Spectral Analysis in Engineering (eBook)

Concepts and Case Studies

Grant Hearn, Andrew Metcalfe (Autoren)

eBook Download: EPUB

1995 | 1. Auflage
320 Seiten
Elsevier Science (Verlag)
978-0-08-054155-6 (ISBN)

Infos

This text provides a thorough explanation of the underlying principles of spectral analysis and the full range of estimation techniques used in engineering. The applications of these techniques are demonstrated in numerous case studies, illustrating the approach required and the compromises to be made when solving real engineering problems. The principles outlined in these case studies are applicable over the full range of engineering disciplines and all the reader requires is an understanding of elementary calculus and basic statistics.
The realistic approach and comprehensive nature of this text will provide undergraduate engineers and physicists of all disciplines with an invaluable introduction to the subject and the detailed case studies will interest the experienced professional.
No more than a knowledge of elementary calculus, and basic statistics and probability is needed
Accessible to undergraduates at any stage of their courses
Easy and clear to follow

Inhalt

Front Cover 1
Spectral Analysis in Engineering Concepts and Cases 4
Copyright Page 5
Contents 6
About the authors 11
Preface 12
Notation and nomenclature 13
Chapter 1. Why understand spectral analysis? 14
1.1 Introduction 14
1.2 Overview 16
Chapter 2. Relationships between variables 21
2.1 Introduction 21
2.2 Discrete bivariate distributions 22
2.3 Continuous bivariate distributions 28
2.4 Linear functions of random variables 40
2.5 Bivariate normal distribution 44
2.6 Confidence intervals for population correlation coefficient 48
2.7 Multivariate normal distribution 49
2.8 Exercises 49
Chapter 3. Time varying signals 52
3.1 Introduction 52
3.2 Why study time series? 53
3.3 Estimation of seasonal effects and trends 54
3.4 Moments of a discrete random process 60
3.5 Stationarity and ergodicity 62
3.6 ARIMA models for discrete random processes 64
3.7 Estimation of parameters of models for random processes 75
3.8 Simulations 86
3.9 Further practical examples 87
3.10 Models for continuous time random processes 96
3.11 Exercises 100
Chapter 4. Describing signals in terms of frequency 104
4.1 Introduction 104
4.2 Finite Fourier series 104
4.3 Fourier series 110
4.4 The Fourier transform 113
4.5 Discrete Fourier transform 118
4.6 Exercises 121
Chapter 5. Frequency representation of random signals 122
5.1 Introduction 122
5.2 Definition of the spectrum of a random process 123
5.3 Estimation of the spectrum from the sample autocovariance function 128
5.4 Estimation of the spectrum from the periodogram 141
5.5 High resolution spectral estimators 149
5.6 Exercises 154
Chapter 6. Identifying system relationships from measurements 156
6.1 Introduction 156
6.2 Discrete processes 158
6.3 Linear dynamic systems 161
6.4 Application of cross-spectral concepts 163
6.5 Estimation of cross-spectral functions 165
6.6 Exercises 170
Chapter 7. Some typical applications 174
7.1 Introduction 174
7.2 Calculating the sample autocovariance function 174
7.3 Calculating the spectrum 175
7.4 Calculating the response spectrum 176
7.5 The spectrum and moving observers 183
7.6 Calculation of significant responses 186
7.7 Exercises 192
Chapter 8. Wave directionality monitoring 197
8.1 Introduction 197
8.2 Background 197
8.3 The technical problem 200
8.4 Reduction of the monitoring problem to a mathematical problem 202
8.5 Application of the mathematical model 205
8.6 The probe arrangements deployed 206
8.7 Analysis of Loch Ness data 207
8.8 MLM-based spectral analysis formulations 210
8.9 Cross-spectral density simulation 211
8.10 Simulation results and alternative probe management 212
8.11 Final comments 219
Chapter 9. Motions of moored structures in a seaway 221
9.1 Introduction 221
9.2 Background 222
9.3 Modelling moored structures 224
9.4 Equations of motion 225
9.5 Determination of time dependent wave force 226
9.6 Evaluation of the quadratic transfer function (QTF) 228
9.7 Simulation of a random sea 229
9.8 Why the probabilistic method of simulation? 231
9.9 Statistical analyses of the generated time series 232
9.10 Sensitivity analysis of a moored tanker and a moored barge to integration time step 234
9.11 Effects of wave damping on the surge motion 234
9.12 Final comments 245
Chapter 10. Experimental measurement and time series acquisition 246
10.1 Introduction 246
10.2 Background 246
10.3 Experimental facilities set-up 248
10.4 Data collection and principles of analysis 249
10.5 Six degrees-of-freedom SELSPOT motion analysis 252
10.6 Data acquisition and SELSPOT calibration 253
10.7 Practical aspects 255
10.8 Some typical results 258
10.9 Final comments 261
Chapter 11. Experimental evaluation of wide band active vibration controllers 266
11.1 Introduction 266
11.2 Background 266
11.3 Techniques for active vibration control 267
11.4 Why use a spectral analyser? 268
11.5 Experimental rig 268
11.6 Some typical results 270
11.7 Final comments 270
Chapter 12. Hull roughness and ship resistance 274
12.1 Introduction 274
12.2 Background 274
12.3 An introduction to surface metrology 275
12.4 Process bandwidth 276
12.5 Surface topography and fluid drag 277
12.6 Measures of texture 280
12.7 Filtering and filter assessment 285
12.8 Final comments 286
Appendices 287
Appendix I: Mathematics revision 287
Appendix II: Inflows to the Font reservoir 296
Appendix III: Chi-square and F-distributions 298
Appendix IV: The sampling theorem 302
Appendix V: Wave tank data 304
Appendix VI: Sampling distribution of spectral estimators 305
References 307
Further reading 312
Index 316

Textauszug

2

Relationships between variables

2.1 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 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 2.1

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 categories quality of waterways in England and Wales : estimates of km in each of 12 categories

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.

Fig. 2.1 Bivariate distribution of waterway classification and water quality

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 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

(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 44 124, and rounding to three decimal places.

Table 2.2

Values of PXY(x,y) for the water quality model

A line diagram for the probability mass function is illustrated in Fig. 2.2. For any bivariate probability mass function

Fig. 2.2 Bivariate probability mass function to model waterway classification and water quality

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

The marginal probability mass function of X is defined by

(2.2)

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

(2.3)

For our example, the marginal probability mass function for X is

The marginal probability mass function for Y is

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

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)

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

If and only if the events X = x and Y = y are independent does Pr{X = x and Y = y} = Pr{X = x} Pr{Y = y}, i.e. Pr{X = x and Y = y} = Px(x)PY(y).

An equivalent definition of independence is, X and Y are independent if and only if PY|x(y|x) =...

Produkt-Details

• Erscheint lt. Verlag: 17.8.1995
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Chemie ► Analytische Chemie
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-054155-0 / 0080541550
• ISBN-13: 978-0-08-054155-6 / 9780080541556