Scholars' Mine Doctoral Dissertations Student Theses and Dissertations Fall 2011 Hilbert Transform applications in signal analysis and non- parametric identification of linear and nonlinear systems Zuocai Wang Follow this and additional works at: https://scholarsmine.mst.edu/doctoral_dissertations Part of the Civil Engineering Commons Department: Civil, Architectural and Environmental Engineering Recommended Citation Wang, Zuocai, "Hilbert Transform applications in signal analysis and non-parametric identification of linear and nonlinear systems" (2011). Doctoral Dissertations. 2012. https://scholarsmine.mst.edu/doctoral_dissertations/2012 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. HILBERT TRANSFORM APPLICATIONS IN SIGNAL ANALYSIS AND NON- PARAMETRIC IDENTIFICATION OF LINEAR AND NONLINEAR SYSTEMS by ZUOCAI WANG A DISSERTATION Presented to the Faculty of the Graduate School of the MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY in CIVIL ENGINEERING 2011 Approved Genda Chen, Advisor William Schonberg Roger A. LaBoube Ian Prowell Xiaoping Du 2011 Zuocai Wang All Rights Reserved iii ABSTRACT Hilbert Huang Transform faces several challenges in dealing with closely-spaced frequency components, short-time and weak disturbances, and interrelationships between two time-varying modes of nonlinear vibration due to its mixed mode problem associated with empirical mode decomposition (EMD). To address these challenges, analytical mode decomposition (AMD) based on Hilbert Transform is proposed and developed for an adaptive data analysis of both stationary and non-stationary responses. With a suite of predetermined bisecting frequencies, AMD can analytically extract the individual components of a structural response between any two bisecting frequencies and function like an adaptive bandpass filter that can deal with frequency-modulated responses with significant frequency overlapping. It is simple in concept, rigorous in mathematics, and reliable in signal processing. In this dissertation, AMD is studied for various effects of bisecting frequency selection, response sampling rate, and noise. Its robustness, accuracy, efficiency, and adaptability in signal analysis and system identification of structures are compared with other time-frequency analysis techniques such as EMD and wavelet analysis. Numerical examples and experimental validations are extensively conducted for structures under impulsive, harmonic, and earthquake loads, respectively. They consistently demonstrate AMD’s superiority to other time-frequency analysis techniques. In addition, to identify time-varying structural properties with a narrow band excitation, a recursive Hilbert Huang Transform method is also developed. Its effectiveness and accuracy are illustrated by both numerical examples and shake table tests of a power station structure. iv ACKNOWLEDGMENTS The author would like to express his sincere gratitude to Dr. Genda Chen for his continuing support, constant encouragement and invaluable advice throughout this research work. He would also like to thank Drs. William Schonberg, Roger A. LaBoube, Ian Prowell, and Xiaoping Du for their serving as Ph.D. committee members and for their continuing interest and encouragement. Financial support to complete this study was provided in part by China Scholarship Council under Award No. NSCIS-2007-3020, by the U.S. National Science Foundation under Award No. CMMI0409420, and by Ameren Corporation. The results and opinions expressed in this dissertation are those of the author only and don’t necessarily represent those of the sponsors. The author appreciates the opportunity provided by the Department of Civil, Architectural, and Environmental Engineering, Missouri S&T, to conduct and complete his research. He would also like to acknowledge the general support and help of Dr. Chen’s graduate students and visitor scholars in the course of research activities. The author wishes to express his special and sincere gratitude to his wife, Qing Zhu, for her love, patience, understanding and assistance whenever needed. Last, but not the least, the author thanks his parents, parents-in-laws, sisters, brothers, and the previous academic advisor for their endless encouragement to pursue his degree. v TABLE OF CONTENTS Page ABSTRACT .................................................................................................................. iii ACKNOWLEDGMENTS ............................................................................................. iv LIST OF ILLUSTRATIONS ......................................................................................... ix LIST OF TABLES ........................................................................................................ xv SECTION 1. INTRODUCTION .................................................................................................1 1.1. STRUCTURAL HEALTH MONITORING AND DAMAGE DETECTION ..1 1.2. STRUCTURAL PARAMETER IDENTIFICATION ......................................2 1.2.1. Frequency Domain. ..............................................................................2 1.2.2. Time Domain........................................................................................3 1.2.3. Time-Frequency Analysis. ....................................................................4 1.3. OBJECTIVES OF THIS STUDY ...................................................................6 1.4. RESEARCH SIGNIFICANCE .......................................................................7 1.5. DISSERTATION ORGANIZATION .............................................................8 2. LITERATURE REVIEW ..................................................................................... 10 2.1. STRUCTURAL DYNAMICAL PARAMETERS ......................................... 10 2.1.1. Natural Frequency. ............................................................................. 10 2.1.2. Mode Shape, Mode Shape Curvature, and Modal Strain Energy. ........ 11 2.1.3. Damping. ............................................................................................ 12 2.1.4. Nonlinear Feature. .............................................................................. 12 2.2. PARAMETER IDENTIFICATION WITH CLOSELY-SPACED MODES .. 12 2.3. TIME-VARYING PARAMETER IDENTIFICATION ................................. 17 2.3.1. Least-Squares Based Method. ............................................................. 18 2.3.2. Wavelet Transform Based Method. ..................................................... 19 2.3.3. Hilbert Transform Based Method. ....................................................... 21 3. ANALYTICAL MODE DECOMPOSITION ....................................................... 24 3.1. HILBERT TRANSFORM AND ANALYTIC SIGNAL ............................... 24 3.2. HILBERT SPECTRAL ANALYSIS ............................................................. 25 3.2.1. Hilbert Spectrum. ............................................................................... 25 vi 3.2.2. Hilbert Spectra of Simple Functions.................................................... 26 3.3. EMD AND HHT .......................................................................................... 28 3.3.1. EMD. ................................................................................................. 28 3.3.2. HHT. .................................................................................................. 30 3.3.3. HHT Issues. ........................................................................................ 32 3.4. HILBERT VIBRATION DECOMPOSITION .............................................. 34 3.4.1. Instantaneous Frequency of the Largest Energy Component. .............. 34 3.4.2. Envelope of the Largest Energy Component. ...................................... 35 3.5. BEDROSIAN THEOREM ........................................................................... 37 3.5.1. Bedrosian Theorem Derivation. .......................................................... 37 3.5.2. Illustrative Examples. ......................................................................... 38 3.6. A NEW SIGNAL DECOMPOSITION THEOREM ...................................... 40 3.6.1. AMD Theorem and Proof. .................................................................. 40 3.6.2. Lowpass and Bandpass Filter Based on AMD Theorem. ..................... 42 3.6.3. Comparison with a Frequency Filtering Technique. ............................ 43 3.7. AMD FOR NONSTATIONARY SIGNALS................................................. 46 3.7.1. AMD Theorem for Non-stationary Signals. ........................................ 47 3.7.2. Role of Transform from Time Domain to Phase Domain. ................... 49 3.7.3. Adaptive Lowpass and Bandpass Filter for Signals with Time- Varying Frequencies. .......................................................................... 53 3.8. SUMMARY ................................................................................................. 55 4. PARAMETER IDENTIFICATION OF TIME INVARIANT SYSTEMS WITH AMD-HILBERT SPECTRAL ANALYSIS.......................................................... 56 4.1. BISECTING FREQUENCY SELECTION ..................................................