Spectral Analysis in Engineering, Concepts and Case Studies

Spectral Analysis in Engineering

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Spectral Analysis in Engineering Concepts and Cases Grant E Hearn Andrew V Metcalfe University of Newcastle upon Tyne

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Elsevier Ltd. Linacre House, Jordan Hill, Oxford OX2 8DP 200 Wheeler Road, Burlington, MA 01803 Transferred to digital printing 2004 9 1995 Grant E Hearn and Andrew V Metcalfe All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronically or mechanically, including photocopying, recording or any information storage or retrieval system, without either prior permission in writing from the publisher or a licence permitting restricted copying. In the United Kingdom such iicences are issued by the Copyright Licensing Agency: 90 Tottenham Court Road, London WlP 9HE. Whilst the advice and information in this book is believed to be true and accurate at the date of going to press, neither the authors nor the publisher can accept any legal responsibility or liability for any errors or omissions that may be made.

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 340 63171 6 1 2 3 4 5

95 96 97 98 99

Typeset in 10/12pt Times by Wearset, Boldon, Tyne and Wear.

Contents

About the authors Preface Notation and nomenclature

xi .o

Xll

Why understand spectral analysis? 1.1 1.2

Introduction Overview

Relationships between variables 2.1 Introduction 2.2 Discrete bivariate distributions 2.2.1 Modelling discrete bivariate populations 2.2.2 Expectation for discrete distributions 2.3 Continuous bivariate distributions 2.3.1 Modelling continuous bivariate distributions 2.3.2 Justification and geometric interpretation for continuous conditional densities 2.3.3 Sample correlation for continuous variables 2.3.4 Expectation for continuous distributions 2.4 Linear functions of random variables 2.5 Bivariate normal distribution 2.6 Confidence intervals for population correlation coefficient 2.7 Multivariate normal distribution 2.8 Exercises

3 Time varying signals 3.1 Introduction 3.2 Why study time series? 3.3 Estimation of seasonal effects and trends 3.3.1 Moving average method 3.3.2 Standardizing method 3.3.3 Multiple regression method 3.3.4 Estimation of trend 3.4 Moments of a discrete random process 3.4.1 The ensemble

8 9 10 12 15 18 20 20 24 27 31 35 36 36 39 39 40 41 42 42 43 43 47 47

vi

Contents 3.4.2 Moments of a discrete random process 3.5 Stationarity and ergodicity 3.5.1 Stationarity 3.5.2 Ergodicity 3.6 ARIMA models for discrete random processes 3.6.1 Discrete white noise (DWN) 3.6.2 Random walk 3.6.3 Moving average processes 3.6.4 Autoregressive processes 3.6.5 ARIMA (p,d,q) processes 3.6.6 Gaussian and non-Gaussian processes 3.6.7 Relationship between MA and AR processes 3.7 Estimation of parameters of models for random processes 3.7.1 Estimation of the autocovariance function 3.7.2 Bias of the autocovariance function 3.7.3 Estimation of the autocorrelation function 3.7.4 Estimation of parameters in ARMA models 3.7.5 Determining the order of ARMA processes 3.8 Simulations 3.9 Further practical examples 3.10 Models for continuous time random processes 3.10.1 The Dirac delta 3.10.2 Autocovariance function 3.10.3 Estimation of the mean and autocovariance function 3.10.4 Wiener process 3.10.5 White noise in continuous time 3.10.6 Linear processes 11 Exercises 11

Describing signals in terms of frequency 4.1 Introduction 4.2 Finite Fourier series 4.2.1 Fourier series for a finite discrete signal 4.2.2 Parseval's theorem 4.2.3 Leakage 4.3 Fourier series 4.4 The Fourier transform 4.4.1 Fourier transform 4.4.2 Generalized functions 4.4.3 Convolution integrals 4.5 Discrete Fourier transform 4.5.1 Discrete Fourier transform for an infinite sequence 4.5.2 Nyquist frequency 4.5.3 Convolution integral results for infinite sequences 4.5.4 The discrete Fourier transform 4.6 Exercises

47 49 49 49 51 51 51 51 53 57 60 61 62 62 62 64 64 70 73 74 83 83 83 84 84 85 86 87 91 91 91 91 94 95 97 100 100 103 104 105 105 106 106 107 108

Contents vii 5 Frequency representation of random signals 5.1 Introduction 5.2 Definition of the spectrum of a random process 5.2.1 The periodogram 5.2.2 The relationship between the sample spectrum and autocovariance function 5.2.3 The spectrum of a random process 5.3 Estimation of the spectrum from the sample autocovariance function 5.3.1 The need for smoothing 5.3.2 Smoothed autocovariance based spectral estimators 5.3.3 Alternative lag windows 5.3.4 Comparing windows 5.3.5 Confidence intervals for the spectrum 5.4 Estimation of the spectrum from the periodogram 5.4.1 Smoothing the periodogram 5.4.2 Segment averaging 5.4.3 Use of the fast Fourier transform to estimate the periodogram 5.5 High resolution spectral estimators 5.5.1 The maximum entropy method 5.5.2 The maximum likelihood method 5.6 Exercises

109 109 110 110 111 114 115 115 119 121 123 127 128 128 129 130 136 136 138 141

6 Identifying system relationships from measurements

143

6.1 Introduction 6.2 Discrete processes 6.2.1 Generalization of the covariance concept 6.2.2 Cross-spectrum 6.3 Linear dynamic systems 6.4 Application of cross-spectral concepts 6.5 Estimation of cross-spectral functions 6.5.1 Estimating cross-correlograms and spectra 6.5.2 Estimation of linear system transfer functions 6.6 Exercises

143 145 145 147 148 150 152 152 153 157

7 Some typical applications 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Introduction Calculating the sample autocovariance function Calculating the spectrum Calculating the response spectrum The spectrum and moving observers Calculation of significant responses Exercises

Wave directionality monitoring 8.1

8.2

Introduction

Background

161 161 161 162 163 170 173 179 184 184 184

viii Contents

8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 9

Motions of moored structures in a seaway

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 I0

Introduction Background Modelling moored structures Equations of motion Determination of time dependent wave force Evaluation of the quadratic transfer function (QTF) Simulation of a random sea Why the probabilistic method of simulation? Statistical analyses of the generated time series Sensitivity analysis of a moored tanker and a moored barge to integration time step Effects of wave damping on the surge motion Final comments

Experimental measurement and time series acquisition

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 II

The technical problem Reduction of the monitoring problem to a mathematical problem Application of the mathematical model The probe arrangements deployed Analysis of Loch Ness data MLM-based spectral analysis formulations Cross-spectral density simulation Simulation results and alternative probe management Final comments

Introduction Background Experimental facilities set-up Data collection and principles of analysis Six degrees-of-freedom SELSPOT motion analysis Data acquisition and SELSPOT calibration Practical aspects Some typical results Final comments

Experimental evaluation of wide band active vibration controllers

11.1 Introduction 11.2 Background 11.3 Techniques for active vibration control 11.4 Why use a spectral analyser? 11.5 Experimental rig 11.6 Some typical results 11.7 Final comments

187 189 192 193 194 197 198 199 206 208 208 209 211 212 213 215 216 218 219 221 221 232 233 233 233 235 236 239 240 242 245 248 253 253 253 254 255 255 257 257

Contents ix

12 Hull roughness and ship resistance 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Introduction Background An introduction to surface metrology Process bandwidth Surface topography and fluid drag Measures of texture Filtering and filter assessment Final comments

Appendices Appendix I:

Appendix II: Appendix III:

Appendix IV: Appendix V: Appendix VI:

261 261 261 262 263 264 267 272 273 274

Mathematics revision Arithmetic series Geometric series Harmonic series Taylor series Even and odd functions Complex numbers Generating pseudo-random numbers Impulse responses Inflows to the Font reservoir Chi-square and F-distributions Chi-square distributions F-distribution Relationship with the t-distribution Comparison of variances The sampling theorem Wave tank data Sampling distribution of spectral estimators Discrete white noise A linear stochastic process Smoothed spectral estimators

274 274 274 275 275 276 277 279 280 283 285 285 286 287 287 289 291 292 292 292 293

References

294

Further reading

299

Index

303

Answers to exercises are available on the Internet at" h ttp://www, ncl. ac. uk/~- ne ngm/publish/

About the authors

Professor Grant E. Hearn BSc 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 is currently Professor of Hydrodynamics and Head of the Department of Marine Technology. Dr Andrew Metcalfe BSc PhD CStat AFIMA is 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 considerable 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 arranging 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 is 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

Newcastle upon Tyne June 1994

Andrew V. Metcalfe

xi

Notation and nomenclature

A summary of the main symbols used in the text is given below. i

Sample (number of data N) .~ 6~ 6r scov r

c(k) r(k) C(to)

Population (usually considered infinite) mean variance standard deviation covariance correlation autocovariance autocorrelation spectrum

a 2 or Var tr Cov p ~,(k) p(k) F(00)

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 used for transforms of time signals, e.g. X(to) for the transform of x(t), but this should be clear from the context. 9 The limits of summations are not explicitly given if the summation is clearly from ltoN. 9 The equals sign is used for definitions, assignments, identities and equations. Which of these is meant should be clear from the context. 9 g is usually gravitational acceleration. 9 The case studies often involve many parameters, and some of the notation used is specific to particular case studies. 9 In the context of sampled signals, frequency to 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 to is in units of 100 radians per second. In the physical context of many of the case studies to is in radians per second. 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. 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 is 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 analysis?

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 10 ~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 Why understandspectralanalysis? 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 is 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 27r radians~because the use of radians eliminates factors of 2zr 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-todigital (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 is 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 approximating 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 is 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

1.2 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 contributions 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 is 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 is 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 understandspectralanalysis? 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 circumstances, 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

1.2 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 is 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 is 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 is known as the lag window and leads to a smoothed spectral estimator. Different windows and the effect of bandwidth are discussed. This general approach is the simplest computational method of spectrum estimation. An alternative is to average neighbouring ordinates in the periodogram. This is made computationally easier by the fast Fourier transform algorithm (FFT), which is an efficient computational technique used to evaluate the discrete Fourier transform of the recorded data. A simple derivation of the FFT method is 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 is 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 is 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 all 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, is not always possible. This is because engineering problems do not generally present themselves as

6 Why understandspectra/analysis? nicely posed questions limited to one particular area of knowledge. The purpose of the case studies presented in Chapters 8 to 12 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 is the danger that each problem will demand so much explanation that the relevance of spectral analysis is 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 p e r 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 is 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 constructively, 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

1.2 Overview 7

(by 10% 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.

2 Relationships between variables

2. I

Introduction

We start by imagining that we are responsible for monitoring the quality of drinking water in a region. To do this we will need to fill bottles with samples of water and analyse the contents. Two questions arise immediately and neither is straightforward. The first is: what measurements should we make during our analyses? The second is: where should we fill our bottles? Some of the items we should measure are acidity, chemical content, including traces of lead or other metals, coloration and number of microbes. The point illustrated, and of principal interest in this chapter, is that we are interested in more than one variate for each bottle analysed. The answer to the second question is that the bottles must be filled according to some random sampling scheme. However, complications arise when we start to define the details of an appropriate random sampling scheme. We could begin by identifying all the kitchen mains supply taps in the region. The simplest procedure would be to number all the taps and use computer generated random numbers to select those from which the bottles would be filled on each occasion. Such a procedure is an example of a simple random sampling scheme. Throughout this book we will assume that samples have been obtained from such a scheme unless we state otherwise. Use of simple random sampling in this case could be criticized because it might lead to all the bottles being filled at taps in a new estate. Although this is unlikely if the sample is large, and most unlikely to occur often in repeated monitoring, unlikely events do occur and such a sample would be unrepresentative in this respect. However, if we generate simple random samples until we obtain a sample we like, we destroy the random nature of the scheme. Randomization is essential so that the sampling can be seen to be 'fair', and as the basis for a measure of the accuracy of estimates. A solution to this dilemma is to divide all the taps into areas and then take simple random samples within each area. This is an example of a stratified random sampling scheme, with the areas forming the strata. It is not necessary for all taps to have the same probabilities of selection provided the results are weighted appropriately. The essential requirement is that they all have a known positive chance of selection. It is easy to think of other examples where we are interested in more than one variate for each unit in a study. If we are manufacturing bonded razor blades we would take frequent random samples to control quality. For each blade we might measure the geometry~for example, the protrusion from the guard bar at both ends~the sharpness, and the number and depth of any microscopic notches in the profile. Whenever we have more than one variate we are considering multi-variate data. When there are only two variates we can use three-dimensional diagrams to illustrate ideas which generalize algebraically. In all situations we must be mindful of distinctions between populations

2.2

Discrete bivariate distributions

9

and samples, and between discrete and continuous variates. These distinctions will be illustrated by looking at some bivariate data sets in detail.

2.2

Discrete bivariate distributions

Example 2. I Table 2.1 provides an estimate of waterway quality for England and Wales in 1985. It has

been taken from the 1987 edition of Social Trends, which is published by the Central Statistical Office. The entries in the table are the number of kilometres (km) of three types of waterway which satisfy certain criteria for quality. Table 2.1 River quality of waterways in England and Wales: estimates of km in each of 12 categories i

Classification of waterway

Classification by quality

Canals

Freshwater rivers

Estuaries

Total

Good Fair Poor Bad Total

955 1271 240 31 2497

26 500 8 463 3 315 619 38 897

1860 652 129 89 2730

29 315 10 386 3 684 739 44 124

i

i|

ill

If we identify a 1 km stretch of water as a 'unit', we can associate two discrete variates with each unit. The variates are the 'waterway classification' and the 'pollutant classification'. If we add over the pollutant classification we have the distribution of waterway classification, given in the bottom line of the table. This is known as a marginal distribution. That is, the distribution of kilometres of waterway according to waterway type irrespective of the quality of the waterways. The term 'marginal distribution' thus represents a distribution in which the influence of one or more known variates has been summed out. The marginal distribution of pollutant classification is given by adding over the waterway classification. The result is shown in the right-hand column of the table. The entries for each of the 12 categories give the bivariate distribution. This is illustrated in Fig. 2.1. The estimates of kilometres of waterway in each category are based on samples. We could use these estimates to postulate a model for the corresponding population. Whilst it is easy to refer to a 'corresponding population' we should give careful thought to its definition. In this case, the population could be all 1 km stretches of waterway in England and Wales. The variates of interest could be the waterway classification and the pollutant classification based on an average throughout 1985. In common with many other engineering examples there is a certain amount of subjectivity in defining the population. The definitions of quality are even more subjective. The populations we define may often be imaginary and infinite, for example, all the items that would be produced if a machine continues indefinitely on its present settings. We must remember that statistical inference is crucially based on the assumption that we have a random sample from the population. Sampling has been used in three respects when constructing Table 2.1. Firstly, not all 1 km stretches of waterway have been monitored. Thus, the results given are based on a

I0

Re/ationshipsbetween variables

Fig. 2.1

Bivariate distribution of waterway classification and water quality

sample of all the waterways in the two countries. Within each stretch of waterway chosen for investigation it is only possible to draw samples from spot locations at any one time. Finally, the samples were taken at some unstated points in time and not continuously throughout the year. There is a great deal of detail behind the construction of Table 2.1, and its soundness depends crucially on the sampling schemes used.

2.2.1

Modelling discrete bivariate populations

Let X and Y be discrete random variables. The bivariate probability mass function is defined by

Pxv(X, Y) = Pr {X = x and Y = y }

(2.1)

If we refer back to Example 2.1 we could define X as taking the values 1, 2 or 3 if a randomly selected 1 km stretch of waterway is a canal, a freshwater river or an estuary respectively. We could define Y as taking the values 1, 2, 3 or 4 according to the classification by pollutant. A probability mass function, which could provide a model for the population from which the data represented by Table 2.1 came, is given in Table 2.2. Each entry has been obtained by dividing the corresponding entry in Table 2.1 by the total number of kilometres of waterway, namely 44124, and rounding to three decimal places. A line diagram for the probability mass function is illustrated in Fig. 2.2. For any bivariate probability mass function

E Z px~x,y) = 1 y

x

and you can check that the numbers in Table 2.2 do satisfy this requirement.

2.2

Table 2.2

Values of

,,

,i

Type of waterway x 2 3

1

,,

0.022 0.028 0.005 0.001

,

II

for the water quality model

Pxy(X,y)

Pollutant y 1 2 3 4

Discrete bivariate distributions

0.600 0.192 0.075 0.014

0.042 0.015 0.003 0.003

The marginal probability mass function of X is defined by (2.2)

Px(x) = Z Pxv(x,y) y

We can define the marginal probability mass function of Y in a similar fashion. We can

also define the conditional probability mass functions of X given y or Y given x. The formal definition of the latter is

Pvl~(y]x) = P r { r = y l X = x} Pr{ Y = y and X = x}

=

P r { X = x}

=

Pxv(x,y) ex(x)

(2.3)

For our example, the marginal probability mass function for X is e x ( 1 ) = 0.056,

e x ( 2 ) = 0.881,

e x ( 3 ) = 0.063

0.8 Jl:

._r

0.7 0.6

~6

E 0.5

i

O.4

E

0.3 -

o

Waterway quality

Fair /

I;

~

~

0.2

/

/

o.1

,7

/

/

/

J /

/

,,

= a d / . . . . ~,~c___y_____/ ./

0.0

/

_~l,,_l___ J~," /" / / .poor/_____~c___ ~ c .... ~,,

~

/

Good . . . . : J J

(

Fig. 2.2

/"

I

/1"

,/I 6

_/"

6

II

f

Canals Rivers Estuaries Waterway classification

Bivariate probability mass function to model waterway classification and water quality

12

Relationshipsbetween variables

The marginal probability mass function for Y is Pr(1) = 0.664,

Pr(2) = 0.235,

e y ( 3 ) = 0.083,

e y ( 4 ) = 0.018

The conditional probability mass function for Y when x = 1 is

evl~(lll)

= 0.393,

P r l l ( 2 l 1) = 0.500,

Pvl~(3l 1) =

0.089,

evl~(411) =

0.018

We see that the conditional probability distribution for Y, when we restrict our attention to canals, is different from the distribution of Y when we consider all waterways. In other words, X and Y are dependent random variables. In general, the random variables X and Y are independent if and only if (2.4)

exr(x, y) = ex(x)er(y)

This follows immediately from the definition of independence in probability theory. To show this we start from the basic definition introduced earlier, namely

Pxy(x,y)

= Pr{X = x and Y = y}

If and only if the events X = x and Y = y are independent does Pr{X = x and Y = y} = P r { X = x} Pr{Y = y}, i.e. P r { X = x and Y = y} = Px(x)Pr(y). An equivalent definition of independence is, X and Y are independent if and only if PYI~(YlX) = PY(Y) for any x. This follows from the formal definition of conditional probability. 2.2.2

Expectation for discrete distributions Expectation is averaging in a population. An average value of a variate is the sum of the values it takes for each item in a sample, or population, divided by the total number of items. In the case of infinite populations this requires careful interpretation. For the water quality example, a meaningful statistic could be the average quality of the water. The water quality categories can be thought of as being on a scale numbered somewhat arbitrarily, as 1, 2, 3 and 4 for good to bad respectively. Turning to Table 2.1 we have 29315 km of water of category 1 which can be thought of as 29315 km units of water quality 1. The average waterway classification would be: 1 x 29315 + 2 x 10386+ 3 x 3684 + 4 x 739 44124

= 1.45

An equivalent method of expressing the left-hand side of this equation is 29315 10386 3684 1x ~ + 2 x ~ +3x~+4x 44124 44124 44124

739 44124

and this is the sum of the possible values of the variate multiplied by the relative frequencies of their occurrences. We now consider the model for the population of all waterways, namely Pxr(x,y). The average value of the random variable Y, which represented pollutant, is called the expected value of Y. The expected value of a random variable is the sum of the values it can take, multiplied by the probability of their occurrences. In this case we have

E[Y] =

1 x Pr(1) + 2 x Py(2) + 3 x ey(3) + 4 x ey(4)

= 1 x 0.664 + 2 x 0.235 + 3 x 0.083 + 4 x 0.018 = 1.45

2.2

Discrete bivariate distributions

13

To summarize, when we average in a sample we take the sum of the products of the values of the variate with their relative frequency. When we average in the population, relative frequencies are replaced by probabilities. This ties in with our common sense interpretation of relative frequencies as estimates of probabilities, and probabilities as some sort of limit of the relative frequencies as the sample size becomes very large. In this example, the sample average equals the population average because the population probabilities were inferred from the sample. Good agreement would, in any case, be expected with such a large sample. We can formally define the expected value of an arbitrary function (g, say) of X and Y for a general bivariate probability mass function as

E[g(X,Y)] = ~, ~, g(x,y) Pxv(x,y) y

(2.5)

x

We can easily verify that this definition is consistent when g is a function of only one of the variables, X say. Then

E [ g ( X ) l = X ~, g(x) Pxy(x,y) y

x

= ~ g(x) Px(X),

as required

x

The expected value of Y is known as the mean of the distribution of Y, written as/~v, thus (2.6)

~ r = E[Y]

Another expected value of considerable interest is the variance of the distribution. The variance is the average value of the squared deviations of Y from its mean. It is written as o'2 and its units are the square of the units of Y. The square root of the variance is called the standard deviation, written as try, and has the units of Y. The reason why the variance is often used as a measure of spread of a distribution is that it is relatively easy to handle mathematically. Formally the variance is defined by

trEy = E [( Y - /~ v ) 2]

(2.7)

We now define the covariance of X and Y, written as Cov(X, Y), as Coy(X, Y) = E [ ( X - I ~ x ) ( r - ~v)]

(2.8)

This quantity is of crucial importance in later chapters. Here, we should note that if X and Y are independent the covariance is zero. This is readily illustrated by formal application of the expectation operator, E. By definition Cov(X, Y) = ~, 2 (x - g x ) ( Y - txr) Pxv(x,y) y

x

and assuming X and Y are independent then

14

Re/adonshipsbetweenvariab/es

~---E [ ( Y -

[Jby)] E [ ( X -

~&x)]

By definition o f / x r and/Xx both expected values are zero and the result follows. It is important to note that the converse is not true, i.e. it is not necessarily true that X and Y are independent if their covariance is zero. It is straightforward to construct a simple discrete distribution to demonstrate this fact. You are asked to do this is in one of the exercises. The covariance has dimensions of the product of the dimensions of X and Y. For many purposes it is more convenient to scale the covariance to make a dimensionless quantity known as the correlation. The correlation of X and Y is defined by

p(X, Y) = Cov(X, Y)/(trxtrr)

(2.9)

and we shall prove later that

- I cos kto as N---> oo

8.

Hint: 2 cos A cos B = cos (A + B) + cos (A - B) Find the acf of the MA(2) process given by 2

x , = Z z,_,/3 i=0

9.

I0.

A company manufactures burners for domestic boilers. A critical dimension is the distance from the front of the burner to the first gas outlet hole (x,). A R M A models have been fitted to data from the past few months and an AR(1) has been found to be a good model for the process {X, }. Find an expression for the mean of a sample of four consecutive readings in terms of O{xand a. In fact, a was estimated as 0.21. Calculate the ratio of the variance of the mean of a sample of four consecutive readings to the mean of a random sample of four. A random process is composed of an ensemble of sample functions, each of which is a sine wave of constant amplitude a and frequency to. The frequency is the same for all samples but the amplitude varies randomly from sample to sample. Each sample also has a constant phase angle ~b, which also varies randomly from sample to sample. A typical sample is therefore represented by the equation x(t) = a sin (tot-4')

where a, to and ~ are all constants for the sample. If the joint PDF for a, 4, is f ( a , ~b) = [1 + ( 2 a / L -

1) cosqb]/27rL

0~< ~b~ M,

where M~< N

A simple example, which is rarely used in practice, is the Bartlett or triangular window defined by

w(k) = (1 -Ikl/M)

Ikl ~<M

This general form of an estimator is a natural development from SAC(to). Both types of estimator only use values of c(k) up to M - 1 ; in SAC(to) the sub-sequences are of length M and it is not possible to calculate c(k) for any larger values, while in C(to) they are excluded by the weighting system. A general feature of lag windows is that they give less weight to c(k) as the modulus of k increases. The estimator SAC(to) also does this, implicitly, because c(k) is estimated in each sub-sequence by the sum of the appropriate M - k products divided by M. The corresponding effect when estimating c(k) up to lag M - 1 from N data is relatively small whenever M is considerably less than N. We now investigate the bias of C(to). The Fourier transform of w(k) occurs as a spectral window in the manner described earlier. The convolution property applied to the definition of C(to) gives

c(o,) = ~1 I " w (o) u c (~ - o) d o where

UC(to) is the

sample spectrum. Taking expectation gives

elC(,o)] = ~1 I " W(OlelUC(,,,- o)1 dO and if N is large this is approximately

] 2"~

I"

W(O)r(oo-O)dO

-~r

We are now in a position to repeat the argument put forward in Section 5.3.1.. If F(~o) is approximately linear over intervals as short as the width of the spectral window, the estimator will be approximately unbiased. The natural window discussed in Section 5.3.1 occurs when c(k) is used to estimate 7(k) because the denominator in the definition of c(k) is N for all lags k. The Bartlett spectral window has the same shape as that shown in Fig. 5.3 with N replaced by M. It is wider by a factor of N/M, so the estimator will inevitably be more biased. Figure 5.4 shows the Bartlett lag and spectral windows when M equals 20. The advantage of using the Bartlett window is that averaging has taken

120 Frequencyrepresentation of random signals

w(o)

"...._.......

2o-

-20

Fig. 5.4

0

20

k

- 8rt . 20

-. 6rt . - 4r~ . - .2~ . 20 20 20

.

2rt . 20

. 4rt . 6n 20 20

8~

0

20

Bartlett lag and spectral windows when M = 20

place, so reducing the sampling variability. To sum up, using a window is an averaging process which reduces sampling error but smooths out peaks and troughs in the spectrum of the underlying random process. The amount of averaging depends on the relationship between M and N and if M is considerably smaller than N, the natural window, which results from using N in the definition of c(k), is negligible compared with the Bartlett window. Sample spectra can be calculated for several values of M, but the usual advice is to start with a value of M approximately equal to 2V'N. The choice of M is analogous to choosing the width of the grouping intervals when drawing a histogram, and the advice given will be interpreted in this context in Section 5.4.1. The formula for the spectral estimator 1

N-I

C(to) = ~

S'.

w(k)c(k) exp(-jwk)

- - 7 r ~ < t o < ~ ,rr

k=-(N-l)

is not in a suitable form for computational purposes because of the complex exponential. If we use the relationship exp (-jtok) = cos (tok) - j sin (tok) and remember that w(k), c(k) and cos(tok) are even functions, sin(tok) is an odd function, w(0) equals unity and w(k) is zero if k exceeds ( M - 1) in absolute value, we obtain the equivalent formula

l

1 c(O)+ 2

2~"

k=l

w(k)c(k) cos(tok)

1

Ir~< to~ M

W(to) has a 'width' of order M-1

three conditions must be satisfied by its Fourier transform, the spectral window. These conditions are presented in Table 5.1. In some ways the most natural lag window might be the rectangular lag window defined by w(k) = 1 for all k between - M and M. This selection corresponds to reducing the formal definition of F(to), namely 1

F(to) = ~

oo

Y~ y(k) exp(-jtok) k--oo

by its approximate finite sum M

=

1 ~, c(k) exp(-jtok)

That is, y(k) has been replaced by its estimate last expression may therefore be written as 1

C(to) = ~

c(k) and only a finite sum considered. This

M

Y~ w(k) c(k) exp (-jtok) k=-M

with

w(k) = 1. However, the results of Section 4.4.3 lead to c(,o) =

w(o)

dO

--I1"

with the spectral window W(O)--

W(O) having the undesirable shape

sin (M0)

MO

This result follows in a similar manner to the results of Section 5.3.2. If this spectral window is plotted it will be observed that W(O) has oscillatory side lobes which include negative values. Hence w(k) = 1 is not a good choice of lag window for practical use. Two commonly used windows, which are a considerable improvement over the rectangular window, and in some ways better than the Bartlett window, are named after Tukey and Parzen. Tukey window The Tukey window is defined by

w(k) = 89 +cos(zrk/M))

Ik} = 0, 1 , . . . , M

(5.9a)

It is also known as the Tukey-Hanning window, and is equivalent to the Hanning

5.3

Estimation of the spectrum from the sample autocovariance function

123

window, which is usually offered as one of the choices for data analysis on commercial spectral analysers. The computational procedure is different because most spectral analysers use the fast Fourier transform which is described in Section 5.4. A good approximation to the spectral form of the window can be obtained in a similar manner to that for the Bartlett window. It is W(O) --" M'tr2 sin (OM )l[(OM) ('rr2 - 02M2)]

(5.9b)

and using a Taylor expansion for the sine function one may demonstrate that W(0) is properly defined and equals M. Parzen window The Parzen window is defined by

w(k)

~1-6(k/M) 2+6(k/M) 3

O 5 ~ = exp ( - (0.0873)2/17.40 x 10 -3) = 0.6453 That is, the probability of 0s exceeding 5 ~ is 0.65 using a more detailed definition of the spectrum than that presented in Table 7.4. Upon using the response spectrum of Table 7.5 with Simpson integration rules, an estimate of the response variance is 16.64 x 10 -3, and this leads to a probability of exceedence value of 0.6325, that is 0.63. In undertaking such a simple example the amount of numerical effort required for one

9080

70 x

60 50 40 30 20

10 0.2

Fig. 7.7

0.4

0.6

0.8

Wave slope spectra

I

1.0

I

I

1.2

..~ r

co

Z4

Ca/cu/aringthe response spectrum

169

case, especially if one includes estimation of the hydrodynamic and hydrostatic data, is sufficient to warrant automated computer calculations. This becomes even more necessary when one realizes that it is usual to consider different forward speeds, different wave headings, different sea spectra, and different ship loading conditions in a full assessment of the behaviour of the ship in a seaway. However, once automated there is the danger that the often elegantly presented results will take on a greater respectability due to their inclusion in a published document, such as this book. In good engineering practice a number of questions related to the illustrative calculation need to be put. First, how well is the physics represented by such a simple model of the roll motion in regular waves. The assumed linear fluid damping implies that viscous roll damping has been omitted. If we include viscous damping then this is proportional to a power of the roll velocity. Often a square law is assumed. Although the non-linear equations can be solved iteratively, the response spectrum-input spectrum relationship used assumes linear dynamic responses. Therefore, the non-linear viscous roll damping is usually linearized by using the first term of its Fourier series. As already indicated, the roll motion of a structure with port-starboard symmetry is not uncoupled from either sway or yaw. Even with fore-aft, as well as port-starboard symmetry, the roll motion remains coupled to sway. The equations of motion, as written, also assume that the mass of the structure and its cargo are arranged symmetrically so that all products of inertia are zero. However, provided the will and the knowledge to calculate all the required additional hydrodynamic and hydrostatic quantities are available, to allow generalization to the dynamic coupled motion equations, the predicted roll transfer function can always be improved. Second, one must ask how representative of the environment are the selected input sea spectra for the intended operational waters of the ship. The manner in which the original time series data were collected, the season and the duration of the wave observation period(s) will obviously influence the final form of the selected spectra. In looking at any specific operation of a ship in a seaway one needs to know its route and the characteristics of the sea in the different regions it crosses. The spectra and their possible characteristics need to be known for each area. Thus, the probability of exceedence calculations just performed only provide a conditional probability. That is, the calculated probability only holds for the spectrum selected. We therefore need to know, or be able to estimate, the probability of occurrence of the selected spectrum. Thus, to estimate realistically the probability of exceedence it is necessary to undertake the calculations implied in the following equations, namely

Pr {Os~> O} =

E

Pr {0~I> 01 selected spectrum} Pr (spectrum)

all spectra

Here 'all spectra' means all spectra relevant to the situation under scrutiny for all the environmental conditions of interest, defined to be mutually exclusive. Next, provided we are confident that the spectrum selected is sufficiently well defined, its moments ought to be calculated to determine the spectral width parameter (see Section 7.6). This would help to decide whether the selected extreme distribution, the Rayleigh distribution, is applicable. The fact that the Rayleigh distribution is traditionally used does not justify our action and we should, in many applications, consider whether a Gaussian or a Cartwright-Longuet-Higgins distribution is more appropriate.

Some typical applications

170

The apparently high conditional probability calculated may be unrealistic for all sorts of different reasons. In general, the volume of work and the apparent sophistication of the methods employed in undertaking such calculations do not by themselves provide validity to the results generated.

7.5

The spectrum and moving observers

Throughout this book it has been implicitly assumed that spectra are determined at some fixed location in space by some 'fixed' device. Consequently, the frequencies associated with the observations are the absolute frequencies of the phenomena under investigation. Suppose that observations are made with respect to a reference point which has a velocity relative to earth equal to Vs. The encounter angle,/3, will be defined by the angle between the direction of wave travel and the heading of the moving observation platform, a ship say. For regular wavefronts this situation is presented in Fig. 7.8. The speed of the ship relative to the waves is Vw-V~ cos/3. The wavelength and the wave velocity are A and Vw respectively. Hence, the wave encounter period, T~, satisfies Te _.,e-

Vw-

cos/3 Tw

V~

l_( )cos

upon noting that Tw = AIVw. In deriving the expression for the encounter period it is Wave crest ,o~/

.8'/~q,

i-

._o

ks = Ship speed

el. 0

Vw= Wave s p e e d /

f/

J,, / /

fJ

Wave crest .

.

.

.

Wave crest

L = Wave length )~e= Encounterwavelength Fig. 7.8

Relative advance of ship and regular wave system

Z5

The spectrum and moving observers 171

simply necessary to realize that the time required by the moving observer (or ship) to travel from one crest to another is the wavelength divided by the speed of travel in the direction of wave propagation. This should make the definition of the wave period T~ equally obvious. From hydromechanics it is known that water waves satisfy the dispersion equation toE = gk tanh (kd)

This states that wave frequency, to, wave-number, k, and water depth, d, cannot be selected arbitrarily. One may therefore choose any two wave parameters, but the third wave parameter is governed by the dispersion equation. For deep water, water depths greater than half the longest wavelength of interest, this relationship reduces to to2 = gk for all frequencies of interest. Hence, the deep water wave velocity is given by

Vw=g-

tO

The wave encounter frequency toe, equal to 2~r/T~, is thus =.,

1-

cos

Since the wave encounter frequency is a function of ship speed, V~, and the encounter angle, fl, it is worth briefly studying the characteristics of toe before considering the relationship between a wave frequency dependent spectrum and an encounter-frequency based spectrum. The encounter frequency is only zero when the velocity of the observer in the direction of wave propagation and the wave velocity are equal. On the other hand, when Vw < g~ cos fl, the encounter frequency is negative and the ship overtakes the waves. Although to the ship-borne observer the waves will appear to come off the bow of the ship, this is not the case. Such waves or seas are described as following seas and occur for 0 ~< fl < 90 ~ and 270 ~< fl < 360 ~ when cosfl is positive. The encounter frequency is positive for Vw > Vs cos fl. Clearly, Vw and V~ cos fl need not be of the same sign. The encounter frequency is maximal when the derivative dto~/dto is zero, that is when Vw satisfies 0.5 Vw = Vscosfl. Thus, when Vw is greater than V~cosfl the waves will approach the ship from behind and will be travelling so fast that to~ is only slightly less than to. If the waves travel slowly, compared with the ship, then to~ will tend to be small. When Vw and V~cosfl are of opposite direction and sign, such as when the waves approach the bow of the ship, then to~ is everywhere greater than to, that is 90~ fl < 270 ~ and Vw + V~ cos fl is positive and large. Generally, we consider such seas as head seas. A more specific definition of sea types is presented in Fig. 7.9. Since the waves can be described in absolute or encounter frequency terms, so too may the spectrum. Suppose F~(to~) is the derived encounter frequency spectrum, then the total energy associated with the elemental frequency band dto~ must equal that corresponding to dto. The energy associated with dto is fixed by definition of F(to), and since each to value maps on to to~, the amount of energy associated with dto and dto~ cannot be changed because of a simple change of independent variable. That is = r

do,

Since we now know that in deep water

172

Some typical applications Beam seas

Quartering

seas'~

I

/

seas B~

Following seas

Head seas

Ouarter,no/

I

seas

\

Bow

seas

Beam

seas

Fig. 7.9 Definition of seas according to direction of approach relative to vessel

(Oe = to 1 - V~ ~ cos g and dol e

dtoe = ~ do

dto

then dtoe = [1 - 2 ~Vst~ COS/3 g

] do~

Hence, the two spectra are related through the equation

r(~) r~(~~ 2~'vg~l cos/3]

Thus, given a particular sea spectrum, F(to), the encounter spectrum as a function of wave encounter frequency, for a given V~ and /3, may be determined using the two derived equations for to~ and Fe(tO~). With some further manipulation, left as an exercise, it is possible to show that

ro(c~

4

1 - - - toe V~ cos/3 g

],/2

Having extended the theory to allow for an encounter spectrum, it is worth looking at

7.6

Calculation of significant responses 173

some of its characteristics and apparent difficulties. For a head-on sea, /3 = 180, the energy in the encounter spectrum is spread out over a wider band of higher frequencies than for a stationary point spectrum. For the corresponding following seas,/3 = 0, and the spectral density Fe(toe) is singular at the wave encounter frequency, toes, defined by 1 g (Des

--

m

4Vs

Here, to~ is positive since V~ is to be treated as a positive quantity irrespective of its direction relative to Vw. However, others may give this same relationship with a negative sign because they prefer to treat V~ as negative under the selected circumstances. Whatever the notation, tO~sis positive. Following-sea spectra may also be double valued where the same encounter frequency exists for two different incident wavelengths. This must be the case since, if to~ is assumed known, then the encounter frequency equation can be treated as a quadratic equation in wave frequency, to. The encounter frequency spectrum of a one-sided wave spectrum F(to) will also extend into the 'negative frequency' range as Vs increases. The negative spectral density does not pose any problems when integrating the spectrum, to determining the total energy say, provided a path of increasing toe values is followed. Since the product Fe(toe)dtoe will always be positive, we may write ~, r (,o) dos = 0

ro (toe) dtoe +

(toe) dtoe mes

and F(to) is a one-sided spectrum. Here, integration is from left to right in the upper half-plane of F~(to~) versus toe for the first encounter frequency integral, and is from right to left in the corresponding lower half-plane second integral. Numerical application is left as an exercise. In practical ship response calculations, the ship speeds would be specified and the wave frequencies corresponding to A/Ls between 0 and 3 would be selected, where L~ is the ship length. Having determined the corresponding encounter frequencies, the transfer functions of interest would be determined for the calculated encounter frequencies. After selecting suitable wave spectra, the response spectra would be determined. From examination, or known properties, of the spectra, the appropriate probability density function(s) would be selected. This allows any required 'exceedence' probability calculations to be undertaken. One particularly useful form of statistic, which is determined from the response spectrum, is described as significant response. The underlying ideas of a significant response and a design related application are considered next.

7.6

Calculation of significant responses

In certain engineering applications it is convenient to consider the mean of the highest 1/nth of the total number of maxima. If N peak-to-peak height measurements are made, then this quantity, Hi#,, is the average of the N/n highest measurements. Thus, as n--)N so H~/, tends to the maximum measurement in the sample. The peak-to-peak amplitude, Xmax, in a given realization may be defined as half the difference in level between a crest (a local maximum) and the preceding (following) trough (a local minimum). This is not to be confused with the apparent amplitude, which is half the difference between the

174 Some typical applications

x(t)~ Local max

/'~

+

[< Tc --~

o0closs:;t / i

Down crossing

V

v

maxtma '-"

7"7

Tz "- apparent zero crossing period Tc --- apparent period or crest period hw --- apparent wave height Xpp = peak to peak amplitude

Fig. 7.10 Definitions of apparent quantities

highest crest (absolute maximum) between two consecutive zero crossings and the preceding (following) lowest trough (see Fig. 7.10). Consequently, the 1/nth highest maxima corresponds to those maxima values Xmax greater than hu., where h~/. is such that Pr {Xmax ~ hi/,, } =

Ioo

h,,.

p(Xmax ) dxmax

1 n

The average value of Xmax for these maxima, Hu., is therefore

H1/. =

I co

Xmax p(Xmax) dXmax

hl/n

Evaluation of these quantities requires specification of the probability density function general probability density function describing the peak or maxima distribution of the random process x(t) has been shown to satisfy

p(Xmax). The

1

p(Xmax) = (2,rrmo)l/2 [e exp( --Xmax 2 /[2moe2])+

( 1 - e 2) u2 (Xmax/mlo/2)exp (--X2m.x/[2mo ])N] where N=

-= exp - ~ x

dx

and the upper limit is X = (Xmax/emo)(1 - e2) !/2

Z6

Calculation of significant responses

175

The quantity e 2 = 1 - m2/(mom4) defines the spectral width parameter of the associated spectrum. In terms of apparent physical oceanographical quantities this parameter is defined as e2= 1-(Tc/Tz) 2 where Tc is the mean crest period and Tz is the mean zero-upcrossing wave period (see Fig. 7.10). Tc is defined as 2~r(m2/m4) 1/2 and T, = 2~r(mo/m2) 1/2. Here, the spectral moments are defined by m. =

0

to'F (to) dto

assuming F(to) is a two-sided spectrum. Thus, the quantity 2mo is the variance of the random signal x(t). The cited probability density function is a weighted mean of the Rayleigh and the normal probability density distributions. Each extreme distribution corresponds to e--~0 and e--~ 1 respectively (see Fig. 7.11). For the narrow banded case,

~e

= 0.0 .--,Rayleigh

0.41-

-3

-2

-1

0

I 1

l 2

3

Fig. 7.11 Probability distribution of the heights of the maxim, 7/= e = 0 (0.2) 1.0, from Cartwright and Longuet-Higgins

x/(2mo)1/2, as

a function of

the maximum wave elevation follows a Rayle~gh distribution and therefore numerical integration of the probability density function may be used to show that

HI/3 = Hs = 4(2mo) 1/2 In some texts this result will be quoted as 4m01/2with mo defined as the variance, that is, the total area under the two-sided spectrum. These factors of 2 (or a half) and hence factors of ~/2 are a recurring problem in those theoretical aspects of spectral analysis associated with providing expressions for H~/,,. This can prove to be quite a problem for student and lecturer alike, especially if one moves from text to text without carefully cross-checking authors' preferred and often implicit, rather than explicit, definitions. Another source of confusion, and hence a factor of 2, is that the authors do not always stipulate whether the spectrum presented uses wave height or wave amplitude as the basic statistic employed in the determination of the spectrum. Assuming the Rayleigh distribution is applicable, the indicated possible variations lead to the following definitions of Hu, for different n (see Table 7.6), with y - 1 or y = 2 according to whether m0 or 2mo defines the total variance of the original signal.

176 Some typical applications Table 7.6 Definitions of banded spectra

H1/,., for narrow-

n

Amplitude

Height

1 3 10 100

1.25 ~/(~/mo) 2.00 x/(ym0 ) 2.55 %/(ymo) 3.34 x/(ymo)

2.50 V(~,mo) 4.00 V'(ymo ) 5.09 V'(yrno) 6.67 V'(ymo)

i

i

,,

Strictly speaking, significant wave height Hs is defined as four times the root mean square of the wave signal. But H1/3, using the earlier definition of H~/,,, has been shown to satisfy H1/3 0.94H~. Therefore, H1/3 4V'(ym0) tends to be treated as a literal definition of significant wave height. This convention has been used in the example that follows. The ideas of significant response are now applied in a small contrived design study. The aim is to investigate four alternative offshore oil rigs with a view to extending their operability. That is, we would like them to continue drilling in larger waves. The sea spectrum chosen is the Pierson-Moskowitz spectrum (see Ochi and Bales 1975) which has the form =

=

og2

F (~o) = - ~ exp

[

g

4

with a = 4,tr3 gT2

and

/3 = 167r3

For the North Sea, a and /3 may be taken as 0.0081 and 0.74 respectively and by manipulating the definitions of a and/3 we have

H1/3 =

2W 2 a ~

g

~

[3

W and

To =

2r

1

g (j~Tr)TM

Here H1/3is the significant wave height (m), W is the wind speed (m s-l), conventionally measured at 19.5 m above the still water, and To is the mean wave period. This spectral form represents a fully developed sea as lower frequencies are present. The four eight-columned rigs have a number of identical features regarding the dimensions of the shape of the submerged pontoons and the lengths of the columns (see Table 7.7). The basic idea behind the design scheme is to try and increase the operability of the rig by increasing the waterplane area. Table 7.7

Some common rig dimensions

Pontoon length (overall) Pontoon breadth (overall) Pontoon depth (overall) Pontoon CB Column length Operating draught Deck height ,

100 m 10 m 10 m 0.9 22.5 m 20.5 m 8.0 m

7.6 Calculation of significant responses 177 Table 7.8

Computed rig particulars (based on Riiser M.Sc Dissertation)

Calculated particulars

.......

Transverse distance between corner columns Column diameters: large small Waterplane area Displacements: columns pontoons total Natural heave period Deck area Weight: equipment steel lightship Lightship vertical position of centre of gravity Max. transit deck load* Max. operating deck load*

Rig 1

Rig 2

Rig 3

Rig 4

47.4 7.0 6.0 267.0 2874.0 18450.0 21 324.0

49.5 8.0 6.0 314.0 3381.0 18450.0 21 831.0

49.4 9.0 6.8 400.0 4302.0 18450.0 22 752.0

50.1 10.0 7.2 477.0 5133.0 18450.0 23583.0

24.2 3643.0 2937.0 5296.0 8283.0

22.5 3910.0 2937.0 6696.0 9682.0

20.3 4027.0 2937.0 7038.0 10025.10

18.5 4205.0 2937.0 7396.0 10383.0

24.55 -ve

25.98 -ve

-ve

23.3

25.95 347.0 1559.0

25.99 1738.9 3167.8

KM transit**

13.27

15.74

18.63

21.75

KM operating (GM = 1.0)

13.27

15.66

17.99

20.34

,

,

,,,

,

* Assuming no loads in column and deck load centre of gravity = 40.5 m ** GM = O, KM corresponds to pontoon submergence condition.

To maintain initial stability, it is necessary to keep the waterplane inertia constant. Since the longitudinal distance between corner column centrelines is to be kept constant at 60 m, and the waterplane areas are gradually increased from 267 m E for Rig I to 477 m 2 for Rig 4 (see Table 7.8), the transverse distances between column centrelines are adjusted for each rig so that the second moments of area about the longitudinal and transverse platform axes are equal. This is achieved by moving the four smaller inner columns. This procedure represents an approximate method of maintaining the initial stability, by offsetting the increased moments of area resulting from the increased displacements, which arise from the increased waterplane areas. The equipment weights were kept constant for all designs. The steel weights, and hence the lightship weights, were evaluated with the hydrostatics, and the maximum permissible deck load capacities (for both the operating condition and the pontoon submergence condition) together with the heave responses using computer programs developed at Newcastle University. Details of computed rig particulars are presented in Table 7.8. It follows from earlier comments that H1/a can be generated by simply selecting different wind speeds. Having calculated H1/3, the corresponding response spectrum can be determined as explained in Section 7.4. By determining the area under the response spectrum one may determine the significant heave response h l/a, say. With h~/3 calculated for each 1-11/3 o n e may plot the results and determine the maximum level of H1/3 for which the rigs may be permitted to drill. The criterion used is that drilling may not continue for significant heave responses, h l/a, greater than 2.5 m. Alternatively, we can consider the

178 Some typical applications 1.4

r

;7, ~,

-

9:

1.0

g

0.8

0.6 0.4 0.2

"'gI

i~

ih

ir

-

_

-

.~~~"! i/':.-.-_-.~ ..,". . . . . . :,~.'?~, i I"

-

0.0 0.2

I

0.3

V~ / 'i' I

0.4

.

.

.

.

I

0.5

"~ I

0.6

I

0.7

0.8

0.9

1

1.0

o~ (rad/s)

Fig. 7.12

Heave response amplitude operators

exercise as seeking to determine how rough the seas may get before drilling must be discontinued! In Fig. 7.12 we observe a considerable difference in the regular wave heave response characteristics of the four rigs. Also, the natural heave period is decreasing with increasing waterplane area. This results in a shift of the secondary peak of the response and a reduction in response near the secondary peak. However, Fig. 7.13 illustrates rather small differences in the significant heave responses of the rigs. Only the significant heave responses for Rigs 1 and 4 are plotted as these envelop the corresponding responses for Rigs 2 and 3. The large differences in the transfer functions of Fig. 7.12 are clearly smoothed out in Fig. 7.13 because of the integrations undertaken. In order to keep so many characteristics of the rigs common, and to try and provide some cause and effect understanding, the generated particulars are slightly artificial. Consequently, Rigs 1 and 2 are impractical. The heave responses were also generated

/

8-

7-

Rig

6 -

e-

3 ~ Maximum permissible

2.5 Fig. 7.13

5.0

/4"

7.5

HII3(m)

10.0

12.5 "-

Significant rig response hl/3 as a function of significant wave height H1/3

7.7

Exercises 179

using quite basic one degree-of-freedom models. This, and the simple basis for generating the alternative figs, clearly makes the design procedure contrived. However, this type of application, based on more sophisticated ideas of design, with more complex hydrodynamic and motion analyses, and the earlier ideas of predicting probability of 'exceedence', represent fairly typical engineering-based applications of spectral analysis. When designing structures which are known to be subject to random loadings, deterministic methods alone are generally not sufficient, as the example demonstrates. This point will be developed further in some of the case studies presented in Chapter 8.

7.7 ,

Exercises Processing a full record of collected wave data (such as that presented in Appendix IV), the observed peak-to-peak wave amplitudes (reading down the columns) are given in Table 7.9. Table7.9 626 6 244 360 371 150 393 67 514 255 90 465 131 162 132 9

Exercise 1 peak data

,,,

3 545 119 458 261 303 163 21 335 23 124 400 123 255 243

144 246 336 219 405 131 161 437 22 8 358 206 240 124 293 ,

,,,

121 132 392 347 167 501 12 216 275 26 303 198 234 63 465

,

87 523 175 170 190 79 473 60 206 75 559 814 495 223 7

343 163 304 394 24 412 235 142 162 380 27 66 326

,

Group the peak wave amplitude data using the intervals 0-99, 100-199, 200-299, 300-399, 4120-499,500-599 and 600-849. Draw a histogram ensuring that the total area equals unity. If the original time series is to produce a narrow bandwidth spectrum then the peak wave amplitude, y, given above should exhibit a Rayleigh distribution, which has the probability density function (PDF)

p(y) = Y--- exp (-y2/[2m o]) mo

Assuming the original wave height data have an estimated variance of 70776 (mm 2) superimpose the Rayleigh PDF on the histogram drawn. Next, calculate the expected frequencies (to 2 decimal points) for each interval, under the hypothesis that we have a random sample of 88 values from a Rayleigh cumulative distribution function, that is Pr {y ~=

, , ~I!ji / ~,,,,~\X

i =~-~1~

0

o.o

0.5

I I 1.0 1.5 2.0 Wave frequency (red/s)

I 2.5

g0.0 3.0

Fig. 7.17 Heave amplitude and phase for device(s) and internal free surface (IFs). (Original calculations by Hearn and Katory 1977)

8 Wave directionality monitoring

8. I

Introduction

This first case study is concerned with designing a piece of equipment to measure correctly the directions of wave travel, and the amount of energy available to a wave energy extraction device being assessed in Loch Ness, Scotland. Given that the engineers involved in designing the wave power device could measure the amount of work done, or energy extracted, to measure the efficiency of such a device, we needed to know how much energy was actually available. This case study will clearly demonstrate that the analysis of data should not be considered as trivial or straightforward. In this case, some of the basic ideas of the initial, imposed analysis, method were an integral part of the design of the energy measuring device. Furthermore, it will be shown that constraining one's thinking to acceptable 'established' norms of analysis can lead to failure. In particular, the original device had to be modified to produce more realistic measurements. Also, given the minimal number of transducers in the measuring device we were forced to look at less conventional spectral analysis methods. Since the final design and the final selected analysis were in conflict with other specialist spectral analyst advice, it was necessary to demonstrate, using simulations, how the modified device and alternative analysis would produce a truer picture of the wave climates. Some of the mathematical detail involved may be considered too advanced by some readers. This should not, however, prevent the reader from appreciating the following points. 1 Spectral analysis cannot always be undertaken as an independent task. 2 Real engineering problems do not respect academic divisions of knowledge. 3 Working in industry often leads to involvement (even responsibility) in projects which require new knowledge. In this case one works with sufficient minimal understanding and full responsibility. The reader may take comfort in the fact that the author involved in this case study knew little about spectral analysis at the start of the project~being a hydrodynamics specialist.

8.2

Background

Here, we need to explain our understanding of the directional wave spectrum and the need for its monitoring. Since we shall refer to the spectral density distribution, or simply the spectrum, as a 184

8.2 Background 185 measure of energy per unit area, let us explain the reasoning behind the interchangeability of these terms. In the case of water waves the total energy in a regular wave is 89 per unit area of sea surface, where ~'~ is the wave amplitude. A one-dimensional (long crested) random sea, ~'(t), may be approximated by a summation of N superposed harmonic wave components with the nth wave component described by cos (tont -I- tonen)

en(l) --'- r

where e,, is the random phase angle. It is obvious (it really is) that E[sr,, ] and hence E[~'] are zero. The mean square value of a sample, 2T in duration, from this random sea is thus E[~.2] = ~ T [ r ~.2(t) dt -T

1

f

T

_ 2T J - r

~'a,, COS(~o,,t+ ~o,,e.

dt

n-1

That is 1

N

=7 nZ= | & because of the orthogonal nature of the trigonometric functions. Thus, the total energy in the sample is 1

N

2 pg ~, r = pg E[~.2] But, we have seen in Chapter 5 that "=1 =

_.

dto = ~

_ ~'2n

or, for a single-sided sea spectrum r ( ~ . ) doJ = ~'2n for a bandwidth do centred on the component frequency to,,. Since the total energy is directly related to the spectrum it is often referred to as the 'energy' spectrum or 'power' spectrum. Water waves travel at various directions 0 relative to the predominant wind direction and therefore their spectral density should be related to direction as well as frequency. The total wave energy is thus not only distributed according to frequency, but also direction. If the total energy is to remain the same, one may assume that r ( , o , o) = r ( c o ) f ( o )

where f (0) is selected to keep the total energy constant. A simple 'spreading function' is

f(O) since

"

2

~ COS 2 0 "/T

186 Wavedirectionafity monitoring

2f ,2

COS 2

0 dO = 1

-ul2

and hence

ffr(~o,o) do dO= fr(a,) da, ff(O) dO= fr(a,) dto with appropriate implicit integration limits. In general, one may select other spreading functions such as f(O) = an cos z" 0 with 1

a,,

f ~r cos2n 0 dO

or, more generally, following Mitsuyasu et al. (1975), we may express the directional spreading as

f(O) = G(n)

O-Oo

11

where G(n) is again a normalizing function satisfying

22" (n!) 2 G ( n ) - 2zr (2n)! for 00, the mean direction of a spectral component; n is a selected integer value and the 'exclamation mark' denotes factorial. We shall use different parameterizations in this study. The spectrum of a directionally spread sea is often referred to as a short crested spectrum. We may now discuss the need for monitoring such short crested seas. The assessment of British wave energy extraction devices was based on projected units of electricity costs and a demonstration of their power take-off capability. Theoretical studies based on inviscid free-surface hydrodynamic theory and generalized frequency domain equations of motion may be used to predict the power take-off of a device subject to regular wave excitation. From our understanding of the energy stored and transported in a regular wave one can determine the theoretical efficiency of the device as a function of the frequency and heading of the incident wave. Knowing the regular wave efficiency transfer function, the irregular sea-state efficiency can be determined using the input-output spectral relationships of Chapter 6. Using two-dimensional hydrodynamics to analyse very long wave energy devices, their length being parallel to the incident wavefront, one can formally prove that the maximum efficiency attainable is 50% if the device has fore-aft symmetry and 100% otherwise. At certain wave frequencies the device appears to focus the waves and so efficiencies greater than 100% are attainable if one assumes that the actual device width, or the resolved width of the device in the direction of the wavefront, is the nominal length of the wavefront available for the extraction of wave energy. In at-sea tests, such as the Salter duck spine studies in Loch Ness, the Cockerell Raft studies in the Solent and the Coventry reservoir tests, it was possible to measure the power produced quite readily. However, rating the efficiency of the device requires a knowledge of the energy available and its directional spread, since most of the then

8.3 The technicalproblem

187

supported wave energy extraction device responses were wave direction sensitive. The Fluids Structural Loading Technical Advisory Group, one of a number of Technical Advisory Groups (TAGs) set up to advise the Department of Energy's Wave Energy Steering Group, therefore agreed to undertake a wave directionality monitoring programme in Loch Ness in 1977 and later in the Solent. The Loch Ness case study will highlight some of the technical difficulties experienced and the personal lessons learnt from real problem solving under commercial and resource limited pressures. This should not be interpreted as criticism or negative reporting, but a personal record of the enhancements achieved in terms of matching a selected monitoring system to a particular wave climate so as to improve the resolution of the spectral energy spread.

8.3 The technical problem The design of the monitoring system requires consideration of such questions as the following.

(i) (ii) (iii) (iv) (v)

What quantity is to be monitored? How is the selected quantity to be measured? How is the required information to be extracted from the measured quantities? How are the measurements to be recorded? How is the monitoring equipment to be located and kept on station?

From Airy wave theory we can establish relationships either between wave height and wave-induced dynamic pressure, or between wave height and the resultant forces or moments experienced by small submerged objects, called 'force balls'. Therefore, in deciding what is to be measured, wave height, pressures or forces and moments, we must also consider on-site calibration and handling of the monitoring device from a small boat as well as its maintenance and survival. Some of the wave-height dependent quantities, such as particle velocity and pressure, are dependent upon the actual water depth and the accurate location of the submerged devices. These difficulties and the lack of consensus regarding the different devices available, and the best form of monitoring, led to the decision to measure the wave height directly using capacitance wire techniques since:

(a) (b) (c) (d) (e)

experience of their use existed in the establishment; it will permit recording without pre-amplification; there is no need for digital logging; it will be simpler to assemble and to replace damaged wires; and software already tested under laboratory conditions existed at another research establishment.

The basic wave height sensor was a polythene-insulated conductor wire held under tension. The multi-channel capacitance sensing transducer electronics provided seven channels of wave height information for reasons explained in Section 8.5. The associated electronics had to be tested to demonstrate linearity over a large range of capacitances (0 to 1000 picofarads, pf), and to provide a locally stabilized voltage and locally RC filter probe data before onward transmission ashore through a multicore cable. Resistive transducers were rejected as they were considered more subject to errors owing to the changing conductivity of the water and its temperature. On-shore transmitted signals

188

Wavedirectionality monitoring

Fig. 8.1

Floating wave monitoring system

would also have to be scaled, filtered and DC levelled to suit the selected FM tape recorder. Having selected the quantity and its means of measurement, and the number of probes, a platform which does not change the wave environment being measured must be devised. We therefore either use a floating device or a bottom mounted device, such as those presented in Figs 8.1 and 8.2. The simpler pole-type device was selected with 'guys' used to stabilize the staff. Both a buckling analysis, to ensure structural integrity, and natural frequency estimates of the mast and the 'plucking' frequencies of the guys, to ensure the wave frequencies of interest and the natural responses of the guys were well separated, were undertaken. The diffraction-borne interference of the wave environment being measured with the more complex floater design was considered problematic. Hence, this design was rejected for being too complicated. The remaining decisions were the number of transducers to be used, their relative positioning with respect to each other and hence the method of analysis to be used to extract the directional information.

8.4

Fig. 8.2

Reduction of the monitoring problem to a mathematical problem

189

Bottom standing wave monitoring system as installed in Loch Ness

8.4 Reduction of the monitoring problem to a mathematical problem The monitoring system was initially required to be sensitive to wave periods of 1-4 seconds at least, and was to provide a measure of the spectral density function and its directional spread. In the adopted approach the spatial distribution of wave height was used to obtain the directional information. Having decided on this particular wave parameter, the configuration of the wave-gauge array must be selected. The choice is either a one-dimensional (linear) array or a two-dimensional (polygonal) array. Line arrays can only be used, without ambiguity, when significant amounts of wave energy arrive from one side of the array, that is, such arrangements are sharply tuned for waves generally travelling normal to the line. It was therefore decided that a polygonal array was preferable for more general use. What was now required was a means of deciding on

190

Wave directionality monitoring /)r Probe J \ \

//

\

/

Wave direction

._~

jf

/ /' , /

D~ // '

~

J,o

I I

I lilY ,

/

I . . . .

i

'~. Wave front

\

,

\

X!

x

q~ -,- wave height at probe I qu --- wave height at probe J

Fig. 8.3

Probe and wave orientation relative to fixed reference system

the shape of the polygon and the spacing of the gauges such that the required sensitivity was attained. The constraint that there already existed an available external analysis package to deal with equilateral triangular arrays (at zero cost to the project) decided the shape. One now required to relate the sizing of the triangles to be used to the required sensitivity of the monitoring system. To do this one must appreciate the underlying mathematical model now outlined. The wave energy is not necessarily concentrated in a finite number of directions and the phasing of the wave components will be random. Consider initially two probes I and J (see Fig. 8.3), and denote the wave elevation at the position (xi, Yl) by qI(xI,yl,t). Assuming that (X, Y) is the position of the Jth probe relative to the Ith probe then xj = xI+X

and yj = yl-t- Y

Then the cross-covariance, cIj, corresponds to the expected value of the product of the wave elevations q~ and qj. The cross-covariance is thus expressed as 2"rrClj -- 2

(-

CO(tO)COS(tot) dto+ 2

0

f" 0

quad (to) sin(tot) dto

(8.1)

where co(to) and quad(to) are the co-spectrum and quadrature-spectrum respectively. That is co (to) =

I

/T

0

r (to, O) cos tx dO

and

(8.2) quad (to) =

f

/t

0

F (to, 0) sin a dO

8.4 Reductionof the monitoring problem to a mathematical problem

191

where a = k(XcosO+ Y sin0), k is the wave-number and F(to,0) is the directional spectral density function. Now, the cross-spectrum, here defined as co(to)-j quad(to), may be obtained as the product of the complex conjugate FFT coefficients of one probe with the FFT coefficient of the other, as argued in Chapter 6. From Equations (8.1) and (8.2) and the identities X = D cos/3 and Y = D sin/3 (see Fig. 8.3), it follows that co + j quad =

F (to, 0) e jkD~~176 ~ dO

(8.3)

0

where 0 denotes the wave heading and/3 is the orientation of the line passing through both probes with respect to the x-axis. We now assume that the directional spectrum can be represented as a finite Fourier series of the form ao

N

F(~o, 0) - ~ + ~ {a,, cos(n0) + b,, sin(n0)} n=l

then, substitution into Equation (8.3) and integrating, yields N

c o + j q u a d = rraoJo(kD)+ Y~ 2N"J,,(kD){a,,cos(nCt)+b,,sin(n~)} ~=~

(8.4)

upon noting that

f 2~cos(n0) 0

eJkDcos( O-13) dO =

2~j~J~(kD) cos (nr

with an analogous result for the sine integration. J~(z) is a Bessel function of the first kind of order n. On expanding the right-hand side of Equation (8.4) it follows, after some manipulation, that coIj(os) = ~r{Ao, --32, - B 2 , A4, B4, 9 9 9} {ao, a2, b2, a4, b 4 , . . . }T and

(8.5) quadIj(to) = ~r{Al, B1, - A 3 , - B 3 , A5, B s , . . . } {al, b l , a3, b3, a5, b5 . . . . }T

where the superscript T denotes the row vector is to be transposed. The coefficients A,, and Bn correspond to A~ = 2 cos nflJ~(kD) t B~ = 2 sin nflJ~(kD) ~ n >I 1 and Ao = Jo(kD). Once D and/3 are specified for each pair of probes then Ao and An and B,, are known coefficients. From our understanding of the evaluation of the cross-spectrum we may determine numerical values for cou and quadij. The only unknowns are therefore the coefficients a~ and b,, of Equation (8.5). Since there are as many cross-spectra between probes as there are pairs of probes, the greatest number of harmonics which can be represented in F(os,0) cannot exceed the number of probe pairs. In particular, if a triangular arrangement of the probes is to be used then N = 3 and the coefficients to be determined are a0, al, a2, a3 and b 1 and bE. For an equilateral triangle D is constant for all pairs.

192

8.5

Wavedirectionality monitoring

Application of the mathematical model

Equation (8.5) provides a means of generating sets of simultaneous equations whereby the spreading characteristics of the wave energy can be determined at any selected frequency. Now, the problem is easier the greater the number of probes used. The factors which limited the number of probes at the time were the financial resources made available for checking out the design, the provision and installation of the indicated hardware and electronics and the constraint of using existing unseen software. At the time, the theoretical basis of the software was unknown to us and was therefore derived from first principles by Hearn (1977). To achieve credibility with the software owners, the mathematical theory outlined was produced and a typed report despatched to them within 12 hours of the contract being awarded, to demonstrate we were capable of developing our own software if necessary. The availability of this software clearly has advantages and disadvantages. Assuming the software is bug free, time, energy and costs are saved. The disadvantages are that it was limited to three probes, we were not sure it would run on the designated analysis computer and we had to ensure that the monitoring system was capable of resolving the specified wave periods of 1-4 seconds. We must therefore try to understand how the theory is to be applied and how the theory can provide dimensions regarding the appropriate separation of the probes. The series representation of Equation (8.4) is a truncated form of the cross-spectral density, and to appreciate the order of magnitude of the truncation one must appreciate the behaviour of the Bessel functions associated with the A,, and B, coefficients. From tabulated values of the Bessel functions we recognize the bounding relationships 1.o

10-2~<Jl(z) ~

100

|

50

n-

O

0.0

5.0

2.5

7.5

10.0

Length (mm) Total profile length analysed = 25 mm Digitised interval = 50pm

125

E E

Before

After

100

E

c .g

75

-~

i.

50

~

2s

.

~o

o

20

ao

40

5o

6o

7o

Wavenumber (rad/mm)

(a) Surface profile before and after 9 months at sea; (b) comparison of spectra

Fig. 12.1

where here YN is the Nyquist wavelength. We may also introduce the partial integrals m .t = m . ( y ' ) =

where

y'

I y' y"F~(y) dy o

may assume any value between zero and YN. These functions were given the

12.6

Measures o f t e x t u r e

269

formal symbol m.(y') because they are functions of wave-number, but they are usually abbreviated to m~, m~, m~, and e', where ~r ._

,n~m~ m~2

Figure 12.2 shows the profile trace, the spectrum and the cumulative integrals of a typical surface profile from a self-polishing copolymer surface after 9 months at sea. Although surface profiles are 'non-stationary', the power in the spectrum diminishes

450 400 350 (a) Profile Spectral moments direct method

450

mo = 762.16 m2 = 2.0840 x 10 -2 m4 = 1.3892 x 10 -5

400 350

~, = morn4 = 24.38

E E 300

mJ

250

v

~ 200 o 150 100 50 0

10

20

30

50

40

"Frequency" wavenumber y = (2~/k rad/mrn) (b) Spectrum 1.0

. . . . m6 ...... m~ . . . . . m;

~ 0.8

t

Ee

t

p

/

s

t

~

"

-

-

-

oooo/~

oo

,'

oC~

-

."

I:

, i

iE

.-'"

._> 0.6 /

E = 0.4

,;

r

,,,

/

,,"

/

,'"

~ 0.2

I"

*'"

o Z

1'

!

:

i

!

1" / ~'~

0.0

I

10-1 Yc

i

i

.....

L ........

5 6789100

,"~

2

""

9. . . . . .

,,--;1 '' " "~"

~1 4 5 6 7 8 9 1 0 1

i

"Frequency" wavenumber y = (2~/k rad/mm) (c) Integrals

Fig. 12.2

I

:3 ,1 i a"/i9102

(a) Profile, (b) spectrum and (c) cumulative integrals

YN

270

Hull roughness and ship resistance Power spectral density function (PSDF) - smoothed FFT periodogram Sampling interval = 50.00 lam. 18 degrees of freedom. 1000 points + 24 added zeros.

400~

~

300

If"

!

~

.A A i-. J

~,~Oli

.....10 5

L.

'

0

~<E=200 ~

150

v,

y .

i ~-

J-., -

l_.^Jj.J

'

1120 15

.,] .

.

l ' ~ ' 30 ''''' 25

Lengthscale(mm)

,., --V I

"

35

40

45

50

,~

,,

0 Yc

,

10

i

. . . . . . .

20 3'0 40 "Frequency" wavenumber,y = (2n/Z.)rad/mm (a) Trend removal

5'0

....

6'0 YN

Power spectral density function (PSDF)- smoothed FFT periodogram Sampling interval = 50.00pm. 18 degrees of freedom. 1000 points + 24 added zeros. r

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

150 125 E

--

= !1

I

l_a]~Jl. ,^ ..J .._ ~ I J L . . L= ~ l .,l~_l AJ._,~. L '~L L ,, ,~

:r,, 1~ '~'' ' ' ' ' * - ' ' ~ 6

5'

10

is

20

~.

~

.

,

100 75

I

' % L . ~ t - . ~ : ~ ,.~_:~.~::l.,.-,---:,. ~.=-~

"~

",",,'~"'"

25

3'0 .... 35

1

" 40

~ 45

50

Lengthscale(mm) ~~

25 i

0. . . . . 0 Yc

10

- .......

,

. . . . . . . . . .

20 30 40 50 "Frequency" wavenumber y = (2n/Z,)rad/mm (b) Moving average

i

60 YN

Fig. 12.3 (a) Trend removal; (b) moving average filters; (c) weighted moving average; (d) infinite impulse response (IIR) filter

sharply at the high wave-number end. It should be noted that reducing the digitizing interval does not alter this trend. Since m~z represents the variance of the profile slope distribution, then the plot of m~ should give an indication of those wavelengths that contribute to hydrodynamic drag. It will be seen that at low wave-numbers, m~ is insignificant, and in fact it does not begin to acquire any appreciable value until a wavelength of about 2 mm. It then grows almost linearly with log(y) from a wavelength

12.6 Measuresof texr:ure

271

Power spectral density function (PSDF) - smoothed FFT periodogram Sampling interval = 50.00 pm. 18 degrees of freedom. 1000 points + 24 added zeros.

=

loo

0g

"

:'v-,'"

v'

5-rl0 "

~,,~ 30

;

35

v-50

Length scale (ram)

0 Yc

10

20

30

....

4() . . . . . . .

5'0

60 YN

"Frequency" wavenumber y = (2niL) rad/mm (c) Weighted moving average Power spectral density function (PSDF)- smoothed FFT periodogram Sampling interval = 50.00pro. 18 degrees of freedom. 1000 points + 24 added zeros.

00

!,__

50

5

10

15

20

25

30

35

Lengthscale(mm)

40

45

50

25

0

0 Yc

10

"

20

,~

30

,

,

a

40

J

I

50

_~

_

i

60 YN

"Frequency" wavenumber y = (2niL) rad/mm (d) Infinite impulse response (IIR) filter Fig. 12.3

(cont.)

of about 1 mm. On the other hand, m'4, which represents profile curvature, remains insignificant until the spectrum has attained 80% of its power at a wavelength of about 1 mm, whereafter it increases sharply. This suggests a number of things. First, it is probably safe to adopt a long wavelength cut-off of 10 or even 5 mm for this type of surface, since at longer cut-offs the height parameters will be dominated by power from wavelengths that have negligible slope and are therefore not hydrodynamically

272

Hull roughnessand ship resistance

significant. Second, m~z would appear to be a stable parameter which can be used with confidence at any short wavelength cut-offs provided that the cut-off is accurately specified. Third, surface features at wavelengths of 125 ~m and less contribute very little to the power in the spectrum and can probably be ignored as being so small that they are submerged in the viscous sublayer. Fourth, curvature is only significant at similarly high wave-numbers and can probably be ignored altogether for the same reason. Finally, both m4 and e are unstable parameters and are best avoided. This analysis also lends some force to the current trend of looking for a single, stable texture parameter based on average slope or wavelength and to avoid multi-parameter measures of hydrodynamic texture. It will also be appreciated that, whatever standards are eventually adopted in this field, the accurate specification of cut-offs and standardization of filters is absolutely essential. Therefore, filtering is now briefly considered.

12.7

Filtering and filter assessment

Surface metrologists have traditionally been content to rely for short wavelength filtering upon specification of the digitizing interval. The stylus radius can also act as a short wavelength filter. In the case of a bench stylus machine having a truncated pyramid diamond stylus, typically 3/~m x 8 ttm, the measurement of a painted surface at a digitizing interval of 25 or 50/xm is likely to have an attenuating effect on the short wavelengths. It has recently been shown that even the 'hull roughness analyser' ball stylus (1.56 mm diameter) removes very little of the short wavelength power from painted surfaces because the slopes are so small (see Byrne et al. 1982). It is common practice to adopt the selection of a digitizing interval as the only short wavelength filter used, although there may be arguments in favour of using a three-point moving average as well. At the long wavelength end, mechanical filters (skids) were and are still used. For the electrical bench stylus machines of the 1960s the 2CR filter was commonplace and is still in use. Its cut-off characteristics are precisely defined in BSl134. Filtering by limiting the length of profile and fitting a single least squares reference line through each cut-off length is the simplest and most common digital method of long wavelength filtering. Unfortunately it does not work very well (see Fig. 12.3(a)). As an improvement to this method, engineers have tended to favour the use of polynomial centre-lines. The problem here is that it is difficult to know what order of polynomial to use. Also, the polynomial function will hardly ever be an intrinsic function of the long wavelengths in the profile. Recent work at Newcastle University has tended to favour methods taken from signal analysis. Moving average filters are the easiest to apply, either unweighted (the simple 'Boxcar' moving average) or weighted with a suitable function (often sin(x)/x). The disadvantage of these methods is that a large window width is required and thus a large length of carefully collected profile at the beginning and end of each trace is effectively thrown away. Computational complexity can also be a problem and here the simple Boxcar moving average filter gains by an order of magnitude. One particular method which we have found lends itself well to surface metrology work is a cascaded phase-corrected bilinear z-transform filter described by Beauchamp and Yuen (1979), known more concisely as the Infinite Impulse Response (IIR) Filter. This is a recursive filter and uses very few ordinates. It thus benefits from both narrow window width and comparative computational simplicity. Phase distortion is avoided by

IZ8

Final commenrs

273

passing the profile through the filter twice, forwards and backwards. Only first and second-order filters are implemented as higher orders are achieved by cascading. Spectral analysis is extremely useful for evaluating the performance of filters. By carrying out a Fourier transform of both the filtered profile and the rejected waveform, the cut-off characteristics can be examined closely and the filter carefully adjusted. In fact, some of the more sophisticated filters produced by signal processing technology incorporate recursive analysis of the Fourier transform of the signal into the filter itself. This level of sophistication is not thought to be necessary for surface metrology work and fixed filters are almost certainly more than adequate. For fixed filters, spectral analysis can be used to help in choosing attenuation rate, window width and to examine phenomena such as side lobes and leakage. Figure 12.3 shows a comparison of the application of four filters to the profile of Fig. 12.2. The first consists of fitting straight lines through each cut-off length only. It can be seen how extraordinarily inefficient this method is. The simple Boxcar moving average works surprisingly well but suffers a little from leakage. The weighted moving average has the best performance of all, but at the expense of a very large window width. The IIR filter has the best combination of performance and window economy and is reasonably simple to program. Whichever method of filtering is adopted, its parameters should be carefully specified and rigidly adhered to.

12.8

Final comments

This study has shown how the ideas of spectral analysis have been transferred in recent years to a discipline which traditionally did not view its data as random. The problems of wide-bandedness and non-stationarity demonstrate, yet again, that to transfer the concepts of spectral analysis to realistic applications requires a lot of effort. One also has to consider the role and application of the technique in the context of the subject matter, a factor borne out in all the case studies presented. The importance of cut-off frequency and its unstabilizing effect upon m4 have been demonstrated. The same problem occurs in water wave spectra generated from time series, where the high frequency cut-off can significantly affect the high frequency tail of the spectra and hence the higher order moments. Spectral analysis has also been used here to assess the appropriateness of differing filtering techniques. The state of the art in terms of internationally accepted ideas can be determined by reading the proceedings of the International Towing Tank Conference held in Madrid in September 1990. Certainly it is now accepted that a correlation between added drag and ship hull surface statistics does exist, and it is important to understand this if ships are to be operated economically.

Appendix I: Mathematics revision

The main text assumes a knowledge of elementary mathematics, including calculus, statistics and probability. The appendix explains and revises some of the results which are crucial to the main argument and which the reader may have forgotten since a first acquaintance, or even managed to avoid meeting during his or her mathematical studies. Calculus and elementary statistics are not, however, included. We expect most readers will be frequent users of the former, but if you are not the book The Essentials o f Engineering Mathematics by A. Jeffrey will help you bring your calculus up to standard in a relaxed manner. A good background in the necessary statistics and probability can be obtained by reading A Basic Course in Statistics by Clarke and Cooke (1992).

Arithmetic series An arithmetic series is a sum of terms which differ by a constant amount d. If we write a for the first term and S,, for the sum of n terms, then S~=a+(a+d)+(a+2d)+.

. .+(a+(n-1)d)

To find S,, in terms of a, n and d is straightforward once you realize the 'trick' of writing it in the opposite order and adding corresponding terms which will all give 2a + ( n - 1)d. That is Sn = a + (a + d ) + (a + 2 d ) + . . . + ( a + ( n -

1)d)

s . = (a + ( n - 1)d) + (a + ( n - 2)d) + (a + (n - 3)d) + . . . + a which add to give 2S,, = (2a + (n - 1)d)n Therefore (2a + (n - l ) d ) Sn =

2

n

Since (2a + (n - 1)d) is the sum of the first and last terms, S,, is the average term multiplied by the number of terms.

Geometric series A geometric series is a sum of terms which differ by a constant factor ratio r. If we write a for the first term and S,, for the sum of n terms, then Sn = a + ar + ar 2 + . . .

+ ar ~-l

To find S,, in terms of a, r and n we multiply both sides of the above equation by r and subtract term by term, to eliminate all terms but the first and last. That is S,, = a + ar + ar 2 + . . .

274

+ am-l

Taylor series 275 and rSn = ar + ar 2 + . . .

+ a m - l + ar n

Subtraction gives Sn - r S , = a - at"

Hence a ( 1 - r n) 'l

- -

1-r If the terms of a general series decrease to zero as n tends to infinity there is a possibility that their sum to infinity may be finite. The expression for S,, shows that this will be the ease for a geometric series whenever the modulus of r is strictly less than 1. Then a S~-

1-r

Despite their mathematical ability, this result was not established by the Greeks, and Zeno's paradox about Achilles and the tortoise was never properly explained. In modern units, Achilles runs ten times as fast as the tortoise, who travels at 1 km hr -I, and gives the tortoise a 1 km start. After Achilles has covered this 1 km the tortoise will have progressed by 0.1 km. After Achilles covers this 0.1 km the tortoise has progressed by 0.01 km and so on. The conclusion that Achilles never catches the tortoise is obviously false. He does so after 1 + 0 . 1 + 0 . 0 1 + 0.001 + . . .

= 10/9 km

Harmonic series It should be noted that the infinite series 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + . . .

does not converge. The terms tend to zero as n tends to infinity so it is sensible to enquire about convergence. Finite sums calculated on a computer only increase very slowly and might be wrongly interpreted as evidence of convergence. In fact, the series can be shown to exceed the sum of an infinite number of halves by grouping the terms as below 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + . . . > 1 + 1/2 + 1/2 + 1/2 + . . .

Taylor series The objective is to find a polynomial approximation to some function f ( x ) . The approximation is constructed to be exact at some point a and a good approximation for values of x near to a. It requires that f ( x ) and its derivatives be evaluated for x equal to a, so f ( x ) has to be differentiable a number of times equal to the degree of the polynomial. Suppose l(x) is a linear approximation to f ( x ) as shown in Fig. A1. It is convenient to write l(x) in the form l(x) = Co + c l ( x - a)

The requirement that l(x) is exact at a can be written as l(a) = f ( a )

276

A p p e n d i x h M a t h e m a t i c s revision y = I(x)

Y

,,y = q(x)

,

f(a)

I . . . . . .

I |

r

a

X

Fig. A.1 Linearand quadratic approximations to a function about the point (a,

f(a))

This implies that Co must equal f(a). A second reasonable requirement is that l(x) and f(x) should have the same gradient when x equals a. That is

l'(a) =f'(a) This implies that cl must equal f'(a). Now consider a quadratic approximation q(x), also shown in Fig. A1. Writing

q(x) = Co+ Cl (x - a) + c2(x - a) 2 equating q(x) with f(a) and the first two derivatives of q(x) with those off(x) evaluated at a leads to

Co = f(a), cl = f'(a) and c2 = f"(a)/2! If we continue in this manner we obtain

f(x) = f ( a ) + f ' ( a ) ( x - a)

+ f"(a) +if(a) 7 . ( x - a) 2 + ' ' " n! ( x - a)"

For any given function f, the right-hand side of the above relationship will only tend towards f(x) as n tends to infinity if x is sufficiently close to a. The interpretation of 'sufficiently close' varies from function to function, and for some crucially important functions such as sine, cosine and exponential, the right-hand side always converges to them. In fact, the Taylor expansions we need are precisely those of sine, cosine and exponential about 0. Replacing a with 0 and evaluating the appropriate derivatives in the Taylor series expansion leads to the following results which are valid for all values of x X2

X3

e"= l+x+--+--+... 2! 3! X3

X5

sinx = x - - - + - - - . . . 3! 5! X2

cosx = 1 - ~

X4

+---... 2! 4!

Even and odd functions An even function is a function which has the same value for both positive and negative values of the argument. Thus, if y = f(x) is the function then it is an even function if and only if

f (-x) =f(x) The geometric interpretation of this property is that the graph of an even function is symmetric about the axis x = 0 . Examples are cosx and x 2.

Complex numbers 277 An odd function is a function which changes sign, but not amplitude, as the sign of the argument changes. Thus, y = f(x) is an odd function if and only if f(-x) = -f(x) Graphically, f ( - x ) can be obtained by rotating f(x) by a half turn about the origin. Examples are sinx and x.

Complex numbers The solution of a quadratic equation by 'completing the square' gives the standard formula for the roots of a quadratic equation. That is if ax2 + b x + c = O

then x = (-b

+ (b E -

4ac)U2)/(2a)

In many physical applications, for example lightly damped vibrations of a second-order system, the discriminant ( b 2 - 4ac) is negative. The advantages of defining a number j as the square root of - 1 are very wide ranging. In particular the quadratic equation will always have two roots, possibly one repeated. If b 2 is less than 4ac they can be written as,

( - b +_j (4ac - bE)U2)/(2a) Any number of the form

x+jy where x and y are real numbers is called a complex number. Real numbers can be thought of as complex numbers with y equal to zero. Complex numbers satisfy the usual rules of algebra together with the definition -1

j2 =

It follows that if zl and z2 are two complex numbers defined by zl = xl + jYl

and

z2 = x2 + jY2

then

azl + bz2 = (axl + bx2) + j(ayl + by2) and

zlz2 = (XlX2--YlY2) + j(xly2 + xzYl ) upon application of the definition of j2. It also follows that if A is a real number then Azl = Axl + jay1 Before demonstrating division, it is useful to define the conjugate of a complex number z~, written z~ and defined by z~ = x l - j y l Thus, defining z2 = z~' we note that zlzT = x 2 + y2 is a real number. Now, division can be defined in terms of multiplication, that is

ZI

Z1Z~ m

Z2

Z2Z~

- Az~z~

where A = 1/(z2z~) is a real number. The real numbers can be represented by points on a line and complex numbers can be represented by their position in a plane, as shown in Fig. A2.

Appendix h Mathematicsrevision

278

Imaginary axis 432-

eZ

t

-4

-3

1-

)

-2

I

-1

I

9z*

0 --1

I

1

I

--

3

J

4

Real axis

-2 --3

1

2

--

-4

Fig. A.2

The complex number z = - 3 + 2j and its conjugate

A typical point, Zl, could also be described in polar form by its distance from the origin r and an angle 0, conventionally measured from the positive real axis with anticlockwise taken as positive. The distance r is known as the m o d u l u s of the number, written Izll. Upon noticing that

IZl) = ziz and that z~ = r cos0 + j r sin 0 we are now in a position to justify the result which we have relied on for much of the main text. That is e j o = cosO + j sin O an immediate consequence of which is Z1 =

re j~

When 0 is zero we note that zl = r; that is, zl is real. For 0 = ~r/2 we have zl = j r and so Zl is purely imaginary. It follows that for each increment of 7r/2 in 0 we, in fact, rotate z~ one quarter turn. We return to this observation below. We assume that Taylor series remain valid for complex numbers, and in particular that e z - 1 + z + z2/2! + z3/3! + . . . cos z = 1 - z2/2! +

z4/4! -

...

and sin z = z - z3/3! + z S / 5 ! - . . . Substituting j0 for z in the series for e z, and remembering that j2 equals - 1 , leads to the result: e j e = cos O+ j sin 0. A famous special case gives Euler's Equation eJr

=0

which unites the five essential numbers of mathematics, 0, 1, ~r, e and j in one elegant identity. The formula for the multiplication of complex numbers is much easier when they are expressed in polar form. If zl and z2 are r e j ~ and pe j* respectively, their product is zlz2 =

rpe j( 0+ ~)

If we choose zl = j, that is, r = 1 and 0 = ~r/2 then z l z 2 = pe j ( ~ ~r and so, from our earlier observations, we can deduce that multiplication by j corresponds to rotation of z2 through a quarter turn.

Example

279

Generating pseudo-randomnumbers Imagine a 'roulette' wheel with ten equal segments and a croupier who always varies the initial conditions. This perfect wheel could produce a sequence of digits from the set { 0 , . . . , 9} such that each digit is equally likely to result from any spin of the wheel. Notice that this definition implies that the outcome for a spin is independent of the outcomes of all other spins. Pseudo-random digits are digits generated by some mathematical algorithm which cannot be distinguished, by empirical tests, from those produced by the 'perfect' roulette wheel. They are, in fact, entirely deterministic, but if we are unaware of the algorithm we can use them as if they are genuine random numbers. The design and testing of such algorithms is a subject in itself. The following example gives an indication of their form but should not be used for serious work.

Example Let u0 equal some arbitrarily chosen prime number and define ui+l = 91 ui (modulo 105) The (modulo 10s) means that we take the remainder after division by 105, that is the least significant five digits. You should check that if you start from 59 you would obtain the following sequence of numbers: 59 5369 88579 6O689 22699 65609 70419 08129 o

.

~

and so on. The leading digits 8, 6, 2, 6, 7, 0 can be taken as a sequence of pseudo-random digits. The numbers 0.88579, 0.60689, 0.22699 . . . . can be used as a sequence of random numbers from a uniform distribution defined on the interval [0,1]. You should ponder why the sequence 885796068922699... cannot be considered a sequence of pseudo-random digits. The following algorithm is recommended by Clarke and Cooke in their book A Basic Course in Statistics ui+l = 16807 ui (modulo 231- 1) Whilst this is not convenient for hand calculations it can easily be programmed. However, for any intensive simulation studies it would be advisable to use some professionally written software, such as a NAG routine, which will have been thoroughly researched and tested. If random numbers from a uniform distribution defined on [0,1] are available it is relatively straightforward to transform them to a random sequence from any distribution with CDF F. Suppose that the random variable X has the distribution F. Let a and b be arbitrary numbers in the domain of F. Then P r ( a < X < b ) = F(b) - F(a) Let R have a uniform distribution on [0,1]. Then, since F(a) and F(b) must be between 0 and 1 Pr(F(a) < R