Noise and Vibration Analysis E-Book

Anders Brandt

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 917

Veröffentlichungsjahr: 2011

Beschreibung

Noise and Vibration Analysis is a complete and practical guide that combines both signal processing and modal analysis theory with their practical application in noise and vibration analysis. It provides an invaluable, integrated guide for practicing engineers as well as a suitable introduction for students new to the topic of noise and vibration. Taking a practical learning approach, Brandt includes exercises that allow the content to be developed in an academic course framework or as supplementary material for private and further study.

• Addresses the theory and application of signal analysis procedures as they are applied in modern instruments and software for noise and vibration analysis
• Features numerous line diagrams and illustrations
• Accompanied by a web site at www.wiley.com/go/brandt with numerous MATLAB tools and examples.

Noise and Vibration Analysis provides an excellent resource for researchers and engineers from automotive, aerospace, mechanical, or electronics industries who work with experimental or analytical vibration analysis and/or acoustics. It will also appeal to graduate students enrolled in vibration analysis, experimental structural dynamics, or applied signal analysis courses.

Leseprobe

Contents

Cover

Title Page

Copyright

About the Author

Preface

Acknowledgements

List of Abbreviations

Notation

1: Introduction

1.1 Noise and Vibration

1.2 Noise and Vibration Analysis

1.3 Application Areas

1.4 Analysis of Noise and Vibrations

1.5 Standards

1.6 Becoming a Noise and Vibration Analysis Expert

2: Dynamic Signals and Systems

2.1 Introduction

2.2 Periodic Signals

2.3 Random Signals

2.4 Transient Signals

2.5 RMS Value and Power

2.6 Linear Systems

2.7 The Continuous Fourier Transform

2.8 Chapter Summary

2.9 Problems

3: Time Data Analysis

3.1 Introduction to Discrete Signals

3.2 The Sampling Theorem

3.3 Filters

3.4 Time Series Analysis

3.5 Chapter Summary

3.6 Problems

4: Statistics and Random Processes

4.1 Introduction to the Use of Statistics

4.2 Random Theory

4.3 Statistical Methods

4.4 Quality Assessment of Measured Signals

4.5 Chapter Summary

4.6 Problems

5: Fundamental Mechanics

5.1 Newton's Laws

5.2 The Single Degree-of-freedom System (SDOF)

5.3 Alternative Quantities for Describing Motion

5.4 Frequency Response Plot Formats

5.5 Determining Natural Frequency and Damping

5.6 Rotating Mass

5.7 Some Comments on Damping

5.8 Models Based on SDOF Approximations

5.9 The Two-degree-of-freedom System (2DOF)

5.10 The Tuned Damper

5.11 Chapter Summary

5.12 Problems

6: Modal Analysis Theory

6.1 Waves on a String

6.2 Matrix Formulations

6.3 Eigenvalues and Eigenvectors

6.4 Frequency Response of MDOF Systems

6.5 Time Domain Simulation of Forced Response

6.6 Chapter Summary

6.7 Problems

7: Transducers for Noise and Vibration Analysis

7.1 The Piezoelectric Effect

7.2 The Charge Amplifier

7.3 Transducers with Built-In Impedance Converters, ‘IEPE’

7.4 The Piezoelectric Accelerometer

7.5 The Piezoelectric Force Transducer

7.6 The Impedance Head

7.7 The Impulse Hammer

7.8 Accelerometer Calibration

7.9 Measurement Microphones

7.10 Microphone Calibration

7.11 Shakers for Structure Excitation

7.12 Some Comments on Measurement Procedures

7.13 Problems

8: Frequency Analysis Theory

8.1 Periodic Signals — The Fourier Series

8.2 Spectra of Periodic Signals

8.3 Random Processes

8.4 Transient Signals

8.5 Interpretation of spectra

8.6 Chapter Summary

8.7 Problems

9: Experimental Frequency Analysis

9.1 Frequency Analysis Principles

9.2 Octave and Third-octave Band Spectra

9.3 The Discrete Fourier Transform (DFT)

9.4 Chapter Summary

9.5 Problems

10: Spectrum and Correlation Estimates Using the DFT

10.1 Averaging

10.2 Spectrum Estimators for Periodic Signals

10.3 Estimators for PSD and CSD

10.4 Estimator for Correlation Functions

10.5 Estimators for Transient Signals

10.6 Spectrum Estimation in Practice

10.7 Multi-channel Spectral Analysis

10.8 Chapter Summary

10.9 Problems

11: Measurement and Analysis Systems

11.1 Principal Design

11.2 Hardware for Noise and Vibration Analysis

11.3 FFT Analysis Software

11.4 Chapter Summary

11.5 Problems

12: Rotating Machinery Analysis

12.1 Vibrations in Rotating Machines

12.2 Understanding Time—Frequency Analysis

12.3 Rotational Speed Signals (Tachometer Signals)

12.4 RPM Maps

12.5 Smearing

12.6 Order Tracks

12.7 Synchronous Sampling

12.8 Averaging Rotation-speed-dependent Signals

12.9 Adding Change in RMS with Time

12.10 Parametric Methods

12.11 Chapter Summary

12.12 Problems

13: Single-input Frequency Response Measurements

13.1 Linear Systems

13.2 Determining Frequency Response Experimentally

13.3 Important Relationships for Linear Systems

13.4 The Coherence Function

13.5 Errors in Determining the Frequency Response

13.6 Coherent Output Power

13.7 The Coherence Function in Practice

13.8 Impact Excitation

13.9 Shaker Excitation

13.10 Examples of FRF Estimation — No Extraneous Noise

13.11 Example of FRF Estimation — with Output Noise

13.12 Examples of FRF Estimation — with Input and Output Noise

13.13 Chapter Summary

13.14 Problems

14: Multiple-input Frequency Response Measurement

14.1 Multiple-Input Systems

14.2 Conditioned Input Signals

14.3 Bias and Random Errors for Multiple-Input Systems

14.4 Excitation Signals for MIMO Analysis

14.5 Data Synthesis and Simulation Examples

14.6 Real MIMO Data Case

14.7 Chapter Summary

14.8 Problems

15: Orthogonalization of Signals

15.1 Principal Components

15.2 Virtual Signals

15.3 Noise Source Identification (NSI)

15.4 Chapter Summary

15.5 Problems

16: Advanced Analysis Methods

16.1 Shock Response Spectrum

16.2 The Hilbert Transform

16.3 Cepstrum Analysis

16.4 The Envelope Spectrum

16.5 Creating Random Signals with Known Spectral Density

16.6 Operational Deflection Shapes — ODS

16.7 Introduction to Experimental Modal Analysis

16.8 Chapter Summary

16.9 Problems

Appendix A: Complex Numbers

Appendix B: Logarithmic Diagrams

Appendix C: Decibels

Appendix D: Some Elementary Matrix Algebra

Appendix E: Eigenvalues and the SVD

E.1 Eigenvalues and Complex Matrices

E.2 The Singular Value Decomposition (SVD)

Appendix F: Organizations and Resources

Bibliography

Index

This edition first published 2011 © 2011, John Wiley & Sons, Ltd

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Library of Congress Cataloguing-in-Publication Data

Brandt, Anders. Noise and vibration analysis : signal analysis and experimental procedures / Anders Brandt. p. cm. Includes bibliographical references and index. ISBN 978-0-470-74644-8 (hardback) 1. Vibration–Mathematical models. 2. Noise–Mathematical models. 3. Acoustical engineering. 4. Stochastic analysis. 5. Signal processing. I. Title. TA355.B674 2011 620.3–dc22 2010039788

A catalogue record for this book is available from the British Library.

Print ISBN: 9780470746448 E-PDF ISBN: 9780470978177 O-Book ISBN: 9780470978160 E-Pub ISBN: 9780470978115

About the Author

Anders Brandt has more than twenty years experience as a consultant and short-course instructor in experimental vibration analysis. During his entire career, he has worked on providing increased understanding of the measurement and analysis procedures used in experimental vibration analysis. Currently, Anders Brandt is an Associate Professor of Experimental Dynamics and Signal Processing at the University of Southern Denmark, where his main research interests are in applied signal processing and operational modal analysis. Anders is a popular short-course instructor and lecturer on the topics covered by this book.

Preface

The material in this book has been developing in my mind for more than twenty years of teaching. During these years I have been teaching over 200 shortcourses for engineers in the industry on techniques for experimental noise and vibration analysis and also on how to use commercial measurement and analysis systems. In addition, in the late 1990s I developed and taught three master's level courses in experimental analysis of vibrations at Blekinge Institute of Technology in Sweden. Noise and vibration analysis is an interdisciplinary field, incorporating diverse subjects such as mechanical dynamics, sensor technology, statistics, and signal processing. Whereas there are many excellent and comprehensive books in each of these disciplines, there has been a lack of introductory material for the engineering student who first starts to make noise and/or vibration measurements, or the engineer who needs a reference in his or her daily life. In addition, there are few textbooks in this field presenting the techniques as they are actually used in practice. This book is an attempt to fill this void.

My aim for this book is for it to serve both as a course book and as supplementary reading in university courses, as well as providing a handbook for engineers or researchers who measure and analyze acoustic or vibration signals. The level of the book makes it appropriate both for undergraduate and graduate levels, with a proper selection of the content. In addition the book should be a good reference for analysts who use experimental results and need to interpret them. To satisfy these rather different purposes, for some of the topics in the book I have included more detail than would be necessary for an introductory text. To facilitate its use as a handbook, I have also included a short summary at the end of each chapter where some of the key points of the chapter are repeated.

This book contains background theory explaining the majority of analysis methods used in modern commercial software for noise and vibration measurement and analysis, with one exception: experimental modal analysis is only briefly introduced, as this is a specialized field with some excellent textbooks already available. This book also includes a number of tools which are usually not found in commercial systems, but which are still useful for the practitioner. With modern computer-based software, it is easy to export data to, e.g., MATLAB/Octave (see below), and apply the techniques there.

Since it is an introductory text, most of the content of this book is of course available in more specialized textbooks and scientific papers. A few parts, however, include some improvements of existing techniques. I will mention these points in the descriptions of the appropriate chapters below.

Signal analysis is traditionally a field within electrical engineering, whereas most engineers and students pursuing noise and vibration measurements are mechanical or civil engineers. The aim has therefore been to make the material accessible, particularly to students and engineers of these latter disciplines. For this reason I have included introductions to the Laplace and Fourier transforms — both essential tools for understanding, analyzing and solving problems in dynamics. Electrical engineering students and practitioners should still find many of the topics in the book interesting.

Signal analysis is a subject which is best learned by practicing the theories (as, perhaps, all subjects are). I have therefore incorporated numerous examples using MATLAB or GNU Octave throughout the book. Further examples and an accompanying toolbox which can be used with either MATLAB or GNU Octave can be downloaded from Internet. More information about this is located in Section 1.6. I strongly recommend the use of these tools as a complement to reading this book, regardless of whether you are a student, a researcher or an industry practitioner.

Chapter 2 introduces dynamic signals and systems with the aim of being an introduction particularly for mechanical and civil engineering students. In this chapter the classification of signals into periodic, random and transient signals is introduced. The chapter also includes linear system theory and a comprehensive introduction to the Laplace and Fourier transforms, both important tools for understanding and analyzing dynamic systems.

In Chapter 3 some fundamental concepts of sampled signals are presented. Starting with the sampling theorem and continuing with digital filter theory, this chapter presents some important applications of digital filters for fractional octave analysis and for integrating and differentiating measured signals.

Chapter 4 introduces some applied statistics and random process theory from a practical perspective. It includes an introduction to hypothesis testing as this tool is sometimes used for testing normality and stationarity of data. This chapter also gives an introduction to the application of statistics for data quality assessment, which is becoming more important with the large amounts of data collected in many applications of noise and vibration analysis.

Chapters 5 and 6 provide an introduction to the theory of mechanical vibrations. I anticipate that the contents of these two chapters will already be known to many readers, but I have found it important to include them because my presentation focuses on the experimental implications of the theory, unlike the presentation in most mechanical vibration textbooks, and because some later chapters in the book need a foundation with a common nomenclature. Chapter 6 also includes an accurate and fast method for computing forced response of linear systems in the time domain which is very attractive, e.g., to produce known experimental signals for testing out signal analysis procedures. This method, developed by Professor Kjell Ahlin, has been presented at conferences, but deserves better dissemination.

In Chapter 7 the most important transducers used for measurements of noise and vibration signals are presented; specifically the accelerometer, the force sensor and the microphone. Because piezoelectric sensors with built-in signal conditioning (so-called IEPE sensors) are widely used today, this technology is presented in some depth. In this chapter I also present some personal ideas on how to become a good experimentalist.

The analysis techniques mostly used in this field are based on the Discrete Fourier Transform (DFT), computed by the FFT. Spectrum analysis is therefore an important part of this book and Chapters 8 through 10 are spent on this topic. Chapter 8 introduces basic frequency analysis theory by presenting the different signal classes, and the different spectra used to describe the frequency content of these signals.

In Chapter 9 the DFT and some other techniques used to experimentally determine the frequency content of signals are presented. The properties of the DFT, which are very important to understand when interpreting experimental frequency spectra, are presented relatively comprehensively.

Chapter 10 includes a comprehensive presentation of how spectra from periodic, random and transient signals, and mixes of these signal classes, should be estimated in practice. Also, I mention a convenient technique for removing harmonics in spectral density estimates using the smoothed periodogram method; which, to my knowledge, has never been presented before. Chapter 10 also includes a comprehensive explanation of Welch's method for PSD estimation, including overlap processing, as this is the method used in virtually all commercial software. The treatment of practical spectral analysis in this chapter should also be of use to engineers outside the field of acoustics and vibrations who want to calculate and/or interpret spectra by using the FFT.

In Chapter 11 the design of modern data acquisition and measurement systems is described from a user perspective. In this chapter both hardware and software issues are penetrated. Chapter 12 addresses order tracking, which is a common technique for analysis of rotating machinery equipment. The chapter describes the most common techniques used to measure such signals both with fixed sampling frequency and with synchronous sampling.

Frequency response functions are important measurement functions in experimental noise and vibration analysis and are used, for example, in experimental modal analysis. Chapter 13 therefore covers techniques for measuring frequency responses for single-input/single-output (SISO) systems. Both impact excitation and shaker excitation techniques are presented in detail. In Chapter 14 the techniques are extended to multiple-input/multiple-output (MIMO) systems. In Chapters 13 and 14 I also present a technique which has not, to the best of my knowledge, been presented before. Using well-known periodic excitation signals, I show that the bias error in frequency response estimates with extraneous noise present in both input and output signals can be eliminated by time domain averaging, for single-input as well as multiple-input systems.

Chapter 15 presents some relatively advanced techniques used for multichannel analysis, namely principal components and virtual signals. These techniques are commonly used for noise path analysis and noise source identification in many of the sophisticated software packages available commercially. I present these concepts in some depth, since they are not readily available in other textbooks.

In Chapter 16 I have collected a number of more advanced techniques that engineers in this field should be acquainted with. This chapter presents, in order, the shock response spectrum, the Hilbert transform with applications, the cepstrum and envelope spectrum, how to produce Gaussian time signals with known spectral density, and finally two very important tools: operational deflection shapes, and experimental modal analysis. The latter is a comprehensive technique and only briefly introduced.

In the Appendix section I have included some fundamentals on complex numbers, logarithmic diagrams and the decibel unit, matrix theory, and eigenvalues and the singular value decomposition. The reader who does not feel confident with some of these concepts will hopefully find enough theory in these appendices to follow the text in this book. The last appendix contains some references to good sources for more information within the noise and vibration community. I hope the newcomer to this field can benefit from this list.

Acknowledgements

This book is inspired partly by class notes I wrote for two classes at Blekinge Institute of Technology, BTH. I am especially grateful to Professor Ingvar Claesson and the Department of Signal Processing at BTH for supporting me in writing these early texts. Also, Timothy Samuels did a great job translating an early manuscript from Swedish to English.

My most sincere appreciation goes to Professor Kjell Ahlin, my colleague and friend for many years. Our many long discussions have strongly contributed to my understanding of this subject and I am grateful for the data provided by Professor Ahlin for examples in Chapter 16.

Dr Per-Olof Sturesson and the noise and vibration group at SAAB Automobile AB have been invaluable resources of feedback and have provided data for Chapters 12 and 15. For this, and many ideas and discussions, I am very grateful. Special thanks also goes to Mats Berggren.

My thanks extend to Professor Jiri Tuma for supporting me with data for Chapter 12 and for kind support through times.

Svend Gade and Brüel and Kjær A/S are acknowledged, along with Niels Thrane, for allowing me to reuse an illustration and an overview description of the Discrete Fourier Transform from an old B & K Technical Review, which I find is of great value for presenting the DFT.

I have always found the many participants at the International Modal Analysis Conference (IMAC), organized by Society for Experimental Mechanics (SEM), an invaluable source of inspiration and knowledge. Special thanks to Tom Proulx, Al Wicks, Dave Brown, and Randall Allemang for their outstanding support and encouragement and continuous willingness to give from their wealth of knowledge.

This book would not be what it is without the professional staff at Wiley, who have been of great help throughout the work. My thanks extend particularly to Debbie Cox and Nicky Skinner who have both been of great help.

Particularly I also wish to thank Dr Julius S. Bendat, Professor Rune Brincker, Knut Bertelsen (in memoriam), and Professor Bo Håkansson for their willingness to always share their knowledge and for inspiring me, to Claus Vaarning and Soma Tayamon for reading parts of the manuscript and offering many good comments, and to all the professional people I have had the opportunity of learning from during my career.

Finally I am, of course, thankful to a great number of people who have inspired and supported me, and to all my students and short-course participants over the years who have taught me so much. And to my family for having endured a long time without seeing very much of me.

List of Abbreviations

2DOFTwo degrees-of-freedom systemACAlternating currentADCAnalog-to-digital converterBTBandwidth-time (product)CSDCross-spectral density functionDACDigital-to-analog converterDCDirect currentDFTDiscrete Fourier transformDOFDegree-of-freedom (point and direction)ESDEnergy spectral densityFEFinite elementFEMFinite element methodFFTFast Fourier transformFIRFinite impulse response (filter)FRFFrequency response functionHFHigh frequencyHPHighpassIDFTInverse discrete Fourier transformIEPEIntegrated electronics piezoelectric (sensor)IFFTInverse fast Fourier transformIIRInfinite impulse response (filter)IRFImpulse response functionLFLow frequencyISOInternational standardization organizationMDOFMultiple degrees-of-freedomMIFMode indicator functionMIMOMultiple-input/multiple-outputMISOMultiple-input/single-outputMPSSMulti-phase stepped sineMrMIFModified real mode indicator functionMvMIFMultivariate mode indicator functionNSINoise source identificationNSRNoise-to-signal ratioODSOperating deflection shapePDFProbability density functionPSDPower spectral densityRMSRoot mean squareRPMRevolutions per minuteSDOFSingle degree-of-freedomSIMOSingle-input/multiple-outputSISOSingle-input/single-outputSNRSignal-to-noise ratioSRSShock response spectrumSVDSingular value decompositionTEDSTransducer electronic data sheet

Notation

< x >Average of x [ ]Fourier transform of [ ] [ ]Hilbert transform of [ ] [ ]Laplace transform of [ ]E [ ]Expected valuea, a(t)Vibration accelerationApqrResidue of mode r, between points p and qAxxAutopower spectrum of xBBandwidth in [Hz]BeEquivalent (statistical) bandwidth in HzBenNormalized equivalent bandwidth (dimensionless)BrResonance bandwidth in HzcpPower cepstrumcrModal (viscous) damping of mode rδ(t) Dirac's unit impulseΔfFrequency increment of discrete Fourier transformΔtTime increment in [s]ɛNormalized errorfFrequency in [Hz]fn, frUndamped natural frequencyg2(f)Virtual coherence functionγyx2Coherence function between x (input) and y (output)γy:x2Multiple coherence of y (output) with all xq (inputs)Gxx(f)Single-sided autospectral density of xGxx′Principal component[Gxx]Single-sided input cross-spectral matrixGyx(f)Single-sided cross-spectral density between x (input) and y (output)[Gyx]Single-sided input/output cross-spectral matrixh(n)Discrete impulse responseh(t)Analog impulse responseH(f)Analog frequency response functionH(k)Discrete frequency response functionH(s)Transfer functionIm [ ]Imaginary part of [ ]jImaginary number, kDiscrete (dimensionless) frequency variablekrModal stiffness of mode rKxKurtosis of xλEigenvalueμx(Theoretical) mean value of xmrModal mass of mode rMnNth statistical (central) momentnDiscrete (dimensionless) time variableφPhase, general random variablepx(x)Probability density of xP(x)Probability distribution of x{ψ}rMode shape vector of mode r[Ψ]rMode shape matrix of mode rQQuality factor (Q-factor)QrModal scale constant of mode rRxx(τ) Autocorrelation of xRyx(τ) Cross-correlation between x (input) and y (output)Re [ ]Real part of [ ]sLaplace operator (in [rad/s])srPole, root to characteristic polynomialσxStandard deviation of xSxSkewness of xSxx(f)Double-sided autospectral density of xSyx(f)Double-sided cross-spectral density between x (input) and y (output)[Gyx]Single-sided input/output cross-spectral matrixtAnalog timeTMeasurement timeτTime delay, time lag variable for correlation functionsTx(k)Discrete transient spectrum of xu, u(t)Vibration displacementv, v(t)Vibration velocityw(n)Discrete time windowx(n)Discrete/sampled (input) signalx(t)Analog (input) signal(t)Hilbert transform of x(t)X(f)(Continuous) Fourier transform of x(t)X′Spectrum of virtual signalX(k)Discrete Fourier transform of x(n)XL(k)Linear (RMS) spectrum of x(n)y(n)Discrete/sampled (output) signaly(t)Analog (output) signalωAngular frequency in [radians/s]ζrRelative (viscous) damping

1

Introduction

This chapter provides a short introduction to the field of noise and vibration analysis. Its main objective is to show new students in this field the wide range of applications and engineering fields where noise and vibration issues are of interest. If you are a researcher or an engineer who wants to use this book as a reference source, you may want to skim this chapter. If you decide to do so, I would recommend you to read Section 1.6, in which I present some personal ideas on how to use this book, as well as on how to go about becoming a good experimentalist — the ultimate goal after reading this book.

I want to show you not only the width of disciplines where noise and vibrations are found. I also want to show you that noise and vibration analysis, the particular topic of this book, is truly a fascinating and challenging discipline. One of the reasons I personally find noise and vibration analysis so fascinating is the interdisciplinary character of this field. Because of this interdisciplinary character, becoming an expert in this area is indeed a real challenge, regardless of which engineering field you come from. If you are a student just entering this field, I can only congratulate you for selecting (which I hope you do!) this field as yours for a lifetime. You will find that you will never cease learning, and that every day offers new challenges.

1.1 Noise and Vibration

Noise and vibration are constantly present in our high-tech society. Noise causes serious problems both at home and in the workplace, and the task of reducing community noise is a subject currently focused on by authorities in many countries. Similarly, manufacturers of mechanical products with vibrations causing acoustic noise, more and more find themselves forced to compete on the noise levels of their products. Such competition has so far occurred predominantly in the automotive industry, where the issues with sound and noise have long attracted attention, but, at least in Europe, e.g., domestic appliances are increasingly marketed stressing low noise levels.

Let us list some examples of reasons why vibration is of interest.

Vibration can cause injuries and disease in humans, with ‘white fingers’ due to long-term exposure to vibration, and back injuries due to severe shocks, as examples.Vibration can cause discomfort, such as sickness feelings in high-rise buildings during storms, or in trains or other vehicles, if vibration control is not successful.Vibration can cause fatigue, i.e., products break after being submitted to vibrations for a long (or sometimes not so long) time.Vibration can cause dysfunction in both humans and things we manufacture, such as bad vision if the eye is subjected to vibration, or a radar on a ship performing poorly due to vibration of the radar antenna.Vibration can be used for cleaning, etc.Vibration can cause noise, i.e., unpleasant sound, which causes annoyance as well as disease and discomfort.

To follow up on the last point in the list above, once noise is created by vibrations, noise is of interest, e.g., for the following reasons.

Excessive noise can cause hearing impairment.Noise can cause discomfort.Noise can (probably) cause disease, such as increased risk of cardiac disease, and stress.Noise can be used for burglar alarms and in weapons (by disabling human ability to concentrate or to cope with the situation).

The lists above are examples, meant to show that vibrations and noise are indeed interesting for a wide variety of reasons, not only to protect ourselves and our products, but also because vibration can cause good things.

Besides simply reducing sound levels, much work is currently being carried out within many application areas concerning the concept of sound quality. This concept involves making a psychoacoustic judgment of how a particular sound is experienced by a human being. Harley Davidson is an often-cited example of a company that considers the sound from its product so important that it tried to protect that sound by trademark, although the application was eventually withdrawn.

Besides generating noise, vibrations can cause mechanical fatigue. Now and then we read in the newspaper that a car manufacturer is forced to recall thousands of cars in order to exchange a component. In those cases it is sometimes mechanical fatigue that has occurred, resulting in cracks initiating after the car has being driven a long distance. When these cracks grow they can cause component breakdown and, as a consequence, accidents.

1.2 Noise and Vibration Analysis

This book is about the analysis methods for analyzing noise and vibrations, rather than the mechanisms causing them. In order to identify the sources of vibrations and noise, extensive analysis of measured signals from different tests is often necessary. The measurement techniques used to carry out such analyses are well developed, and in universities as well as in industry, advanced equipment is often used to investigate noise and vibration signals in laboratory and in field environments.

The area of experimental noise and vibration analysis is an intriguing field, as I hope this book will reveal. It is so partly because this field is multidisciplinary, and partly because dynamics (including vibrations) is a complicated field where the most surprising things can happen. Using measurement and analysis equipment often requires a good understanding of mechanics, sensor technology, electronic measurement techniques, and signal analysis.

Vibrations and noise are found in many disciplines in the academic arena. Perhaps we first think of mechanics, with engines, vehicles, and pumps, etc. However, vibrations are also found also in civil engineering, in bridges, buildings, etc. Many of the measurement instruments and sensors we use in the field of analyzing vibrations and noise are, of course, electrical, and so the field of electrical engineering is heavily involved. This makes the initial study of noise and vibration analysis difficult, perhaps, because you are forced to get into some of the other fields of academia. Hopefully, this book can help bridge some of the gaps between disciplines.

If many academic disciplines are involved with noise and vibrations, the variety in industry is perhaps even more overwhelming. Noise and vibration are important in, for example, military, automotive, and aerospace industries, in power plants, home appliances, industrial production, hand-held tools, robotics, the medical field, electronics production, bridges and roads, etc.

1.3 Application Areas

As evident from the first sections of this chapter, noise and vibration are important for many reasons, and in many different disciplines. Within the field of noise and vibration, there are also many different, more specialized, disciplines. We need to describe some of these a little more.

Structural dynamics is a field which describes phenomena such as resonance in structures, how connecting structures together affect the resonances, etc. Often, vibration problems occur because, as you probably already know, resonances amplify vibrations — sometimes to very high levels.

Environmental engineering is a field in which environmental effects (not to be confused with the ‘green environment’) from such diverse phenomena as heat, corrosion, and vibration, etc., are studied. As far as vibrations are concerned, vibration testing is a large industrial discipline within environmental engineering. This field is concerned with a particular product's ability to sustain the vibration environment it will encounter during its lifetime. Sensitive products such as mobile phones and other electronic products are usually tested in a laboratory to ensure they can sustain the vibrations they will be exposed to during their lifetime. Producing standardized tests which are equivalent to the product's real-life vibration environment, is often a great challenge. Transportation testing of packaging is a closely related field, in which the interest is that, for example, the new video camera you buy arrives in one piece when you unpack the box, even if the ship that delivered it encountered a storm at sea.

Fatigue analysis is a field closely related to environmental engineering. However, the discipline of fatigue analysis is usually more involved with measuring the stresses on a product and, through mathematical models such as Wöhler curves etc., trying to predict the lifetime of the product, e.g., before fatigue cracks will appear. From the perspective of experiments, this practically means it is more common to measure with strain gauges rather than accelerometers.

Vibration monitoring is another field, where the aim is to try to predict when machines and pumps, for example, will fail, by studying (among many things) the vibration levels during their lifetime. In civil engineering, a somewhat related field, structural health monitoring attempts to assess the health of buildings and bridges after earthquakes as well as after aging and other deteriorating effects on the structure, based on measurements of (among many things) vibrations in the structures.

Acoustics is a discipline close to noise and vibration analysis, of course, as the cause of acoustic noise is often vibrations (but sometimes not, such as, for example, when turbulent air is causing the noise).

1.4 Analysis of Noise and Vibrations

There are several ways of analyzing noise and vibrations. We shall start with a brief discussion of some of the methods which this book is not aimed at, but which are crucial for the total picture of noise and vibration analysis, and which is often the reason for making experimental measurements.

Analytical analysis of vibrations is most commonly done using the finite element method, FEM, through normal mode analysis, etc. In order to successfully model vibrations, usually models with much greater detail (finer grid meshes, correctly selected element types, etc.) need to be used, compared with the models sufficient for static analysis. Also, dynamic analysis using FEM requires good knowledge of boundary conditions etc. For many of these inputs to the FEM software, experiments can help refine the model. This is a main cause of much experimental analysis of vibrations today.

For acoustic analysis, acoustic FEM can be used as long as the noise (or sound) is contained in a cavity. For radiation problems, the boundary element method, BEM, is increasingly used. With this method, known vibration patterns, for example from a FEM analysis, can be used to model how the sound radiates and builds up an acoustic field.

FEM and BEM are usually restricted to low frequencies, where the mode density is low. For higher frequencies, statistical energy analysis, SEA, can be used. As the name implies, this method deals with the mode density in a statistical manner, and is used to compute average effects.

1.4.1 Experimental Analysis

In many cases it is necessary to measure vibrations or sound pressure, etc., to solve vibration problems, because the complexity of such problems often make them impossible to foresee through analytical models such as FEM. This is often referred to as trouble-shooting. Another important reason to measure and analyze vibrations is to provide input data to refine analytical models. Particularly, damping is an entity which is usually impossible to estimate through models — it needs to be assessed by experiment.

Experimental analysis of noise and vibrations is usually done by measuring accelerations or sound pressures, although other entities can be measured, as we will see in Chapter 7. In order to analyze vibrations, the most common method is by frequency analysis, which is due to the nature of linear systems, as we will discuss in Chapter 2. Frequency analysis is a part of the discipline of signal analysis, which also incorporates filtering signals, etc. The main tool for frequency analysis is the FFT (fast Fourier transform) which is today readily available through software such as MATLAB and Octave (see Section 1.6), or by the many dedicated commercial systems for noise and vibration analysis. Methods using the FFT will take up the main part of this book.

Some of the analysis necessary to solve many noise and vibration problems needs to be done in the time domain. Examples of such analysis is fatigue analysis, which incorporates, e.g., cycle counting, and data quality analysis, to assess the quality of measured signals. For a long time, the tools for noise and vibration analysis were focused on frequency analysis, partly due to the limited computer performance and cost of memory. Today, however, sophisticated time domain analysis can be performed at a low cost, and we will present many such techniques in Chapters 3 and 4.

1.5 Standards

Due to the complexity of many noise and vibration measurements, international standards form an important part of vibration measurements as well as of acoustics and noise measurements. Acoustics and vibration standards are published by the main standardization organizations, ISO (International Standardization Organization), IEC (International Electrical Committee), and, in the U.S., by ANSI (American National Standards Institute). The general recommendation from many acoustics and vibration experts is that, if there is a standard for your particular application — use it. It is outside the scope of this book, and practically impossible, to summarize all the standards available. Some of the many standards for signal analysis methods used in vibration analysis are, however, cited in this book.

1.6 Becoming a Noise and Vibration Analysis Expert

The main emphasis in this book is on the signal analysis methods and procedures used to solve noise and vibration problems. To be successful in this, it is necessary to become a good experimentalist. Unfortunately, this is not something which can be (at least solely) learned from a book, but I want to make some recommendations on how to enter a road which leads in the right direction.

1.6.1 The Virtue of Simulation

As many of the theories of dynamics, as well as those of signal analysis, are very complex, a vital tool for understanding dynamic systems and analysis procedures, is to simulate simplified, isolated, cases, where the outcome can be understood without the complicating presence of disturbance noise, complexity of structures, non-ideal sensors, etc. I have therefore incorporated numerous examples in this book which use simulated measurement data with known properties. A practical method to create such signals is presented in Section 6.5. The importance of this cannot be overrated. Before making a measurement of noise or vibrations, it is crucial to know what a correct measurement signal should look like, for example. The hidden pitfalls in, particularly, vibration measurements are overwhelming for the beginner (and sometimes even for more experienced engineers). The road to successful vibration measurements therefore goes through careful, thought-through simulations.

Another important aspect of good experiments, is to make constant checks of the equipment. In Section 7.21.1 I present some ideas of things to check for in vibration measurements. In Section 7.8.1 I also present a by no means new technique, but nevertheless a simple and efficient one (mass calibration, if you already know it) to verify that accelerometers are working correctly. These devices are, like many sensors, sensitive and can easily break, and unfortunately, they often break in such a way that it can be hard to discover without a proper procedure to verify the sensors on a known signal. Single-frequency calibration, which is common for absolute calibration of accelerometers, usually completely fails to discover the faults present after an accelerometer has been dropped on a hard floor.

Having written this, I want to stress that good vibration measurements are performed every day in industry and universities. So, the intention is, of course, not to discourage you from this discipline, but simply to stress the importance of taking it slowly, and making sure every part of the experiment is under your control, and not under the control of the errors.

1.6.2 Learning Tools and the Format of this Book

If you anticipated finding a book with numerous data examples from the field by which you would learn how to make the best vibration measurements, you will be disappointed by this book. The main reasons for this are twofold; (i) for the reasons just given in the preceding section, real vibration measurements are usually full of artifacts from disturbance noise, complicated structures, etc.; and (ii) each structure or machine or whatever is measured, has its own vibration profile, which makes ‘typical examples’ very narrow. If you work with cars, or airplanes, or sewing machines, or hydraulic pumps, or whatever, your vibration signals will look rather different from signals from those other products.

I have instead based most examples in this book on simplified simulations, where the key idea of discussion is easily seen. These examples will, hopefully, provide much deeper insights into the fundamental signal analysis ideas we discuss in each part of the book. They are also easily repeated on your own computer, which leads us to the next important point.

I believe that signal analysis (like, perhaps, all subjects) is far too mathematically complicated to understand through reading about it. Instead, I believe strongly in simulation, and application of the theories by your own hands. I have therefore throughout the book given numerous examples using the best tool I know of — MATLAB. This software is, in my opinion, the best available tool for signal analysis and therefore also for the vibration analysis methods we are concerned with in this book. If you do not already know MATLAB, you will soon learn by working through the examples.

The drawback of MATLAB may be that it is commercial software, and therefore costs money. If you find this to be an obstacle you cannot overcome, you can instead use GNU Octave, which is free software published under the GNU General Public License (GPL) and can be freely downloaded from http://www.gnu.org/software/octave/. Octave is to a large extent compatible with MATLAB in the sense that MATLAB code, with some minor tweaks, can run under Octave. I have made sure that all examples in this book run under both MATLAB and Octave, so you are free to choose whichever of the two software tools.

In addition to the examples in this book, there will be a free accompanying toolbox for MATLAB/Octave made available by me to aid your learning. There will also be more examples than could fit this book. More information about this toolbox and examples for instructors, etc., can be found at the book website at www.wiley.com/go/brandt.

2

Dynamic Signals and Systems

Vibration analysis, and indeed the field of mechanical dynamics in general, deals with dynamic events, i.e., for example forces and displacements which are functions of time. This chapter aims to introduce many of the concepts typical for dynamic systems, particularly for mechanical and civil engineering students who may have little theory at their disposal for understanding this subject. We will start with some rather simple signals, and later in this chapter introduce some important concepts and fundamental properties of dynamic signals and systems. This chapter also covers basic introductions to the Laplace and Fourier transforms — two very important mathematical tools to describe and understand dynamic signals and systems.