Fundamentals of Sound and Vibration, Second Edition

510 Pages·2014·6.375 MB·English

by  Fahy, Frank, Thompson, David

Acoustical Engineering Fahy
Thompson
Fundamentals of Fundamentals of
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Sound Vibration u Sound Vibration
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and and
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SECOND EDITION SECOND EDITION
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This introductory text covers the physical, mathematical and technical
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foundations of sound and vibration at audio frequencies. Acoustics,
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vibration and the associated signal processing are presented in a single

volume at a level suitable for a graduate student or practising engineer o
having no prior formal training in the field. f

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The book is a coherent textbook based on the first semester of the Masters
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programme in Sound and Vibration Studies at the internationally acclaimed
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Institute of Sound and Vibration Research at the University of Southampton.
It has been extensively revised and updated from its previous edition, with n
a new introductory chapter and new chapters on the measurement of d
sound and vibration. Other chapters include fundamentals of acoustics,

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fundamentals of vibration, signal processing, noise control, human
response to sound and human response to vibration; many of these have n
been substantially revised. Example problems and answers for self-study d
are included.
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Frank Fahy is former professor of engineering acoustics at the Institute of i
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Sound and Vibration Research (ISVR) of the University of Southampton, UK.
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David Thompson is professor of railway noise and vibration at the ISVR.
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i
o
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SECOND
EDITION Edited by Frank Fahy and David Thompson
Y105324
ISBN: 978-0-415-56210-2
90000
9 780415 562102
Y105324_Cover_final.indd 1 3/5/15 10:12 AM
Fundamentals of

Sound Vibration
and
SECOND EDITION

Fundamentals of

Sound Vibration
and
SECOND EDITION
Edited by
Frank Fahy
(Formerly) Institute of Sound and Vibration Research
University of Southampton, UK
David Thompson
Institute of Sound and Vibration Research
University of Southampton, UK
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2015 by Institute for Sound and Vibration Research
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Version Date: 20141024
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Contents
Preface xix
Acknowledgements xxi
Contributors xxiii
1 Introduction 1
Frank Fahy
1.1 Sound and noise 1
1.2 Vibration 2
1.3 Signal processing and analysis 3
1.4 Experimental methods and equipment 3
1.5 Need for noise and vibration control technology 4
1.6 Challenges of noise and vibration control technology 6
1.7 Can software solve the problem? 7
1.8 An exciting field 8
Reference 8
2 Fundamentals of acoustics 9
DavID thompson anD phIlIp nelson
2.1 Introduction to sound 9
2.1.1 Sound pressure 10
2.1.2 Harmonic motion and frequency 11
2.1.3 Decibel scale 13
2.1.4 Decibel arithmetic 15
2.1.5 Complex exponential notation 16
2.2 One-dimensional wave equation 17
2.2.1 Propagation of acoustic disturbances 17
2.2.2 Fluctuating fluid properties 19
2.2.3 Equation of mass conservation 20
v
vi Contents
2.2.4 Equation of momentum conservation 21
2.2.5 Relation between pressure and density 21
2.2.6 Wave equation 22
2.2.7 Solutions of the one-dimensional wave equation 23
2.2.8 Sound speed and specific acoustic impedance 26
2.2.9 Linearity and the superposition principle 27
2.2.10 Sound produced by a vibrating
piston at the end of the tube 29
2.3 Acoustic energy density and intensity 30
2.3.1 Energy density 30
2.3.2 Acoustic intensity 31
2.3.3 Power radiated by a piston source
in a semi-infinite tube 33
2.4 The wave equation in three dimensions 35
2.4.1 Derivation of the wave equation in three dimensions 35
2.4.2 Spherically symmetric solutions of the
three-dimensional wave equation 37
2.4.3 Point sources of spherical radiation 40
2.4.4 Acoustic power output from a pulsating sphere 41
2.4.5 Two-dimensional solutions of the wave equation 43
2.5 Multipole sources 44
2.5.1 Point dipole source 44
2.5.2 Point quadrupole sources 48
2.5.3 Sources near a rigid surface 50
2.5.4 Rayleigh integral 53
2.6 Enclosed sound fields 55
2.6.1 One-dimensional standing waves
in a closed uniform tube 55
2.6.2 One-dimensional standing waves in
a uniform tube with an open end 58
2.6.3 Enclosed sound fields in three
dimensions at low frequencies 60
2.7 Room acoustics 63
2.7.1 Enclosed sound fields in three
dimensions at high frequencies 63
2.7.2 One-sided intensity 66
2.7.3 Power balance for an enclosure 68
2.7.4 Direct and reverberant field in an enclosure 70
2.7.5 Schroeder frequency 71
2.8 Further reading 72
Contents vii
2.9 Questions 72
References 74
3 Fundamentals of vibration 77
BrIan mace
3.1 Introduction 77
3.1.1 Some terminology and definitions 77
3.1.2 Degrees of freedom 78
3.1.3 Methods of rigid body dynamics 79
3.1.3.1 Newton’s second law of motion 80
3.1.3.2 Work–energy 80
3.1.3.3 Impulse–momentum 81
3.1.4 Simple harmonic motion 81
3.1.5 Complex exponential notation 82
3.1.6 Frequency-response functions 83
3.1.7 Stiffness and flexibility 85
3.2 Free vibration of an SDOF system 88
3.2.1 Undamped free vibration 88
3.2.2 Energy: effective mass and stiffness 92
3.2.3 Viscously damped SDOF system 95
3.2.4 Estimation of SDOF-system parameters
from measured free vibration 97
3.3 Time-harmonic forced vibration of an SDOF system 98
3.3.1 Estimation of SDOF-system
parameters from measured FRFs 103
3.3.2 Vibration isolation 104
3.3.3 Other applications 106
3.3.4 Damping and loss factor 107
3.4 Response of an SDOF system to general excitation 108
3.4.1 Transient excitation: impulse response 109
3.4.2 Response to general excitation: convolution 110
3.4.3 Periodic excitation 111
3.4.4 Response to general excitation: Fourier
transform and frequency response 112
3.4.5 Response to random excitation 113
3.5 Multiple degrees of freedom systems 116
3.5.1 Undamped free vibration of 2DOF systems 116
3.5.2 Matrix notation: n arbitrary 119
3.5.3 General free vibration: modal superposition 120
3.5.4 Orthogonality 121
viii Contents
3.5.5 Modal decomposition: uncoupling
the equations of motion 122
3.6 Forced response of MDOF systems 123
3.6.1 Damped MDOF systems 124
3.6.2 Response to harmonic forces: FRF 125
3.6.3 Reciprocity 127
3.6.4 Characteristics of FRFs 127
3.6.5 Vibration absorber 129
3.7 Continuous systems 131
3.7.1 Axial vibration of a rod 131
3.7.2 Bending vibration of a beam 134
3.7.3 Waves in structures 138
3.8 Concluding remarks 139
3.9 Questions 140
References 144
4 Fundamentals of signal processing 145
Joe hammonD anD paul whIte
4.1 Introduction 145
4.1.1 Data classification, generation and processing 146
4.1.1.1 Objectives of data analysis 147
4.1.1.2 Methods of data analysis 147
4.2 Fourier analysis of continuous time signals 147
4.2.1 Preliminary: delta function 147
4.2.2 Fourier series and Fourier integral 149
4.2.2.1 Random signals and nonstationarity 150
4.3 Some results in signal and system analysis 151
4.3.1 Effect of delay 151
4.3.1.1 Echoes 152
4.3.2 Convolution 152
4.3.3 Windowing 154
4.3.4 Uncertainty principle 157
4.3.5 Hilbert transform 158
4.3.6 Instantaneous frequency and group delay 159
4.4 Effects of sampling 160
4.4.1 Analogue-to-digital conversion 160
4.4.2 Fourier transform of a sequence 160
4.4.2.1 Impulse train modulation 161
4.4.2.2 z-transform 162
Contents ix
4.4.3 Aliasing 163
4.4.4 Discrete Fourier transform 164
4.4.5 Fast Fourier transform 167
4.4.5.1 Radix 2 FFT 167
4.4.6 Practical issues associated with
analogue-to-digital conversion 169
4.5 Random processes 170
4.5.1 Introduction 170
4.5.2 Probability theory 170
4.5.3 Random variables and
probability distributions 171
4.5.3.1 Probability distributions for
discrete random variables 171
4.5.3.2 Probability distribution function 171
4.5.3.3 Continuous distributions 172
4.5.3.4 Probability-density functions 172
4.5.3.5 Joint distributions 173
4.5.4 Expectations of functions of a random variable 173
4.5.4.1 Moments of a random variable 173
4.5.4.2 Normal distribution
(Gaussian distribution) 174
4.5.4.3 Higher moments 175
4.5.4.4 Bivariate processes 176
4.5.5 Stochastic processes 176
4.5.5.1 Ensemble 177
4.5.5.2 Probability distributions associated
with a stochastic process 177
4.5.5.3 Moments of a stochastic process 178
4.5.5.4 Stationarity 179
4.5.5.5 Covariance (correlation) functions 179
4.5.5.6 Cross-covariance
(cross-correlation) function 180
4.5.5.7 Summary of the properties
of covariance functions 181
4.5.5.8 Ergodic processes and time averages 181
4.5.5.9 Examples 182
4.5.6 Spectra 185
4.5.6.1 Comments on the power-
spectral density function 187
4.5.6.2 Examples of power spectra 187