,216,221-2 compact bodies 222 cylinders 225-6, 460-1 definition 203-4 dipole-type sound sources 223-4 finite structural elements 221-7 infinite flat plate 204-5 monopole-type sources 223 pipes 226-7 plates 224-5 radiation ratio curves 222-3

spherical sound source 205-6, 223 structural components 537

random signals 22-3, 344-7 random error problems 370-2 random excitation 37-52 random loads on beams see beams, response of

continuous systems to random loads random noise burst reverberation decay measuring

technique 412, 414, 416 ranking see noise source ranking ray acoustics 133 Rayleigh integral 195 reactive power 54-5 real (resistive) power 54-5 receptance (displacement/force) 28 reciprocity principle

basic concept 227-8 quiet and loud machines example 228-30 with SEA 391 vibrating piston example 195

reflection coefficient, sound 313-14 resolution bandwidth 370-1 resonance

acoustic enclosures 307-8 amplitude 26-7 concept 2, 12 phase resonance 25-6

retardation time, aerodynamic sound 168, 178 reverberant ( diffuse) sound fields 283-7 reverberation radius 3 19 reverberation time 286, 315 reverberation room absorption coefficient

measurement 313, 315, 316 rods and bars, quasi-longitudinal vibrations 72-7

boundary conditions 74, 76 damping 75 longitudinal displacement of a bar element 72-3 wave impedance (characteristic mechanical

impedance) 73 and wave velocity 74 wavenumber 75

rolling-contact bearing damage 550-4 auto-spectrum of vibration examination 550-1 envelope power spectrum of vibration 55 1-4

room constant 286 room to room sound transmission 301-3 rotor and shaft vibration 514, 516-18

Sabine absorption coefficient 286 safety monitoring 560 sample record 38 SEA see statistical energy analysis semi-reverberant-field sound measurement techniques

287-9 shaft and rotor vibration 514, 516-18 shells see cylindrical shells, dynamic stress/strain short time Fourier transforms (STFTs), as a diagnostic

tool 512 signal analysis as a diagnostic tool see diagnostic tools signal analysis techniques and functions 342-82

analogue signal analysers/analysis 365-6 auto-correlation functions 351-2 auto-spectral density functions 352

-----. 628 Index -. -------------------------------------

signal analysis (cont.) cepstrum analysis 353-6 coherence functions 362-4, 378-9 cross-correlation functions 40, 355, 357, 379-80 cross-spectral density functions 358-60 deterministic and random signals 344-7 digital signal analysers/analysis 366-70 direct Fourier transforms (DFTs) 366-9 dual signal analysis 355-64 fast Fourier transforms (FFTs) 352, 366 forward Fourier transform 353-5, 367 frequency domain analysis 342-4, 352-5 frequency response functions (transfer functions)

28,47-9,52,358-62 Gaussian distributions 350 Gumble distributions 350 impulse response functions 357-8, 362 inverse Fourier transform 353, 367 lag windows 374-6 magnitude analysis 347-50 MaxweII distributions 350 power cepstrum 353-4 power flow techniques 360 probability density functions 348-9, 499 probability distribution function 38-9, 348, 497-8 spectral analysis 342-3, 374 statistical error problems 370-7 time domain analysis 342-4, 351-2 WeibuII distributions 350 see also diagnostics using noise and vibration

analysis; diagnostic tools; errors in signal analysis; sound

measurement; sound power; statistical energy analysis (SEA)

skewness 350 solid bodies in the flow (effects of) 177-80 solid structures, interactions with sound waves

193-253 see also discontinuities, sound radiation in close

proximity to; finite plate-type structures, sound radiation from; infinite plates, sound radiation

wave!boundary matching concepts sone 258 sound

definition of 128 directional characteristics 158, 278-9 energy density 146,285,304 pressure 143-4, 148-9, 158 radiation from an infinite plate 197-203 radiation from free bending waves in finite

plate-type structures 207-16 spherical waves 147-51 speed of 141 see also homogeneous wave equation;

inhomogeneous wave equation; plane waves; sound intensity;

sound measurement; sound power sound-absorbing materials and techniques 313-21

absorption coefficients for typical materials 321, 602

Helmholtz resonators 316-19 impedance tube measurements 313,314 measurement techniques 313-16

panel absorbers 317-20 porous an~ fibrous materials 313, 316 reverberauon room measurements 283-7 313 3 space absorbers 319 ' , 15

sound energy density 145-6 sound intensity 144-5,279

analysis as a diagnostic tool 504-7 cylindrical line sources 165 dipoles 153, 179 mapping 505 measurement techniques 290-4 noise source identification 504-6 piston in an infinite rigid baffle 158 plane waves 144-5 quadrupoles 179 sound power measurement techniques

290-4 spherical waves (monopoles) 149, 156, 179

sound measurement 256-9, 282-94 anechoic chambers 282 free-field techniques 282 measurement instrumentation see microphones measurement errors 377-80 objective scales 256-7

decibels 256 sound power 256-7 sound pressure levels 256

reverberant (diffuse) sound fields 283-7 semi-reverberant-field techniques 287-9 sound intensity measurement 290-4

closely spaced sound pressure microphones 292-3

dual channel signal analysers 293 sound level meters 269-70 subjective scales 257-9

loudness level (phon) 258 loudness scale (sone) 258 subjective response of humans 258 weighted networks 258-9

see also decibels; frequency analysis bandwidths; sound power; vibration measurement

sound power 146,273-9 dipoles 153-4 industrial noise sources 294-5 lateral quadrupoles 162 line force excitation (drive-line) 219 longitudinal quadrupoles 163-4 models - constant power and volume sources

279-82 piston in an infinite rigid baffle 162 plane waves 146 point force excitation (drive-point) 218 spherical waves (monopoles) 149, 156 measurement

free-field techniques 282 reverberant-field techniques 283-7 semi-reverberant-field techniques 287-9 sound intensity techniques 290-4

see also acoustic source models sound pressure levels 273-8

pascals (units) 256 sound reduction index 232 sound source ranking see noise source ranking

629 Index ~

sound sources/radiation/propagation industrial noise sources 294---{) pressure level/power/intensity relationships

and directivity factors and indices 278-9 from large plane surface sources 277-8 from line sources 275-7 from point sources 274-5

see also acoustic source models; sound measurement

sound strength 148 sound transmission

between rooms 301-3 panels and partitions 230-44 transmission coefficient 232,290, 475---{) typical TL values 600-1

sound transmission coefficients see transmission coefficients of materials

sound wave fundamentals aerodynamic sound 128-9 basic concepts 4, 128-31 jet nozzle noise 130-2 speed of sound 141 structure-borne sound 128 wave equations, homogeneous and inhomogeneous

129 see also homogeneous acoustic wave equation;

inhomogeneous acoustic wave equation; acoustic wave equation;

sound waves in solid structures see solid structures, interactions with sOI,md waves

sources of noise and vibration see fault detection, prediction and source identification

space absorbers 319 space averaging 16, 424-7 spectral analysis 342-3

for bearing fault detection 522 spectral density functions 41-2, 45, 49,352-3,360-4,

367-8,374,453,459 spectral window functions 375 spectrum see spectral density functions speed of sound 141 spoilers 468, 469 spring theory

damping 8-9 energy concepts 15 human body as a system of damped springs 9-10 mass/spring system 14 mass/spring/damper models 9-10

with harmonic excitation 22 modelling 8-9 potential energy 15-16 spring stiffness 8

springs, metal, as vibration isolators 330-1 standard deviation 39 standards

bearing vibration limits (CDA/MS/NVSH 107) 540 machinery vibration severity (ISO 2372, VDI 2056,

BS 4675) 539 standing waves 4, 11 standing wave ratio 313 statistical energy analysis (SEA) 383-440

applications to coupled systems 423-30 about SEA 383-4

acoustic radiation loss factors 410-11 assumptions and procedures 387-8, 390 basic concepts 384-7 coupled oscillators and energy flow 388-90 dynamic stress/strain/structural vibration

relationships 433-5 energy flow concepts 388-9 energy flow relationships 387-97 heat energy flow/vibration analogy 384---{) ill-situ estimation procedures 393-5 modal density 387 multiple subsystems 395-7 non-conservative coupling/coupling damping

430-1 pipeline system example 386-7 power dissipation concepts 388-9, 391 characterising structural component SEA

parameters using noise and vibration signals 537

structural loss factors 408-10 three-subsystem model 427-30 total loss factor concept for estimation of sound

radiated 431-3 two subsystem model 391-3 wave transmission analysis 417 see also coupling loss factors; energy concepts;

internal loss factors and SEA; modal density statistical errors with signal analysis see errors with

signal analysis steady-state energy flow measuring technique 412-14 steel pipelines see dynamic stress/strain; pipe flow

noise and vibration; statistical energy analysis stiffness

complex 31 partitions and panels 233-4 springs 8

strings, vibration in 64-72 boundary considerations 66 complete general equation of the wave motion 65 drive-point mechanical impedance 69-70 equation of motion in the lateral direction 65 evaluation of complex constants 68-9 standing and travelling waves 10-12 wavenumber concept 67

Strouhal number 444 diffusers 468 grilles 469 jets 468 and pipework design 477-8 spoilers 468, 469 turbulent mixing 470 vortex shedding 471

structural loss factor 31,53,75,83,92,408-10 for some common materials 410 see also damping

structural damping 2, 30 see also damping

structure-borne sound 193-249 definition of 128, 194 fluid-structure interactions 194-201 radiation ratios from structural elements 221-7 wave!boundary matching concepts 197-201 see also sound transmission

..

630 Index iIIIIIIIIIIIIII

subjective noise measurement 257-9 synchronous time-averaged signals 496

temporal waveform recovery, as a diagnostic tool 510-11

test cases cabin noise on a load-haul-dump vehicle 541-7 gas pipeline flow induced noise and vibration 554-7 induction motor noise and vibration 547-50 racing sloop (yacht) flow induced noise and

vibration 557 rolling-contact bearing damage 550-4

three-degree-of-freedom system 59-60 time-averaging 53

time-averaged power 29, 54 time domain analysis 342-4, 351-2

for fault diagnosis 494-50 I time record averaging 377 transfer functions (frequency response functions) 28,

47-9,52,358-62 transient excitation 33-6 transmissibility of vibration 323, 329-30 transmission coefficient 232,290,475-6

see also transmission loss, sound transmission transmission loss

between rooms 303, 427-30 panels and partitions 232, 235-8 pipe walls 471-6 see also transmission coefficient; sound

transmission transmission matrix modelling conventions

(acoustics) 184-6 travelling waves 10-12 travelling wave equations (solutions to the equation

of motion) beams, bending 81 beams, longitudinal 74 cylindrical shells I 14 one-dimensional sound waves 143 one-dimensional sound waves with mean flow 183 plate bending 109 string 65-8

turbulent mixing, Strouhal number 470 turbulence 130-1, 178,442-5,455 two-degree-of-freedom system 56-9

undamped free vibration 12-5 underdamped motion 19-21 units and conversion factors 603-4 unit impulse 33-6

valve noise 469-70, 478-9 variance 39 velocity see group velocity; particle velocity; phase

velocity; volume velocity vector theory, with acoustic velocity potential 141-2 velocity potential 147 vibration

amplitude resonance 26-7 basics concepts 1-3 as complex quantities 14-15,28-9 damping ratio 25 forced and free 2

free vibrations with viscous damping 16-21 gas pipeline flow induced. noise and vibration

554-7 industrial vibration sources 294-5 magnification factor 27 modes of vibration 10 quality factor 27 steady-state solutions 25 undamped free vibrations 12-15 see also dynamic stress/strain; fluid loading on

vibrating structures; forced vibration; strings vibration in; vibration measurement '

vibration control with machines and engines 320-35 audio frequency range - vibration isolators

327-32 basic principles 320-2 dynamic absorption/absorbers 332-4 isolation efficiency 324 low frequency isolation - multiple-degree-of_

freedom 325-7 low frequency isolation - sing1e-degree-of-freedom

322-5 damping 323-4

transmissibility 323 see also damping

vibration intensity measurement techniques 536 vibration isolators 327-32

air springs!bags 331 cork pads 330 felt pads 330 fibrous glass pads 330 inertia blocks 331-2 metal springs 331 rubber 330

vibration measurement 259-61 choosing displacement, velocity or acceleration

259-60 decibel notation 260-3 frequency considerations 259-60 vibration transducers

accelerometers 270-3 eddy current probes 270 moving element velocity pick-ups 270

vibration severity guides 539-41 viscoelastic damping 334-5 viscous damping

critical viscous-damping coefficient 18 free vibrations 16-21 in real systems 30-2, 452 see also damping

volume velocity 148, 185 vortex shedding 468, 470, 471 vOl1ex sound 167, 180-2

waterfall plots 501 wave acoustics 131, 133 wave!boundary matching concepts 197-201 wave equation

acoustic 140 bending waves in beams (Euler beam equation)

79-80 bending waves in plates 95 rods and bars 73

Q

j

1

1

!

r ! 631 Index ~

vibration string 64-5 see also homogeneous acoustic wave equation;

inhomogeneous acoustic wave equation wave impedance (characteristic mechanical

impedance) 73, 78 wave transmission analysis 417 wave-mode duality concept I, 10-12,70 wa~enumber

acoustic 200, 448,462-3, 454-5 axial pipe 448, 462-3, 460 bending 80-1, 95, 200 circumferential pipe 454-5, 462-3, 460 concept 5-6, 200 continuous systems 67, 75 diagrams 209-10 vectors 210 wavenumber curves, cylindrical shells

117-18 see also dispersion curves, in cylindrical shells;

dispersion relationships waves

basic concepts 3-7 compressional waves 4-5 dispersive/non-dispersive waves 7

propagation 64-7 propagation in solids 134 Rayleigh waves 84 sound waves 4 standing waves 4, 11-12 transmission and reflection 77-9 travelling waves 10-12 velocity 5-6, 65, 74, 80 wave-mode duality concept I wavelength 6 wave types in structures 84 see also bending waves; compressional

(longitudinal) waves; group velocity; impedance; phase velocity;

quasi-longitudinal vibrations wavenumber transform approach 217 Weibull distributions 350 weighted noise levels

basic concept 258-9 industrial sources 295-6 tables 266-7

windowing 374-7

zoom (passband) auto-spectral density 501-2