Hilbert Transform Applications in Mechanical Vibration E-Book

Michael Feldman

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

Seitenzahl: 455

Veröffentlichungsjahr: 2011

Beschreibung

Hilbert Transform Applications in Mechanical Vibration addresses recent advances in theory and applications of the Hilbert transform to vibration engineering, enabling laboratory dynamic tests to be performed more rapidly and accurately. The author integrates important pioneering developments in signal processing and mathematical models with typical properties of mechanical dynamic constructions such as resonance, nonlinear stiffness and damping. A comprehensive account of the main applications is provided, covering dynamic testing and the extraction of the modal parameters of nonlinear vibration systems, including the initial elastic and damping force characteristics. This unique merger of technical properties and digital signal processing allows the instant solution of a variety of engineering problems and the in-depth exploration of the physics of vibration by analysis, identification and simulation.

This book will appeal to both professionals and students working in mechanical, aerospace, and civil engineering, as well as naval architecture, biomechanics, robotics, and mechatronics.

Hilbert Transform Applications in Mechanical Vibration employs modern applications of the Hilbert transform time domain methods including:

• The Hilbert Vibration Decomposition method for adaptive separation of a multi-component non-stationary vibration signal into simple quasi-harmonic components; this method is characterized by high frequency resolution, which provides a comprehensive account of the case of amplitude and frequency modulated vibration analysis.
• The FREEVIB and FORCEVIB main applications, covering dynamic testing and extraction of the modal parameters of nonlinear vibration systems including the initial elastic and damping force characteristics under free and forced vibration regimes. Identification methods contribute to efficient and accurate testing of vibration systems, avoiding effort-consuming measurement and analysis.
• Precise identification of nonlinear and asymmetric systems considering high frequency harmonics on the base of the congruent envelope and congruent frequency.
• Accompanied by a website at www.wiley.com/go/feldman, housing MATLAB®/ SIMULINK codes.

Leseprobe

Contents

Cover Page

Title Page

Copyright

List of Figures

List of Tables

Preface

1: Introduction

1.1 Brief history of the Hilbert transform

1.2 Hilbert transform in vibration analysis

1.3 Organization of the book

Part I: Hilbert Transform and Analytic Signal

2: Analytic signal representation

2.1 Local versus global estimations

2.2 The Hilbert transform notation

2.3 Main properties of the Hilbert transform

2.4 The Hilbert transform of multiplication

2.5 Analytic signal representation

2.6 Polar notation

2.7 Angular position and speed

2.8 Signal waveform and envelope

2.9 Instantaneous phase

2.10 Instantaneous frequency

2.11 Envelope versus instantaneous frequency plot

2.12 Distribution functions of the instantaneous characteristics

2.13 Signal bandwidth

2.14 Instantaneous frequency distribution and negative values

2.15 Conclusions

3: Signal demodulation

3.1 Envelope and instantaneous frequency extraction

3.2 Hilbert transform and synchronous detection

3.3 Digital Hilbert transformers

3.4 Instantaneous characteristics distortions

3.5 Conclusions

Part II: Hilbert Transform and Vibration Signals

4: Typical examples and description of vibration data

4.1 Random signal

4.2 Decay vibration waveform

4.3 Slow linear sweeping frequency signal

4.4 Harmonic frequency modulation

4.5 Harmonic amplitude modulation

4.6 Product of two harmonics

4.7 Single harmonic with DC offset

4.8 Composition of two harmonics

4.9 Derivative and integral of the analytic signal

4.10 Signal level

4.11 Frequency contents

4.12 Narrowband and wideband signals

4.13 Conclusions

5: Actual signal contents

5.1 Monocomponent signal

5.2 Multicomponent signal

5.3 Types of multicomponent signal

5.4 Averaging envelope and instantaneous frequency

5.5 Smoothing and approximation of the instantaneous frequency

5.6 Congruent envelope

5.7 Congruent instantaneous frequency

5.8 Conclusions

6: Local and global vibration decompositions

6.1 Empirical mode decomposition

6.2 Analytical basics of the EMD

6.3 Global Hilbert Vibration Decomposition

6.4 Instantaneous frequency of the largest energy component

6.5 Envelope of the largest energy component

6.6 Subtraction of the synchronous largest component

6.7 Hilbert Vibration Decomposition scheme

6.8 Examples of Hilbert Vibration Decomposition

6.9 Comparison of the Hilbert transform decomposition methods

6.10 Common properties of the Hilbert transform decompositions

6.11 The differences between the Hilbert transform decompositions

6.12 Amplitude—frequency resolution of HT decompositions

6.13 Limiting number of valued oscillating components

6.14 Decompositions of typical nonstationary vibration signals

6.15 Main results and recommendations

6.16 Conclusions

7: Experience in the practice of signal analysis and industrial application

7.1 Structural health monitoring

7.2 Standing and traveling wave separation

7.3 Echo signal estimation

7.4 Synchronization description

7.5 Fatigue estimation

7.6 Multichannel vibration generation

7.7 Conclusions

Part III: Hilbert Transform and Vibration Systems

8: Vibration system characteristics

8.1 Kramers–Kronig relations

8.2 Detection of nonlinearities in frequency domain

8.3 Typical nonlinear elasticity characteristics

8.4 Phase plane representation of elastic nonlinearities in vibration systems

8.5 Complex plane representation

8.6 Approximate primary solution of a conservative nonlinear system

8.7 Hilbert transform and hysteretic damping

8.8 Nonlinear damping characteristics in a SDOF vibration system

8.9 Typical nonlinear damping in a vibration system

8.10 Velocity-dependent nonlinear damping

8.11 Velocity-independent damping

8.12 Combination of different damping elements

8.13 Conclusions

9: Identification of the primary solution

9.1 Theoretical bases of the Hilbert transform system identification

9.2 Free vibration modal characteristics

9.3 Forced vibration modal characteristics

9.4 Backbone (skeleton curve)

9.5 Damping curve

9.6 Frequency response

9.7 Force static characteristics

9.8 Conclusions

10: The FREEVIB and FORCEVIB methods

10.1 FREEVIB identification examples

10.2 FORCEVIB identification examples

10.3 System identification with biharmonic excitation

10.4 Identification of nonlinear time-varying system

10.5 Experimental Identification of nonlinear vibration system

10.6 Conclusions

11: Considering high-order superharmonics. Identification of asymmetric and MDOF systems

11.1 Description of the precise method scheme

11.2 Identification of the instantaneous modal parameters

11.3 Congruent modal parameters

11.4 Congruent nonlinear elastic and damping forces

11.5 Examples of precise free vibration identification

11.6 Forced vibration identification considering high-order superharmonics

11.7 Identification of asymmetric nonlinear system

11.8 Experimental identification of a crack

11.9 Identification of MDOF vibration system

11.10 Identification of weakly nonlinear coupled oscillators

11.11 Conclusions

12: Experience in the practice of system analysis and industrial application

12.1 Non-parametric identification of nonlinear mechanical vibration systems

12.2 Parametric identification of nonlinear mechanical vibrating systems

12.3 Structural health monitoring and damage detection

12.4 Conclusions

References

Index

This edition first published 2011

© 2011 John Wiley & Sons, Ltd.

Registered office

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

MATLAB¯ and Simulink¯ are trademarks of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion of MATLAB¯ or Simulink¯ softwares or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB¯ or Simulink¯ softwares.

Library of Congress Cataloging-in-Publication DataFeldman, Michael, 1951- Hilbert transform applications in mechanical vibration / Dr. Michael Feldman. p. cm. Includes bibliographical references and index. ISBN 978-0-470-97827-6 (hardback) 1. Vibration--Mathematical models. 2. Hilbert transform. I. Title. TA355.F35 2011 620.301′515723--dc22 2010051079

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

Print ISBN: 9780470978276 (H/B) E-PDF ISBN 9781119991649 O-book ISBN 9781119991656 E-Pub ISBN 9781119991526

List of Tables

Table 4.1 Typical examples of the central frequency and the spectral bandwidth of random vibration (Feldman, 2009b)

Table 6.1 Application of the HT decompositions for typical vibration signals

Table 10.1 Extreme and mean values of the envelope and the IF of the biharmonic signal

Table 10.2 Model parameters

Preface

The object of this book, Hilbert Transform Applications in Mechanical Vibration, is to present a modern methodology and examples of nonstationary vibration signal analysis and nonlinear mechanical system identification. Nowadays the Hilbert transform (HT) and the related concept of an analytic signal, in combination with other time--frequency methods, has been widely adopted for diverse applications of signal and system processing.

What makes the HT so unique and so attractive?

It solves a typical demodulation problem, giving the amplitude (envelope) and instantaneous frequency of a measured signal. The instantaneous amplitude and frequency functions are complementary characteristics that can be used to measure and detect local and global features of the signal -- in the same way as for classical spectral and statistical signatures.The HT allows us to decompose a nonstationary complicated vibration, separating it into elementary time-varying components -- preserving their shape, amplitude, and phase relations.It identifies and has an ability to capture -- in a much faster and more precise way -- the dynamic characteristics of system stiffness and damping, including their nonlinearities and the temporal evolution of modal parameters. This allows the development of more adequate mathematical models of tested vibration structures.

The information obtained can be further used in design and manufacturing to improve the dynamic behavior of the construction, to plan control actions, to instill situational awareness, and to enable health monitoring and preventive surplus maintenance procedures. Therefore, the HT is very useful for mechanical engineering applications where many types of nonlinear modeling and nonstationary parametric\break problems exist.

This book covers modern advances in the application of the Hilbert transform in vibration engineering, where researchers can now produce laboratory dynamic tests more quickly and accurately. It integrates important pioneering developments of signal processing and mathematical models with typical properties of mechanical construction, such as resonance, dynamic stiffness, and damping. The unique merger of technical properties and digital signal processing provides an instant solution to a variety of engineering problems, and an in-depth exploration of the physics of vibration by analysis, identification, and simulation. These modern methods of diagnostics and health monitoring permit a much faster development, improvement, and economical maintenance of mechanical and electromechanical equipment.

The Hilbert Vibration Decomposition (HVD), FREEVIB, FORCEVIB, and congruent envelope methods presented allow faster and simpler solutions for problems -- of a high-order and at earlier engineering levels -- than traditional textbook approaches. This book can inspire further development in the field of nonlinear vibration analysis with the use of the HT.

Naturally, it is focused only on applying the HT and the analytic signal methods to mechanical vibration analysis, where they have greatest use. This is a particular one-dimensional version of the application of HT, which provides a set of tools for understanding and working with a complex notation. HT methods are also widely used in other disciplines of applied mechanics, such as the HT spectroscopy that measures high-frequency emission spectra. However, the HT is also widely used in the bidimensional (2D) case that occurs in image analysis. For example, the HT wideband radar provides the bandwidth and dynamic range needed for high-resolution images. The 2D HT allows the calculation of analytic images with a better edge and envelope detection because it has a longer impulse response that helps to reduce the effects of noise.

HT theory and realizations are continually evolving, bringing new challenges and attractive options. The author has been working on applications of the HT to vibration analysis for more than 25 years, and this book represents the results and achievements of many years of research. During the last decade, interest in the topic of the HT has been progressively rising, as evidenced by the growing number of papers on this topic published in journals and conference proceedings. For that reason the author is convinced that the interest of potential readers will reach its peak in 2011, and that this is the right time to publish the book.

The author believes that this book will be of interest to professionals and students dealing not only with mechanical, aerospace, and civil engineering, but also with naval architecture, biomechanics, robotics, and mechatronics. For students of engineering at both undergraduate and graduate levels, it can serve as a useful study guide and a powerful learning aid in many courses such as signal processing, mechanical vibration, structural dynamics, and structural health monitoring. For instructors, it offers an easy and efficient approach to a curriculum development and teaching innovations.

The author would like to express his utmost gratitude to Prof. Yakov Ben-Haim (Technion), Prof. Simon Braun (Technion), and Prof. Keith Worden (University of Sheffield) for their long-standing interest and permanent support of the research developments included in this book.

The author has also greatly benefited from many stimulating discussions with his colleagues from the Mechanical Engineering Faculty (Technion): Prof. Izhak Bucher, Prof. David Elata, Prof. Oleg Gendelman, and Prof. Oded Gottlieb. These discussions provided the thrust for the author's work and induced him to continue research activities on the subject of Hilbert transforms.

The book summarizes and supplements the author's investigations that have been published in various scientific journals. It also reviews and extends the author's recent publications: Feldman, M. (2009) “Hilbert transform, envelope, instantaneous phase, and frequency”, in Encyclopedia of Structural Health Monitoring (chapter 25). John Wiley & Sons Ltd; and Feldman, M. (2011) “Hilbert transform in vibration analysis” (tutorial), Mechanical Systems and Signal Processing, 25 (3).

The author is very grateful to Donna Bossin and Irina Vatman who had such a difficult time reading, editing, and revising the text. Of course, any errors that remain are solely the responsibility of the author.

Michael Feldman

Part I

Hilbert Transform and Analytic Signal

2

Analytic signal representation