Spectral Analysis in Engineering. Concepts and Cases

309 Pages·1995·5.62 MB·English

by  Grant E Hearn and Andrew V Metcalfe (Auth.)

About the authors
Professor Grant E. Hearn cSB MSc CMath FIMA CEng FRINA trained as an industrial
mathematician, graduating from Bath University of Technology. He also studied at
Sheffield University and was a Research Fellow in Applied Mathematics before joining
Strathclyde University as the DTI Research Fellow in Naval Architecture. At BSRA he
was promoted to Head of Mathematics Group and joined Newcastle University from
BSRA. He has worked in the aircraft industry, telecommunications industry and glass
industry before committing himself to ship and offshore related hydrodynamics and
design. He si currently Professor of Hydrodynamics and Head of the Department of
Marine Technology.
Dr Andrew Metcalfe cSB PhD CStat AFIMA si a Senior Lecturer in the Department of
Engineering Mathematics which he joined in 1978. His current research involves the
applications of statistical methods to engineering problems, but past work has included
theoretical and practical investigation of active vibration controllers. He has consider-
able experience of acting as a statistical consultant to industry, through personal
consultancies and Teaching Company schemes.
Preface
Twelve years ago, we suggested a course called 'Probability and Spectral Techniques' as
part of a Science and Engineering Research Council programme of postgraduate training
for engineers. We thought the ability to model wind and wave forces dynamically, rather
than as a static wave, was an essential skill for modern engineering. Other important
applications ranged from the design of vibration controllers to the measurement of
surface finish. We thoroughly enjoyed giving the course, at the University of Newcastle
upon Tyne, on several occasions, and hope the participants learnt as much as we did.
This book is based on the course notes, which have evolved over the years. Peter
Gedling, who then worked for the British Ship Research Association, was a guest
lecturer. His clear exposition, and physical insight, are the basis of the sections on the
maximum entropy and maximum likelihood methods for estimating spectra. One of the
engineers on the first course, John Medhurst, has contributed the case study on
measuring ship hull roughness, and Bob Townsin gave us helpful comments on our
presentation of this work. We thank them for permission to use their work. We also
thank Ann Satow for organizing the courses so efficiently, Richard Carter for valuable
library support, and Diane Sevenoaks for all her secretarial work.
The Newcastle short courses were followed by a special 'in-house' version for BP
International in London. We thank the Company for taking such an interest, and Colin
McFarlane, now Professor of Subsea Engineering at Strathclyde University, for arrang-
ing this. Our own undergraduates were, and still are, taught about spectral analysis
during their second year, and Professors Caldwell and Jeffrey, then our respective Heads
of Departments, suggested we write a book on the subject. This we followed up, and we
thank them for their support. We also thank John Roberts of Edward Arnold for his
enthusiasm for the project.
In the first half of the book we mix the physical and mathematical development with
practical applications. The reader needs no more than a knowledge of elementary
calculus, and basic statistics and probability, to follow this material, which is accessible to
undergraduates at any stage of their courses. We thank many people for the data and
permission to base examples on their work. The later chapters describe case studies.
These are valuable, because they demonstrate the type of thinking, and the sort of
compromises that have to be made, to solve real engineering problems. We do not
expect readers to be faced with identical challenges. What we do hope si that this book
will help them solve new problems in their careers as engineers.
Preparing the text took longer than we expected, so we also have to thank two
subsequent editors, David Mackin and Russell Parry, for their forbearance. We are also
grateful to all our undergraduate students for their perceptive comments, to Brenda
Robson for typing the manuscript, and to the production staff at Arnold.
Grant E. Hearn Andrew V. Metcalfe
Newcastle upon Tyne
June 1994
xi
Notation and nomenclature
A summary of the main symbols used in the text is given below.
i
Sample Population
(number of data N) (usually considered infinite)
~. mean
6 ~ variance a 2 or Var
r6 standard deviation tr
scov covariance Cov
r correlation p
c(k)
autocovariance )k(,~
r(k) autocorrelation p(k)
C(to) spectrum )00(F
(cid:12)9 Random variables and their values in a particular case, data, are usually distinguished
by upper and lower case letters respectively, e.g. X and x. Upper case letters are also
x(t),
used for transforms of time signals, e.g. X(to) for the transform of but this should
be clear from the context.
(cid:12)9 The limits of summations are not explicitly given if the summation is clearly from
ltoN.
(cid:12)9 The equals sign is used for definitions, assignments, identities and equations. Which of
these is meant should be clear from the context.
(cid:12)9 g is usually gravitational acceleration.
(cid:12)9 The case studies often involve many parameters, and some of the notation used is
specific to particular case studies.
(cid:12)9 In the context of sampled signals, frequency ot is dimensionless and expressed as
radians per sampling interval. If the sampling interval is 1 second, then 00 is in radians
per second, whereas a sampling interval of 0.01 second implies that ot is in units of 100
radians per second. In the physical context of many of the case studies ot is in radians
per second.
(cid:12)9 For those not familiar with the motions of rigid bodies (aeroplanes or ships), the terms
surge, sway and heave, and the terms roll, pitch and yaw, refer to oscillatory
translations and rotations of the body, respectively.
(cid:12)9 Water waves in the engineering context are designated: incident, scattered or
diffracted, and radiated waves. An incident wave is the wave approaching the
structure. The scattered, or diffracted, wave is the wave resulting from the interaction
of the incident wave with a floating or fixed structure. The radiation waves are
generated by the motions of the structure, and there si one radiation wave system for
each degree of freedom. The incident and scattered waves provide the wave excitation
forces and moments. The radiation waves provide the reactive forces and moments.
Second order forces are designated drift forces or added resistance forces, according
to the absence or presence of forward speed, respectively.
xii
Why understand spectral ?sisylana
I. I Introduction
If you can hear the difference between Chopin and Beethoven you are responding to the
different frequency compositions of changes in air pressure. As well as our hearing, our
sight is also very sensitive to the frequency, and amplitude, of signals. Visible light is
electromagnetic radiation between frequencies of approximately 6 x 10 4~ and 01 5~ cycles
per second. Within this frequency range our eyes can detect all the colours of the
rainbow and their mixtures. A rainbow is a common and beautiful sight, and is readily
explained as refraction of sunlight in water droplets. Since the amount of refraction is
greater for shorter wavelengths (higher frequencies) the sunlight is split into its different
components--a spectrum. Most other creatures are responsive to light and sound
signals~bats are a good example of the latter. We are also sensitive, as far as our health
is concerned, to frequencies of electromagnetic radiation that we cannot sense directly.
For example, the higher frequencies, including X-rays and gamma rays, are known to be
dangerous in anything but very small doses.
Physical systems are also highly sensitive to signal frequency, and we refer to such
systems throughout this book. The techniques described here can be used to distinguish a
repeating pattern in a signal from background noise. However, we concentrate on using
these techniques to describe an average frequency composition of non-deterministic
disturbances, such as wind gusts and wave motion. Anyone who doubts the need to
understand such phenomena should think about the collapse of the Tacoma Narrows
Bridge. In 1940, four months after the bridge opened, a mild gale set up resonant
vibrations along the half-mile centre span of the bridge, which caused it to collapse
within a few hours. However, this was not the first case of resonance causing a bridge to
collapse. In 1831, a column of soldiers marched across the Broughton Bridge near
Manchester, UK and set up a forced vibration whose frequency was close to a natural
frequency of the bridge. The Broughton Bridge collapsed, and the advice to break step
when marching across bridges is now traditional.
An oscillatory motion, under a retarding force proportional to the amount of
displacement from an equilibrium position, is known as simple harmonic motion. Very
many natural oscillations are well modelled in this way. Some examples are: a mass on a
spring which is given an initial displacement and then released, a pendulum making small
oscillations, and a spindly tree branch after a bird lands on it. The displacement of a body
undergoing simple harmonic motion, plotted against time, is a sine curve (also known as
a sinusoid or harmonic). The bird on the branch will also be subject to considerable
natural damping, which reduces the amplitude of the oscillation quite rapidly. If this is
modelled by assuming damping is proportional to velocity, then the resulting motion is a
2 yhW ?sisylanalartcepsdnatsrednu
harmonic whose amplitude decays exponentially. The essential harmonic nature of the
disturbance is retained. Another example is radio transmitters, which emit signals that
are the sum of harmonic waves.
In spectroscopy, elements are excited at their natural frequency and emit harmonic
radiation of a 'pure' colour. One reason for taking a harmonic as the standard shape for a
wave in spectral analysis is its frequent natural occurrence. A harmonic signal si
completely described by its amplitude, its frequency and, when it is being considered
relative to some time origin or other signals, its phase. The most common (SI) scientific
unit for frequency is now cycles per second, hertz, but old wireless sets are usually
calibrated in terms of wavelength. (If f is the frequency in hertz the wavelength in metres
is c/f where c is the speed of light in m s-~; that is, approximately 3 x 108ms-~.)
However, from a mathematical point of view, it is more convenient to use radians---one
cycle is r72 radians~because the use of radians eliminates factors of rz2 from arguments
of exponential functions in the formulae. A further remark about units is that not all
waves occur in time. Measurement of surface roughness, from aeroplane runways to
painted surfaces on ship hulls, is an important application of spectral analysis and the
appropriate units are cycles (or radians) per unit length.
An alternative approach to describing a harmonic signal, and any other signal, is to
give its value over time. In principle, this could be a continuous trace~as recorded by an
oscilloscope~but it is now more usual to give sampled values. The sampling interval
must be small compared with the time taken for a single cycle at the highest frequency
present in the signal, so that we avoid ambiguity in our analyses. The discrete sequence
resulting from the sampling process is known as a 'time series'. The time series can be
analysed on a digital computer and is the starting point for the techniques described in
this book. Electronic devices that sample a continuous signal are known as analogue-to-
digital (A/D) converters, and speeds up to 250000 samples per second are usual on
standard equipment. The physical systems described in this book have their natural
frequencies well below 100 Hz and can be analysed with such equipment. The theoretical
concepts are also relevant for specialist applications such as radio and radar. Although
the sampling interval is of crucial importance, it is convenient to work in radians per
sampling time unit and thereby avoid introducing the sampling interval explicitly into the
formulae. The final results can always be rescaled into radians per second or hertz.
For some purposes, notably the design of digital controllers for systems ranging from
inter-connected reservoirs to industrial robots, the time history is usually used. For other
purposes a 'frequency domain' description, which excludes the phase information, may
be more appropriate. This si provided by calculating a function of frequency, known as
the 'spectrum'. A brief explanation of the spectrum is needed if one is to appreciate why
it can be a more useful way of looking at data than the data's original time order.
Jean Baptiste Joseph Fourier (1768-1830) first investigated the possibilities of approx-
imating functions by a sum of harmonic components. These ideas can be applied to a
time-ordered sequence consisting of an even number (N) of data. If this 'time series' is
plotted, datum against sample number, we have N points equally spaced in the
horizontal direction but, in general, with no obvious pattern in the vertical direction.
However, it is possible to construct a function, which si the sum of a constant and N/2
harmonics of frequencies 2"rr/N, 47r/N, 67r/N ..... (N/2)2zr/N radians per sampling
interval, whose graph passes through all N points. This function requires a unique
specification of the amplitudes and phases of the N/2 harmonics. Throughout this book
the time series will be thought of as one of an infinite number that could have been
2.1 Overview 3
generated by some underlying random process. The precise form of the function is
therefore irrelevant, particularly as the specific frequencies used depend only on the
record length. The sample spectrum is calculated by averaging the squares of amplitudes
of the harmonics over sensibly chosen frequency intervals, and it consists of contribu-
tions to the variance of the original time series over a continuous frequency range. The
sample spectrum is an estimate of the unknown spectrum of the supposed underlying
random process.
Thus, the spectrum highlights the frequency composition of a signal. This may be of
scientific interest in itself, for example sunspot cycles and periodicities in other data
obtained from space exploration. However, the main justification for spectral analysis is
its practical importance, emphasized by examples throughout the text and the case
studies which form the later part of the book.
Two applications, which demonstrate the possible advantages of calculating the
spectrum, rather than relying on the original time series, are mentioned here. The first is
'signature analysis' of rotating machinery. A sudden change in the spectrum of a
vibration signal from machinery can provide early warning of a breakdown. A policy of
rectifying faults before catastrophic breakdowns contributes to safety and can result in
considerable financial savings. If the change in the spectrum is a shift it might not be
detected by simply monitoring the amplitude of the time series. However, an additional
peak in the spectrum would be linked to an increase in variance of the time series, even
though this would be more difficult to detect at an early stage.
The second example concerns the design of offshore structures, such as drilling
platforms. These, and many other structures, can reasonably be modelled as linear~at
least within certain limits. A linear structure responds to a harmonic disturbance by
vibrating at the same frequency. The amplitude of the induced vibration si proportional
to the amplitude of the disturbance, and the constant of proportionality depends on the
frequency. The frequencies at which the structure is most responsive are known as its
natural frequencies, and these can be calculated theoretically from the design. If the
spectra of typical seas are estimated, the structure can be designed so that its natural
frequencies are distanced from likely peaks in the sea spectra, and its response can be
predicted. It would, admittedly, be feasible to model the response of the structure to
typical time series, but this would not give any insight into the design. If the response was
excessive it would not be possible to decide which natural frequencies to try and move,
unless the spectrum of the response signal was itself calculated. Even if the response
appeared satisfactory, the proximity of a natural frequency to possible peaks in the sea
spectra would go unnoticed. The design of a vehicle suspension for typical road or
runway surfaces is a similar problem.
The objective of this introduction has been to show that, whilst the spectrum is
calculated from a time history and cannot contain any additional information, it presents
the frequency content very clearly. For many engineering and scientific purposes this si
exactly what is required. This is not to deny the value of analyses in the time domain,
which are also covered in their own right and as a prelude to spectral analysis.
1.2 Overview
This section provides a brief overview of the contents of the following chapters.
Chapter 2, 'Relationships between variables', assumes some background knowledge of
4 Why dnatsrednu ?sisylanalartceps
probability and statistics and concentrates on joint probability distributions. The ideas of
covariance and correlation (which can be thought of as measures of linear association
between random variables) are particularly relevant to what follows. A closely associated
method is regression analysis, where we investigate the distribution of the random
variable Y for fixed values of x; that is, the conditional distributions of .Y This is a widely
used and useful technique.
Chapter 3 is entitled 'Time varying signals'. Throughout this book a time series is
considered as a realization of some underlying random (stochastic) process. A full
description of a random process is usually very complex and we will concentrate on the
first and second moments, known as the mean and the autocovariance function. The
latter plays an essential part in spectral analysis. It is particularly important to
understand clearly the concepts of stationarity and ergodicity. A random process is
second-order stationary if its mean and variance do not change with time, and the
covariance depends only on the time lag between variables and not on absolute time. The
'ensemble' is the hypothetical population of all possible time series that might be
produced by the underlying random process. A random process is ergodic if time
averages tend to averages over the ensemble. Loosely speaking, ergodicity means that a
sufficiently long stretch of one record will be representative of the ensemble. It is usual to
define the ensemble so that the assumption of ergodicity is reasonalJle. In some situations
it may be possible to obtain several realizations, in which case the assumption of
ergodicity is open to investigation, although there is always the problem that each
realization may not be sufficiently long. An example of this situation might be signals
received from mobile pipeline scanners on different occasions. In many other circum-
stances, such as river flow records, there may only be the one realization. Before carrying
out a spectral analysis it is assumed that the underlying process is stationary. This may
require some preprocessing of the data to remove any trends or seasonal variability.
Non-stationarity can be checked to some extent by looking through the one time series
for obvious trends or changes in variability.
Fourier analysis is covered in Chapter 4, 'Describing signals in terms of frequency'. A
finite Fourier series is a finite sum of sine waves with periods and phases chosen so that it
coincides exactly with a finite number of datum points. A Fourier series is an infinite sum
of such waves which converges to a signal defined for all time on a finite interval. In both
the above cases the Fourier series is periodic, with period equal to the time interval of the
signal. The Fourier transform is obtained by considering the time interval in the (infinite)
Fourier series tending to infinity. Finally, the discrete Fourier transform is defined for an
infinite sequence. The complex forms of these results are much easier to handle
algebraically. The various statements of Parseval's Theorem are crucial to the arguments
that follow and the convolution results are also often used. In spectral analysis, the usual
situation is that a sequence of data is available that can be considered a sample from a
continuous signal. A potential pitfall is the phenomenon of aliasing. For example, with a
sampling interval of 0.2 s a frequency of 4 Hz appears identical to one of frequency 1 Hz,
yet a system may respond to a vibration of 1 Hz but be relatively unaffected by the higher
frequency vibration. Once a signal is digitized, aliasing cannot be avoided if frequencies
higher than the 'Nyquist frequency' are present. To avoid aliasing, the original
continuous (analogue) signal must have the higher frequency components filtered out by
electrical means, or the sampling interval must be chosen so that the Nyquist frequency is
higher than any frequencies present in the signal.
Chapter 5, 'Frequency representation of random signals', begins with a justification of
2.1 Overview 5
the definition of the spectrum of a stochastic process as the Fourier transform of the
autocovariance function. The following sections deal with estimation of the spectrum. As
the sample size increases, more ordinates are estimated in the unsmoothed sample
spectrum, but the variability of the estimates does not decrease. The fidelity increases
but the stability does not. Therefore, some smoothing procedure si necessary. One
approach is to consider dividing the record into sections, calculating the sample spectrum
for each section and averaging the results. The more sections the record is divided into
the more stable the estimate will become, but this will be at the expense of smoothing out
peaks and troughs. This approach si shown to be a special case of smoothing a sample
spectrum estimator by giving decreasing weight to the autocovariances as the lag
increases. The weighting function si known as the lag window and leads to a smoothed
spectral estimator. Different windows and the effect of bandwidth are discussed. This
general approach si the simplest computational method of spectrum estimation. An
alternative is to average neighbouring ordinates in the periodogram. This si made
computationally easier by the fast Fourier transform algorithm (FFT), which si an
efficient computational technique used to evaluate the discrete Fourier transform of the
recorded data. A simple derivation of the FFT method si given. The ideas behind the
more recent maximum entropy method (MEM) and maximum likelihood method
(MLM) of estimating the spectrum are also described. These are valuable with short data
sets and for certain spatial problems.
Up to this point all the analyses have been of a single time series. Most practical
applications involve the relationships between the inputs and outputs of physical
systems. In this book the emphasis si on linear systems. For a linear system, the response
to a sum of inputs is the sum of the responses to the individual inputs. Furthermore, the
response to a harmonic signal will be at the same frequency, with the amplitude
multiplied by some factor, and a phase shift. Both the factor and the phase shift depend
on the frequency, but are constant for any given frequency, and are summarized by the
'transfer function'. Linear models provide good approximations to many physical
systems. Their theory si particularly important because they are also often used as 'local
approximations' to the theoretically more complicated non-linear systems. Chapter 6,
'Identifying system relationships from measurements', includes the theory behind
transfer functions and their estimation. This involves looking at two random processes,
or time series, and investigating the relationship between them. Concepts such as
cross-covariance, cross-spectra and coherency are natural extensions of the previous
work. At this stage lla the essential theory has been recovered.
Chapter 7, 'Some typical applications', includes all the detailed working required to
arrive at a sample spectrum. With an estimate of the spectrum available and some
knowledge of a structure's transfer functions the response spectra may be determined.
This may then lead to assessment of the probability of the responses exceeding some
design or operational threshold. Transformation of the spectrum from one form to
another and the ideas of significant responses are introduced and applied to the design of
offshore oil rigs.
In the first part of the book, examples have been used to highlight specific points
associated with the development of the theory, and to illustrate the application of
methods to relatively simple situations. The second part of the book deals with
engineering case studies. In addressing engineering problems the convenient partitioning
of knowledge into subjects, which form the basis of specific courses, si not always
possible. This si because engineering problems do not generally present themselves as
6 yhW ?sisylana/artcepsdnatsrednu
nicely posed questions limited to one particular area of knowledge. The purpose of the
case studies presented in Chapters 8 to 21 is to give some insight into the application of
spectral analysis to actual engineering problems, mostly tackled by the authors, and
simultaneously to provide some appreciation of the variety of roles spectral analysis will
play in different situations. In attempting such presentations there si the danger that each
problem will demand so much explanation that the relevance of spectral analysis si lost,
or its role cannot be appreciated because of the knowledge required to address other
integral aspects of the problem. In each case study we shall therefore provide some
general background on the engineering problem, identify a number of questions that
must be answered via analysis to solve the problem, explain the role of spectral analysis
in the solution and then proceed with the applications.
Because the problems solved are real, and the authors human, errors of judgement will
have sometimes been made in the initial analysis. In these cases we explain how initial
methods of analysis have been modified, or additional analyses undertaken, to quantify
the errors or lack of resolution in the engineering quantities of interest. At the time it was
carried out, much of this work was innovative. However, subjects develop, and the
computing power now available may facilitate more sophisticated analyses. Even so, we
think the solutions are still adequate answers to the problems posed. Furthermore, the
solutions are included here because they provide insight into how to go about solving
problems, not to provide the latest state of forward thinking regarding analysis methods
per se.
The different case studies presented can be read in any order, depending upon the
reader's needs and interest. For those involved in the various disciplines of Marine
Technology the ordering of the marine related case studies si both logical and deliberate.
However, the book is meant to be of general interest to all engineers. To provide case
studies related to other specific disciplines would require us to obtain wider experiences.
The lessons learnt in honestly reporting the case studies should be transferable to other
situations.
The first case outlines the mathematical modelling required behind the design of a
wave monitoring system. The system described was actually built and here we consider
the use of the FFT method to estimate the spreading of the wave energy and the
problems of resolutions which were overcome using the MLM approach. The second
case study considers the simulation of moored offshore structures subject to random
excitation and the associated problem of generating realizations representative of a
specific spectral form and characteristics. The method was used to show that certain
simplified hydrodynamic models for predicting low frequency damping are totally
inappropriate for realistic simulations of moored structures. The third case study
discusses an investigation of the low frequency damping forces of moored tankers and
barges. This case study highlights the difficulties of extracting the required time series
from the recorded data and the importance of including the analysis techniques as an
integral part of the experimental design. These two studies also, rather more construc-
tively, allowed judgements to be made about the appropriateness of the mooring lines'
configuration and materials, and their ability to keep the structure on station even if one
or two failed.
The fourth case study describes the use of a spectral analyser to investigate the
performance of active vibration controllers on a test rig. The final case is concerned with
the characterization of the roughness of paint surfaces on the outer hulls of ships. A
moderate level of wetted hull roughness may significantly increase a shipowner's fuel bill
2.1 Overview 7
(by %01 or more) as a result of the increased resistance caused by the roughness. A
considerable amount of research effort has been devoted to measuring and reducing
roughness of ship paint finishes.