Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems 3658424796, 9783658424794

Table of contents :
Acknowledgements
Contents
List of Notations
Abbreviations
Mathematical Notations and Symbols
List of Figures
List of Tables
1 Introduction
1.1 Background and Motivation
1.1.1 Multivariate Signal Processing Based Fault Diagnosis
1.1.2 Nonlinear Frequency Analysis Model Based Fault Detection
1.2 Objectives of this Work
1.3 Outline of this Thesis
2 The Basics and Kinematics of Gears
2.1 Types of Gears and their Operation
2.2 Gear Mesh Stiffness and Dimensioning Geometry of Gears
2.2.1 Gear Mesh Stiffness and Contact Geometry
2.2.2 Dimensioning Geometry of Gears
2.2.3 Definition of Gear Failure Mode
2.2.4 Gear Features in Condition Monitoring
2.3 Wind Turbine Gearbox Kinematics and Fault Feature Frequencies
2.4 Signal Model Characteristics of Planetary Gearbox
2.4.1 Localized Fault Signal Model at Meshing Point
2.4.2 Transmission Path Influence on Vibration Signal Model
2.4.3 Observed Faulty Vibration Signal Model from Planetary Gearbox
2.4.4 Numerical Example and Model Validation
2.5 Concluding Remarks
3 Vibration Signal-Based Analysis for Gear Faults
3.1 Time Domain Analysis
3.2 Frequency Domain Analysis
3.3 Joint Time-Frequency Domain Analysis
3.3.1 Short Time Fourier Transform
3.3.2 Wavelet Analysis
3.3.3 Wigner-Ville Distribution
3.3.4 Minimum Entropy Deconvolution Technique
3.3.5 Empirical Mode Decomposition
3.4 Other Approaches
3.4.1 Order Tracking
3.4.2 Time Synchronous Average
3.4.3 Noise Cancellation Based Adaptive Signal Processing
3.5 Concluding Remarks
4 Frequency Domain Analysis for Nonlinear Systems
4.1 Nonlinear Systems Representation Based Polynomial Model
4.1.1 Introduction
4.1.2 Nonlinear System Modeling and Frequency Analysis Based Fault Diagnosis
4.2 Frequency Analysis Representation of Nonlinear Systems
4.2.1 The Functional Based Volterra Time Series Model Approach
4.2.2 Linear Versus Nonlinear Frequency Response Functions
4.2.3 The Output Frequency Response Based GFRFs
4.2.4 Methods for Computation of GFRFs
4.3 Characteristics of Output Frequencies for Nonlinear Systems
4.3.1 Generation of Frequencies in Linear and Nonlinear Systems Response
4.3.2 Output Frequencies Under Harmonics Inputs for Nonlinear Systems
4.3.3 Output Frequencies Under General Inputs for Nonlinear Systems
4.4 The Analysis of Nonlinear Systems in the Frequency Domain Based NOFRFs
4.4.1 Nonlinear Output Frequency Response Functions
4.4.2 Numerical Estimation of NOFRFs
4.5 Concluding Remarks
5 Development of Advanced Signal Processing Based Fault Diagnosis
[DELETE]
5.1 Problem Statement
5.2 Multivariate Signal Processing Based Feature Extraction
5.2.1 Theoretical Background of Multivariate Extension of EMD
5.2.2 Projections and Direction Vectors
5.2.3 Problems of Single and Multichannel Data Analytics with EMD
5.2.4 Improved Form of Multivariate EMD
5.2.5 Simulation Results
5.3 Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis
5.3.1 Filter Bank Property of MEMD
5.3.2 Selection of Significant Multivariate IMFs
5.3.3 Demodulation Analysis Aided Fault Feature Extraction
5.3.4 Framework of Multivariate Signal Processing Based Gear Fault Detection
5.3.5 Experimental Results
5.4 Concluding Remarks
6 Estimation of NOFRFs Based Parametric Characteristic Analysis
[DELETE]
6.1 Problem Formulation
6.2 Parametric Characteristic Analysis
6.2.1 Preliminaries
6.2.2 Separable Function
6.2.3 Coefficient Extractor
6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis
6.3.1 Computation of Parametric Characteristic of GFRFs with Respect to Model Nonlinear Parameters
6.3.2 Effects of Nonlinearity on the GFRFs From Different Types of Nonlinearity
6.3.3 Computation of NOS Based on the Parametric Characteristic Analysis
6.3.4 Computation of NOS Based on Specific Nonlinear Parameters
6.3.5 Numerical Example
6.4 Estimation of NOFRFs Using Parametric Characteristic Analysis for Damage Detection
6.4.1 A Proposed Method for Computation of NOFRFs
6.5 Concluding Remarks
7 Conclusions and Future Work
[DELETE]
A Bibliography

Citation preview

Hogir Rafiq

Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems

Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems

Hogir Rafiq

Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems

Hogir Rafiq Frankfurt am Main, Germany Von der Fakultät für Ingenieurwissenschaften, Abteilung Elektrotechnik und Informationstechnik der Universität Duisburg-Essen zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Dissertation von Hogir Rafiq aus Sulaimani, Irak. 1. Gutachter: Prof. Dr.-Ing. Steven X. Ding, Universität Duisburg-Essen, Deutschland 2. Gutachter: Prof. Dr. Zi-Qiang Lang, Universität Sheffield, Großbritannien Tag der mündlichen Prüfung: 15.05.2023

ISBN 978-3-658-42479-4 ISBN 978-3-658-42480-0 (eBook) https://doi.org/10.1007/978-3-658-42480-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

To my parents, my sisters and my brothers

Acknowledgements

This work was done during my Ph.D. study in the Institute for Automatic Control and Complex Systems (AKS) in the Faculty of Engineering at the University of Duisburg-Essen, Germany. I would like to express my most sincere gratitude to Prof. Dr.-Ing. Steven X. Ding, my valuable mentor and one of the pioneers in fault detection, for his guidance to my scientific research work. I am very grateful for all his support, encouragements and insightful discussions on this work during my Ph.D. study. I am also deeply indebt to Prof. Dr.-Ing. Zi-Qiang Lang, who came from the University of Sheffield-UK, for his interest in my work and also insightful discussions as well as inspiring comments on the manuscript. My sincere thanks also go to Prof. Dr. Daniel Erni, Prof. Dr.-Ing. Niels Benson, and Prof. Dr. Andreas Stöhr for their valuable comments. I would also like to thank Dr.-Ing. Birgit Köppen-Seliger, Dr.-Ing. Chris Louen, and a former AKS colleague Dr.-Ing. Minjia Krüger for their earnest advices and valuable supports in supervision of master’s student. A speical thank to the former AKS colleague Dr.-Ing. Changsheng Hua who offered enormous help and support and also insightful discussions during my study, I wish him all the best. I owe a great debt of gratitude of a former AKS colleagues Dr.-Ing. Yunsong Xu, Dr.-Ing. Qian Lu, Dr.-Ing. Ting Xue, Dr.-Ing. Fredrich Hesselmann Dr.-Ing. Yuhong Na, M.Sc. Abdul Latif for their valuable supports. I would like to extend my thanks also to M.Sc. Micha Obergfell, M.Sc. Deyu Zhang, M.Sc. Christopher Reimann, M.Sc. Caroline Zhu, M.Sc. Yannian Liu, M.Sc. Tieqiang Wang, M.Sc. Jiarui Zhang. A very special thank to Mrs. Sabine Bay, Dipl.-Ing. Klaus Göbel and Mr. Ulrich Janzen for their kind technical and administrative assistances.

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Acknowledgements

I would also like to appreciate the financial support for my Ph.D. research study provided by the German Academic Exchange Service (DAAD), Bonn, Germany. I would like to warmly thank to a National Renewable Energy Laboratory (NREL) for providing the Wind Turbine Gearbox Vibration Condition Monitoring Benchmark Datasets. Last but not least, I would like to dedicate this work to my family, especially my parents, my sisters and my brothers for their ever present support, even from far away. Duisburg May 2023

Hogir Rafiq

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Multivariate Signal Processing Based Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Nonlinear Frequency Analysis Model Based Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objectives of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Basics and Kinematics of Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Types of Gears and their Operation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Gear Mesh Stiffness and Dimensioning Geometry of Gears . . . . 2.2.1 Gear Mesh Stiffness and Contact Geometry . . . . . . . . . . . . 2.2.2 Dimensioning Geometry of Gears . . . . . . . . . . . . . . . . . . . . 2.2.3 Definition of Gear Failure Mode . . . . . . . . . . . . . . . . . . . . . 2.2.4 Gear Features in Condition Monitoring . . . . . . . . . . . . . . . . 2.3 Wind Turbine Gearbox Kinematics and Fault Feature Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Signal Model Characteristics of Planetary Gearbox . . . . . . . . . . . . 2.4.1 Localized Fault Signal Model at Meshing Point . . . . . . . . 2.4.2 Transmission Path Influence on Vibration Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Observed Faulty Vibration Signal Model from Planetary Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Numerical Example and Model Validation . . . . . . . . . . . . . 2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 10 10 13 13 15 16 17 18 18 21 26 27 30 32 36 39

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Contents

3 Vibration Signal-Based Analysis for Gear Faults . . . . . . . . . . . . . . . . . 3.1 Time Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Joint Time-Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . 3.3.1 Short Time Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Wigner-Ville Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Minimum Entropy Deconvolution Technique . . . . . . . . . . . 3.3.5 Empirical Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . 3.4 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Order Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Time Synchronous Average . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Noise Cancellation Based Adaptive Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 45 48 49 50 52 53 58 62 62 63

4 Frequency Domain Analysis for Nonlinear Systems . . . . . . . . . . . . . . . 4.1 Nonlinear Systems Representation Based Polynomial Model . . . 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Nonlinear System Modeling and Frequency Analysis Based Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . 4.2 Frequency Analysis Representation of Nonlinear Systems . . . . . . 4.2.1 The Functional Based Volterra Time Series Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Linear Versus Nonlinear Frequency Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Output Frequency Response Based GFRFs . . . . . . . . 4.2.4 Methods for Computation of GFRFs . . . . . . . . . . . . . . . . . . 4.3 Characteristics of Output Frequencies for Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Generation of Frequencies in Linear and Nonlinear Systems Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Output Frequencies Under Harmonics Inputs for Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Output Frequencies Under General Inputs for Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 71

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72 73 73 76 80 81 89 90 91 94

Contents

4.4 The Analysis of Nonlinear Systems in the Frequency Domain Based NOFRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Nonlinear Output Frequency Response Functions . . . . . . . 4.4.2 Numerical Estimation of NOFRFs . . . . . . . . . . . . . . . . . . . . 4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Development of Advanced Signal Processing Based Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Multivariate Signal Processing Based Feature Extraction . . . . . . . 5.2.1 Theoretical Background of Multivariate Extension of EMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Projections and Direction Vectors . . . . . . . . . . . . . . . . . . . . 5.2.3 Problems of Single and Multichannel Data Analytics with EMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Improved Form of Multivariate EMD . . . . . . . . . . . . . . . . . 5.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Filter Bank Property of MEMD . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Selection of Significant Multivariate IMFs . . . . . . . . . . . . . 5.3.3 Demodulation Analysis Aided Fault Feature Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Framework of Multivariate Signal Processing Based Gear Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Estimation of NOFRFs Based Parametric Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Parametric Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Separable Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Coefficient Extractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Computation of Parametric Characteristic of GFRFs with Respect to Model Nonlinear Parameters . . . . . . . . . .

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95 95 97 98 101 101 102 103 106 106 109 110 114 115 115 117 120 121 131 133 133 135 136 136 138 145 145

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Contents

6.3.2 Effects of Nonlinearity on the GFRFs From Different Types of Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Computation of NOS Based on the Parametric Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Computation of NOS Based on Specific Nonlinear Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Estimation of NOFRFs Using Parametric Characteristic Analysis for Damage Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 A Proposed Method for Computation of NOFRFs . . . . . . 6.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150 150 157 162 167 167 171

7 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

List of Notations

Abbreviations AM CE CM DESA EEMD EMD FDD FFT FM FRF GFRF GMF HSS HT IMF ISS LS LSS MED MEMD NA-MEMD NARX NDE NOFRF

amplitude modulation coefficient extractor condition monitoring discrete energy separation algorithms ensemble empirical mode decomposition empirical mode decomposition fault detection and diagnosis fast Fourier transform frequency modulation frequency response function generalized frequency response function gear meshing frequency high speed shaft Hilbert transform intrinsic mode functions intermediate speed shaft least square low speed shaft minimum entropy deconvolution multivariate empirical mode decomposition noise assisted multivariate empirical mode decomposition nonlinear auto-regressive with eXogenous input nonlinear differential equation nonlinear output frequency response function

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NOS NREL PCA PS SNR STFT TKEO WPT WTB WT

List of Notations

nonlinear output spectrum national renewable energy laboratory parametric characteristic analysis planetary stage signal-to-noise ratio short-time Fourier transform Teager Kaiser energy operator wavelet packet transform wind turbine wavelet transform

Mathematical Notations and Symbols ∀ ∈ ⊆ =⇒ ⇐⇒ ≈ = ∝ lim max(min)   ∅ fc fG f hss f iss fm fp fr f sun f sunr ot c p,q (k1 , . . . , k p+q ) Hn ( jω1 , . . . , jωn )

for all belong to subset imply equivalent to approximately equal not equal proportional to limit of a function maximum (minimum) intersection of two sets union of two sets empty set carrier speed frequency gear shaft frequency high speed gear pinion intermediate gear pinion frequency gear mesh frequency planet gear tooth defect frequency ring gear tooth defect frequency sun gear tooth defect frequency sun shaft rotational frequency nonlinear model parameter n th order GFRFs

List of Notations

C E(·) C E(Hn ( jω1 , . . . , jωn )) f n ( jω1 , . . . , jωn ) Y ( jω)   C E Y ( jω) G n ( jω) Zc Z hi Z ho Z mi Z mo Zs Zp ⊗ ⊕ ω R C Z Z+

Rn Rn×m

(·) χ χT χ −1 det(χ) 

xv

coefficient extraction operator parametric characteristics of the nth-order GFRF correlative function of C E(Hn ( jω1 , . . . , jωn ) nonlinear output spectrum parametric characteristics of the Y ( jω) n th order nonlinear output frequency response function number of ring gear teeth number of gear teeth in high speed shaft number of pinion gear teeth in high speed stage number of drive gear teeth in intermediate speed shaft number of pinion gear teeth in intermediate speed shaft number of sun gear teeth number of planet gear teeth reduced Kronecker product reduced vectorized summation frequency (in Hz) the set of all real numbers the set of complex numbers the set of all integer numbers set of all positive integers greater than zero space of n-dimensional vectors space of n by m matrices for all cases , this means that ( jω1 , . . . , jωn ) a matrix transpose of χ inverse of χ determinant of χ summation

List of Figures

Figure 1.1 Figure 1.2 Figure 2.1 Figure 2.2

Figure 2.3 Figure 2.4

Figure 2.5

Figure 2.6 Figure 2.7

Figure 2.8 Figure 2.9

Scheme of model-based fault diagnosis [47] . . . . . . . . . . . . Organization of the thesis chapters (dashed lines shows the suggested order of reading) . . . . . . . . . . . . . . . . . Gear types (a) Spur gear, (b) Helical gear, (c) Bevel gear, (d) Worm gear [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the mesh behaviour of a spur gear pair (1 < C R < 2): (a) mesh process, (b) gear mesh stiffness curve [34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helical gear teeth faults: (a) Scuffing, (b) Pitting [6] . . . . . Graphical representation of typical vibration spectra: (a) Healthy gear (low sidebands), (b) Faulty gear (with higher sidebands) [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . Wind turbine system:(a) WTB drive train, (b) planetary gear stage, (c) three stages WTB gearbox [257] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal structure of the WTB gearbox [228] . . . . . . . . . . . . A planetary gear stage with simultaneous mesh design: (a) with normal operation, (b) with an imaginary rotation [154] . . . . . . . . . . . . . . . . . . . Three different possible transmission paths to sensor in planetary gearbox [62] . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated faulty planet gear signal without the effect of transmission path (a) time-domain, (b) Frequency spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 12 14

16 18

20

22 22

24 31

38

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List of Figures

Figure 2.10

Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4

Figure 3.5 Figure 3.6

Figure Figure Figure Figure

3.7 3.8 3.9 3.10

Figure Figure Figure Figure

4.1 5.1 5.2 5.3

Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9

Simulated faulty planet gear signal with the effect of transmission path: (a) time-domain, (b) Frequency spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear vibration signal spectrum for Healthy and Faulty gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear vibration based signal processing using envelope analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AR-MED filtering process [55] . . . . . . . . . . . . . . . . . . . . . . . MED processing results (a) faulty gear signal, (b) its power spectrum, (c) gear signal after MED filtering, (d) power spectrum of MED filtered signal . . . . . The vibration signal and its five decomposed IMFs using EMD method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal reconstruction by combining IMFs 1–3 (a) 1500 sample points , (b) selected 300 sample points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time synchronous averaging process . . . . . . . . . . . . . . . . . . Adaptive noise canceller . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-adaptive noise canceller . . . . . . . . . . . . . . . . . . . . . . . . . Denoising of simulated vibration signal using adaptive filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System frequency response (a) linear, (b) nonlinear . . . . . Low discrepancy point sets for projection [229] . . . . . . . . . Example of mode mixing in EMD approach . . . . . . . . . . . . Decomposition results of multivariate signal using standard EMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition results of multivariate signal using the original MEMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition results of multivariate signal using imptoved MEMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Framework of development of advanced signal processing based multi-stage gear fault diagnosis . . . . . . . . A schematic diagram of NREL Dynamometer Test Rig and the sensor location [206] . . . . . . . . . . . . . . . . . . . . . Decomposition results of multivariate signal using MEMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IMFs alignment using MEMD as a dyadic filter bank structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 46 48 56

57 61

61 65 67 67 69 92 105 107 111 112 113 121 122 124 125

List of Figures

Figure 5.10 Figure 5.11 Figure 5.12

Figure 5.13

Figure 5.14

xix

Decomposition results of multivariate signal using improved MEMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IMFs alignment using improved MEMD as a dyadic filter bank structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Healthy signal’s IMF 2, 3, 4 of AN4-Low speed sensor (column 1), instantaneous frequency (column 2), and amplitude envelope (column 3) . . . . . . . . . Faulty signal’s IMF 2, 3, 4 of AN4-Low speed sensor (column 1), instantaneous frequency (column 2), and amplitude envelope (column 3) . . . . . . . . . . . . . . . . . . . . Comparison between EMD, MEMD and proposed approach for IMF3 (AN7-HSS sensor)—faulty case . . . . . .

126 127

129

129 130

List of Tables

Table 1.1 Table 2.1 Table 3.1 Table 3.2 Table Table Table Table

4.1 5.1 5.2 5.3

Comparison between signal and model based fault diagnosis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters for the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical features definition of a vibration signal . . . . . . . . . . Correlation coefficient for each IMF to select the best IMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinearity effects in the frequency domain . . . . . . . . . . . . . The teeth number of multiple gears in WTB gearbox . . . . . . Characteristic frequencies and the GMFs . . . . . . . . . . . . . . . . . Correlation Fault Factor (CFF) of each IMF for improved MEMD approach . . . . . . . . . . . . . . . . . . . . . . . . .

9 37 44 60 91 122 123 128

xxi

1

Introduction

1.1

Background and Motivation

Condition Monitoring (CM) and Fault detection and diagnosis (FDD) help to ensure the reliability and low-cost operation of industrial facilities in engineering systems. CM can provide early detection of machine faults so that appropriate actions can be taken before that fault causes breakdown and, possibly, a catastrophic failure [162]. The success of CM is due in part to: • Minimal operational and environmental variability associated with this type of monitoring; • Well-defined damage types that occur at known locations; • Large databases that include data from damaged systems; • Well-established correlation between damage and features extracted from the measured data; Rotating machinery is an important mechanical component of industrial infrastructure [52]. Major rotating machinery applications include aircraft engines, automotive equipment, and wind turbines (WTB). The key common component of all the aforementioned machinery is the gearbox. Due to its continual nature of operation, an efficient and fault-free performance is a major requirement. Faults, especially if they are un-anticipated, can be costly and cause significant financial losses. Furthermore, due to relatively tough operating conditions, rotating machinery components are prone to early damage, leading to reduced service life of the operating unit or shut down in severe conditions. Gearboxes, which are used to step-up or step-down the rotating speeds of the shafts and to transfer the power, are considered as one of the most important parts of © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_1

1

2

1

Introduction

rotational machines. They are complex and inseparable rotating parts of machines. In addition, gear transmission systems are widely used in a variety of industry applications, such as automotive, the WTB, mining, marine and industrial power transmission. Hence, the reliability of the whole system depends on smooth operation of the gearboxes. Development of faults in the gearboxes should be monitored efficiently and at early stages of occurrences in order to avoid costly failures. Therefore, the gearbox is a critical component in any mechanical process involving rotating machines such as WTB; it should be monitored effectively and efficiently in order to prevent any damage in the whole system [142][134][211][53]. For example, in a WTB system, typically the gearbox is a complicated rotating part and responsible for around 15–20% of its maintenance and downtime costs [75]. The average preventive maintenance and repair cost of a gearbox in a WTB costs around $50,000 in every 7–8 years of its operational life [167]. The cost of unexpected gearbox failure is even higher. As a specific example, according to the World Wind Energy Association, a gearbox of the WTB developed a severe fault in one of the wind farms in Canada in 2004 because of no condition based maintenance program in place. The repair and replacement cost was around $426000 [14]. This catastrophe could have been avoided with proper predictive and condition monitoring systems in place. On the other hand, similar to CM of rotational machines, structural health monitoring (SHM) is also critical. It is analogues to CM, but instead of identifying fault in rotating and reciprocating machinery, addresses the damage identification in a stationary part of mechanical and civil engineering structures such as bridge, buildings, beams, etc. Additionally, it is an engineering service to guarantee the performance of these systems so that they can detect the damage at the earliest possible time and predict the remaining useful life of the engineering system to prevent failures and optimize resource allocation. Thus, the potential economic and life-safety implications of early damage detection in aerospace, civil and mechanical engineering systems have motivated a significant amount of research in SHM [59][72]. This damage identification problem is fundamentally different from that of rotational machinery because the damage location is not known a priori and the majority of the structure is not readily accessible for a measurement. Therefore, to attain safe and reliable operation with high equipment performance and meanwhile the lowest possible maintenance costs, it is crucial to timely detect the faults or anomalies of engineering systems because if a crack propagates continuously and is not detected, abrupt failure, may occur and finally lead to a catastrophic failure with enormous costs in down time, consequential damage to equipment and potential injury to personnel. Consequently, fault detection and diagnosis of both rotational and stationary part of machines has been increasingly popular and drawn

1.1 Background and Motivation

3

an attention by researchers in this field during the past decades. The aim of this chapter is to review some of the significant developments in this area. The objectives and thesis outline are also presented.

1.1.1

Multivariate Signal Processing Based Fault Diagnosis

Signal processing methods are a key technique of the condition based maintenance (CBM) of rotational machines especially gearboxes and rolling bearings. The signal acquired from accelerometer are analysed through different signal processing techniques. Moreover, gearbox vibration signals are inherently complex due to contribution from a large number of moving parts and huge background noise. Thus, a wide range of signal processing techniques have been developed in the last two decades. In this section, a concrete review in this scope is addressed and the main reason of using this technique in fault diagnosis is studied. In the classical approach for CBM, vibration signals are considered in their raw form and signal processing techniques are directly applied on them with the application of signal enhancement techniques [93]. These techniques utilize methods for power spectrum estimation, fast Fourier transform (FFT), cepstrum analysis, and envelope spectrum analysis, etc., and have been found to be effective in gear fault detection. However, these methods are based on the assumption of stationarity and linearity of the vibration signal and hence are limited in their application. Gear fault signatures are time-localized transient events and hence non-stationary by nature. At an early stage, faults manifest themselves as impulsive events and for an early diagnosis, it is necessary to utilize methods aimed in tracking frequency contents of the signal [260]. Dealing with non-stationary and nonlinear signals requires the use of time-frequency analysis techniques such as the short-time Fourier transform (STFT) [40], the wavelet transform (WT) [217][242] or wigner-ville distribution (WVD) [120]. the continuous wavelet transform [40][41], has been successfully used in non-stationary vibration signal processing and fault detection. The filter bank implementation of the WT, namely the discrete wavelet transform and wavelet packet transform(WPT), have also been successfully applied [41][163]. Signal decomposition methods like empirical mode decomposition (EMD) employed on single channel measurements have also found applications in gear-fault diagnosis [135][138]. The main advantage of such techniques is the ease and simplicity of the approach. EMD in particular is a powerful tool that is applicable to a majority of signal types encountered in practice. The main drawbacks of EMD include lack of mathematical structure, inability to separate closely spaced modes even with the use of linear filters. Furthermore, EMD approach suffers from different more issues including

4

1

Introduction

mode mixing and poor noise performance especially when applied to multivariate signals. Therefore, the above signal based approaches are only suitable to analyse univariate signal and thus they are not effective with multivariate or multi-sensor signal analysis. In order to process the multivariate signals acquired from different sensors at different locations of machines for the purpose of extracting enough fault information and avoid the loss of information, the multivariate signal processing can be used to analyse the multichannel signals in order to achieve useful information to extract the fault features. The main purpose of using multivariate signal analysis is to detect faults in differents locatoins of the machine at the same time. The use of multiple sensors at different locations to acquire vibration data are commonly used in WTB gearboxes. This is because WTB gearboxes are regarded as one of the complicated systems consisting of multi-stage systems. Therefore, gear failure will cause the vibration of other components, and therefore the collected gearbox vibration signals are mixed with vibration signal of other parts such as bearings and shafts, etc. and noise. To address these problems, the earliest multivariate EMD approaches [223][8][190][233] have been limited to both bivariate and trivariate signals. These techniques provide in the discovery of the projection method, which involves integrating multiple data channels to extract similar scales across them. The data channels are projected onto several direction vectors. In 2010, Rehman et al. [188] proposed a general method for n-variate EMD (multivariate EMD). After that, multivariate EMD (MEMD) technique has been considerably utilized for analysing multivariate signals in various field such biology, etc. For instance, by using standard single and multiple channel EEG signals, EMD-based emotional recognition methods were developed [176][195][168]. Furthermore, due to the idea of multiple channel approach, the researchers whose works are cited in performed emotional state recognition using MEMD [168][230][160]. However, MEMD itself suffers from some problems including mode mixing problems between different channels. Motivated to this problem, an improved MEMD is proposed in this work to reduce the effect of mode mixing and applied in WTB gearbox fault diagnosis to enhance extracting a useful fault feature information. Subsequently, using the multivariate signal processing such as improved MEMD to simultaneously analyse the multivariate signal is beneficial to extract fault information, especially for weak fault characteristics during the period of early failure and thereby detection faults in multi-stage of gear simultaneously. As a result, signal based techniques are powerful in diagnostics of faults in rotational machines under the non-stationary and linearity assumption. However, they are basically model-free, which extract only the characteristics of signal but not mirror the system characteristics. This is due to the ignorance of the input signal by anlyzing only the measurement output. Besides,

1.1 Background and Motivation

5

several techniques presently available require a good consideration of human intervention and expertise to successfully apply them in practise [122]. In this regard, alternative approaches should be developed for gear fault detection and diagnosis.

1.1.2

Nonlinear Frequency Analysis Model Based Fault Detection

In signal based fault diagnosis, only measurement signals are analysed to extract the fault information and thus the effects of input and/or the system behaviour are not considered. However, the system dynamics, which can be noticed by taking the effects of input/output or system model, has a vital role in detection of faults both in the time and frequency domain for different application purposes. Thus, the model based fault diagnosis provides an effective and powerful fault diagnosis compared to signal based methods. The model based fault diagnosis can be understood as a twostep procedure: residual generation model-based and nonlinear frequency domain model based fault detection as described below. Model-based residual generation Model-based fault detection (FD) approaches, in contrast to signal-based methods, establish a general input-output or state space model based on the mathematical and physical knowledge of the process, and use the measurement data for identifying the model. Based on this model identification, well-established model-based techniques can be established to design a powerful fault diagnosis system. These model-based fault detection methods are often identified as an analytical or software redundancy based approaches. The idea of software redundancy schemes is building a software redundant model which runs parallel to the process; fault detection thus can be obtained with the difference between the measured outputs of the process and the model outputs (i.e. their software redundancy), this difference is called residual signal. The procedure of estimating the process outputs and generating the difference between the estimated and actual process outputs is commonly known as residual generation [47]. A fundamental residual generation based fault diagnosis system as shown in Fig.1.1 includes a residual generator, an evaluation function and a detection logic. Currently, model-based fault diagnostic approaches have been developed into a well-established theoretical framework and several approaches have been established after a rapid and dynamic development [47][92][33]. Based on the survey paper of Frank [69], the model based FD approaches have been classified in to three different categories: parameter identification based methods, parity space methods

6

1

Introduction

and observer-based methods. Failure detection filter (FDF), which was proposed by Beard [18] and Jones [104] as the first model-based fault detection system, has been greatly influenced by the developed observer theory in the early 1970s. Since then, the residual generation in dynamic systems has been mostly estimated by using observers and Kalman filtering methods [16][70]. Chow and Willsky [37] were the first to propose the parity space approach, whose main idea is the parity relation rather than an observer. The parity relationship between the model and the available process input and output measurements is checked in order to achieve fault detection. It is worth pointing out that the interconnections between the parity space and observer based residual generators, which are very useful in industrial applications, were established in the early 90s [47]. Furthermore, the parameter identificationbased approaches have been studied for detecting unmeasurable parameter drift [91], where fault decision was carried out using an online parameter estimation. Many effective applications in industrial processes and automatic control systems have proven the performance of model-based fault diagnostic systems [77][170][207]. Therefore, it is considered as one of the powerful solutions to the shortcomings of signal based fault diagnosis in linear systems. However, for nonlinear systems, the model based method is very complicated especially when disturbance and noise exist. Thus, to deal with the dynamics of system behaviour due to nonlinearity effects and analyse nonlinear systems in the frequency domain for the purpose of damage detection, under steady state condition or time invariant system, an alternative possible solution which is model based fault detection in the context of nonlinear frequency analysis in the next section will be proposed.

Output

Input Process

-

Residual

Process Model

Residual generation

Residual Evaluation

Decision logic

Residual evaluation

Model-based fault diagnosis system

Figure 1.1 Scheme of model-based fault diagnosis [47]

Knowledge of faults

1.1 Background and Motivation

7

Nonlinear frequency analysis model based fault detection Unlike the residual generation model based fault diagnosis, this concept of fault diagnosis is based on the system analysis in the frequency domain with respect to the nonlinearity influence. In this approach, the effects of nonlinearity in the system including nonlinear model parameters can be analysed in the frequency domain under the assumption that the nonlinear system is time invariant with no uncertainty in this study. Nonlinear analysis is a popular topic in the literature because it plays a significant role in system analysis in practice, from engineering issues to biological systems. For this reason, a variety of techniques are available, including harmonic balance, the perturbation approach, the averaging method, and describing functions [107][159][78][202][164][171][101][96][101][98][97][189][250][48][194]. It is also possible to conduct nonlinear analysis in the frequency domain. In practice, in order to deal with the analysis and design of linear systems, several approaches and techniques were well established including the Bode diagram, the root locus, and the Nyquist plot [165]. Frequency domain techniques have thus been widely used in engineering practice because they frequently offer more intuitive insights into the linear dynamics of a system or its dynamic properties. For instance, no matter how the system model is transformed by any linear transformations, the transfer function of a linear system is always equivalent description and a coordinate-free; for a linear system, the instability is typically related to at least one pole in the right-half-plane of the system transfer function; the peak of the frequency response of the system output often occurs around the system natural resonance frequency, and so on. Hence, frequency domain analysis of engineering systems is frequently one of the most popular approaches in practice and is the subject of in-depth theoretical and practical research. However, nonlinear systems analysis in the frequency domain is not straightforward and thus it is challenging. The dynamic response and output frequency characteristics of nonlinear systems, which can transfer system energy between frequencies to produce outputs at some frequency components that may be very different from the input frequency components, are typically very complicated . Examples of these output frequency characteristics generated in the frequency response due to nonlinearity include harmonics, inter-modulation, chaos, and bifurcation. For this reason, the frequency domain methods for linear systems cannot be simply applied to nonlinear system applications because these phenomena make it difficult to analyse nonlinear systems in the frequency domain. Since the 1950s of the last century, researchers have studied the frequency domain analysis of nonlinear systems. A traditional approach was initiated by studying of the stationary point global stability within the context of absolute stability theory, followed by the development of frequency domain techniques for the investiga-

8

1

Introduction

tion of stationary set stability, the existence of cycles and homoclinical orbits, the dimension attractors estimation, etc. [128]. In practice, describing functions or harmonic balance approaches in the frequency domain are typically used to analyze the nonlinear characteristics or behavior of a specific nonlinear unit or nonlinear part in a system. The describing function approach is a very powerful mathematical method for the design and analysis of the nonlinear systems behaviour with a single nonlinear component [11]. It can be successfully utilized to analyse an oscillation and limit cycle of nonlinear systems in which the nonlinearity is independent to the frequency and generates no sub-harmonics and other frequency components. Several publications have been reported in the applications to controller designs based on descriptive function analysis [236][226]. The descripting function techniques, however, have some drawbacks. For instance, the erroneous of the limit cycle prediction for a set of nonlinear systems using describing functions [56]. Additionally, several improved techniques were proposed [198][54][164]. The harmonic balance is another elegant technique for nonlinear systems analysis in the frequency domain(see examples in [210][105]). Under the assumption that a Fourier series can describe the steady state solution, this method approximates the amplitude of the steady state periodic response of a nonlinear system. For both the time domain and frequency domain responses, it can handle more widespread nonlinear system issues including subharmonics and jump behavior, etc. There are other results for nonlinear system analysis in the frequency domain described in literature in addition to these well-known and obvious methods. For instance, the study of frequency domain techniques for the synthesis and analysis of time-varying systems or uncertain systems was researched based on the frequency domain techniques for linear systems, such as Bode diagrams, singular value decomposition, and the principle of varying eigenvalues or varying natural frequencies [166][79][204][139]; and Pavlov in have presented a frequency response function for convergent systems excited by harmonic inputs [171]. Frequency domain analysis can also be carried out for a class of nonlinear systems that have a convergent Volterra series expansion using the idea of a generalized frequency response function [76][201][194]. A Volterra series of a sufficiently high order can be used to represent nonlinear systems that are causal, time invariant, and have fading memories, according to [28]. The results in [197][196] illustrate that even nonlinear system with time varying, under certain conditions, have such a locally convergent Volterra series expansion. Therefore, a considerably large class of nonlinear systems from this type of frequency domain analysis approaches can be investigated since these approaches can be subject to any input signals and also do not restrict to consider a specific nonlinear component, and thus they are more general approaches. Although the study of frequency domain methods based Volterra systems has been carried out for many decades since the middle of last century, sev-

1.1 Background and Motivation

9

eral problems related to the theoretical and application issues still remain unsolved. For this reason, in this thesis, this approach is utilized in fault detection of structure beams in a context of model based fault detection in the nonlinear frequency analysis as an extension for the signal based FD. The main comparison among different fault diagnosis approaches can be summarized in the Table 1.1.

Table 1.1 Comparison between signal and model based fault diagnosis methods FAULT DIAGNOSIS METHODS

KEY APPROACHES

Signal processing based method

• directly analyse the • no need system measured data to model; extract the fault • easy to analyse features. quantitatively; • effective to extract faults in non-stationary signal.

Model based method-residual generation

• use different approaches such as state and parameter estimation, residual generation to identify the changes in the state or model parameters. • directly map non-linear model parameters to system output frequency response to detect faults.

Model based method-nonlinear frequency analysis

ADVANTAGES

• concrete mathematical modelling; • easy to implement,analysis,and fault isolation; • powerful in real time monitoring • mathematically elegant; • Suitable for large class of nonlinear systems including strong nonlinear behaviours; • provides GFRFs which are very similar to transfer functions of linear systems; • holds for any bounded input signals.

DISADVANTAGES

• difficult for detecting unforeseen and small faults; • ignore the effect of inputs and less analytical computations; • only powerful in off-line monitoring. • affected by model uncertainty (noise and disturbance); • requires a good model and sufficient input excitation.

• need a high precision time domain model; • powerful in off-line monitoring; • only effective in time-invariant systems.

10

1.2

1

Introduction

Objectives of this Work

Motivated by the aforementioned features and state of the art of both signalprocessing based and nonlinear frequency model-based fault diagnosis techniques, the main objective of this thesis is to develop the advanced signal based fault diagnosis for gearbox system in one hand, and to compute a nonlinear frequency response functions with respect to the nonlinear model parameters of interest and applied to fault detection of beam structure systems. More specifically, the thesis goals are as follows: • Study the gearbox fault detection and diagnosis issues by applying advanced signal- processing based techniques. • Develop a signal-based fault detection scheme to enhance the extraction of fault features and its energy level in WTB multi-stage gearboxes. • Investigate the nonlinearity effects in the frequency domain by considering the input-output system characteristics. • Enhance and propose a nonlinear output spectrum (NOS) and nonlinear output frequency response functions (NOFRFs) scheme based parametric characteristic analysis (PCA) with respect to the nonlinear model parameters of interest, which can be used in fault detection of beam structure systems.

1.3

Outline of this Thesis

This thesis consists of seven chapters, which are structured as demonstrated in Fig. 1.2. The major objectives and contributions of each chapter are briefly summarized as follows. Chapter 1: Introduction This chapter presents the background, motivations, objectives, contributions and the organization of this thesis. Chapter 2: The basics and kinematics of gears This chapter presents the basics and kinematics of gears, where the overview on types, geometry, and components of gears are first given for a rudimentary understanding. Then, kinematics of gears, gear features in CM and related fault characteristic frequency including gear meshing frequency (GMF) which is essential frequency in gear faults are discussed, which reveal some defect laws of gears. In addition, the structure of the WTB gearboxes which are focused in this thesis are

1.3 Outline of this Thesis

11

studied. To achieve a deeper insight into the mechanism of faults and the generated signals caused by faulty gears, the faulty signal modelling of planetary gearbox is presented. In the end, the validity of the signal model is verified using numerical examples. Chapter 3: Vibration signal-based analysis for gear faults Various signal processing techniques in this chapter as the most common ways in dealing with gearbox fault detection issues, are reviewed based on the time domain analysis, frequency domain analysis, joint time-frequency domain analysis and some other approaches. Many of them demonstrate a good result in obtaining a reduction of noise and highlighting the interested features of the signal. However, it is worth mentioning that almost all of these techniques extract only the characteristics of the signal but rarely mirror the system characteristics, which is the main shortcoming of signal processing approaches. Chapter 4: Frequency domain analysis for nonlinear systems In this chapter, as an extension to the signal-based fault diagnosis, a model based fault diagnosis in the context of nonlinear frequency analysis by considering inputoutput system analysis is introduced. A nonlinear representation based on the polynomial model which is called Volterra model representation is discussed. Then, a generalized frequency response functions (GFRFs) as an extension of the frequency response function (FRF) or transfer function for a linear system is studied. Several methods for computation of GFRFs including analytical methods are explained in detail. Moreover, the nonlinearity influence on the system characteristics in frequency domain and how these nonlinearity effects generate new frequencies in the output frequency responses are investigated. Finally, the NOFRFs which provide a concept to extend the well-known concept FRF of linear systems to the nonlinear case, are investigated and numerically estimated. Chapter 5: Development of advanced signal processing based fault diagnosis In this chapter, a multivariate signal processing is devoted to study as a development of signal based fault diagnosis based on Chapter 2 and 3 for the purpose of extracting fault features in the multi-channel signals by addressing the mode mixing problems between channels. A multivariate signal processing techniques called MEMD is applied to multivariate signal to analyse and extract fault features in different locations of machines simultaneously. A problem of mode mixing in the MEMD technique is addressed. Then, an improved MEMD as a possible solution to reduce the mode mixing issue is proposed. In order to enhance the performance of extracting fault features, a combined improved multivariate signal processing is

12

1

Introduction

CHAPTER 1 Introduction CHAPTER 2 The basics & kinematics of gears CHAPTER 3 Vibration signal-based analysis for gear faults

CHAPTER 4 Frequency domain analysis for nonlinear systems

CHAPTER 5 Development of advanced signal processing based fault diagnosis

CHAPTER 6 Estimation of NOFRFs based parametric characteristic analysis

CONTRIBUTION 1

CONTRIBUTION 2

CHAPTER 7 Conclusions & future work

Figure 1.2 Organization of the thesis chapters (dashed lines shows the suggested order of reading)

implemented. Finally, a fault detection of WTB multi-stage gearbox is utilized as a case study to verify the effectiveness of the proposed method. Chapter 6: Estimation of NOFRFs based parametric characteristic analysis In this chapter, the computation of NOFRFs based parametric characteristic analysis is proposed. A powerful mathematical operator, which is called coefficient extractor (CE) operator, is applied with the help of separable function definition to extract the nonlinear parameters of interest and analyse their effects in the frequency domain. This NOFRF computation based PCA approach consists of two steps. In the first step, the nonlinear output spectrum is analytically computed based on the parametric characteristics of the GFRFs which considerably reduce the computational cost compared to the available result. In the second step, the NOFRFs based parametric characteristics, under certain condition, are computed in terms of only specific nonlinear parameters of interest which have most influence on the system behaviour in the frequency domain. The result can be used to the application of the civil structure crack detection. Chapter 7: Conclusions and future work This chapter concludes the thesis and discusses the future scope.

2

The Basics and Kinematics of Gears

To obtain a general understanding and background of gearboxes first, this chapter is devoted to introduce basic knowledge of gearboxes. It mainly contains different types of gears, gear mesh stiffness, components and geometry of the gearbox and the typical gear faults. Then, an overview regarding WTB multi-stage gearboxes and their characteristic frequencies are demonstrated. After that, a signal model characteristics of planetary gearbox and the transmission path effects will be discussed. Finally, the characteristics and properties of a WTB planetary gearbox and vibration responses of planetary stage to different localized fault types are presented.

2.1

Types of Gears and their Operation

The evolution of gear technology for manufacturing several types of gears gives the ability to transmit power and motion between rotating shafts regardless of whether they rotate about parallel axes, non-parallel axes, or intersecting axes. Compared to the other power transmission systems, a gear system can be utilized to transmit high power with generally low space requirements with a high reliability. The gear system can also be entirely enclosed, ensuring that it is not exposed to the environment [143]. These characteristics and some others have made the gear wheel a critical component in nearly every machine in use today. Nowadays gears are being widely used in home appliances, toys, naval vessels, automobiles, industrial and other applications. In the field of aircraft, gears are required to drive propellers, pumps and many other accessories. Gears are a critical component in helicopters to drive the main and tail rotors. Also, in the WTB, a gearbox is used to raise the rotational speed from a low-speed rotor to a higherspeed electrical generator. For instance, with a rate between (11–16.7) rpm input from the rotor to 1500 rpm output for the generator, a common ratio is around 90:1. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_2

13

14

2 The Basics and Kinematics of Gears

The most common types of gears used in industry are helical, spur, bevel, and worm gears as shown in Fig. 2.1. They all have a driven and driving wheel, but differ in the shape and arrangement of the wheels. Spur gears have teeth which are normal to the face of the gear (parallel to the shaft) and when a change of direction between two shafts is required, it cannot be employed [35][144][50]. The Helical gear, which is similar to a spur gear but has teeth which are at an angle to the shaft and form a helix. As a result, the teeth on a helical gear are longer than those on a spur gear of equal width. Helical gears run smoother than spur gears, but they are more expensive, hence they are used less commonly. Also the spur gears are slightly more efficient that helical gears of the same size [50]. The force on a helical gear is dispersed across a larger tooth area at any given time than on a spur gear, resulting in reduced wear and there is less noise and vibration than spur gears due to the more gradual engagement of the teeth during meshing. Because the teeth of a helical gear are not parallel to the shaft, a thrust is generated along the gear shaft during operation, requiring the employment of thrust bearings to absorb the strain and preserve gear alignment [251]. Helical gears can transfer power across shafts that are not at a 90-degree angle to each another [243]. Although the higher transmission

(a)

(b)

(c)

(d)

Figure 2.1 Gear types (a) Spur gear, (b) Helical gear, (c) Bevel gear, (d) Worm gear [2]

2.2 Gear Mesh Stiffness and Dimensioning Geometry of Gears

15

ratio in the gearbox requires the use of multiple stages, Helical gearboxes provides reduction ratios of between 3 and 500 [71]. Another type of gears is called bevel gear. When shafts are at 90 degrees, it is frequently employed to transfer power from one shaft to another. Teeth on bevel gears are all cut straight and parallel to the line leading to the apex of the cone on which they are built. They cannot be utilized for parallel shafts, though, and they are noisy at high speeds. Worm gears are also another type of gears which can be used to drive spur or helical gears and mesh two nonintersecting and perpendicular shafts. The irreversibility of worm gear meshes is an important feature. When a worm gear rotates, the meshing spur rotates as well, but the spur does not rotate the worm gear. The result is a ratcheting mechanism which is called “self-locking”. Worm gears could be employed to convert rotational motion to a translational motion as they can tolerate large loads. They can also have very large pitch diameters [50]. To transmit power without slippage, gears usually operate with the teeth of one engaging the teeth of the other. When the teeth are meshed, driving one gear causes the other to turn, transferring power while allowing the direction of the rotation and the rotational speed to be changed. The pinion is the gear with the fewest teeth. When the gear drives the pinion, the rotation speed is raised; when the pinion drives the gear, the rotation speed is reduced. The ratio of the speed reduction can be calculated as Z1 R= , (2.1) Z2 where Z 1 is a pinion teeth number and Z 2 is a gear teeth number. For the case of the WTB gearbox in this thesis, depending on the type of the WTB design, the gearbox may have also the secondary function of supporting the main shaft bearings [144]. In this second case the gearbox is called Integrated Gearbox since the gearbox housing provides the bearing supports. In WTB gearboxes, the spur and helical gears are commonly used.

2.2

Gear Mesh Stiffness and Dimensioning Geometry of Gears

In the previous section, the different types of gears are briefly introduced. In this section, the main components and geometry of a WTB gearbox as a multi-stage gearbox and some important dimension parameters for each stage are presented for a basic understanding of the WTB gearboxes.

16

2 The Basics and Kinematics of Gears

2.2.1

Gear Mesh Stiffness and Contact Geometry

One of the main excitations that causes unwanted vibration and noise in gear transmission systems is time-varying gear mesh stiffness and contact teeth geometry. The contacting characteristics of a pair of helical and spur gears are different from each other. The motion is transmitted more smoothly and gradually between the mating gears. In case of a spur gear drive, the contact exists along a straight line which is parallel to the gear axis. The contact starts abruptly over the full face width at the initial point of the tooth meshing cycle and also ends suddenly at the end. Unlike the spur gears, the contact between a pair of helical gears initiates at the first tooth face end as a point. The contact expands from a point to a line of continuously increasing length as the gears rotates, moving over the tooth flank and extending until it reaches the second tooth face. At the second tooth face end, the contact line length shortens and finishes as a point. The contact line runs diagonally between the face ends of the helical teeth. Furthermore, when using helical gear drives, at least two pairs of teeth are always in contact. The periodic shift in the number of tooth pairs in the mesh zone causes its amplitude to fluctuate. Fig. 2.2(a) illustrates a spur gear pair mesh behaviour with contact ratio (CR) between 1 and 2. The zone of the gear mesh (over the line of action) A–E is separated into three sections: A–B, B–D and D–E section. During the A–B and D–E sections, there are double tooth pairs in mesh at the same time, whereas during the B–D segment, there is only one. As a result, the magnitude of gear mesh stiffness will be higher at the

(b)

(a) Gear

Base circle

Addendum Circle Addendum Circle Pitch circle Base circle

Gear mesh stiffness

Pitch circle

Pinion A

B

C

D

E

Figure 2.2 Schematic of the mesh behaviour of a spur gear pair (1 < C R < 2): (a) mesh process, (b) gear mesh stiffness curve [34]

2.2 Gear Mesh Stiffness and Dimensioning Geometry of Gears

17

double mesh zone and lower at the single mesh zone, as shown in Fig. 2.2 (b). The periodic change of the tooth pairs number in mesh causes the periodic variation of the gear mesh. This causes an inherent excitation in the gear system, which means the vibration problem of the gear always happens even when the gears are perfectly and correctly machined and assembled. This is the distinct and prominent dynamic feature of gears compared to the other common rotational machinery components such as bearings and shafts [131][34][13].

2.2.2

Dimensioning Geometry of Gears

Gearbox fault detection and identification require knowledge of gear dimensions and mechanics. Correctly dimensioning the gearbox is a critical responsibility for reducing incipient failures and boosting inherent reliability. The dimensioning of the gears is typically carried out by the manufacturers. The torque to be transmitted is the most significant parameter. The torque of the rotor is not a constant value but it is subjected to a large variation, depending on the technical system design concept. The rated torque and the quotient of the equivalent torque TE are called application factor K A according to DIN 3990 (German National Standards) and can be defined as TE KA = , (2.2) TN where TN denotes as the rated torque to be transmitted which can be very simply obtained by dividing the rated power of mechanical rotor by the rotor speed. Two other criteria at least are used in gearbox technology to characterize the external load situation for the transmission. The so-called service factor is employed, which is defined in American Gear Manufacturing Association (AGMA) standards. Due to the different definitions, the WTB system designer must reach an agreement with the gearbox manufacturer on the dimensioning parameters to be used [30]. The breaking strength of the WTB gearboxes is typically designed to be at least three times the rated torque. The “generator short circuit load” is the only case that can result in a larger torque peak in the drive train. In most situations, overload clutches are integrated into the high-speed shaft to protect the gearbox and rotor shaft from this situation.

18

2 The Basics and Kinematics of Gears

2.2.3

Definition of Gear Failure Mode

Gears similar to several other machine components are supposed to be faulty when they can no longer carry out their intended function with a particular level of efficiency. The definition of fault in gear teeth covers a wide range of causes such as fatigue, impact, wear, etc. Fatigue is regarded as one of the most common modes of failure in gear applications [6]. Pitting, scuffing as shown in Fig. 2.3, and to some extent mild wear are likely to be categorized as significant mechanisms of tooth failure from a tribological standpoint, in which the surfaces of mated teeth are expected to be separated by a very thin layer of lubricant. These defects primarily refer to the lubricating performance of the gears during operation [209]. In some applications, such as the WTB and helicopters, a breakdown in the lubricating film thickness, for whatever reason, can lead to catastrophic failure, and failure must be prevented at all costs. Pitting area

(a)

(b)

Scuffing area

Figure 2.3 Helical gear teeth faults: (a) Scuffing, (b) Pitting [6]

2.2.4

Gear Features in Condition Monitoring

The meshing action of the gears is the main cause of vibration in a geared transmission system. The vibration signatures of all non-defective gears are determined by their design characteristics and operating conditions. The cyclically variable gear meshing forces are responsible for the distinctive characteristics of gear vibrations. Vibrations are generated by these meshing forces and are transmitted to the gearbox case, where they can be monitored and measured. The vibration behaviour is strongly influenced by the geometry of the gear profile. The GMF, defined as the product of the number of teeth on the gear and its turning speed, is the rate at which

2.2 Gear Mesh Stiffness and Dimensioning Geometry of Gears

19

gear teeth engage together in the pitch point. In the case of a parallel helical stage, the basic meshing frequency, f m , is generated by a gear with Nt teeth on a shaft rotating at a frequency f r ot [235]: f m = Nt × f r ot.

(2.3)

The GMF of any two meshing gears must be the same. A perfect single tone vibration spectrum would be implied by a perfectly sinusoidal meshing force. The surfaces of the real gears are not ideally smooth and the tooth shape is not perfect, As a result, harmonics of the basic meshing frequency will always be present in the vibration spectrum. n f m = n Nt × f r ot , ∀n ∈ {2, 3, 4, ...}. (2.4) Eqs. (2.3) and (2.4) describe a line spectrum generated by a gear with teeth precisely uniform, or identical, profile and pitch. The real shape of meshing force determines the amplitude of each line. In practice, even when the tooth profiles are perfect, a meshing error is introduced as the teeth deform under stress. Furthermore, there are geometric variations from the perfect profiles due to gear manufacturing errors [147][186]. All gears are manufactured to strict tolerances, with tiny variations in profile and pitch from tooth to tooth and roughness on the tooth surfaces. These mechanisms produce the AM and FM of the meshing forces resulting in small amplitudes called side-bands that appear in pairs around the tooth meshing frequency and its harmonics and also are periodic with the rotation gear frequency [58][46]. The side-bands are equally spaced frequency components around the center frequency namely carrier frequency. The tooth meshing frequencies and their harmonics, as well as the modulation side-bands generated by mean geometric errors on the tooth profiles, manufacturing errors, and wear, are the most essential components in gear vibration spectra, as shown in Fig. 2.4. The amplitude changes, in the GMF and its harmonics when faults develop, are not always detectable compared to the base line or healthy vibration measurements. The mesh frequency, in other words, can be essentially unaffected by the development of a fault. Extraction and trending mesh frequency and harmonic amplitudes have been used in the past to monitor gears for ongoing deterioration. Since it has a low sensitivity to developing faults, this approach has been determined to be ineffective. However, as faults develop in a gear mesh, the modulation phenomenon in the vibration signal occurs by the defect. The fault engages in the gear mesh, modulating the amplitude and frequency of the GMF. In the frequency spectra, this leads in typical amplitude increases in the modulation side-bands. In good condition gearboxes, the side-bands level tends to stay constant with time. As a result, an increase

2 The Basics and Kinematics of Gears

(a) Pinion

Amplitude

20

1x Gear 1x GMF 1x Pinion

Spur: radial Helical: axial 2x GMF

(b)

3x GMF

Frequency

Gear

1x GMF

Amplitude

1x Gear 1x Pinion

Sidebands at 1x Gear or 1x Pinion 2x GMF

3x GMF

Frequency

Figure 2.4 Graphical representation of typical vibration spectra: (a) Healthy gear (low sidebands), (b) Faulty gear (with higher sidebands) [1]

in the number and amplitude of such side-bands around the GMF may indicate a defect condition [42] [240]. If a certain tooth is worn, broken, or missing in part, an impact between the gears will occur once for each revolution of the faulted gear. A single damaged tooth on a pinion or gear directly connected to a shaft causes a hit or impact, resulting in erratic contact behaviour each time the defective tooth meshes. The local tooth damage produces short-duration impacts that add the AMFM effects to the meshing vibration, resulting in a higher level of side-bands around the mesh harmonics. Furthermore, the spacing of the side-bands is connected to their source, providing valuable diagnostic information [183]. The side-bands spacing is the same as the modulation frequency, which is the same as the rotating speed of the shaft where the fault occurs. The localized modulation effect, in particular, occurs only during the engagement of the faulty teeth, also is repeated once in each gear rotation. As a result, the spectrum shows a significant number of side-bands spaced by the faulty gear rotational frequency of the teeth meshing frequency and its harmonics. For instance, the spectrum will indicate the mesh frequency with sidebands spaced at the pinion shaft speed if a gear pinion develops a damaged tooth. Multiple harmonics will appear in the side-bands, both below and above the center frequency or meshing frequency. Typically, the more fault that happens, the more energy the side-bands have [80]. In particular, monitoring the second harmonic of the gear mesh and its side-bands has been proven to provide early diagnosis of gear wear in the literature on vibration signature analysis [161][81].

2.3 Wind Turbine Gearbox Kinematics and Fault Feature Frequencies

2.3

21

Wind Turbine Gearbox Kinematics and Fault Feature Frequencies

From the mechanical point of view, a WTB drive train consists mainly of the three main parts which are input rotor main shaft, a speed-increasing gearbox, a coupler and a generator. The WTB, similar to other turbine electricity conversion systems, converts the rotor rotational kinetic energy into electric energy via a generator. The rotor of a utility-scale WTB is designed to rotate at a very low speed, roughly 1/3 Hz, to maximize wind energy capture over a wide range of wind speeds. The rotational speed rotor is frequently raised by a gearbox to improve the efficiency of electricity conversion. The doubly fed induction generator is the most widely used system connected with a multi-stage gearbox [179]. Gear failure occurs when a gear is working under extreme stress conditions [6][208]. In terms of vibration analysis of rotational machines, there are two types of gear and bearing faults, localized faults and generalized faults [212]. Generalized faults include wear, deformations or inappropriate installation that occur during manufacturing which is also known as distributed faults. Broadband vibrations will be generated as a result of these faults, and typically some special approaches are required to detect them. Local gear faults which include tooth breakage, pitting and scoring are the most dangerous since they develop quickly once they are started and usually have a significant influence on power transmission. If not detected as early as possible, there could be dramatic consequences. For that reason, increasing research is focused on local faults on both for gears and bearings. The most typical feature of local faults of gears lies in three typical characteristics namely periodic gear meshing due to the rotating nature, impulsive as the components induced by the faults and transient in the amplitude. The WTB gearbox, which is a multi-stage gearboxes as shown in Fig. 2.5, is considered as a complex transmission system. It consists of three different stages, low speed stage (LSS) or planetary stage (PS), intermediate speed stage (ISS), and high-speed stage (HSS). The PS contains planetary arm, planetary gear (three planet gears in total), ring gear, sun gear, and sun shaft (sun rotation). Two other stages namely intermediate shaft and high-speed shaft are parallel stages. For all the three stages, the number of the teeth, as illustrated in Fig. 2.6, is denoted as the following, Z s is the number of sun gear teeth, Z p is the number of planet gear teeth, Z c is the number of ring gear teeth, Z mi is the number of drive gear teeth in the ISS, Z mo is the number of pinion gear teeth in intermediate shaft, Z hi is the number of gear teeth in high speed stage, and Z ho is the number of pinion gear teeth in high speed stage. In vibration-based fault diagnosis, the fault characteristic frequencies and gear meshing for the three stages should be computed first based on kinematics and geometrical design of the WTB gearboxes as following; The

22

2 The Basics and Kinematics of Gears

(a)

Parallel stages

(b)

Planet gear

(c)

Output shaft Sun gear

Input (driving) shaft Ring gear Planet gear

(annulus)

Figure 2.5 Wind turbine system:(a) WTB drive train, (b) planetary gear stage, (c) three stages WTB gearbox [257] PS

ISS

HSS

Casing

Generator

Rotor

Gearbox

Figure 2.6 Internal structure of the WTB gearbox [228]

2.3 Wind Turbine Gearbox Kinematics and Fault Feature Frequencies

23

number of teeth sharing the load changes over time when two gears mesh. The tooth deflection varies as a result of this. The torque varies because to the variable deflection, which is caused by an inevitable mean variation from the ideal tooth shape. This provides a dominant vibration at the GMF, f m , and multiple of its harmonics. Thus, for the two parallel meshing gear pairs, the relationship between GMF and shaft frequency is straightforward and is simply the gear tooth numbers multiplied by the corresponding shaft speed, therefore, can be calculated as f m = f G × Z mi = f P × Z mo ,

(2.5)

where f m is the GMF, f G is the gear shaft speed, Z mi is tooth number of the gear, f P is the pinion shaft speed, and Z mo is tooth number of the pinion. For the planetary gear stage as shown in Fig. 2.5, the relationship is much more complicated. To obtain a simple method for the relationships between the tooth meshing frequencies and the rotation frequencies is to assume that the planet carrier is brought to rest by rotating the entire gearbox in the opposite direction at the carrier rotation frequency f c , as illustrated in Fig. 2.7(b), (adopted from [154]). The tooth meshing frequency f m is related to shaft rotation frequencies and the gear teeth as a result of this transformation and can be computed as f m = Z c × f c = Z p ( f p + f c ) = Z s ( f s − f c ).

(2.6)

Therefore, all the gearbox shaft frequencies could be then related to the input shaft speed f c or f L SS as   Zc fp = − 1 × fc , (2.7) Zp   Zc + 1 × fc , (2.8) fs = Zs   Z mi Z c + 1 × fc , (2.9) f L SS = Z mo Z s   Z hi Z mi Z c − 1 × fc . (2.10) f H SS = Z ho Z mo Z p A definition of two imaginary shafts with relative shaft frequency is also beneficial as Zc f P LT r  ( f p + f c ) = × fc , (2.11) Zp

24

2 The Basics and Kinematics of Gears

(a)

(b)

Ring Gear Fixed

Ring Gear Moving

Planet Gear

Planet Gear

−

Sun Gear

Sun Gear

Carrier fixed Sun Gear

Figure 2.7 A planetary gear stage with simultaneous mesh design: (a) with normal operation, (b) with an imaginary rotation [154]

f sunr ot  ( f s − f c ) =

Zc × fc , Zs

(2.12)

where f P LT r is the relative speed of the planet, f p , with respect to the rotor input speed, f c , and f sunr ot is the relative speed of the sun, f s , with respect to the rotor input speed f c . The gear fault, for a parallel gear mesh, is typically showing up as gear mesh response amplitude modulated by the shaft rotation period. In the frequency domain, this phenomenon is expressed as the GMF and its harmonics with one or multiple side-bands at the frequency interval of the corresponding shaft speed. The relationship between the fault features and the shaft frequencies, for an epicyclic gear, are more involved. With N N P the number of planet gears in the PS, the gear component fault signatures, namely, the ring frequency fr , planet frequency f p , and sun gear frequency f s for a simultaneous mesh planetary gear as an example shown in Fig. 2.7. • For calculating the planetary gearbox stage fault characteristic frequency, the ring gear fault characteristic frequency ( fr ) is expressed as fr = f c × N N P .

(2.13)

2.3 Wind Turbine Gearbox Kinematics and Fault Feature Frequencies

25

Because each planet that passes across the ring gear fault location causes an additional impulsive vibration. Also, it will run in to the ring gear and sun gear once every planet related rotation in the planet relative shaft. As a result, the frequency of single tooth fault on the planet ( f p ) is computed as f p = 2( f p + f c ) = 2 × f P LT r ,

(2.14)

similarly, the fault feature of single tooth sun gear frequency ( f s ) is also expressed as (2.15) f s = N N P ( f s − f c ) = 2 × f sunr ot . In summary, the fault characteristic frequencies of planetary gearbox stages can be computed using eqs. (2.13), (2.14) and (2.15). • For calculating intermediate and high speed parallel gearbox stage fault characteristic frequency, N I SS , (2.16) 60 N H SS , (2.17) f H SS = 60 where N I SS and N H SS denote the speed of intermediate and high stage of the rotational shaft respectively. f I SS =

• Also, the GMF for all three stages could be calculated as 1) The GMF for planetary gearbox f m(L SS) = Z c f c = Z p ( f p + f c ) = Z s ( f s − f c ).

(2.18)

2) The GMF for the other two parallel stages could be expressed as f m = f G Z mi = f p Z mo .

(2.19)

26

2 The Basics and Kinematics of Gears

2.4

Signal Model Characteristics of Planetary Gearbox

Planetary gear systems, as compared to the fixed shaft gearboxes, provide a larger transmission ratio in a more compact way and they are beneficial for machines with a high requirement of output power. Because of its significant advantages over typical fixed axis transmissions, the planetary gear train is widely employed in transmission design for automobiles, helicopters, aircraft engines and WTB. Torque capability is boosted since the load is distributed among multiple planets. Another advantage of a planetary gear is that its special combination of compactness and high power transmission efficiency. The main advantage of planetary gearboxes over the parallel gearbox stage is its ability to • • • •

considerably increase low speeds and the efficiency. achieve high torque within a small space and deliver a high reduction ratio. install in a few installation spaces due to its compact size and light weight. attain a high reliability due to a proper and correct distribution of stress among various bearing components.

Despite the advantages, analysing the complex dynamic forces that occur between the sun, planet, and ring gears for sources of vibration is difficult. It is also more challenging to mathematically represent and model a vibration signal from a planetary gear with multiple meshing stages. Planetary gears, nevertheless, prone to significant wear and impact damage of essential components such as gears, shafts, and bearings due to their high load and tough operating environment. In the community of signal-based fault diagnostics, spectral analysis is quite common. More effective fault diagnosis using spectrum analysis would require a detailed understanding of the spectral structure of planetary gearbox vibration signals as well as the fault symptoms of each type of gear (sun, planet, and ring). All main factors, in planetary gearbox vibration signal modelling, will be considered including the effects of the AM-FM generated as a result of both gear fault and periodical fluctuation in working conditions such as operating speed and load variation, and the influence of AM due to time varying vibration transmission paths. In this section, the vibrations signals acquired from the planetary gear stage are modelled using the same AM-FM process, induced by either distributed or local gear fault. We do not distinguish between the local and distributed faults models as they both generate an AM-FM effect in the gear meshing vibration, the only difference is that the characteristic frequencies of the two gear fault scenarios are different. In the following, the faulty vibration signal at the meshing point is modelled, then,

2.4 Signal Model Characteristics of Planetary Gearbox

27

the influence of vibration transmission paths on sensor observed signals will be analysed. Finally, the effects of transmission paths is considered and a numerical example is demonstrated to verify the effects of transmission path effects.

2.4.1

Localized Fault Signal Model at Meshing Point

Firstly, when the damaged part goes into meshing, the induced vibration faulty signals at the meshing point are focused on without the transmission paths effects on the vibration signals. In this situation, the signal model can be used to model any gear system, including fixed shaft and planetary gear systems. Assume that a local damage on the single tooth gear surface occurs or distributed fault across all teeth. Sudden changes will be produced for such damage in the vibration signal as the mating gear meshes with the faulty gear tooth. The amplitude and instantaneous frequency of vibration signals will both be modulated by these changes. The AM and FM functions can be represented by discrete Fourier series with the characteristic frequency of the faulty gear as the fundamental frequency since the modulation is periodic with the fault feature frequency of the gear. The vibration signal induced by gear fault at the meshing point, under such assumptions, can be modelled as an AM-FM signal, with the gear meshing frequency or its harmonics as the signal carrier frequency, and the characteristic defect frequency of the faulty gear or its harmonics as the modulating frequency [183][152][61][63][64]: x(t) =

K 

ak (t)cos[2πk f m t + bk (t) + θk ] ,

(2.20)

k=0

where ak (t) and bk (t) are the functions of AM and FM signal respectively, and K denotes the highest order of AM-FM signal, also f m represents the GMF, and θ is the initial phase. The local gear fault produces the modulation effects to the GMF vibration, and the modulation is periodic with the characteristic frequency of the faulty gear. Therefore, the functions of AM and FM can be expanded as the Fourier series with the characteristic frequency of the faulty gear as the fundamental frequency

28

2 The Basics and Kinematics of Gears

ak (t) =

N 

Akn cos(2πn f character stic t + φkn )

n=0



=c 1+

N 

(2.21) 

Akn cos(2πn f character stic t + φkn ) ,

(2.22)

n=1

bk (t) =

L 

Bkl sin(2πl f character stic t + ϕkl ) ,

(2.23)

l=0

where Akn > 0 and Bkl > 0 represent the magnitude of the AM and FM signal respectively, c is the signal amplitude which is constant (without losing generality, for simplicity it is set to 1), f character stic refers to the gear fault characteristic frequency, the φ and ϕ are the initial phases of AM and FM signal respectively and N and L denote the highest order of the considered AM and FM signal. In eq. (2.21), if n = 0, Ak0 ≡ 1 and φ ≡ 0, without loss of generality, here only the fundamental characteristic frequency of the damaged gear and the GMF are considered, then the vibration signal model of a faulty gear becomes x(t) = [1 + Acos(2π f character stic t + φ)] cos[2π f m t + Bsin(2π f character stic t + ϕ) +θ ] ,   



AM signal by faulty gear rotation

FM signal by faulty gear rotation

(2.24) according to the identity, exp( j zsin(θ )) =

∞ 

Jm (z) exp( jmθ ) ,

(2.25)

m=−∞

where Jm (z) denotes the Bessel function of the first kind with integer order m and argument z, eq. (2.24) could be expanded as a sum of infinite harmonics [3]:

x(t) = [1+ Acos(2π f character stic t +φ)]

∞ 

Jm (B)cos[2π( f m +m f character stic )t +mϕ+θ ].

m=−∞

(2.26) In addition, based on the identities of trigonometric functions, eq. (2.26) can be rewritten as

2.4 Signal Model Characteristics of Planetary Gearbox

29

A [h( f character stic , f character stic , φ) 2 + h( f character stic , − f character stic , −φ)] ,

x(t) = h( f character stic , 0, 0) +

(2.27)

where the intermediate function can be expressed as ∞ 



h( f character stic , f , φ) =



Jm (B)cos(2π [ f m + m f character stic t + f ]t + mϕ + θ + φ).

m=−∞

(2.28) Then, taking the Fourier transform of eq. (2.27) yields A [H ( f character stic , f character stic , φ) 2 + H ( f character stic , − f character stic , −φ)] ,

X ( f ) = H ( f character stic , 0, 0) +

(2.29)

where the intermediate function is 

H ( f character stic , f , φ) =

∞ 



Jm (B)δ( f −[ f m +m f character stic + f ]) exp[ j(mϕ+θ +φ)].

m=−∞

(2.30) The signal will be observed in the from of eq. (2.29). The faulty vibration signal generated at the meshing position, without considering the transmission path effect, has side-bands around the meshing frequency f m , with a side-band spacing equal to the characteristic defect frequency of the faulty gear f character stic . The faulty generated vibration signals at the meshing point, for both the planetary gear set and fixed shaft gears, have the same form. Even though they operate at the same speed, the characteristic frequencies of defective gears in a planetary gearbox differ from those of fixed shaft gears. Furthermore, because the meshing location of fixed shaft gears is fixed with regard to the vibration sensor mounted on the gearbox casing, the signal model in this section is appropriate to represent the damage induced signal of fixed shaft gear systems. This signal model, however, is insufficient to characterize the gear faulty-induced vibration in planetary gear systems. The influence of vibration transmission paths must be considered, because the meshing point of planet-ring or sun-planet gear pairs with respect to the fixed sensor are time-varying [152][132][58][61][261][252][63][64].

30

2.4.2

2 The Basics and Kinematics of Gears

Transmission Path Influence on Vibration Signal Model

The ring gear of a planetary gearbox in most cases is stationary, and the sensors, which are directly connected to the ring gear, are mounted on the gearbox casing. This will be the only case which is considered in this section. As illustrated in Fig. 2.8, assume that the damage occurs on a planet gear, the damage-induced vibration signal at the meshing point can take three different paths from its source to the sensor through solid mechanical components and their contacts. For transfer path 1, the damage-induced vibration signal firstly passes through the ring gear, then to the gearbox case, and ultimately to the sensor. While for the second path, the vibration signal passes through a longer path. Firstly, it passes from its source to the sun gear, then from the sun gear to the shaft attached to the sun gear. After that, it passes from the sun gear shaft to the supporting bearing, then from the bearing to the gearbox casing, and finally reaches the sensor. For the third path, is the same as the second one: the vibration signal firstly passes to the planet gear carrier from its origin through the bearing and shaft supporting the planet gear. After that, it passes from the planet carrier to the planet carrier shaft, then from the planet carrier shaft to the bearing which supports the planet carrier shaft. Then, it passes from the bearing to the gearbox casing and finally to the sensor. The properties of the second and third transmission paths, in the planetary gearbox, do not vary with time, if all the gears are isotropic circumferentially. However, all the gears are rotating except the ring gear. When the sun gear and the planet carrier rotate, the propagating distance for the fault induced signal from its source to the sensor remains constant, therefore the transmission paths only create a scaling influence rather than an AM effect during the propagation of the faulty induced vibration signal. Without loss of generality, the observed signal of a sensor from the second and the third paths could be simplified as eq. (2.24), and also it has a similar structure of the spectrum as revealed by eq. (2.29): the perceived sensor signal due to gear fault coming from transmission paths 2 and 3 has side bands around the GMF f m , and a side band spacing equal to the characteristic frequency of faulty gear f character stic . However, for the longer transfer or transmission path, the induced faulty vibration signal will be more damped during the propagation of the second and the third paths, especially for the bearings damping due to the complexity of the transmission path inside, and the observed sensor signal will very certainly be of insignificant amplitude. However, during the propagation through the second and the third paths, the faulty induced vibration signal will be damped more by the longer transfer path, especially for the bearings damping due to the complexity of the transmission path inside, and The observed sensor signal will more likely to have insignificant

2.4 Signal Model Characteristics of Planetary Gearbox

31

magnitudes. Hence, the scenarios for the second and the third transmission paths will be further considered. The first path, on the other hand, is shorter than the second and third transmission paths. Because the bearings, which cause signal attenuation, do not involve. The observed sensor signal from the first path will have significantly more information about the damaged gear. Because the signal propagating distance is time-varying when the sun gear and the planet carrier rotate, the transmission path for the faulty vibration signal, which is going through the first path, is time-varying except for the ring gear fault. For this reason, the sensor perceived signal model will be simulated based on eq. (2.24) for the cases of faults on each gear in the following sections. Meanwhile, lubricating oil inside the planetary gearboxes may also transfer a small amount of gear vibration. The time variation of the transmission path through oil, on the other hand, is the same as the time variation of the first path through solid mechanical components and their contacts. The transmission path for the faulty ring gear induced vibration to reach the sensor through oil is fixed in the case of ring gear damage, regardless of whether the fault location is in the oil or not. The transmission path through oil then only has a scaling influence on the faulty vibration signal from the ring gear. For the case of faulty sun gear and planet gear, the transmission path for the faulty induced gear vibration to reach the sensor through oil also changes

Vibration sensor Transmission path 2 Ring gear Transmission path 1 Sun gear shaft bearing

Sun gear shaft

Planet gear shaft bearing

Planet gear shaft Transmission path 3

Planet carrier shaft Planet carrier shaft bearing

Sun gear Planet carrier Planet gear

Gearbox casing

Figure 2.8 Three different possible transmission paths to sensor in planetary gearbox [62]

32

2 The Basics and Kinematics of Gears

periodically with the rotation of sun gear or planet carrier. As a result, the effect of AM of the time-varying transfer path through oil can be expressed in the same form as the first path through solid mechanical components and their connections (to be considered in Sections 2.4 and 2.5). Furthermore, during transmission through oil, the gear faulty-induced vibration signal will be heavily damped by the lubricating oil, and the observed signal from the sensor will most likely lose the majority of the energy of the vibration signal. Hence, because of the following two reasons, the effects of the transmission path through oil will not be considered [62][254]: 1) Because it has a similar effect to the first path through solid mechanical components and their contacts, its impact will be inextricably included in the consideration of the first path in Sections 2.3−2.5. 2) The faulty vibration gear signal through oil due to the damping oil is negligible.

2.4.3

Observed Faulty Vibration Signal Model from Planetary Gearbox

Because of the complexity of the planetary stage of the WTB gearbox, the observed faulty signal from each parts of the ring, planet and sun gear can be represented as the AM-FM equation. These modulations, in the vibration spectrum, appear as loworder side-bands around the gear meshing harmonics. The side-band frequencies are f sidebands = n f m ± k × f sidebands ,

∀n, k ∈ {1, 2, 3, 4, ...} ,

(2.31)

where k is meshing harmonic number and n is a number of side-bands, contributions from multiple gears operating at different shaft frequencies add up to the total modulation in multi-stage gearboxes. This results in side-bands for non-integer orders as well. • Observed vibration sensor faulty signal model from ring gear Assume that the ring gear is damaged. The transmission path for the faulty induced vibration signal which propagates from the faulty ring gear tooth to the sensor does not vary since the vibration sensor is fixed and mounted on the gearbox casing. As a result, during the propagation of the faulty-induced vibration signal, the planet carrier rotation does not cause AM effect. The same form as eq. (2.24) will be

2.4 Signal Model Characteristics of Planetary Gearbox

33

observed in the sensor observed vibration signal due to the faulty ring gear and can be written as x(t) = [1 + Acos(2π fring t + φ)] cos[2π f m t + Bsin(2π fring t + ϕ) +θ ] ,   



AM related to ring gear rotation

FM by ring gear rotation

(2.32) where fring is the ring gear fault characteristic frequency which can be calculated as described in eq. (2.13). Following the same procedure as from eq. (2.24)−(2.29), the Fourier transform of eq. (2.32) can be computed as X ( f ) = H ( fring , 0, 0)+

A [H ( fring , fring , φ)+ H ( fring , − fring , −φ)]. (2.33) 2

From eq. (2.33), the vibration signal of a faulty planetary gearbox with a faulty ring gear has side-bands around the meshing frequency f m , with a side-band spacing equal to the faulty ring gear characteristic frequency fring . For the cases when the signal passes through both the second and the third transmission paths, because the transmission paths have a scaling effect during the propagation of the faulty induced vibration signal, the sensor observed signals have the same form as eq. (2.24) and eq. (2.32), and their spectral form are also the similar to eq. (2.33). When the AM-FM effects are considered with the higher orders of faulty ring gear characteristic frequency n fring as the modulating frequency and the higher orders of meshing frequency k f m as the signal carrier frequency, peaks will appear in the Fourier spectrum at the frequency locations k f m ±n × fring . Since the spectral peaks of the faulty induced vibration signal are related to the characteristic frequency of the faulty ring gear, the detection of the ring gear fault will be expected by monitoring the presence or amplitude increase of spectral peaks at the frequency locations of k f m ± n × fring ,

∀n, k ∈ {1, 2, 3, 4, ...} ,

(2.34)

where k is meshing harmonic number and n is a number of side-bands. • Observed vibration sensor faulty signal model from planet gear When a planet gear is faulty, the meshing point of the faulty area with the mating gear (either ring gear or sun gear) relative to the fixed mounted sensor will change with the planet carrier rotation. As a result of the relative motion of the meshing location, the transmission path for the faulty produced vibration propagating from the meshing

34

2 The Basics and Kinematics of Gears

position to the sensor will change, resulting in an AM impact on the vibration signal during propagation. Assume that the faulty planet gear begins meshing at the farthest point from (i.e. opposite) the sensor. The observed vibration due to fault increases as the planet gear approaches the sensor, reaching a maximum when the faulty planet gear meshes at a place closest to the sensor. Similarly, the observed vibration signal produced due to the fault will decrease as the planet gear goes away from the sensor, it reduces to a minimum when the faulty planet gear meshes at a location farthest from (opposite to) the sensor. A Hanning function can be used to model the AM effect [89][90]. As a result, an additional multiplicative term is then added to eq. (2.24) due to the planet carrier rotation induced AM effect, and the sensor observed signal due to fault in the planet gear can be modelled as x(t) = [1 − cos(2π f carrier t] [1 + Acos(2π f planet t + φ)] cos[2π f m t + Bsin(2π f planet t + ϕ) +θ] ,  

  



AM by carrier rotation

AM related to planet gear rotation

FM by planet gear rotation

(2.35) where f carrier is the planet carrier rotating frequency and f planet is the faulty planet gear characteristic frequency. The calculation of the fault characteristic frequency of planet gear f planet has been explained in Section 2.3. The Fourier transform of eq. (2.35) can be expanded as X ( f ) = H ( f planet , 0, 0) +

A [H ( f planet , f planet , φ) + H ( f planet , − f planet , −φ)] 2

A [H ( f planet , f carrier , 0) + H ( f planet , − f carrier , 0)] 2 A − [H ( f planet , f planet + f carrier , φ) + H ( f planet , f planet − f carrier , φ)] 4 A − [H ( f planet , − f planet + f carrier , −φ) + H ( f planet , − f planet − f carrier , −φ)] . 4 −

(2.36)

Eq. (2.36) can be interpreted as the following. The sensor observed vibration signal for a planetary gearbox with a faulty planet gear has side bands around three centers: the meshing frequency f m , the meshing frequency minus and plus the planet carrier rotating frequency f m ± f carrier , with a side-band spacing equal to the faulty planet gear characteristic frequency f planet . For the cases that are the signal goes through the second and the third transmission paths, since the transmission paths have a scaling effect during the propagation of the faulty induced vibration signal, the sensor observed signals have the similar form as eq. (2.24), and their spectral form are also the same as eq. (2.29). Side-bands

2.4 Signal Model Characteristics of Planetary Gearbox

35

show up around the meshing frequency f m , with a side-band spacing equal to the characteristic frequency of faulty planet gear f planet . If the AM-FM effects are considered with the higher orders of planet gear faulty characteristic frequency n f planet as the modulating frequency and with the higher orders of meshing frequency k f m as the signal carrier frequency, peaks will show up in the Fourier spectrum at the frequency locations of k f m ± n × f planet and k f m ± f carrier ± n × f planet Since the spectral peaks of planet gear damage induced vibration signal are related to the faulty planet gear characteristic frequency, planet gear fault can be detected and identified by monitoring the existence or increasing magnitude of spectral peaks at the frequency locations above. • Observed vibration sensor faulty signal model from sun gear When fault occurs on the sun gear, the meshing point of the faulty area with any mating planet gear will also vary with the sun gear rotation. This will generate an AM effect on the faulty induced vibration during its propagation too. Similarly, an additional multiplicative term is then added to eq. (2.24) due to the sun gear rotation induced AM effect, and the observed sensor signal due to sun gear fault can be modelled as x(t) = [1 − cos(2π f sunrot t] [1 + Acos(2π f sun t + φ)] cos[2π f m t + Bsin(2π f sun t + ϕ) +θ],   

 



AM by sun rotation

AM by fault of sun gear

FM by sun gear rotation

(2.37) where f sunr ot is the rotating frequency of the sun gear and f sun denotes the faulty sun gear characteristic frequency. The calculation of f sun for distributed and local fault cases has been explained in Section 2.3. Following the same procedure as mentioned in eq. (2.24)−(2.27), and eq. (2.37) can be expanded as x(t) = h( f sun , 0, 0) +

A [h( f sun , f sun , φ) + h( f sun , − f planet , −φ)] 2

A [h( f sun , f sunr ot , 0) + h( f sun , − f sunr ot , 0)] 2 A − [h( f sun , f sun + f sunr ot , φ) + h( f sun , f sun − f sunr ot , φ)] 4 A − [h( f sun , − f sun + f sunr ot , −φ) + h( f sun , − f sun − f sunr ot , −φ)] . 4 (2.38) −

36

2 The Basics and Kinematics of Gears

Taking the Fourier transform of eq. (2.38) yields X ( f ) = H ( f sun , 0, 0) +

A [H ( f sun , f sun , φ) + H ( f sun , − f planet , −φ)] 2

A [H ( f sun , f sunr ot , 0) + H ( f sun , − f sunr ot , 0)] 2 A − [H ( f sun , f sun + f sunr ot , φ) + H ( f sun , f sun − f sunr ot , φ)] 4 A − [H ( f sun , − f sun + f sunr ot , −φ) + H ( f sun , − f sun − f sunr ot , −φ)] . 4 −

(2.39)

From eq. (2.39), the following conclusion can be achieved. For a faulty sun gear case in a planetary gearbox, the sensor observed vibration signal has side-bands around three centers which are the meshing frequency f m and the meshing frequency minus and plus the rotating frequency of sun gear f m ± f sunr ot , with a side-band spacing equal to the characteristic frequency of faulty sun gear f sun . For the cases that are the signal goes through the second and third transmission paths, since the transmission paths have a scaling effect during the propagation of the faulty induced vibration signal, the observed sensor signals have the similar form as eq. (2.24), and their spectral form is also the same as eq. (2.29). Side-bands will appear around the meshing frequency f m , with a side-band spacing equal to the characteristic frequency of faulty sun gear f sun . If the AM-FM effects are considered with the higher orders of faulty sun gear characteristic frequency f sun as the modulating frequency and with the higher orders of meshing frequency k f m as the signal carrier frequency, peaks will appear at the frequency locations of k f m ± n × f sun and k f m ± f sunr ot ± n × f sun in the Fourier spectrum. Since the spectral peaks of sun gear faulty induced vibration signal are related to the faulty sun gear characteristic frequency, a sun gear fault can be detected and identified by monitoring the existence or increasing magnitude of spectral peaks at the frequency locations mentioned above.

2.4.4

Numerical Example and Model Validation

In this subsection, the planetary gearbox faulty vibration signal model described above is simulated in order to validate the above theoretical representation regarding the transmission path effects on observed vibration response and demonstrate the spectral form of planetary gearbox vibration signals. For this reason, we first generate and analyse two signals for simulating the vibration produced by local fault on a

2.4 Signal Model Characteristics of Planetary Gearbox

37

Table 2.1 Parameters for the simulation AM magnitude A

FM magnitude B

Meshing frequency of planet carrier(Hz)

Rotating frequency (Hz)

Characteristic frequency of faulty planet gear(Hz)

1

0.5

11.97

0.1478

0.3861

planet gear tooth: the first one is according to eq. (2.24) which mimics the local fault induced vibration signals at meshing point without a transmission path effect, and the other is based on eq. (2.35) which includes the AM effect due to planet carrier rotation. Table 2.1 lists the parameters utilized in the simulation. Both of the signals are sampled at a frequency of 256 Hz. Fig. 2.9 and Fig. 2.10 demonstrates the timedomain waveforms simulated faulty planetary gearbox vibration signal and their frequency spectrum, respectively. For the first signal case without considering the AM effect (or so called transmission path effects) caused by planet carrier rotation, the spectral peaks show up as side-bands around the GMF, at the frequencies of k f m ± n × f planet1 ,

∀k ∈ {1, 2, 3, 4, ...} .

(2.40)

This behaviour is consistent with the expectation from eq. (2.24). For the second simulated signal which includes the AM effect (transmission path effect) caused by planet carrier rotation, more peaks in the spectrum appear as side-bands in Fig. 2.10(b) compared to the Fig. 2.9(b). These side-bands center around the meshing frequency and the meshing frequency plus and minus the planet carrier rotating frequency at the frequencies of k f m ± n × f planet1 and k f m ± f carrier ± n × f planet1 in the Fourier spectra. This behaviour is consistent and symmetrical with the expectation from eq. (2.36). From this simulation example, the effects of transmission paths has been clearly noticed in the frequency spectrum.

38

Amplitude (m/s2)

(a)

2 The Basics and Kinematics of Gears 2

1

0

-1

-2 0

2

4

6

8

10 Time[s]

12

14

16

18

20

(b) 0.7 +

0.5 0.4 +2

−

0.3 −2

Magnitude

0.6

0.2 0.1 0 11

11.2

11.4

11.6

11.8

12

12.2

12.4

12.6

12.8

Frequency (Hz)

Figure 2.9 Simulated faulty planet gear signal without the effect of transmission path (a) time-domain, (b) Frequency spectra

2.5 Concluding Remarks

(a)

39

Amplitude (m/s2)

4

2

0

-2

-4 0

2

4

6

8

10

12

14

16

18

20

Time[s]

(b)

+

+2

+

+2

+2

+

− +

−2

−

0.2

−

+

0.3

+

−2

−

−

0.4

−

−

−

0.5 −2

Magnitude

+

0.6

+

0.7

0.1 0 11

11.2

11.4

11.6

11.8

12

12.2

12.4

12.6

12.8

13

Frequency (Hz)

Figure 2.10 Simulated faulty planet gear signal with the effect of transmission path: (a) time-domain, (b) Frequency spectra

2.5

Concluding Remarks

In order to discuss and study gearbox condition monitoring, fault detection and diagnosis, it is necessary and required to understand the principles and basics of gears operation and the characteristic features of the vibration signals when faulty gears in operation. For this reason, this chapter introduces a basic knowledge and understanding of gears and the WTB gearbox, which give us a foundation for subsequent

40

2 The Basics and Kinematics of Gears

studies. Introduction on types, gear mesh stiffness, dimensioning and geometry of the gears provides a rudimentary understanding of gears. Kinematics of the gears and fault characteristic frequency reveal some failure laws of gearbox, which become the key points for gearbox fault detection and diagnosis. A vibration signal modelling of planetary gearbox systems is demonstrated in order to gain insights into the mechanism of fault happening and the signals generated by faulty gears, particularly planetary gears, which are a complicated type of a gear stage. Besides, the effects of the transmission paths to the vibration response is discussed. In addition, this vibration faulty signal modelling can also provide relatively accurate data under different fault conditions for the planetary gear stage in the absence of laboratory testing. A numerical example of the faulty vibration signal regarding the effect of the transmission paths on detecting and location of the faults is given and the validity of the signal model is verified by comparing the simulation results for two different cases, namely, with and without the transmission path effects.

3

Vibration Signal-Based Analysis for Gear Faults

A variety of fault diagnosis technologies have been developed for detecting faults in gears, depending on the sensor types utilized or the information about the physical variables collected (for example, temperature, oil sample, acoustics and vibration, etc.). These approaches include temperature analysis [45][9], wear or oil debris analysis [172][51], acoustic-based condition monitoring [224][182], and vibrationbased fault detection [187][5] [31], etc. Among these techniques, vibration signature analysis becomes the most successful and popular one for detecting faults in rotational machinery, particularly gearboxes and bearings. The key reasons are that vibration signals are easy to collect, with a high precision, and are extremely sensitive to several defects. In general, there is a considerable background noise in the acquired vibration signals, not only from measurement itself but also from the normal operation condition of the machines, which would have an influence on the decision-making process for successful and powerful fault detection and diagnosis. Especially in the early stage of the fault, small or weak fault signals are commonly buried in high background noise. As a result, noise reduction pre-processing is critical and an important issue in signal processing for achieving reliable fault detection and diagnosis. For that, several techniques based on the aforementioned methodologies have been improved and developed in order to improve the signal-to-noise ratio (SNR) and thereby enhance the fault-related features in the signal. In this chapter, the operating mechanisms and performance of these key techniques in achieving noise reduction by increasing SNR and highlighting features of the signal of interest will be briefly reviewed in more detail, which would be conducive to developing more advanced techniques for achieving powerful fault detection and diagnosis especially for mechanical systems and rotational machinery.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_3

41

42

3.1

3 Vibration Signal-Based Analysis for Gear Faults

Time Domain Analysis

The time domain analysis of the collected vibration signal is one of the simplest fault detection and diagnosis methods. A time series vibration signal can be employed to extract a large number of features that reflect the present state of the machines. While this can be as easy as visually inspecting the vibration signal, other more sophisticated techniques [4], such as time domain parameters, are considered as one of the simplest methods of feature extractions. Furthermore, additional analysis is usually performed so that essential characteristics that are not easily appeared can be highlighted. The amplitude and temporal information contained in the gear vibration time signal are used in conventional time-domain analysis to detect gear faults. The magnitude of the signal can be utilized to identify the presence of a fault, and the periodicity of the vibration can then be utilized to indicate the location of the fault [216]. When periodic vibration is observed, defects produce wideband frequencies due to periodic impulses, and in that case the time domain techniques are acceptable. Changes in the vibration signature induced by defects can be recognized using the waveform, but determining the source of faults is difficult. Mechanical systems such as a gearbox with a fault reveal features which cannot be predicted over time. The random or non-deterministic properties of such systems cannot be accurately estimated, but they can be inferred using statistical parameters, which can then be used to predict fault propagation [148]. The indices obtained directly from the time domain waveforms of vibration signals are known as time domain parameters. The Peak Value, Root Mean Square, Kurtosis, and Crest Factor are statistical indicators based on the time-domain waveform which are extensively employed for mechanical fault identification [66][119]. The term “condition indices” is also used to describe these indicators [129]. A gearbox vibration signal is analysed, and a single value is returned to indicate whether or not the condition is within normal operating parameters. The condition index should increase as the fault becomes more severe; indicating that the condition of the gearbox is deteriorating. This analysis can sometimes be performed simply using visual inspection of the vibration signal, however, the timedomain signal is more likely to be processed to provide a statistical parameter (feature) with a known connection to vibration severity. The root mean square (RMS) value and the peak value are mutual dimensional characteristics that can directly reflect the highest amplitude and energy level of vibrations [44].

3.1 Time Domain Analysis

43

• For a discrete signal, the RMS, peak values and standard deviation can be mathematically expressed as 1 x pk = (max(x(t)) − min(x(t))) , (3.1) 2   Ns  1  xi2 , (3.2) xr ms =  Ns i=1

xstd =

  Ns   (xi − x)2   i=1 Ns − 1

,

(3.3)

where x pk , xr ms , and xstd represent the peak value, RMS value, and standard deviation of the discrete vibration signal x(t) respectively, x represents the mean value and Ns is the number of samples. From this definition, xr ms is clearly insensitive to isolate short-duration peaks in the signal, hence, it is often insufficient to detect incipient tooth failure. xr ms is a measure of the total vibration level of the system that becomes more powerful as tooth fault progresses. Therefore, it is regarded as a crucial indicator of gearbox overall condition. It is also sensitive to changes in load and speed from operational conditions [237]. • Crest factor and kurtosis are the most common non-dimensional time domain parameters, which can be utilized to identify the spikiness of the signal associated with fault-induced impulses because it focuses more on the shape description of the time domain waveform. The crest factor C F is calculated by normalizing the peak value by the RMS value, which is expressed as x pk . (3.4) CF = xr ms The kurtosis xkur , defined as the fourth standardized moment, emphasizes a large amplitude deviation from the usual level, which is described as

xkur

  N s   (xi − x)4   i=1 = . 4 (Ns − 1)xstd

(3.5)

44

3 Vibration Signal-Based Analysis for Gear Faults

The shape of the signal, rather than the overall vibration level, is a much better indicator of fault in a various types of gear faults. Crest factor is often used on the raw vibration signal to detect changes in the signal pattern caused by impulsive vibration sources such as tooth crack in the gear. During meshing, a discrete fault on a gear will theoretically produce an impulsive (short duration) signal. Because the time over which the impulse acts does not greatly increase, the peak value of the impulsive signal will increase as the damage increases, while the RMS level of the overall vibration will change only slightly [222]. All the statistical time domain features can be computed and summarized in Table 3.1. The kurtosis value of a vibration signal collected from a healthy gear is normally about 3.0 or lower, and the pattern is usually uniform. Kurtosis grows as the gear defect progresses, showing that the vibration distribution is no longer Gaussian. This is mostly owing to the damaged gear impulses (single peaks with large amplitude) [220]. Consequently, kurtosis is frequently employed as a measure to detect gear faults [218]. Clearance factor, C L F, is another time domain statistical functions that have been used to extract features from vibration signals. The clearance factor is determined by the ratio of the peak value of the vibration signal to the root mean square of the absolute value of the vibration signal and can be represented as shown in Table 3.1. Note: in Table 3.1, xi represents a signal with i = 1, 2, ..., Ns , and Ns is the number of data points of a raw signal. The shape factor, S F, is determined by the ratio of the RMS value to the average of the absolute value of the vibration signal and can be defined as shown in Table 3.1. The shape factor is useful in measuring the change resulting in the vibration signal due to unbalance and misalignment defects. Table 3.1 Statistical features definition of a vibration signal Features

Equation

Features

s xm = N1 xi s i=1 Ns  xr ms = N1 xi2

2. Standard deviation

Equation  N  s  (xi −xm )2  xstd = i=1 N −1

4. Peak value

x pk = max|xi |

N

1. Mean 3. Root Mean Square

s

i=1

N s

5. Skewness 7. Crest factor

9. Shape factor

(xi −xm )3 xske = i=1 3 (Ns −1)xstd x k C F = xrpms

SF =

xr ms N 1 s √|x | i Ns i=1

6. Kurtosis 8. Clearance factor

10. Impulse factor

s

N s (xi −xm )4 i=1 xkur = 4 (Ns −1)xstd x pk CLF = N s √ ( N1 |xi |)2 s i=1 x pk IF = N 1 s |x | i Ns i=1

3.2 Frequency Domain Analysis

45

The impulse factor, I F, is defined as the ratio of the peak value to the average of the absolute value of the vibration signal and can be represented as in Table 3.1. The impulse factor is also useful in measuring the fault impact generated in the vibration signal. To summarize, the advantages of time domain indicators for vibration monitoring are their simplicity and ease of implementation. Nevertheless, it is impossible to detect all available faults only by utilizing these simple parameters. The existence of noise and changes in operation condition constrain some of these parameters, as shown in [149]. Furthermore, when the signals monitored are combined with many other disturbed sources besides impulse-like or modulation responses produced by gear defects, these indicators are unable to clearly and properly identify the cause of the fault.

3.2

Frequency Domain Analysis

The Fourier transform is utilized in frequency domain analysis, which is widely employed in rotational machinery fault diagnostics [156] [7]. The ability to simply identify and isolate certain frequency components created by different defects is the fundamental advantage of frequency-domain analysis over time-domain analysis. When using time domain analysis, for instance, the signal from an incipient fault developing in a gear will be buried in a huge background noise, whereas in the frequency domain, the signal from the incipient fault will have a dominant peak in the spectrum at the characteristic fault frequency. When gear dimensions, teeth numbers and shaft rotation speed are available, detailed knowledge about gear fault frequencies including the GMF and its side-bands can be obtained for gear fault and diagnosis, as explained in Section 2.3. Identifying and highlighting the fault characteristic frequencies from a high number of frequency components is a key point for many frequency domain signal analysis techniques which may appear based on the fault mode, machine class, and operating environment [184][212], and then to make a reliable and correct decision. For rotating equipment condition monitoring and fault diagnosis, spectral analysis is a widely used and straightforward frequency domain technique [7]. In the frequency domain, it depicts the distribution of the original vibration signal and gives feature information more directly than in the time domain. Power spectrum is often employed to identify the locations of the fault by separating the characteristic fault frequencies from the fundamental frequency components in the spectrum. As a result, analysts may easily concentrate on these frequencies, which are useful for fault diagnostics [66]. The spectrum is a measure of vibrations over a large number

46

3 Vibration Signal-Based Analysis for Gear Faults

of discrete narrow frequency bands, whereas the overall vibration is a measure of vibrations produced over a wide band of frequencies. Hence, a popular technique to vibration CM is to transform the vibration signal to the frequency domain using the FFT method. This technique is completely acceptable if the measured signal does not change in spectral content over time (i.e. with no speed fluctuations in the rotational machine). The vibration frequencies generated by each component in a machine can be estimated for machines working at a known constant speed (see Chapter 2). Consequently, any variation in vibration level within a specific frequency band can be linked to a specific component. The relative vibration levels in different frequency bands can be analysed to provide some diagnostic information. As shown in Fig. 3.1, local faults, such as gear tooth damage, are frequently detected in the vibration spectrum by variations in the amplitude of particular frequencies (side-bands appear around the GMF f m and their harmonics have higher amplitude in the faulty case compared to healthy gear). Side-bands produced by the AM or FM of the vibration signal can provide useful information about gear faults [206]. The tooth breakage or eccentricity of the gear or shaft causes the AM signal, with a damaged tooth generating pulses at a rate equal to the gear speed. the FM, on the other hand, is caused by gear manufacturing faults such as non-uniform tooth spacing. The first three gear meshing harmonics and their side-bands, according to Randall [183], give enough information for successful gear fault diagnosis. Hence, variations in the amplitude of a particular frequency peak or a signal side-bands of the GMF can be a reliable indicator of possible gear failure. In practice, because the spacing of the side-bands is dependent on the periodic qualities of the loading and the transmission path as discussed in previous chapter, extracting relevant features

Figure 3.1 Gear vibration signal spectrum for Healthy and Faulty gears

3.2 Frequency Domain Analysis

47

from vibration spectra using only an FFT is difficult. If the SNR is low and the vibration spectrum includes a large number of frequency components due to the system complexity. Thus, distinguishing fault peaks from peaks from other sources becomes nearly difficult. This is the most difficult challenge and problem with the FFT-based fault detection approach. To deal with this problem and to enhance the performance of FFT method, envelope analysis has become one of the popular vibration signal processing techniques for detection and diagnosis of incipient fault in rotational machinery [86]. As mentioned, local faults such as tooth crack and teeth surface wear generate an AM-FM with an impact in the vibration signal. The periodic excitation and the AM of a faulty signal can be extracted using envelope analysis [86]. Envelope analysis, which extracts the envelope of a signal, is used to obtain the envelope spectrum utilized to detect rotational machinery fault frequency components. For amplitude demodulation and extracting the envelope of a signal, Hilbert transform (HT) algorithms are commonly utilized [185]. The envelope analysis technique, which will be studied in detail in Chapter 5 using the HT method, was described in detail by Ho and Randall [84]. This technique is commonly used for a frequency area with the maximum SNR, such as the high frequency area. However, accessible vibration signals are significantly contaminated by high amounts of background noise, which encompasses all other vibration sources in the machine, and the envelope spectrum is still subject to discrete and random noise masking. As a result, Ho and Randall [84] and Williams et al. [247] have used adaptive noise cancellation (ANC) and adaptive line enhance (SANC) combined with envelope analysis to reduce interference from discrete frequencies (for example, removing gear-induced frequency components from the band before demodulation), this will be discussed in the next subsection. To demonstrate the envelop analysis approach for detecting and diagnosing gearbox faults, envelop analysis is carried out for the faulty experimental vibration signals to detect gear faults. The GMF is 270 Hz and the rotational gear shaft frequency is 8 Hz. Fig. 3.2 demonstrates the process of envelop analysis for fault detection. The envelop line can be generated by using the HT technique as illustrated in Fig. 3.2(a). From the envelope spectrum in Fig. 3.2(b), distinctive peaks of the GMF and its sidebands in the spectrum are better highlighted compared to the original signal, where the main dominant peaks around the GMF are related to the gear fault characteristic frequency. This means that a fault in the gear teeth exists. The main advantage of the envelop analysis is that the high frequency components can be filtered out which are uncorrelated with fault frequency, while the pure periodic and modulation information can be extracted. This approach is effective for both the gear and bearing incipient fault detection and diagnosis, since the modulation and periodicity are the

48

3 Vibration Signal-Based Analysis for Gear Faults

Sidebands

Sidebands

Figure 3.2 Gear vibration based signal processing using envelope analysis

key characteristic to distinguish the cases of gear tooth damage and bearing inner race, outer race fault and normal operation. However, this method becomes less efficient when the impulse responses, the AM and FM are no longer visible, which often happens to severe gear and bearing faults. Furthermore, during the envelop analysis process, a significant quantity of information about the gearbox system is neglected, and also envelop analysis is typically employed as a first step in the diagnosis of gearbox faults.

3.3

Joint Time-Frequency Domain Analysis

Signal characteristics for each domain are produced by analysing vibration signals in the time domain and frequency domain separately. When a time-domain signal is converted to the frequency domain, the detailed information regarding the time domain is lost. Therefore, one of the disadvantages of frequency domain analysis is that when converting to the frequency domain, several non-stationary or transient characteristics, such as trends, drifts, abrupt changes, and the end and beginning effects, which are often important parts of the signal, are lost. Consequently, both techniques have drawbacks and limitations. It is also worth mentioning that the Fourier transform has limitations that apply only to stationary signals. The characteristics of signals over both time and frequency are represented by time-frequency analysis in order to highlight failure modes more clearly for better and more precise diagnostics. Recently, several studies have been conducted on vibration signal analysis in the time-frequency domain, with the goal of combining this with frequency domain analysis to provide a complete description of a vibration signal. Time-frequency analysis gives information on the spectral contents of the sig-

3.3 Joint Time-Frequency Domain Analysis

49

nal change over time, allowing transient features such as impacts to be extracted. In recent years, joint time-frequency representations such as the STFT [241], WignerVille Distribution (WVD) [155][243][213], and the WT [214][221][173] have commonly used as a technique to address for the lack of either time domain or frequency domain analysis. Atles et al. [12] provide an overview of the joint time frequency application for machine fault diagnostics. The main differences between all these transforms are their respective time and frequency resolutions. WT analysis has been proved to be an effective method for gear condition monitoring. The WT technique, unlike the STFT, employs wide time windows at low frequencies and narrow time windows at high frequencies [219][255]. Therefore, it is a very powerful tool for the analysis of non-stationary and transient signals. Discrete and continuous wavelet transformations [244] can be used to detect abnormal transients caused by early stage gear defects. Despite the fact that the discrete WT provides a highly efficient signal representation with no redundancies, the resulting time-scale map is constrained and not very informative. Lin et al. proposed a linear wavelet transform idea, in which the wavelet map was normalized based on signal amplitude rather than energy [133]. Boulahbal et al. [27] studied cracked and chipped tooth faults using both the WT amplitude and frequency at the same time, and recommended polar representation as a valuable technique for detecting the damage location of the gear in the WT maps. Description and the importance of these techniques especially for the machinery fault diagnostics will be more explained in the following sections.

3.3.1

Short Time Fourier Transform

The STFT is one of the common time-frequency analysis techniques that involves applying a Fourier transform to a short segment signal separated by a window that moves continuously along the time axis. The STFT captures frequency characteristics as functions of time by using sliding windows in time. A 2D time-frequency distribution is constructed by evaluating the frequency content of the signal as the time window moves [73]. Two steps are required to achieve time-localization: the first step is the signal windowing, in which only a small section (well-localized) of the signal is cut off at time x(t). Then, the second step is by applying the Fourier transform to this segment of the signal. To achieve this, the time domain vibration signal is multiplied by the sliding window w(t − τ ), at each fixed time of interest,(t). The signal is emphasized at time t, and the Fourier transform of the resulting windowed signal is calculated, yielding the continuous STFT [241]

50

3 Vibration Signal-Based Analysis for Gear Faults

 ST F T (τ, f ) =

∞

∞

w(t − τ )x(t)e−i2π f t dt ,

(3.6)

a Gaussian window of the following form was utilized as the sliding time window function: 2 w(t) = e−(t/a) , (3.7) where a is constant which represents the width of the window used. Therefore, eq.(3.6) can be rewritten as b ST F T (τ, f ) =

x(t)e−(t−τ/a) e−i2π f t dt . 2

(3.8)

a

The Fourier transform of the function x(t) windowed by w(t) centered at time τ will be obtained using this formula. It is possible to generate a description of how the frequency content of the signal varies with time by repeatedly executing the same analysis at several translated locations τ . Forrester introduced the STFT method to the detection of gear failures, which was one of the first applications of time-frequency methods to gear defects [66][65][68]. The main drawback of the STFT application is that during the entire analysis, the window width remains constant. This is a restriction, when better resolution is necessary to detect sudden changes over time. The width of the widow cannot be changed once it has been selected during the transform [232]. The energy distribution of a signal x(t) over time-frequency space can be used to detect some gear defects. For vibration signal analysis, Wang et al. [244] has used a combination of STFT and the Gaussian window function. The wavelet transform is similar to the STFT in that it generates a time-frequency map of the signal being studied. WT is superior to STFT because it can attain higher frequency resolutions with sharper time resolutions.

3.3.2

Wavelet Analysis

The wavelet transform technique can be utilized as an alternative method to the STFT. The wavelet transform approach compares numerous components of the vibration signal at various resolutions, whereas the STFT method measures the local frequency content of the signal. The wavelet transform is a decomposition of a signal into a set of basis functions called wavelets, which are derived from a signal

3.3 Joint Time-Frequency Domain Analysis

51

prototype wavelet by dilations, scaling, and shifts [192]. In the STFT technique, the sine and cosine signals are multiplied by a fixed sliding resolution. The window in the WT technique is already oscillating and is known as a mother window. Rather of being multiplied by sine and cosine, the mother wavelet is expanded and contracted based on the value of the dilation parameter. The mother wavelet then generates more wavelets, which is the foundation of wavelet analysis. In recent years, numerous WT analysis-based signal processing approaches have been widely used for rotating machinery condition monitoring. It is possible to determine not only the existing frequencies in the signal but also the length of each individual frequency in time by decomposing a time series into time-frequency space [221][173]. This is particularly useful when analysing vibration signals from faulty rotational machines, where small or large scale variations in the vibration might occur, depending on whether it is local or distributed fault [242]. In particular, WT is widely used to identify all possible transients in vibration signals produced by various faults when monitoring the condition of a gearbox. Different types of gear damages can be highlighted with a single time-scaled distribution analysis from the transform because WT has multiple resolutions for localization of the short time components [191]. In addition, several researchers have described their use for detecting local faults in gears and bearings [214][193]. As mentioned, the WT decomposes a signal into the time-scale domain using wavelets a,b (t), which is a set of linearly independent functions produced from a single mother wavelet with dilation and translation, i.e. 1 t −b a,b (t) = √ ( ), a a

(3.9)

where (t) represents the mother wavelet function, a denotes the scale parameter (dilation parameter) and b defines the time shift parameter (translation parameter). The continuous wavelet transform of a signal x(t) can be defined with respect to wavelets as  ∞ 1 t −b x(t) √  ∗ ( W Tx (a, b) = ), (3.10) a a ∞ where ∗ is the complex conjugate, and the factor √1a is used to ensure energy preservation. W Tx (a, b) defines the wavelet transform coefficients, This is a measure function that displays how linked the signal x(t) is to the wavelet family  of scale a and a specific time b. The more the signal feature component is similar to the wavelets, the bigger is the corresponding wavelet coefficient. The effectiveness of wavelet transforms is found by the creating and choice of wavelet basis function.

52

3 Vibration Signal-Based Analysis for Gear Faults

There are various types of wavelet functions which can be selected to best match the features of signal of interest [137]. Yang and Ren [256] demonstrated that the Morlet wavelet is more efficient than other wavelet functions for analysing signals with periodic impulsive features, which are common in signals from gearboxes due to faulty teeth. Wavelet analysis, due to its flexibility and efficient computational implementations particularly its effectiveness in processing non-stationary vibration signals, is widely used in machine fault diagnosis [43], including the time-frequency analysis of signals, fault feature extraction, denoising and extraction of weak signals, and so on. During the first denoising procedure, WT, on the other hand, requires prior knowledge of the noise characteristics. This knowledge is typically unknown in practice. As a result, wavelet analysis reveals a strong relationship between wavelet scale and frequency [219][238]. This enables for the detection of extremely short transient signals in the time dimension, allowing non-stationary vibration signals to be detected [130]. Wavelet packets [163] is a newly proposed approach based on the wavelet transform for detecting gear localized faults. The wavelet packet transform (WPT) is a wavelet transform version with constant time and frequency resolution and has been utilized in signal processing and fault diagnosis applications [118][125].

3.3.3

Wigner-Ville Distribution

The Wigner-Ville Distribution (WVD) is a more precise approach for analysing Joint time-frequency signals. It is a widely used time-frequency analysis approach because it has a high resolution that allows for accurate representation in both the time and frequency domains. WVD, on the other hand, is particularly vulnerable to aliasing-related errors and nonlinear behaviour during computing. The original real signal must be transformed into a more complicated analytical signal in order to avoid aliasing [120]. Wigner created the Wigner distribution in 1932 with the goal of studying statistical equilibrium in quantum physics. The Wigner distribution was first employed in signal processing in 1947 by Ville. It became later, among the signals processing community, known as the Wigner-Ville Distribution (WVD).

3.3 Joint Time-Frequency Domain Analysis

53

In terms of time, the Wigner-Ville Distribution, W V D(t, f ), can be expressed as [120]:  τ −i2π f τ τ ∗ x t− e dτ , (3.11) W V D(t, f ) = x t + 2 2 where x∗ is the complex conjugate of the analytic signal x which can be determined by using HT of the time domain signal x(t). The ability of the WVD approach to overcome the limitations of the STFT method by improving both time and frequency resolution is one of its main advantages. To detect gear failures, Forrester used the WVD approach [67]. In this work, the WVD has been shown to be capable of detecting various gear faults, such as a tooth crack and pitting. McFadden and Wang, employed normal and weighted WVD techniques for detecting gear faults [158]. Later, Baydar and Ball showed how to identify various gear failures using a smoothed version of WVD [17]. Visual inspection of the WVD contour plots or monitoring of changes in the statistical properties of the distribution are used to diagnose faults. The major disadvantage of employing WVD is that it has a non-linear behaviour that creates severe interference (cross-terms) between distinct signal components. For example, the WVD of a signal with two frequencies at 300 Hz and 900 Hz will have a spurious frequency measurement at 600 Hz due to a cross term. The interpretation of the energy distribution is greatly complicated by cross-term interference, and signal recovery can be difficult due to aliasing components. Meanwhile, signal resampling can be used to reduce aliasing. It is often required, before evaluating the WVD, to weight the signals using a window function for computational reasons. With respect to the frequency variable, the pseudo-Wigner distribution is employed as a smoothed version of the original WVD. The FFT and other more efficient algorithms are used to compute this updated approach efficiently [10].

3.3.4

Minimum Entropy Deconvolution Technique

In a planetary stage of the WTB gearbox, consider a sun gear with a faulty tooth such as (pitting, spalling, etc.), when the faulty sun gear tooth meshes with the planet gear, fault-related vibrations are generated due to the abnormality in meshing stiffness. The gearbox behaves like a healthy one when the faulty gear tooth is not engaged in meshing with any planet gear; no fault related dynamic features can be noticed from sensor acquired data till the faulty gear meshes with planet gear again next time.

54

3 Vibration Signal-Based Analysis for Gear Faults

Faulty gear vibration signal x mainly contains regular vibrations u caused by (normal time varying gear meshing stiffness), impulse fault related sequences d (when fault occurs), noises n and impulse response function (IRF) of the signal transmission path effects, as explained in Chapter 2, which can be represented as h. Therefore, a faulty gear vibration signal x can then be modelled as x = h ∗ (u + d + n) ,

(3.12)

where ∗ represents convolution, also ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ u1 d1 n1 x1 ⎢ u2 ⎥ ⎢ d2 ⎥ ⎢ n2 ⎥ ⎢ x2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x = ⎢ . ⎥, u = ⎢ . ⎥, d = ⎢ . ⎥, n = ⎢ . ⎥ . ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ x Ns u Ns d Ns n Ns ⎡

Fault related impulses d are a type of transient signal that can be easily submerged by regular components. For instance, detecting planet gear fault associated impulses was a challenge in the previous chapter, making it difficult to diagnose planet gear faults effectively. The sun gear, which is placed in the center of the planetary gearbox, has a significantly longer signal transmission path compared to planet gear, and the sensor acquired vibration signal due to sun gear defect is much weaker. As a result, in order to diagnose a sun gear defect, fault feature enhancement is required. This is often carried out by reducing the noise in the faulty vibration signal by increasing the signal to noise ratio. To address this problem, MED (minimum entropy deconvolution) as a filter is employed to extract fault associated impulses. The MED technique has been firstly proposed by Wiggins [249], when seeking to extract comprehensive pulse-like signal information from amplitude anomalies in reflection seismic recordings. The main goal of MED is to design an inverse filter that minimizes the entropy of the signal, or in other words, maximizes the impulsiveness (kurtosis) of the signal. This filter will then be used to deconvolve (rather than convolve) the signal, recovering the fault related impulses in a relatively clear manner [15]. The MED is designed to remove the influence of the transmission path , which was discussed in the previous chapter, by optimizing the finite impulse response (FIR) filter. Assume the input vibration signal with the number of measured samples, Ns , is represented as a sequence xi (i = 1, 2, 3, .....Ns ), which contains a white Gaussian noise and other interference signal components including impulses and harmonics.

3.3 Joint Time-Frequency Domain Analysis

55

The output signal yi achieved by the input signal vibration signal xi and an FIR filter with L coefficients as follows yi =

L 

fl xi−l ,

(3.13)

l=1

where

  f = f1 f2 · · · f L ,

here f 1 , f 2 , · · · , f L are filter coefficients and also yi should match the impulsive signal generated by the gear or bearing defect as closely as possible. Consider an impulsive fault signal with a high kurtosis, the chosen MED filter must maximize the output signal kurtosis. The kurtosis of the output signal yi with zero mean is expressed by Ns  yi4 Kf =

i=1 Ns  (yi2 )2

.

(3.14)

i=1

The filter that optimizes the kurtosis of the output signal is the optimal solution f opt can be given by f opt = { f opt : K ( f opt ) ≥ K ( f )), ∀ f ∈ R L } , where R L is the L− dimensional Euclidean space; K represents the Kurtosis defined in eq.(3.14) and f is any L−sized filter. The standard MED technique differentiates eq.(3.14) with respect to f and sets it to zero in order to optimize the filter solution. The iterative local maximum solution can be formulated as Ns 

f =

(X 0 X 0T )−1 X 0 [y13 y23 · · · y N3 s ]T ,

i=1 Ns  i=1

where

yi2 yi4

(3.15)

56

3 Vibration Signal-Based Analysis for Gear Faults

⎡

x1 ⎢0 ⎢ ⎢0 ⎢ X0 = ⎢ 0 ⎢ ⎢. ⎣ .. 0 

x2 x1 0 0 .. . 0

x3 x2 x1 0 .. . 0

··· ··· ··· ··· .. . ···  Ns

⎤

x Ns x Ns −1 x Ns −2 x Ns −3 .. . x Ns −L+1

⎫ ⎪ ⎥⎪ ⎪ ⎥⎪ ⎬ ⎥⎪ ⎥ ⎥ L ⎥⎪ ⎥⎪ ⎪ ⎪ ⎦⎪ ⎭ 

The iterative procedure begins with an initial filter and then uses eq.(3.15) to update the filter coefficients. The details of this process are described in [151], which includes the derivation of the iterative formula and the stopping criteria of the iteration algorithm. Remark 3.1 What is actually extracted from the MED method by maximizing kurtosis as the cost function is not the actual fault impulse d, but rather a pulse-like signal that simply pulses at the exact same time as the actual fault impulse. This signal is useful for fault detection because it has the same fault characteristic frequency as the actual fault impulse, but it has nothing to do with the estimation of the fault (Fig. 3.3). To validate the MED filtering process in gearbox vibration signal, we use the vibration datasets from national renewable energy laboratory (NREL) experimental gear vibration model, it can be seen that after the MED process in the time domain,

Noise =

Fault impulses in original form

∗( + )

ෝ≈ AR filter

Transmission path effect

Removes periodic Gear mesh

( Periodic components like gear mesh (Normal operation) Gear vibration signal components

Figure 3.3 AR-MED filtering process [55]

∗

)

MED filter

Filter coefficients optimized to extract the impulsiveness of the signal

(d)

(c)

Figure 3.4 MED processing results (a) faulty gear signal, (b) its power spectrum, (c) gear signal after MED filtering, (d) power spectrum of MED filtered signal

(b)

(a)

3.3 Joint Time-Frequency Domain Analysis 57

58

3 Vibration Signal-Based Analysis for Gear Faults

moments of sudden changes appear, i.e. the gear fault area. Nevertheless, the effectivity of this method almost without exception is affected by noise. Fig. 3.4(a,c) illustrates the deconvolution result of the original signal before and after MED denoising process in the time domain, the impacts appear after MED methods indicates the faults in the gear. Also, in Fig. 3.4(d), the spectrum of the signal after MED has a clear GMF and its side-bands especially in the second gear mesh compared to the noisy signal before the MED method in Fig. 3.4(b). This indicates that the fault occurs in the gear as the amplitude of the frequency spectral are more dominant compared to the original gear signal which means that the MED filter improves the detectability of gear fault at early stage. As a result, this technique is commonly utilized to improve impulse definition. The MED filter and the filtered signal will inevitably lose their physical meaning in this circumstance. Regardless of whether an impulse exists in the original signal, the MED can always produce a signal with a significant kurtosis. Furthermore, the resultant impulse signal cannot be correlated with the original impulse signal.

3.3.5

Empirical Mode Decomposition

In recent decades, empirical mode decomposition (EMD) has been proposed by researchers in the field of rotational machinery fault diagnosis as a novel approach which was considered as one of the most powerful and effective time-frequency analysis techniques. It is based on the time scales local characteristic of the signal and decomposes the signal into a set of complete and almost orthogonal components known as intrinsic mode functions (IMFs). The IMFs represent inherent oscillation mode of the signal and act as basis functions that are calculated by the signal itself rather than pre-defined kernels. Therefore, it can be considered as a self-adaptive signal processing technique which is convenient with nonlinear and non-stationary data. For that reason, in recent years, researches on EMD have become popular for fault detection of rotating machinery [123][126][127]. Huang et al. [87] were the pioneers of the EMD approach. The idea is that, EMD method decomposes a signal into IMF components, under the assumption that each signal consists of several simple IMFs representing simple oscillatory mode embedded in the signal. The decomposed IMFs from the signal should satisfy the following two conditions based on the definition of EMD [87] • The number of extrema and the number of zero-crossings, in the whole dataset, must either equal or differ at most by one;

3.3 Joint Time-Frequency Domain Analysis

59

• At any point, the mean value of the envelope estimated by the local maxima and the envelope defined by the local minima is zero. The signal x(t) is decomposed by the EMD decomposes into several IMFs in the following steps [87][74]: 1) Identify all of the local minima and maxima of the signal , then use cubic spline lines to interpolate the local minima and maxima to build lower and upper envelops, respectively. and then compute the mean of two envelops as m 1 . 2) The difference between the signal x(t) and m 1 is computed as a candidate of the first IMF (3.16) h 11 = x(t) − m 1 . If h 11 meets the IMF requirements above, it is considered as the first IMF c1 (t); otherwise, h 11 is treated as the original signal, and the processes above are repeated until the candidate h 1k meets the IMF conditions, at which point h 1k is designated as the first IMF c1 (t) c1 (t) = h 1k .

(3.17)

3) The first IMF c1 (t) is then separated from the x(t) by x(t) − c1 (t) = r1 (t) ,

(3.18)

where r1 (t) is the residue which is considered as the next original signal and the other IMFs, c2 (t), c3 (t), . . . , cn (t) can be obtained by using the same process above, which satisfy r1 (t) − c1 (t)

= r2 (t)

.. .

(3.19)

rn−1 (t) − cn (t)

= rn (t) ,

The decomposition process is stopped until rn (t) becomes monotonic. The decomposed signal x(t) can then be reconstructed by summing up both eq.(3.18) and eq.(3.19) yields n  ci (t) + rn (t) . (3.20) x(t) = l=1

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3 Vibration Signal-Based Analysis for Gear Faults

The IMFs c1 (t), c2 (t), c3 (t),. . ., cn (t) contain various frequency bands ranging in descending order. Each frequency band has multiple frequency components that change with the variation of the signal x(t), whereas the residual rn (t) defines the central tendency of the signal x(t). As it can be noticed, the EMD approach performs quite well in processing nonlinear and non-stationary signals, despite having such a simple, intelligible, and easyto-implement algorithm. EMD is also a self-adaptive signal processing approach, in contrast to wavelet analysis. Thus, in the past few years, research on EMD applied to fault diagnostics of rotating machinery has become popular. Every year, a large number of academic publications on this topic are published. Lei et al. [126] reviewed and summarized recent EMD research and development in the field of rotating equipment fault diagnostics [124][205][140]. Despite the fact that the EMD approach presents a good performance in the processing of non-stationary data, the algorithm itself has some shortcomings, such as a lack of theoretical mathematical foundation, the IMFs are not strictly orthogonal each other, end effects issue, sifting stop criterion, extremum interpolation, etc. Various theoretical evaluations have been conducted, and many improved EMD methods have been developed, in order to address these problems, including WPTEMD (Wavelet packet transform EMD), EEMD (Ensemble EMD), BS-EMD (Bspline EMD), Multivariate EMD, and so on, which will go into details in the next chapter. Despite various improved EMD approaches, the problem of a lack of theoretical mathematical foundation has remained unsolved so far. The reason why EMD approach is so elegant on the noise reduction and fault diagnosis issues cannot be demonstrated and explained using a mathematical theory.

Table 3.2 Correlation coefficient for each IMF to select the best IMF IMF order

1

2

3

4

5

Correlation coefficient

0.6885

0.2701

0.3642

0.2432

0.2205

To demonstrate the effectiveness of the EMD method, an example of a gear vibration dataset in [206] is used. Based on EMD method, 5 IMFs are extracted from the signal as shown in Fig. 3.5. Several methods have been utilized to determine the most effective IMFs related to the original vibration signal. One of the common ways is by calculating the correlation coefficients between the signal and the extracted IMFs as shown in Table 3.2. It can be observed that c1 and c2 have the highest correlation coefficients. This indicates that these two IMFs with the high-

3.3 Joint Time-Frequency Domain Analysis

61

Figure 3.5 The vibration signal and its five decomposed IMFs using EMD method

est correlation coefficient have the most fault information or the impulses of the simulated faulty signal. EMD can also be used as a noise filter, extracting the IMFs of interest while eliminating the useless IMFs in order to reduce the noise [26]. Fig. 3.6 demonstrates that the reconstructed signal obtained by extracting the IMFs c1 −c4 is quite similar to the original simulated vibration signal, indicating that the EMD method is effective for noise reduction for the purpose of better gear system fault detection. However,

(a)

(b)

Figure 3.6 Signal reconstruction by combining IMFs 1–3 (a) 1500 sample points , (b) selected 300 sample points

62

3 Vibration Signal-Based Analysis for Gear Faults

because the EMD approach is empirical method, prior knowledge plays a crucial role for extracting IMFs. The EMD approach, for example, performs effectively in the above simulation vibration gear signal because a good critical condition was chosen, namely that the correlation coefficients of the decomposed signals should be higher than a critical value. Normally, such a critical condition is task-oriented and varies based on the several demands. As a result, the EMD method for the FD performance is greatly dependent on the influence of human interaction, which is one of the major drawbacks of this method.

3.4

Other Approaches

There are some other signal processing approaches include order tracking, time synchronous average (TSA) and adaptive signal processing which are also critical for gear fault diagnosis and signal denoising in the literature. In the following, each of these techniques are briefly discussed and explained.

3.4.1

Order Tracking

Multi-stage gearboxes are considered as a complicated part of the rotational machinery and the condition monitoring of such rotational machines can be carried out by measuring their vibration signals. The number of revolutions per minute (RPM) determines the waveform of vibration signals generated by these machines. The majority of rotational machinery operates at varying rotational speeds. This considerably complicates the traditional spectrum analysis of vibration signals with constant sampling frequency. Spectral analysis may be inadequate for early detection of gear failures in gearbox condition monitoring, especially in the case of local faults that largely influence side-bands in the spectrum. As a result, determining the spacing and evolution of side-band families in a spectrum may be difficult. Thus, order analysis is frequently utilized [110]. The vibration signal is re-sampled in Order Analysis from equal time to equal phase increments. Consequently, a signal with constant angular frequency components is produced, which can be used with TSA in the next section. A synchronously recorded reference signal is utilized to calculate the equal phase spacing used to resample the signal. A reference signal from a tachometer directly linked to the rotating shaft of interest is typically used for order analysis [25]. Order Analysis permits the sampling rate during data acquisition to be synchronized with the rotation speed of the shaft

3.4 Other Approaches

63

in order to lock the sampling locations to constant shaft angles. As a result, even if the speed changes, the sampling will take place at the same shaft angle, and the frequency content of the measured signal will be a function of order instead of time. This can be obtained, during data acquisition to record the shaft position, by attaching an encoder to the shaft. The data is then transformed into order domain using the Fourier transform analysis, with the result being the amplitude of the sine waves at fixed multiples of the basic rotation frequency, i.e. the order amplitude.

3.4.2

Time Synchronous Average

Time synchronous averaging (TSA), sometimes known as time domain averaging, is a widely used signal processing technique for extracting periodic waveforms from noisy inputs [85][19]. It is considered as an effective algorithmic tool for analysing vibration signals in studying the condition of rotating equipment [153]. Furthermore, while the TSA is an effective method for isolating synchronous gear and shaft components, it fails to isolate random vibration components because subtracting a single TSA signal from the original signal results in a mixture of random signals and other shaft-synchronous components. A frequency domain solution was developed to overcome this problem by performing the FFT of the full signal and simply eliminating spectral peaks at discrete harmonics of known periodic signals [157]. Multiple resampling and subtraction of time synchronous averages from a signal were also tried in the time domain method [38]. Both of these problems, however, have shortcomings. Except in the very exceptional scenario where the number of samples per period and the number of periods are powers of two, the frequencydomain technique must deal with the issue that the frequency of the harmonics leaks into adjacent windows. The periodic signal will not be totally removed if discrete peaks at shaft harmonics are removed. The time-domain technique described in has the drawback that certain timedomain properties are shared by the time synchronous average of two or more shafts, such as the signal corresponding to the GMF. This method is used to describe the time-domain of the vibration signal produced by the meshing of the gear teeth throughout one complete revolution. TSA of the vibration signal can help remove all frequencies from the gear vibration signal except the fundamental and harmonics of the teeth meshing frequency. In this manner, visual inspection may typically identify the variation in the vibration signal produced by a single gear tooth [153]. In practice, this carried out by averaging a set of signal segments, each of which corresponds to one period of a synchronizing signal. Given a periodic signal x(t), and x(t) = x(t + T ). Measurement signal y(t) contains signal x(t) and noise signal

64

3 Vibration Signal-Based Analysis for Gear Faults

w(t). Without losing generality, assume Ns number of sample points and and T the averaging period are available for analysis, i.e. y(t) = x(t) + w(t), t ∈ (0, Ns T ] .

(3.21)

To calculate a synchronous average signal y(t) from a time-based signal x(t), an additional trigger signal, such as pulse trains from a shaft encoder, is required, as shown in Fig. 3.7. TSA signal can be formulated as yT S A (t) =

Ns −1 1  y(t + i T ) Ns i=0

Ns −1 1  = x(t) + w(t + i T ), t ∈ (0, Ns T ] . Ns

(3.22)

i=0

This can also be represented or modelled as the convolution of y(t) with a train of Ns delta functions displaced by integer multiples of the periodic time T [19], y(t) = c(t) ∗ x(t) ,

(3.23)

where c(t) defines a train of Ns impulses each of amplitude 1/Ns , the impulses are spaced at intervals T = 1/ f , and can be given by c(t) =

Ns −1 1  δ(t + i T ) . Ns

(3.24)

i=0

In the frequency domain, this is identical to multiplying X ( f ) and C( f ), which are the Fourier transforms of the signal and impulse train, respectively [29], i.e. Y ( f ) = C( f ) ∗ X ( f ) ,

(3.25)

where ∗ represents the convolution, also C( f ) = (1/Ns )

sin(π Ns T f ) , sin(i T f )

(3.26)

which is basically a comb filter. The only frequencies that are passed for a large number of averages Ns are precise multiples of the trigger frequency f (t). In addition,

3.4 Other Approaches

Gear vibration signal ( )

65

TSA Data input

( )

Trigger Input

T

Tachometer Pulses Figure 3.7 Time synchronous averaging process

signal processing with TSA is frequently too complicated and complex for reliable analysis in real-world situations. This is due to the challenges of obtaining reliable trigger signals, either from hardware such as shaft encoders or from a demodulating meshing component for a phase signal. TSA can successfully suppress noise in general, but it cannot define nonlinearity. Further, it is often difficult to detect clear symptoms of any defect in the gear if TSA is used in isolation. The technique may also fail to detect and differentiate between faults, particularly if multiple faults are present simultaneously in multiple gears within the gearbox. A wide variety of different techniques have been explored over the years to further process the TSA method to make it more sensitive to early fault detection. Over the years, a variety of different techniques have been investigated to improve the TSA method and make it more sensitive to early fault detection [244].

3.4.3

Noise Cancellation Based Adaptive Signal Processing

The vibration signal acquired from a large machine containing several components might be unable to detect the sources when represented in the time domain. modulations and Impulses generated by the rotational machines such as gears and bearing faults are typically buried in other signal components from the machine and background noise, making it difficult to clearly identify the impulse responses from the gear or bearing vibration signals. Noise reduction requires more advanced time domain approaches, such as adaptive filters. Classical adaptive filters are linear digital filters (usually finite impulse response) which set their parameters or coefficients to improve some performance criterion [121].

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3 Vibration Signal-Based Analysis for Gear Faults

For the sake of presentation, the real-time signal to be filtered is assumed to be as x(k) = s(k) + n(k) ,

(3.27)

where s(k) is the fault-related signal of interest that has been contaminated by additive noise and interference signal n(k). Two adaptive filter techniques, namely adaptive noise cancellation (ANC) and self adaptive noise cancellation (SANC), typically utilized for gear and bearing fault detection under low SNR circumstances, will be described in the following. The configuration of the two types of adaptive filters are shown in Fig. 3.8 and Fig. 3.9. 1) Adaptive noise cancellation Adaptive noise cancellation is obtained by using two measurements from a “primary” transducer and a secondary “reference” transducer as illustrated in Fig. 3.8. The signal carrying the information about the fault condition which is called primary signal x(k) is mixed with uncorrelated background noise. The reference input signal v(k) should be uncorrelated with the signal of interest but correlated version of the corrupting noise. The filter modifies its coefficients automatically to minimize a cost function, usually the mean squared error [E[e2 (k)] [121][86]. We can utilize a FIR filter with the tap weights adapted using the LMS algorithm for the adaptive filter. The filter utilizes the reference input to estimate (at its output) the noise and interfering signal present in the primary input. As a result, the influence of interference and noise is reduced by subtracting the adaptive filter output from the primary input [82]. Applying the LMS algorithm, the tap-weight update of the filter is expressed by means of the equations y(k) =

M−1 

wˆ i (k)v(k − i)δ(t + i T ) ,

(3.28)

i=0

e(k) = x(k) − y(k) ,

(3.29)

and wˆ i (k + 1) = wˆ i (k) + μv(k − i)e(k), i = 0, · · · , M − 1 ,

(3.30)

where M represents the length of the FIR filter and the constant μ denotes the step-size parameter. The issue with ANC is that it requires a reference signal that is highly correlated with the noise signal n(k), but not correlated with the signal of interest. The availability and quality of the reference signal are usually the limitation for the performance of an ANC scheme.

3.4 Other Approaches

Primary input ( )

67

Gear + Bearing

( )

+

−

Reference input ( )

Adaptive Filter

( )

Figure 3.8 Adaptive noise canceller

Primary input ( )

Gear + Bearing

( )

+

−

Delay −

Reference Input ( − )

Adaptive Filter

( )

Figure 3.9 Self-adaptive noise canceller

Algorithm 3.1 shows the ANC procedure. In adaptive filtering, the step size and filter length are important parameters. The value of the step size influences the convergence speed of the adaptive filter, stability and steady state error. Low steady state error and a slower convergence speed are ensured by a short step size. While a big step size increases convergence speed, it also increases the instability. The computational resource needs, convergence speed, and steady-state error are all affected by the filter length. A trial-and-error procedure determines the optimal filter length. However, to select the most effective filter length, an adaptive filter simulation is recommended. The number of significant taps in the unknown system’s impulse response must be larger than the length of the filter. The steady-state error can be minimized by using a lengthy filter length. It is highly desirable to optimize the filter length to meet the needs of the application [82].

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3 Vibration Signal-Based Analysis for Gear Faults

Algorithm 3.1 ANC procedure STEP 1: Optimal selection of the order of filter coefficient M, step size μ, and the initial filter coefficient wˆ 0 (k); STEP 2: Evaluate output signal from the adaptive filter y(k) =

M−1 

wˆ i (k)v(k − i)δ(t + i T ) ;

i=0

STEP 3: Measurement of the error signal as e(k) = x(k) − y(k) ; STEP 4: Updating of the adaptive filter coefficient using LMS algorithm wˆ i (k + 1) = wˆ i (k) + μv(k − i)e(k), i = 0, · · · , M − 1 .

2) Self-adaptive noise cancellation The self-adaptive noise cancellation (SANC), as shown in Fig. 3.9, in fact is a degenerate version of the adaptive noise canceller since it only requires a single input signal and replaces its separate reference signal with a delayed version of the primary input signal. The delay d is set to a big enough value to eliminate the correlation between the noise n(k − d) in the reference signal and the noise n(k) in the original input signal [231]. The reference signal x(k − d) is processed by an FIR filter to yield an error signal e(k), represented as the difference between the SANC’s output y(k) and the actual input x(k). The LMS algorithm adjusts the tap-weight parameters f the FIR filter to reduce the mean-square value of the error signal. Similar to ANC, the tap-weight parameters of the filter are adaptively updated by means of the following equations y(k) =

M−1 

wˆ i (k)x(k − d − i) ,

(3.31)

i=0

e(k) = x(k) − y(k) ,

(3.32)

and wˆ i (k + 1) = wˆ i (k) + μx(k − d − i)e(k), i = 0, · · · , M − 1 .

(3.33)

3.4 Other Approaches

69

SANC appears to be very similar to ANC in terms of both algorithm and configurations, with the exception that SANC only requires one single input signal. However, another significant difference is that ANC achieves the fault-related signal of interest by subtracting the adaptive filter output y(k), which offers a noise and interfering signal estimation, from the primary input x(k). Finally e(k) represents the signal of interest as illustrates in Fig. 3.8. While the SANC output signal y(k) is the desired fault related signal achieved by processing the delayed form of actual input x(k − d) by the FIR filter, and as shown in Fig. 3.9, e(k) represents only the error between the actual input x(k) and output signal y(k). SANC is a self-tuning adaptive filter that can separate stochastic and periodic components in a signal with a short correlation length. Because gear and bearing fault vibrations have a tiny random fluctuation, the SANC can suppress any random components, allowing the gear and bearing fault frequency components to be highlighted [247][84]. In order to demonstrate the denoise effect of adaptive signal processing, SANC is applied to filter the simulated gear vibration signal from the model in Section 2.4. In this simulation example, the original signal is combined with Gaussian noise of amplitude 0.4. The length of the filter length has been set to 25. Fig. 3.10 depicts the original vibration signal with additive noise, as well as a comparison of the filtered signal.

Figure 3.10 Denoising of simulated vibration signal using adaptive filter

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3 Vibration Signal-Based Analysis for Gear Faults

From the above example, it can be seen that the additive noise can be removed from the signal after few iterations by using adaptive filter. The correlated component of signals has been identified, and the uncorrelated components, such as noise and random interference, have been removed. This method, however, is dependent on the assumption that the signal is strictly periodic, which is not necessarily satisfied in practice. However, due to the influence of noise, some associated components might be missing after filtering as illustrated in Fig. 3.10.

3.5

Concluding Remarks

In this chapter, different signal processing techniques commonly employed in gear defect detection are discussed and introduced. In these signal based methods, the fault detection issue is regarded as a signal characteristics extraction and enhancement problem despite the fact that they are based on different mathematical models. All of these signal processing methods are dealing with the univariate signal processing and one sensor signal analysis based fault diagnosis. It is worth noticed that, even with such a valid signal model, there are few researches on multi-sensor signal analysis for mechanical fault diagnosis. In which faults are detected and diagnosed in all multi-stage gearboxes at the same time based on multivariate signal processing. Some difficulties still exist in univariate signal-based fault detection. For example, gear faults will cause the vibration of other components through the transmission shaft, and therefore the collected gear vibration signals are mixed with vibration signal of other parts and noises. Using multiple sensors collecting signals at different locations on the machine to obtain multivariate signals can avoid the loss of local information. Subsequently using the multivariate signal processing approaches to simultaneously analyse the multivariate signal is beneficial to extract fault information by reducing the influence of the noise for multi-stage gearboxes especially for weak fault characteristics during the period of early failure. All of these reasons make traditional univariate signal processing approaches difficult to use to detect different faults in multi-stage gears. As a result, univariate signal-based approaches to fault detection and diagnosis of gears have become mainstream. As an alternative solution, in Chapter 5, the multivariate signal-processing based techniques in the form of MEMD and its improved form will be proposed and the case study for the multi-stage wind turbine gearbox is investigated to validate the effectiveness of the proposed results.

4

Frequency Domain Analysis for Nonlinear Systems

In Chapter 3, signal-based fault diagnosis has been studied, in which only vibration signal acquired from the sensor has been analysed in order to extract some features belonging to the fault. This means that only the output measuring signal is considered in detecting faults and the effects of the input excitation, which is also critical in many engineering applications, has not been taking into account. In order to deal with nonlinearity effects in the frequency domain, model-based fault diagnosis based nonlinear frequency analysis is studied. Therefore, this chapter introduces the fundamental aspects of the nonlinear frequency analysis methods and reviews recent achievements in the field in order to give the theoretical framework upon which the results of this research are based.

4.1

Nonlinear Systems Representation Based Polynomial Model

4.1.1

Introduction

Recently, due to a great demand in dealing with practical environments of complex nature, modelling and analysis of nonlinear systems received an increasing attention. Nonlinear systems are frequently split into classes for which specific techniques are designed for the aim of analysis and design, due to the variety of nonlinear problems that exist and the difficulty of constructing universal procedures for analysing general scenarios. The Volterra series model is a traditional method to a specific but broad class of nonlinear systems. It can be used to describe moderate nonlinear behaviour which is often observed in a variety of real-world scenarios, such as generation of inter-modulation and harmonics. The explicit description of the system output in terms of polynomial-based representations of the input signal © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_4

71

72

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Frequency Domain Analysis for Nonlinear Systems

is the fundamental feature of this method, allowing some of the well-established frequency domain approaches to be extended from linear systems analysis to the nonlinear case. This can be very beneficial and applicable in FDD/CM issues, since monitoring typical nonlinear phenomenon in the frequency domain, such as intermodulation and output harmonics, is already known as a consolidated technique, although it is commonly utilized from a pure data-analysis perspective. In this context, a system analysis technique can provide additional details about the problem and, as a result, can be used as a potential tool to design fault diagnosis systems in the frequency domain.

4.1.2

Nonlinear System Modeling and Frequency Analysis Based Fault Diagnosis

Faults, in engineering practice, can often be described by a disturbance to the system physical parameters. For instance, in mechanical and civil structures, the Young’s modulus1 of a material can be used to estimate its condition in structure health monitoring (SHM). However, as these physical quantities may not be directly measurable, directly using them is often not feasible. Evaluating changes in a representation e.g. FRF or transfer function of the system characteristics can considerably aid to solve these issues because changes in these representations can reflect changes in system model parameters. For this reason, in mechanical and civil engineering, calculating natural frequencies and mode shapes has proven to be more practical than other methods of assessing the condition of structural systems. By using input-output data of a system, evaluation of the FRFs can be carried out for a linear system modelling, for which a wide variety of methods are available. Once the FRF has been achieved, an automated method, for example, comparison with a priori known patterns of FRF, can be employed to determine whether there have been any changes to the system parameters. The method described above needs linear systems, which may not always be the case in real-world applications. When the nonlinearity effects of system cannot be ignored, nonlinear system modelling can be obtained by using well-known techniques explained in [24]. However, because the FRF notion is no longer relevant, the corresponding frequency analysis is more complex. Although approaches such as describing functions [164] and generalized frequency response functions (GFRFs) 1

Young’s modulus, which is defined as the ratio of stress to strain, is a measure of the stiffness of an elastic material. structures that have low Young’s modulus tend to be ductile while those that have high Young’s modulus tend to be brittle [215].

4.2 Frequency Analysis Representation of Nonlinear Systems

73

[76] have been used to extend the FRF concept to nonlinear case, these extensions are only applicable to specific classes of nonlinear systems.

4.2

Frequency Analysis Representation of Nonlinear Systems

4.2.1

The Functional Based Volterra Time Series Model Approach

The term functional is basically related to a map between different functions. The focus of this work, however, will be reduced to a more specific definition in which functional can be characterized as natural extensions of polynomial functions, a concept first proposed by the Italian mathematician Vito [239]. This concept is useful in applied mathematics and engineering because it enables for the development of a formal theory to approximate operators between vector spaces, which is necessary for a better and deep understanding of the relationships between input and output signals. To illustrate further regarding this idea, consider a multivariate polynomial function of degree n in variables u i , i ∈ {1, ..., k}, k is a finite number, which can be written as k k   ... h i1 ,...,in u i1 ,...,in , (4.1) yn (u 1 , ...., u k ) = i 1 =1

i n =1

where the n−dimensional coefficients h i1 ,...,in can be expressed as the kernel coefficients and i p = {1, ..., k}, where p = {1, ..., n}. The concept of the functional appears when k −→ ∞. The extrapolation of eq.(4.1) in this case involves replacing summations with integrals, the functions u 1 , ..., u k by a single variable and the summation indices i 1 , ..., i n in by integration variables, becomes  yn =

b1

a1

 ...

bn

an

h n (τ1 , ..., τn )

n 

u(τi )dτi ,

(4.2)

i=1

where the intervals [ai , bi ] are the integer ranges. The kernel coefficients have also been replaced by an n-dimensional function h n (τ1 , ..., τn ), sometimes known as the n-th order kernel. Eq.(4.2) is commonly written in a different form to explain memory effects and dynamical behaviour, where the time t is defined and an clear dependency of yn (t) on the past values of the input is assumed. Assuming there is

74

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Frequency Domain Analysis for Nonlinear Systems

a Volterra series that can be identified, the response up to order N , i.e. y N (t) ∈ R of such system to an input u(t) ∈ R is given by  yn (t) =

∞

−∞

 ...

∞ −∞

h n (τ1 , ..., τn )

n 

u(t − τi )dτi ,

(4.3)

i=1

y N (t) =

N 

yn (t) ,

(4.4)

n=1

where h n (τ1 , ..., τn ) : Rn  −→ R is the n th order Volterra kernel and yn (t) represents the corresponding n th order output, and N is the maximum degree of nonlinearity. The Volterra series model in eq.(4.3) can be defined for the class of Volterra systems as follows. Definition 4.1 (VS : Volterra system). A dynamic system is known as Volterra system if there exists a Volterra series representation eq.(4.3) that conveys uniformly around a giving operating point, with probability 1 to y(t) for all bounded inputs u(t) ∈ G (a set of Gaussian signals), i.e. τ E{| y(t) − y N (t) |} = 0 , ∀τ ∈ R ≥ 0 , lim N →∞

0

where the expected value E{·} defines the ensemble average over the considered class of random inputs ( see: [201] [203]) . Eq.(4.3) can be interpreted as a polynomial function of the past values of the input, weighted by the kernel function for each set of delays (τ1 , ..., τn ), However, because the delays might take any value on a continuous range, this polynomial function must be expressed using continuous summations (i.e. integrals) rather than discrete summations. Because it is created only by terms of degree n, eq.(4.3) is said to be homogeneous with regard to u(t). It is also worth noting that eq.(4.3) is a multidimensional convolution integral which is an extension of the well-known convolution integral for a linear case, therefore if n = 1 in eq.(4.3)  y1 =

∞ −∞

h(τ )u(t − τ )dτ .

(4.5)

4.2 Frequency Analysis Representation of Nonlinear Systems

75

As a result, eq.(4.3) can be defined as a generalization of the convolution operator or linear system representation of eq.(4.5). Another interesting feature of the functional-polynomial analogy is that any single variable continuous function f(u) can be approximated by a polynomial series, according to the Stone-Weirstrass theorem [83] as f (u) ≈ a0 +

N 

an u n ,

(4.6)

n=1

where a0 , a1 , ... denotes appropriate constants and N is the degree of a polynomial function series also an u n represents the dominant terms of the polynomial function. Hence, it is reasonable to assume that any system operator: y(t) = H [u(t)] ,

(4.7)

where H denotes a system operator which maps from the input to the output function space. This could also be approximated using a functional series y(t) ≈ y0 +

N 

yn (t) ,

(4.8)

n=1

where yn (t) is defined as eq.(4.3) and y0 is a constant value which is usually set to 0 to make the formula easier to understand. In this case, the Volterra series is referred to a “Taylor series with memory” [201]. This is the case when H represents a fading memory system2 , as demonstrated in [28]. It is the same as claiming that the system is stable around an equilibrium point and the output is independent on the remote past of the input, which appears in many practical cases. Furthermore, for n  2, observe that eq.(4.3) is nonlinear with respect to u(t), implying that eq.(4.8) can be used to describe nonlinear systems. The work of Wiener [248], who developed a theory for investigating nonlinear circuits and modeling the spectra of electroencephalograms, provided more attention to the use of functionals to describe nonlinear systems. The inverse operators [200] and the theory of orthogonal functionals [199] and were two of the many studies that followed the work of Wiener. For discrete-time systems, eq.(4.3) and functional series eq.(4.8) can also be defined where t ∈ Z (set of integer numbers), the output and input samples are 2

Fading memory means that the response of the system depends only on recent inputs but independent to the remote past inputs. It is analogous to the fact that the nonlinear system is stable about an equilibrium, thus, holds in a wide range of real-world scenarios.

76

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Frequency Domain Analysis for Nonlinear Systems

denoted by y(t) and u(t), respectively. When H in eq.(4.7) is fading memory, the output can be expressed as the series eq.(4.8), where functionals yn (t) are now defined as yn (t) =

∞  τ1 =−∞

...

∞ 

h n (τ1 , ..., τn )

τn =−∞

n 

u(t − τi )dτi ,

(4.9)

i=1

under the condition ∞  τ1 =0

...

∞ 

| h n (τ1 , ..., τn ) | < ∞ ,

n = 1, 2, 3....,

τn =0

which is a sufficient condition for the stability of a discrete-time system or Volterra series model. The Volterra series has been used in a wide range of various fields, including nonlinear circuits [246], nonlinear filtering and image processing [150], channel equalization [32], and some others.

4.2.2

Linear Versus Nonlinear Frequency Response Functions

System analysis in the frequency domain based on integral transformation such as the Laplace, Fourier and Z-Transforms, is the theoretical fundamentals of several different fields e.g. signal processing and control system engineering. It is commonly known that linear system analysis and synthesis in the frequency domain are well established. Several approaches and techniques, such as the Bode diagram, root locus, and Nyquist plot, have been developed to deal with the analysis and design of linear systems in practice [165]. Frequency domain approaches are widely used in engineering practice because they can frequently provide more intuitive insights on system linear dynamics or dynamic features. For instance, whatever the system model is transformed by any linear transformations, the transfer function of a linear system is always a coordinate-free and equivalent description; the instability of a linear system is usually refers to at least one right-half-plane pole of the system transfer function; the peak of the system output frequency response occurs frequently around the natural resonance frequency of the system, and so on. The ability to transfer linear issues defined in the time domain by the convolution integral into an algebraic formulation in the frequency domain allows this theory to be applied to real applications successfully. The Volterra series approach offers the fundamental foundation for expanding these results for a class of nonlinear systems since polyno-

4.2 Frequency Analysis Representation of Nonlinear Systems

77

mial functionals are an extension of the convolution operator. Although alternative transforms can produce similar results, the Fourier transform is more widely utilized in the frequency domain analysis of functional series formulation. The Fourier transform of a time domain function x(t) is expressed as X ( jω) = F (x(t)) and defined in the following definition. Definition 4.2 (X ( jω) : Fourier transform). Consider a signal x(t). Then its Fourier transform X ( jω) is given by:  X ( jω) =

∞ −∞

x(t)e− j2π ωt dt ,

(4.10)

√ where ω ∈ R is the angular frequency in [H z] and j = −1 is the imaginary unit. The inverse Fourier transform can be used to reconstruct the time domain function x(t) from its unique transform: x(t) =

1 2π



∞

−∞

X ( jω)e j2π ωt dω ,

(4.11)

Based on the definition 4.2, let Yn ( jω) be the Fourier transform of n th order Volterra series function in eq.(4.3) and let U ( jω) be the Fourier transform of u(t), i.e. yn (t) =

1 2π

u(t) =

1 2π



∞

−∞



∞

−∞

Yn ( jω)e j2π ωt dω ,

(4.12)

U ( jω)e j2π ωt dω .

(4.13)

By substituting both eq.(4.12) and eq.(4.13) into eq.(4.3), the relationship between the n th order output spectrum Yn ( jω) and the input spectrum U ( jω) is achieved as shown in [111]:  Yn ( jω) =

ω=ω1 +ω2 +···+ωn

Hn ( jω1 , . . . , jωn )

n 

U ( jωi )dσn,ω ,

(4.14)

i=1

where dσn,ω defines the area of a minute element on the hyperplane ω = ω1 + ω2 + · · · + ωn . Due to the computational complexity of calculating eq.(4.14) numerically or in closed form, especially for large n, it is not much used in practical application. For this reason, system properties are typically investigated through the analysis of

78

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Frequency Domain Analysis for Nonlinear Systems

Hn ( jω1 , . . . , jωn ) function which will be discussed in the next section or notions derived from eq.(4.14). •

The Generalized Frequency Response Functions

The n-dimensional Fourier transform of the n th order kernel h n (τ1 , ..., τn ) is denoted by the function Hn ( jω1 , . . . , jωn ) or simply Hn (·) and can be summarized in the following definition. Definition 4.3 (Hn (·) : Generalized Frequency Response Function (GFRF)). Consider a Volterra system according to definition 4.1. Then its n th order GFRF is denoted by Hn (·) : Rn  −→ C, where (·) = ( jω1 , . . . , jωn ) ∈ Rn . The GFRF is defined as the n−dimensional Fourier transform of the n th order volterra kernel in eq.(4.3), i.e.  Hn ( jω1 , . . . , jωn ) =

∞

−∞

 ...

∞ −∞

n 

h n (τ1 , ..., τn )

e− jωi τi dτi ,

(4.15)

i=1

(see: [76][57] [201] [23]) . Eq.(4.15) is the nth order GFRF and can also be written in this form Hn ( jω1 , . . . , jωn ) =

 ∞ −∞

...

 ∞ −∞

h n (τ1 , ..., τn )e− j(ω1 τ1 ,...,ωn τn ) dτ1 , . . . , dτn , n = 1, 2, ....

(4.16) The GFRF is a natural extension of the well-known concept of FRF or transfer function for linear systems under the assumption that the nonlinear systems can be represented by a convergent Volterra series. This assumption is valid if the condition of fading memory is satisfied by the system [28], which is equivalent to that the nonlinear system is stable about an equilibrium and therefore holds in very general practical situations. For a linear system, if n = 1 in eq.(4.14), the output frequency response can be achieved as Y1 ( jω) = H1 ( jω)U ( jω),

(4.17)

where the H1 ( jω) can be determined from eq.(4.17) as  H ( jω) = H1 ( jω) =

∞

−∞

h(τ )e− jωτ dτ =



∞

−∞

h(τ1 )e− jωτ1 dτ1 .

(4.18)

4.2 Frequency Analysis Representation of Nonlinear Systems

79

In contrast to the linear case, the function of GFRFs in the composition of the output spectrum for nonlinear systems is very complicated, since it requires the analysis of multi-dimensional frequency spaces. For instance, in case of linear case Y ( jω) = H ( jω)U ( jω) and the support of the output spectrum Y ( jω), i.e. the frequencies ω for which | Y ( jω) | = 0, is the similar support of the input spectrum. This is not always the case in the nonlinear scenario, which allows for a significantly richer range of support, often exceeding the input bandwidth. This was more studied in deep in [112][253], where different complicated methods were presented to compose the output spectrum for any arbitrary input spectrum. The GFRFs can be used to describe system properties in the frequency domain for a time-invariant nonlinear system. This has been widely applied in practice, e.g. in [169], a GFRF based approach employed to investigate a dam buttress. GFRFs were utilized by [225] to characterize three types of defects in a rotor-bearing system. However, due to the multidimensional structure of the GFRF, there are various difficulties with this technique. For example, the realization of eq.(4.14) is not unique; Hn ( jω1 , . . . , jωn ) is normally an asymmetric function, and a permutation of variables in eq.(4.14) can usually be carried out without affecting the value of Y n( jω). This essentially means that the system can support several asymmetric n th order GFRFs. This ambiguity can be eliminated by symmetrizing the GFRFs, which composed of substituting Hn ( jω1 , . . . , jωn ) in eq.(4.14) with the symmetrical GFRF obtained from the original function as sym

Hn

( jω1 , . . . , jωn ) =

1 n!



Hn ( jω1 , . . . , jωn ) ,

all permutations of{1,2,...,n}

(4.19)

Hn (− jω1 , . . . , − jωn ) = Hn∗ ( jω1 , . . . , jωn ) , this is called conjugate symmetry. When n is large, computing Hn ( jω1 , . . . , jωn ) for every possible permutation of parameters becomes prohibitively time consuming and infeasible. Another significant drawback of GFRF-based analysis is the difficulty in performing graphical analysis. In linear system situation, FRF or Transfer function analysis by using traditional methods like Bode and Nyquist plots is effective because typical system parameters like bandwidth, resonance frequencies, and phase margin can be clearly comprehended from graphical displays of the FRF, which are often plane figures. This is still possible for 2nd order cases, as H2 ( jω1 , jω2 ) can be analyzed via surface plots against ω1 and ω2 , For example, the analysis is not as straightforward as in the linear case since the structure of the output spectrum should be determined from plane slices of the type ω1 + ω2 = ω. For the 3rd

80

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Frequency Domain Analysis for Nonlinear Systems

order case, although some attempts have been made to build more rigorous analysis processes, graphical displays become much more complex [258]. Special computational procedures are required for this, but the complexity of the algorithms make these techniques infeasible.

4.2.3

The Output Frequency Response Based GFRFs

As previously mentioned in eq.(4.3), a class of nonlinear system, under certain condition, can be represented by a Volterra series model as ⎧ N  ⎪ ⎪ ⎪ y(t) = yn (t) , ⎪ ⎨ n=1  ∞  ∞ n  ⎪ ⎪ ⎪ (t) = ... h (τ , ..., τ ) u(t − τi )dτi , y ⎪ n n 1 n ⎩ −∞

−∞

(4.20)

i=1

where u(t) and y(t) represent the input and output of the system respectively. Also yn (t) refers to the n th order nonlinear system response. In the frequency domain, the nonlinear output spectrum with related to the GFRFs were derived in [111], the following results were obtained for a general input: ⎧ N  ⎪ ⎪ ⎪ ⎪ Y ( jω) = Yn ( jω) ⎪ ⎨ n=1 √ ⎪ 1/ n ⎪ ⎪ ⎪ ⎪ Yn ( jω) = (2π)n−1 ⎩

for ∀ω, 

ω=ω1 +ω2 +···+ωn

Hn ( jω1 , . . . , jωn )

n 

U ( jωi )dσωn ,

i=1

(4.21) where N defines the highest order of system nonlinearity, Y ( jω) and Yn ( jω) are the nonlinear output spectrum and n th order nonlinear output spectrum respectively. Moreover, Y ( jω), Yn ( jω) and U ( jω) are the Fourier transform of y(t), yn (t) and u(t) respectively. Also, the GFRFs, Hn ( jω1 , . . . , jωn ) is the multi-dimensional Fourier transform of h n (τ1 , ..., τn ) and dσωn is the area of a minute portion on the hyperplane ω = ω1 + ω2 + · · · + ωn . Eq.(4.21) is a representation of the nonlinear output frequency spectrum derived by Lang and Billings [111]. This expression provides a more physically meaningful description of the output frequency response of nonlinear systems which defines the relationship between the nonlinear output spectrum or nonlinear systems frequency response, the system GFRFs, and the input

4.2 Frequency Analysis Representation of Nonlinear Systems

81

excitation for a general continuous input U ( jω). For the case where the input is a multi-sinusoidal input excitation to the systems as R 

R  Ai



R  Ai jωi t , e 2 2 i=1 i=1 i=−R (4.22) where Ai∗ is a conjugate of Ai , A− i = Ai∗ , and ω− i = −ωi . Then, the system nonlinear output spectrum is [111]

u(t) =

|Ai |cos (ωi t + ∠Ai ) =

⎧ N  ⎪ ⎪ ⎪ Y ( jω) = Yn ( jω) ⎪ ⎪ ⎨

e jωi t +

Ai∗ − jωi t e 2

=

∀ω,

n=1

n   ⎪ 1 ⎪ ⎪ Yn ( jω) = n Hn ( jωi 1 , . . . , jωi n ) A(ωi k ), i k ∈ {±1, ..., ±R}, k = 1, .., n, ⎪ ⎪ 2 ⎩ ω=ωi +ωi +···+ωi n k=1 1 2

(4.23) where A(ωik ) denotes the Fourier transform of u(t) can be defined as  A(ωik ) = where

4.2.4

if ω ∈ (ωik , i k = ±1, ..., ±R), |Aik |e jsgn(ik )Aik 0 otherwise

⎧ ⎨ 1 sgn(i k ) = 0 ⎩ −1

ik > 0 ik = 0 ik < 0 .

for i k ∈ R

Methods for Computation of GFRFs

As explained, achieving frequency-domain kernels or GFRFs, is a key step in Volterra series-based analysis. In most applications, however, the system is not explicitly represented as a functional series, hence the kernels/GFRFs must be generated from alternative representations. Since these are the most prevalent types of a priori information about the system, this section will present and briefly explain some typical approaches for deriving the system GFRFs from the available system models achieving from input-output data and differential/ difference equation models. Two types of GFRF representations will be addressed in this discussion: nonparametric and parametric approaches. For the first approach, there are several problems and difficulties to estimate GFRFs including the computational cost imposed

82

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Frequency Domain Analysis for Nonlinear Systems

by the need to process a considerable amount of data in order to obtain the required precision for predicting a reasonable number of Volterra kernels. For instance, to describe the first-order kernel h 1 (t), the first three Volterra kernels may require typically the estimation of 30 points “t”, and to describe the second order kernel h 2 (t1 , t2 ) require “30 × 30” points and in the same way, to describe the third-order kernel h 3 (t1 , t2 , t3 ), a “30 × 30 × 30” points is required. As a result, no matter what technique is utilized, big data sets will always be a challenge. A non-parametric method is defined by a finite number of complex values of Hn ( jω1 , . . . , jωn ) calculated at specific points in the n-dimensional frequency space. The parametric based model method, on the other hand, is defined by the explicit algebraic formulations used to describe Hn ( jω1 , . . . , jωn ), such as a recursive implicit representation or an n-variable rational function, i.e. when Hn ( jω1 , . . . , jωn ) is expressed as a function of Hk ( jω1 , . . . , jωn ), with k < n. This method employs a parametric model of the nonlinear system, which can be identified from sampling input—output data or generated from first principles based on the physics of the underlying system. After that, these models are analytically transferred into the frequency domain [20] [194] [23]. Because parametric models have fewer parameters than Volterra kernels, they just require short data lengths and no specific inputs are required. The advantage here is that only the unbiased Nonlinear Auto-Regressive with eXogenous input (NARX) model is mapped to the GFRFs, this means that the averaging is avoided and noise models can be estimated as part of the Nonlinear Auto-Regressive Moving Average models with eXogenous inputs (NARMAX) modeling estimation process and then discarded. The GFRFs can be calculated of any order and both a plot and a formula of the GFRF can then be achieved. For two reasons, this is important: Firstly, a GFRF of the same order as the input nonlinearity, for systems that are nonlinear in the input, is sufficient to completely represent the behaviour of the system. However, many systems in real world include nonlinearities in the output, and output nonlinearities usually involve a large number of terms in the Volterra series, thus it is crucial to identify a couple of the higher-order GFRFs to describe such systems. Secondly, the time- and frequency-domain behaviours is lost, if a non-parametric approach is utilized. In the parametric techniques, the exact influence of each time domain model term on each GFRF can be easily understood and seen since each GFRF can be written as an explicit analytical formulation or described using a symbolic algebraic algorithm. • Harmonic Probing Method The harmonic probing method is related to the parametric model computation of the GFRFs and refers to a series of approaches for estimating the GFRFs from a

4.2 Frequency Analysis Representation of Nonlinear Systems

83

known model. The major goal is to come up with a collection of algebraic equations which will allow one or more GFRFs to be computed in terms of their lower order counterparts i.e. Hn as a function of Hn−1 , Hn−2 . . . H1 . This gives the GFRFs a relatively compact representation while still allowing them to be computed explicitly if needed. The computation of the GFRFs is obviously involved in the output spectrum of a nonlinear system. The “harmonic probing” method [194], which may be traced back to [20] or earlier, can be used to derive the GFRFs given the parametric model of a nonlinear system. In addition, computing the nth order GFRFs for both NARX (discrete-time) and NDE (continuous time) models can be obtained using parametric approaches. Different parametric models can commonly be used to describe Volterra type nonlinear systems such as NARX and NDE models. A parametric model of the nonlinear system under investigation can be used to compute the GFRFs. The NARX and the Nonlinear Differential Equation (NDE) are the two parametric models that are discussed in this thesis. The NARX model, which is considered as a discrete time parametric model representation of Volterra type nonliear systems, gives a natural and unified representation for a wide range of nonlinear systems. It can be described as [103] y(t) =

M 

ym (t) ,

m=1 m 

ym (t) =

K 

c p,q (k1 , . . . , k p+q )

p=0 k1 ,k p+q =1 p+q=m

p  i=1

p+q 

y(t − ki )

u(t − ki ) ,

(4.24)

i= p+1

where ym (t) represents the m th order output response of NARX model; and K K K    (·) = (·) . . . (·) ; p+q refers to the nonlinear degree of parameter k1 ,k p+q =1

k1 =1

k p+q =1

c p,q , which is related to the ( p+q)−degree nonlinear terms

p  i=1

y(t −ki )

p+q  i= p+1

u(t −

ki ) for instance, in case y(t − 1) p u(t − 1)q , p is orders in terms of the output y(t) while q is orders in terms of the input u(t), and also ki is the lag of the i th output when i ≤ p or the (i − p)th input when p < i ≤ m with the maximum lag K ; and c p,q (·) for c0,1 (·) and c1,0 (·) of nonlinear degree 1 are referred to as linear parameters as p = 1 and q = 1, and all the other model parameters which are described

84

4

Frequency Domain Analysis for Nonlinear Systems

by c p,q (·) for p > 1, q > 1, are referred to as nonlinear parameters. The model includes all the possible nonlinear combinations in terms of y(k) and u(k) with the maximum order M. Furthermore, the NDE model is considered as a continuous-time version of the NARX model, which is often derived by physical modelling and can be described as [103] y(t) =

m M  

K 

c p,q (k1 , . . . , k p+q )

m=1 p=0 k1 ,k p+q =0 p+q=m

p p+q  d ki y(t)  d ki u(t) , (4.25) dt ki dt ki i=1

i= p+1

k = x(t), and for convenience, all other notations in NDE model where d dtx(t) k k=0 take similar definitions and forms to those for the NARX model. But for the NDE model, K represents the maximum order of the derivative, and c p,q (·) for p + q > 1 are related to the nonlinear parameters corresponding to nonlinear paramter terms p p+q  d ki y(t)  d ki u(t) , e.g.y p (t)u q (t). in the model of the form dt ki dt ki i=1

i= p+1

• Analytical computation of the nth order GFRFs In this part, a recursive approach for the computation of the nth order GFRFs for both discrete time i.e. NARX model, and continuous time i.e. NDE model are explained using the so called Harmonic probing method. First, for discrete time case (NARX model), a recursive approach to compute the nth-order GFRF in terms of model parameters can be determined [106][99]:

K  

1− c1,0 (k1 ) exp − j(ω1 + · · · + ωn )k1 .Hn ( jω1 , · · · , jωn ) =

k1 =1 K 

c0,n (k1 , · · · , kn ) exp − j(ω1 k1 + · · · + ωn kn ) k1 ,kn =1 q K n−1  n−q   

+

ωn−q+i k p+i Hn−q, p ( jω1 , · · · , jωn−q )

c p,q (k1 , · · · , k p+q ) exp − j

+

q=1 p=1 k1 ,kn =1 K n  



i=1

c p,0 (k1 , · · · , k p )Hn, p ( jω1 , · · · , jωn ) ,

p=2 k1 ,k p =1

(4.26) where

4.2 Frequency Analysis Representation of Nonlinear Systems

Hn, p (·) =

n− p+1 

85

 Hi ( jω1 , · · · , jωi )Hn−i, p−1 ( jωi+1 , · · · , jωn ) exp − j(ω1 + · · · + ωi )k p

i=1

 (4.27)

Hn,1 ( jω1 , · · · , jωn ) = Hn ( jω1 , · · · , jωn ) exp − j(ω1 + · · · + ωn )k1 . (4.28) Moreover, define H0,0 (·) = 1, Hn,0 (·) = 0 for n > 0 , Hn, p (·) = 0 for n < p and let

 q  1 q = 0, p > 1 exp − j ε( p) = 0 q = 0, p ≤ 1 , i=1

where ε( p) is a function of p, also let in eq.(4.26)

K 

 1− c1,0 (k1 ) exp − j(ω1 + · · · + ωn )k1 = L n (ω1 , · · · , ωn ) , k1 =1

then, eq.(4.26) can be concisely written as Hn ( jω1 , · · · , jωn ) =

n n−q 1   L n (·)

K 



q  c p,q (k1 , · · · , k p+q )) exp − j ωn−q+i k p+i

q=0 p=0 k1 ,k p+q =1

i=1

×Hn−q, p ( jω1 , · · · , jωn−q ) .

(4.29) Therefore, eq.(4.29) is a recursive computation of the n th order GFRFs. Moreover, it is noted that eq.(4.27) can be also written in this form Hn, p ( jω1 , · · · , jωn ) =

n− p+1 

p 

r1 ,··· ,r p =1 ri =n

i=1

Hri ( jω X +1 , · · · , jω X +ri )

× exp − j(ω X +1 + · · · + ω X +ri )ki where X =

i−1 

(4.30) 

,

rx .

x=1

To further illustrate how to compute the nth order GFRFs for NARX model by using the harmonic probing method based on [106], the following example is studied. Example 1: Consider a discrete time model which is a specific case of NARX model eq.(4.24) with K = 1 (Max. Lag), M = 3 (max degree of nonlinearity). y(k) = au(k − 1) + by(k − 1) + cy 3 (k − 1) ,

(4.31)

86

4

Frequency Domain Analysis for Nonlinear Systems

where the linear parameters are a = c0,1 (1) and b = c1,0 (1), and the nonlinear parameter is c = c3,0 (1, 1, 1), all the other nonlinear parameters are equal to 0 i.e. c p,q (·) = 0. The main goal is to compute the n th order GFRFs with related to the model parameters using the harmonic probing algorithm, let n = 5, i.e. 5th order GFRFs will be computed using this method. By applying Harmonic probing input u(k) = e jω1 k + · · · + e jωn k ,

(4.32)

when n = 5 ; then eq.(4.26) which represents a recursive method for computation of GFRFs of discrete time nonlinear systems [24] becomes   1  

c1,0 (k1 ) exp − j(ω1 + · · · + ω5 )k1 · H5 ( jω1 , · · · , jω5 ) = 1− k1 =1

5 1  

c p,0 (k1 , · · · , k p )H5, p ( jω1 , · · · , jω5 )

p=2 k1 ,k p =0



 1 − c1,0 (1) exp − j(ω1 + · · · + ω5 ) · H5 ( jω1 , · · · , jω5 ) = c3,0 (1, 1, 1)H5,3 ( jω1 , · · · , jω5 ).

By substituting the model parameters yields c · H5,3 ( jω1 , · · · , jω5 )

 . H5 ( jω1 , · · · , jω5 ) =  1 − b exp − j(ω1 + · · · + ω5 )

(4.33)

In order to compute H5 ( jω1 , · · · , jω5 ) in eq.(4.33), first H5,3 ( jω1 , · · · , jω5 ) need to be calculated using eq.(4.27) with n = 5 and p = 3, yields

H5,3 (·) =

5−3+1 

 Hi ( jω1 , · · · , jωi )H5−i,3−i ( jωi+1 , · · · , jω5 ) exp − j(ω1 + · · · + ωi ) ,

i=1

(4.34) this can be more expanded,

4.2 Frequency Analysis Representation of Nonlinear Systems

87

  ⎧ ⎫ ⎨ H1 ( jω1 )H4,2 ( jω2 , · · · , jω5 ) exp − j(ω 1 ) + ⎬  H5,3 (·) = H2 ( jω1 , jω2 )H3,2 ( jω3 , jω4 , jω5 ) exp− j(ω1 + ω2 ) +  , ⎩ ⎭ H3 ( jω1 , jω2 , jω3 )H2,2 ( jω4 , jω5 ) exp − j(ω1 + ω2 + ω3 ) (4.35) Similarly, using eq.(4.27) with the same procedure to compute the terms H4,2 ( jω2 , · · · , jω5 ), H3,2 ( jω3 , jω4 , jω5 ) and H2,2 ( jω4 , jω5 ) inside eq.(4.35), yields   ⎧ ⎫ ⎨ H1 ( jω2 )H3,1 ( jω3 , jω4 , jω5 ) exp− j(ω2 ) +  ⎬ H4,2 ( jω2 , · · · , jω5 ) = H2 ( jω2 , jω3 )H2,1 ( jω4 , jω5 ) exp− j(ω2 + ω3 ) +  , ⎩ ⎭ H3 ( jω2 , jω3 , jω4 )H1,1 ( jω5 ) exp − j(ω2 + ω3 + ω4 )

  (4.36) H1 ( jω3 )H2,1 ( jω4 , jω5 ) exp− j(ω3 ) +  . H3,2 ( jω3 , jω4 , jω5 ) = H2 ( jω3 , jω4 )H1,1 ( jω5 ) exp − j(ω3 + ω4 ) (4.37) In addition, H3,1 ( jω3 , jω4 , jω5 ), H2,1 ( jω4 , jω5 ), and H1,1 ( jω4 , jω5 ) inside eq.(4.35) can be also computed using eq.(4.28), yields 

  H3,1 ( jω3 , jω4 , jω5 ) = H3 ( jω3 , jω4, jω5 ) exp − j(ω  3 + ω4 + ω5 ) , H2,1 ( jω4 , jω5 ) = H2 ( jω4, jω5 ) exp  − j(ω4 + ω5 ) , H1,1 ( jω5 ) = H1 ( jω5 ) exp − j(ω5 ) . Finally, all the results above are combined together in order to obtain the 5th order GFRFs yields 

3c exp − j(ω1 + · · · + ω5 )

 H5 ( jω1 , · · · , jω5 ) =  1 − b exp − j(ω1 + · · · + ω5 )  H1 ( jω1 )H1 ( jω2 )H3 ( jω3 , jω4 , jω5 )+ × H1 ( jω1 )H2 ( jω2 , jω3 )H2 ( jω4 , jω5 ) . It can be noticed from the above equation that the nonlinear model parameter “c” is a function of the 5th order GFRFs. Similarly, for continuous time model or NDE model, the nth order GFRFs in terms of model parameters can be computed recursively as [22][97]:

88

4

Frequency Domain Analysis for Nonlinear Systems K 

L n (ω1 , · · · , ωn ).Hn ( jω1 , · · · , jωn ) = + +

n−1  n−q 

c0,n (k1 , · · · , kn )( jω1 )k1 + · · · + ( jωn )kn

k1 ,kn =1

 q c p,q (k1 , · · · , k p+q ) (ωn−q+i )k p+i Hn−q, p ( jω1 , · · · , jωn−q )

K 

q=1 p=1 k1 ,kn =0 K n  

i=1

c p,0 (k1 , · · · , k p )Hn, p ( jω1 , · · · , jωn ) ,

p=2 k1 ,k p =0

(4.38) where L n (ω1 , · · · , ωn ) = −

K 

c1,0 (k1 ) jω1 + · · · + jωn )k1 .

k1 =0

Furthermore, Hn, p (·) in equation can be written as Hn, p (·) =

n− p+1 

Hi ( jω1 , · · · , jωi )Hn−i, p−1 ( jωi+1 , · · · , jωn )( jω1 + · · · + jωi )k p ,

i=1

Hn,1 ( jω1 , · · · , jωn ) = Hn ( jω1 , · · · , jωn )(− jω1 + · · · + jωn )k1 .

(4.39) (4.40)

Moreover, it is noted that eq.(4.39) can be also written in this form Hn, p ( jω1 , · · · , jωn ) =

n− p+1 

p 

r1 ,··· ,r p =1 ri =n

i=1

Hri ( jω X +1 , · · · , jω X +ri ) (4.41)

×( jω X +1 + · · · + jω X +ri )ki , where X =

i−1 

rx .

x=1

Similarly, define the following for the convenience in the discussion H0,0 (·) = 1, Hn,0 (·) = 0 for n > 0 , Hn, p (·) = 0 for n < p , and let

q  i=1

 (·) =

1 0

q = 0, p > 1 q = 0, p ≤ 1 .

Eq.(4.38), which represents the recursive computation of nth order GFRFs for the NDE system model, can also can be concisely written in this form

4.3 Characteristics of Output Frequencies for Nonlinear Systems

Hn ( jω1 , · · · , jωn ) =

89

n n−q K    1 c p,q (k1 , · · · , k p+q ) 

n q=0 p=0 k1 ,k p+q =0 Ln j ωi

i=1 

q × (ωn−q+i )k p+i Hn−q, p ( jω1 , · · · , jωn−q ) . i=1

(4.42) It is assumed that the GFRFs above for both the NDE and NARX models are to be asymmetric. Generally different permutations of the frequency variables ω1 , · · · , ωn may lead to different values of Hn ( jω1 , · · · , jωn ). The symmetric GFRFs can be achieved as sym

Hn

( jω1 , . . . , jωn ) =

1 n!



Hn ( jω1 , . . . , jωn ) .

(4.43)

all permutationsof{1,2,...,n}

However, for nonlinear output spectrum computation i.e. Y ( jω) in eq.(4.21), asymmetric GFRFs suffice.

4.3

Characteristics of Output Frequencies for Nonlinear Systems

The requirement to study the relationship between the system input and output frequencies is one of the most significant aspects of the frequency domain system analysis. For the case of linear systems, the system frequency response function establishes a straightforward linear relationship between the output and input frequency spectrum. The input frequencies pass through the system separately in this case; in other words, at steady state, an input at a specific frequency generates an output at the same frequency with a different phase and amplitude, but no new frequency components are generated and no energy is transferred to the output or produce any other frequency components. However, this is not the case for the nonlinear systems, in which the output frequency components of nonlinear systems can be significantly richer than the comparable input frequencies. In the nonlinear scenario, the input frequencies pass in a coupled fashion through the system; that means, depending on the dynamics of the nonlinearity, an input at a certain frequency might result in a wide range or very complicated output frequencies, appearing as harmonics, inter-modulation, and other effects. This makes it difficult, for most nonlinear systems, to establish a general and explicit expression for the relation of input and output frequencies. Many researchers have investigated out-

90

4

Frequency Domain Analysis for Nonlinear Systems

put frequencies of nonlinear systems, for this, several results have been obtained [246][111][21][113][245][253][100][178]. All of these studies are focused on the nonlinear system output frequencies, which are represented by a Volterra series model and can be divided into two different categories: output frequencies of nonlinear systems excited by general inputs or a multi-tone or sinusoidal harmonic inputs.

4.3.1

Generation of Frequencies in Linear and Nonlinear Systems Response

Nonlinear systems unlike linear systems could produce complicated effects in the frequency domain, besides various new frequencies might be generated caused by system nonlinearities. As shown in Fig.4.1(a), a linear system excited by two sinusoidal inputs at frequencies f 1 and f 2 generates outputs at exactly the same frequencies in steady state. However for the case of nonlinear systems, this is different because new frequencies in different forms such as harmonics3 , intermodulations and even bifurcation and chaos and etc. could be generated due to the effect of nonlinearity. Table 4.1 describes possible generated frequencies in the output response caused by nonlinearity behaviour. To further illustrate the nonlinearity effects on the output frequency response, consider a simple nonlinear system scenario with only static linear, quadratic, and cubic nonlinearities which is excited by the same mentioned two sinusoidal signal as inputs with frequencies f 1 and f 2 . The frequency content of both the system input and output is illustrated in Fig.4.1(b). The components coming from linear behavior are indicated by the term marked (1). The terms (2) and (3) are produced by the quadratic and cubic nonlinearities respectively. As it is shown in Fig.4.1(b), yields 13 different frequencies including harmonics and intermodulations at the output response despite the fact that there are only two frequency components at the input, it can be noted that this is for a simple case of nonlinear system with only time invariant system i.e. no dynamics. Therefore, energy can be transferred between frequencies, and these effects must be understood in order to unravel the behaviour of the nonlinear systems in the frequency domain. The input in the above example was simple and it was easy to interpret the result as the system model was known. In real application, multi-dimensional pictures which is known as GFRF have to be investigated to interpret these phenomena for general systems. 3 This includes both sub-harmonics and super-harmonics. The first one can be defined as a fraction of a fundamental frequency while the latter represents an integer multiple of that of another; an overtone.

4.3 Characteristics of Output Frequencies for Nonlinear Systems

91

Table 4.1 Nonlinearity effects in the frequency domain Nonlinearity effects

description

Harmonics

generation of frequency components at multiples of the fundamental input frequency. frequencies combine to produce new frequencies which contains two or more different frequencies, caused by nonlinearities. gain variation for changing in the amplitude of input. special frequencies generated due to strong nonlinear behaviours.

Intermodulation

Gain compression/expansion Bifurcation and Chaos

4.3.2

Output Frequencies Under Harmonics Inputs for Nonlinear Systems

The system output spectrum is represented by eq.(4.23), when it is subject to a multi-tone input. The output frequencies are consisted of the frequencies contributed by all the orders of system nonlinearities included in the system. The frequencies contributed by the nth-order system nonlinearity are composed of all ω s that can be computed from the input frequencies −ω R , . . . , −ω1 , ω1 , . . . , ω R as ω = ωi1 + · · · + ωin ,

i k ∈ {±1, . . . , ±R}, k = 1, · · · , n .

(4.44)

Let m ik represents the number of times the frequency ωik showing up in a particular frequency mix to generate a specific output frequency produced by the nth-order system nonlinearity. A frequency mix vector can then be used to represent the frequency mix as m = {m −R , · · · , m −1 , m 1 , · · · , m R } , (4.45) which satisfies the following constraint m −R + · · · + m −1 + m 1 , · · · + m R = n .

(4.46)

At the nth-order frequency mix vector is referred to as vector m in eq.(4.45), and the corresponding produced output frequency can be written as  

ωm = m 1 − m −1 ω1 + · · · + m R − m −R ω R .

(4.47)

92

4

Frequency Domain Analysis for Nonlinear Systems

(a) Input

0

1

.

2

I/P

1

0

O/P

Linear

Output .

2

Linear

(b) Output

(2)

(1) (1) (3)

(3)

(2)

+

I/P

(2)

(2)

Quadrac

(3)

2

−

1

2

2

1

1

+

2

3

1

O/P

+

(3)

(3)

1

2

(3)

(2)

Cubic 0

1

2

2

+

3

. 1

Figure 4.1 System frequency response (a) linear, (b) nonlinear

Consequently, the output frequencies produced by the nth-order system nonlinearity are interpreted as those frequencies that might be generated by all possible choices of the m s such that the above eq.(4.46) is then satisfied. Basically, the output frequencies generated by the nth-order system nonlinearity can be computed from eq.(4.47) and all m s that satisfy the eq.(4.46). This is the technique which has been studied in [246] and other relevant literature. The complexity of producing the frequency mix vector m is a challenge with this method, while considerable research has been done recently to try to overcome this issue [178]. Several effective approaches have been presented to overcome the limitations of using the frequency mix vector to determine all feasible system output frequencies [111][245]. The basic concepts that have been used to construct these algorithms for determining the available frequencies in a nonlinear system output are outlined below. Using eq.(4.44), all the possible output frequencies generated by the nthorder system nonlinearity are produced by a combination of the input frequencies −ω R , . . . , −ω1 , ω1 , . . . , ω R . When n = 1, the possible generated non-negative output frequencies for the simplest case can be calculated using the vector W1 = [ω1 , . . . , ω R ]T as Y1 =

4.3 Characteristics of Output Frequencies for Nonlinear Systems

93

s W1 , where s denotes a set and X s represents a set whose elements consist of all the various entities of the vector X by applying absolute values. When n = 2, the following vector can be used to determine the possible generated non-negative output frequencies as W2 = [ω1 −ω R . . . ω1 −ω1 , ω1 +ω1 . . . ω1 +ω R . . . ω R −ω R . . . ω R −ω1 , ω R +ω1 . . . ω R +ω R ]T .

Moreover, W2 can also be written as W2 = W1 ⊗ I2R + I R ⊗ W , s where W = [ω R , . . . , −ω1 , ω1 , . . . , ω R ]T , I R = [1, . . . , 1]T as Y2 = W2 .    R

It can be noticed that, from these results, for n = 2, the possible generated non-negative output frequencies are computed s from the following vector Wn = Wn−1 ⊗ I2R + I R(2R)n−2 ⊗ W as Yn = Wn . This result provides all the possible non-negative output frequencies that might be generated by the nth-order system nonlinearity. In order to find all the possible non-negative output frequencies generated by all the system nonlinearities, [39] proved under certain conditions, and [112] proved under general conditions that the output frequencies from the nonlinear systems only consist of the frequencies that are contributed by the N th- and (N − (2 p ∗ − 1))th-order nonlinearities. As a result, by combining the possible output frequencies contributed by the N th- and (N − (2 p ∗ − 1))th-order nonlinearities, all conceivable frequencies in the system output can be found. The value to be taken by p ∗ in this case could be {1, 2, . . . , N /2}, where N /2 denotes taking the integer equal or less than N /2. The specific value of p ∗ is defined by the orders of the current system nonlinearities, and can be calculated so that if the system GFRFs H N −(2i−1) (·) = 0, for i = {1, 2, . . . , q − 1} but H N −(2q−1) (·) = 0 then p ∗ = q. As a consequence, based on the above procedure, a set containing all potential generated system output non-negative frequencies Y , can be achieved as follows using the following generic result for all the cases: ⎧ T ⎪ ⎪W1 = [ω1 , ω2 , . . . , ω R ] , for n = 1 ⎪ ⎪ ⎨W = W n n−1 ⊗ I2R + I R(2R)n−2 ⊗ W , for n  2 s ⎪ Yn = Wn ⎪ ⎪ ⎪ ⎩ =    , Y

YN

Y N −2 p∗ +1

(4.48)

94

4

Frequency Domain Analysis for Nonlinear Systems

where Yn and Y represent the generated non-negative frequency ranges(bands) of Yn ( jω) and Y ( jω) respectively under harmonic inputs.

4.3.3

Output Frequencies Under General Inputs for Nonlinear Systems

Assume that the general input frequency spectrum band follows a generic and realistic assumption  U ( jω) if ω ∈ [a, b] , U ( jω) = (4.49) 0 else . It can be deduced from eq.(4.21) that the non-negative output frequency band or range f Yn generated by the nth-order system nonlinearity for the general inputs is calculated using the expression ω = [ω1 , . . . , ωn ]

with ωi ∈ [−bi , ai ] or [−ai , bi ], i = 1, . . . , n , (4.50) where 0 ≤ a < ∞ , a ≤ b < ∞. The range of the output frequency for the entire system, f Yn , is the union of the frequency ranges generated by each order of the nonlinearities in the system, i.e. fY =

N 

f Yn ,

(4.51)

n=1

where f Yn and f Y represent the non-negative generated frequency ranges of Yn ( jω) and Y ( jω) respectively. In case if a = 0 in eq.(4.49), the non-negative possible generated output frequency range provided by eq.(4.50), as a result, is f Yn = [0, nb], N f Yn = [0, N b]. However, the range of output frequencies for so that f Y = n=1 general situations becomes more complicated when a < b is any non-negative number. Lang and Billings in [111][112] established a series of theoretical results on this topic and proposed an efficient approach for determining the non-negative possible generated output frequency range of nonlinear systems. One of these effective theoretical results for determining the non-negative output frequency range for a nonlinear system can be summarized in the following proposition.

4.4 The Analysis of Nonlinear Systems in the Frequency Domain Based NOFRFs

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Proposition 4.1 [112] The non-negative possible output frequency range (band) of the whole system f Y can be analytically estimated using the following result when the frequency range of the input is [a,b] and considering the parameter n and p∗ related to system nonlinearities orders available with the system:  f Y N −2 p∗ +1 fY = fYN ⎧ ∗  i −1 ⎪ ⎪ na nb ⎪ ⎪ + Ik when ⎪ ⎨ (a + b) (a + b) f Y N = k=0  i∗ ⎪  ⎪ na nb ⎪ ⎪ I when + k ⎪ ⎩ (a + b) (a + b) k=0  na i∗ = +1 (a + b)

α N −1 ¯ . . . > α1 > 0 (to achieve a non-singular matrix AU ( jω)). Also u ∗ (t) is the signal input that NOFRFs of the systems under which are estimated √  n  1/ n Hn ( jω1 , . . . , jωn ) U ( jωi )dσωn n−1 (2π ) i=1 ω=ω1 +ω2 +···+ωn √  n  1/ n = αn H ( jω , . . . , jω ) U ∗ ( jωi )dσωn = α n U ∗ ( jω). n n 1 (2π )n−1

Un ( jω) =

ω=ω1 +ω2 +···+ωn

i=1

(4.59) ¯ Step 2: calculation of the N¯ output frequency responses Y 1,..., N under N¯ times excitation as

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  ¯ ¯ Y 1,..., N ( jω) = AU 1,..., N ( jω) G ∗ ( jω) , where

(4.60)

! "T ¯ Y 1,..., N ( jω) = Y 1 ( jω), · · · · · · , Y N¯ ( jω) , ⎡ ⎤ α1 U1∗ ( jω) · · · α1N U N∗ ( jω) .. .. .. ⎢ ⎥ ¯ A U 1,..., N ( jω) = ⎣ ⎦ . . . . N ∗ ∗ α N¯ U1 ( jω) · · · α N¯ U N ( jω)

Step 3: consequently, using the least square approach, the NOFRFs G ∗1 ( jω), · · · , G ∗N ( jω) can be estimated as  ∗   T G ( jω) = G ∗1 ( jω), · · · · · · , G ∗N ( jω) ) *T ) *−1 ) *T ¯ ¯ ¯ ¯ A¯ U 1,..., N ( jω) A¯ U 1,..., N ( jω) Y 1,..., N ( jω) . = A¯ U 1,..., N ( jω)

(4.61) This algorithm for the computation of NOFRFs requires simulation or experimental data for the system under N¯ various input excitations αi u ∗ (t), i = 1, . . . , N¯ . The method can be applied when either a simulation model such as a finite element, mathematical, or NARMAX model is available, or when experiments in practice for multiple inputs can be carried out on the system.

4.5

Concluding Remarks

The essential notions discussed in this chapter, as well as other publications, show that the Volterra series is a consolidated tool for nonlinear system analysis. The convolution integral is generalized by homogeneous polynomial functionals, which allows the functional components to be analyzed in the frequency domain. This provides an explanation regarding how phenomena such as harmonics and intermodulation arise. Both continuous and discrete-time formulations are accessible, allowing major applications in a variety of fields to be developed. The GFRFs are the most used method to perform frequency analysis in nonlinear systems. Despite the fact that significant work based on GFRFs has been studied, there are still a number of concerns that need to be addressed. The first issue is how to extract the GFRFs from a specific problem scenario. This can be accomplished either by directly using input-output data or by fitting a model and extracting the GFRFs from the model equations afterwards. This task can be accomplished in a variety of ways, and the most common ones have been reviewed in this chapter. The second and possibly

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most difficult aspect with GFRF-based nonlinear system analysis is finding how to extract multidimensional features and use them to understand system behaviour. Alternative FRF formulations, on the other hand, appear to be more attractive due to the huge computational efforts required to cope with these challenges. In this context, the concept of NOFRFs, which can be considered as a potential tool for analysing nonlinear systems in the frequency domain in practice, has been proposed. The characteristic of NOFRFs provides a convenient way to study nonlinear systems in the frequency domain, which has been verified by recently published studies for different applications. However, problems regarding computing NOFRFs using numerical estimation based input-output data still need to be addressed, as the existing approach can cause numerical conditioning issues, it cannot be used to solve the issues. To address this problem, a new algorithm of estimation NOFRFs based on the parametric characteristic analysis will be proposed in Chapter 6.

5

Development of Advanced Signal Processing Based Fault Diagnosis

In Chapters 2 and 3, in order to detect the fault in gearbox systems, different signalbased fault detection approaches with the effects of transmission path between different parts is studied with the aid of advanced signal processing techniques. These methods are all univariate signal processing to analyse only one sensor signal. As mentioned in Chapter 2, a wind turbine (WTB) has a multi-stage gearboxes. The measured vibration signal of one stage at any part linked to other stages of the gear and other parts of the rotating machine through the shaft which merges feature information when fault occurs, differing only by energy levels. Gear faults in one stage will cause the vibration of other components, and therefore the collected gear vibration signals are mixed with vibration signals of other parts and noises. In this case, univariate signal processing may lead to information loss. To address this problem, using multiple sensors to collect signals at different locations on the machine to obtain multivariate signal can avoid the loss of local information and enhance a fault feature extraction from different stages of the wind turbine gearbox. In this study, a combined improved multivariate signal processing is proposed to deal with this problem and to enhance the fault diagnosis of the WTB multi-stage gearbox.

5.1

Problem Statement

The feature extraction of non-stationary vibration signals acquired from multi accelerometer sensors is critical to the accuracy of the mechanical system fault detection and condition monitoring. Due to the signals being masked by a huge

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_5

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background noise, extracting fault characteristics from such complicated vibration signals is a challenging task. In order to extend EMD for multi-channel signals and make it suitable for processing multivariate signals, the MEMD technique was recently proposed. It has been proven that, similar to the standard EMD, MEMD acts as a dyadic filter bank for the multivariate input signal on each channel. MEMD, on the other hand, aligns the same intrinsic mode functions (IMFs) over the same frequency range from different channels better than EMD, which is critical in realworld applications. The MEMD still shows the degree of mode mixing problem which affects the accuracy of extracting fault features. To address this problem, in this chapter, an enhanced MEMD, namely, noise-assisted MEMD (NA-MEMD) is proposed for extracting the most relevant multivariate IMFs by adding uncorrelated white Gaussian noise (WGN) in different channels, under certain conditions, to improve the decomposed multivariate IMFs by minimizing the mode mixing problem. The most effective multivariate IMFs connected to faults are then selected using a new technique called correlation fault factor analysis (CFF). A suggested NA-MEMD approach is then combined with a competent non-linear Teager-Kaiser Energy Operator (TKEO) as an energy separation algorithm to optimize the performance of extracting vibration fault features, resulting in superior fault diagnostic performance. Both a synthetic analytic signal and experimental WTB benchmark vibration datasets are used and validated to demonstrate the efficacy of the proposed work. The results in this study demonstrate that the proposed method is suited for capturing a significant fault feature in the WTB multi-stage gearboxes, thus providing a viable multivariate signal processing tool for multi-stage gearbox condition monitoring.

5.2

Multivariate Signal Processing Based Feature Extraction

Basically, several parameters of the same physical system are often recorded for the study across the sample domain in real-world complex systems. These variables might or might not be correlated to each other. For instance, in an oceanic system, a correlation between wind velocity and temperature readings could reveal information about the underlying system dynamics. Such a correlation would surely reveal in the modal decomposition of the data as well. As a result, being able to decompose several variables (process parameters) in a dataset at the same time is critical. In statistics, this is known as multivariate decomposition which is derived

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from multivariate analysis. In the previous chapter, we only dealt with the time series (one-dimensional) data which means the univariate signal processing; in this section, we will discuss about multidimensional and multivariate data at the same time.

5.2.1

Theoretical Background of Multivariate Extension of EMD

To obtain the local mean value, standard EMD uses an interpolation approach to determine the mean value of the upper and lower envelop curves of a univariate signal. The value of the local maximum and minimum, on the other hand, cannot be precisely specified, and the IMF concept established by the vibration mode is unclear. To address these issues, the earliest multivariate EMD approaches [223][8][190][233] were limited to bivariate and trivariate signals. These methods aided in the discovery of the projection method, which involves integrating multiple data channels to extract similar scales across them. The data channels are projected onto several direction vectors in this method, as shown in Fig.5.1. The term “projection” refers to the process of treating each data channel as a component of a vector (with dimensions equal to the number of channels) and computing the dot product of the vector including all channels with a specific direction. The multivariate data is reduced to a univariate time series using this method. [223][190] provide more information on the projection approach. In 2010, Rehman et al. [188] proposed a general method for n-variate EMD (multivariate EMD). This approach considers an n-variate signal n  2 to be an n−dimensional time series and selects the appropriate direction vectors in n−dimensional space. To form the projection signal, the n−variate signal is projected to the selected direction vectors, and the envelopes of each projection signal are calculated by interpolating only maxima of the projections, then averaging them all to obtain the local mean of the multivariate signal. The IMF groups can then be determined using the standard EMD methods, resulting in the multivariate EMD decomposition of the original multivariate signal. The point sets of a (n − 1) dimensional sphere can be regarded as the direction vector sets of n−dimensional space, hence, the challenge of computing the direction vector sets of the n−dimensional space can be related to the problem of computing uniform sample point sets on a (n − 1) dimensional sphere.

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The uniform sampling point sets in multivariate EMD can be realized using one of two methods: (1) Uniform angular sampling method. In this method, the two poles of the sphere have a high density, which makes uniform projection difficult; (2) Generating the point sets on a (n − 1) dimensional sphere using quasi Monte Carlo lower deviation sequences (Hammersley and Halton sequences) [175], which can provide more uniform direction vectors. In the following, the Hammersley sequence will be introduced as an example of the quasi Monte Carlo lower deviation sequence. The discrepancy can be utilized as an indicator of quantitative analysis to find the uniformity of the vectors distribution. The calculating method of the Hammersley sequence [60] is defined as follows: for any non-negative integer k and a prime number p k = a0 + a1 p + a2 p 2 + · · · + ar pr ,

(5.1)

where ai ∈ [0, p − 1] is an integer, i ∈ [0, r ], r ≥ 0, the function φ p of variate k is represented by φ p (k) =

a0 a2 ar a1 + 2 + 3 + · · · + r +1 . p p p p

(5.2)

Let d be the dimension of the sampling space, then function sequences φ p1 (k), φ p2 (k), φ p3 (k), · · · , φ pd−1 (k), can be defined by utilizing a group of prime number sequences p1 , p2 , p3 , · · · , pd−1 . In this case, k− Hammersley points of d−dimensions can be defined as k

 , φ p1 (k), φ p2 (k), φ p3 (k), · · · , φ pd−1 (k) ,

(5.3) n where k = 0, 1, 2, · · · , n − 1 , p1 < p2 < · · · < pd−1 and n is the total number of the Hammersley sequence points. The direction vectors are then formed by mapping Hammersley points on spherical surfaces. As a result, the Hammersley points are first mapped in a cylindrical domain as follows k

 , φ p (k) → (φ, t),

n where (φ, t) ∈ [0, 2π] × (1, −1).

(5.4)

5.2 Multivariate Signal Processing Based Feature Extraction

z

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Figure 5.1 Low discrepancy point sets for projection [229]

The unit cylinder is then projected along radius to map into the unit sphere   (φ, t) → ( (1 − t 2 )cosθ, (1 − t 2 )sinθ, t)T .

(5.5)

Hammersley sequence is used to produce uniformly distributed direction vectors. The multidimensional envelop is defined by the maxima and minima of projections along direction vectors. The local mean of the multivariate signal is calculated by taking the average of these various envelopes. In contrast to the univariate EMD, the stopping criteria does not use equality of zero crossings or extremes. Consider a n-dimensional vectors sequence s(t) = [s1 (t), s2 (t), · · · , sn (t)], representing n-components multivariate signal, xθ k = x1k , x2k , · · · , xnk for a set of direction vectors k = 1, 2, · · · , K along the direction given by the angles  θk = θk1 , θk2 , · · · , θkn−1 in Rn . The detailed algorithm for the MEMD which provides the analysis of general non-stationary multivariate time series is shown in Algorithm 5.1. In addition, sifting process is an iterative procedure for producing IMFs, and the iteration conditions of multivariate EMD and EMD are nearly identical. Currently, the sifting iterative process has three types of stopping criteria [108]: (1) When a Cauchy-type convergence condition is applied, the iteration will end if the difference between adjacent sifting results is less than a threshold value, which is typically in the range of 0.2−0.3 ; (2) When the two sequential sifting processes have the same number of zero crossing points and continuous extreme points for l times, the iteration will come to an end ( l is computed by the practical application); (3) The energy difference tracking approach ensures that the energy difference

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between distinct IMFs is as little as feasible, ensuring that EMD orthogonality is approximated.

5.2.2

Projections and Direction Vectors

Some explanations will be made at this point in order to build a clear image of projections and direction vectors of MEMD. The projection of an n-variate signal s(t) onto a predetermined direction xθk is just a weighted sum of the channels of s(t). Consider a trivariate signal with three components (channels): s1(t), s2(t), and s3(t). We need to select a point set for sampling on the 2-unit sphere (n = 3, see Fig.5.1) to take its projection. As a result, a point on the 2-unit sphere will produce a 3-dimensional direction vector. Let us choose one of the directions, for the sake of demonstration, to be that of the vector [49][88]. In the Cartesian frame of ˆ The projection reference, the unit direction vector is given by √1 iˆ + √2 jˆ + √3 k. 14 14 14 is represented by 1 2 3 pθ k (t) = s1 (t). √ iˆ + s2 (t). √ jˆ + s3 (t). √ kˆ , 14 14 14

(5.6)

the n-variate signal will naturally be projected in the direction given by an n−dimensional vector. By interpolating these points on each individual channel to obtain a directional envelope curve for that particular direction, the maxima of this projected signal can be obtained. The mean envelope obtained by averaging the envelopes of all K projection directions is the actual mean envelope. This concept can also be generalized and used to any number of channels with multiple modes.

5.2.3

Problems of Single and Multichannel Data Analytics with EMD

Because of its data-driven nature, the EMD method is highly adaptable. However, because it is an empirically based method, the algorithm has several drawbacks, such as mode mixing and end artifacts. In the following, we briefly explain both of them.

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Algorithm 5.1 Multivariate EMD STEP 1: Generate and choose a suitable point set for sampling (n − 1)-sphere based on Hammersley sequences; STEP 2: Obtain the direction vector xθ k , k = 1, 2, ....., K where K is the number of the direction vectors; STEP 3: Calculate the projections pθ k (t) of the input signal s(t) along the direction vectors  K xθ k , for all k (the whole set of direction vectors), giving pθ k (t) k=1 as the set of projections; K  STEP 4: Find the time instants tθi k (t) k=1 corresponding to the maxima of the set of projected  K signals pθ k (t) k=1 ;  K STEP 5: Interpolate [tθi k (t), s(tθi k )] to obtain the multivariate envelope curves eθ k (t) k=1 ; STEP 6: For a set of K direction vectors, calculate the envelope mean M(t) =

K 1  eθ k (t) ; K k=1

STEP 7: Compute the residual components r (t) = s(t) − M(t), if r (t) fulfills the stoppage criterion for multivariate IMFs, defines r (t) as an multivariate IMFs and repeat the above steps to s(t) − r (t), until next order IMF is isolated, else repeat the above steps to r (t) until it meets the stoppage criterion.

x(t)

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Figure 5.2 Example of mode mixing in EMD approach

t [s]

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• Mode mixing When scales of multiple frequencies are included in a single IMF, this is referred to as mode-mixing. This happens when the energy of a signal is spread out over a large range of spectral bands. Fig.5.2 illustrates an example case of mode mixing in EMD on a harmonic signal x(t) = x1 (t) + x2 (t) + x3 (t). When we take a Fourier decomposition of the signal, we can notice that it is made up of three separate sinusoidal signals, two of which have identical angular frequencies. The EMD technique illustrated in Fig.5.2(b), on the other hand, comprises only two IMFs, c1 (t) and c2 (t). When there are widely varying scales superposed on each other, mode mixing also occurs. The addition of a small amount of white noise to the signal has shown to be the most successful solution to the mode mixing problem. A single true decomposition is obtained by combining several instances of the noise-added signal into an ensemble of realizations and then averaging them. Because the noise spans the whole spectral domain of the signal, it effectively permits all of the scales to be recognized during the sifting process. Another method to understand the noise adding effect is to consider EMD as a quasi-dyadic filter bank. The activation of all the filters in the banks is ensured by small, finite-amplitude noise. IMFs spanning the whole spectrum range, including frequencies not present in the signal itself, will be found in the ensuing decomposition. IMFs with frequencies not present in the signal, on the other hand, will have a much smaller amplitude and may be easily identified and ignored. It should be emphasized that the computational cost of this method increases linearly as the number of ensembles increases. • End effects Another challenge with the EMD process is how to deal with the end effects issue. Interpolation of smooth envelope curves comes to an abrupt end, causing artifacts. These artifacts can spread via the sifting process, resulting in the extraction of false modes. One solution is to mirror the domain at the boundaries, although the situation is not always so straightforward. End artifacts at the end domain are happened and visualized in Fig.5.3 and Fig.5.4 in the next section.

5.2 Multivariate Signal Processing Based Feature Extraction

5.2.4

109

Improved Form of Multivariate EMD

To address the problem of MEMD which is mode mixing, we have proposed an improved form of MEMD, namely, the NA-MEMD method [181] which makes use of the quasi-dyadic filter bank properties of MEMD on white noise, and demonstrated that for classes of signals where the quasi-dyadic filter bank structure is effective, it is capable of greatly minimizing the mode mixing problem. In this study, additive WGN is used as an additional noise channel. Additionally, the noise is additive where it is statistically independent of the signal and the received signal equals the transmit signal plus some noise. Hence, the NA-MEMD first runs by forming a multivariate signal containing one or more input data channels and adjacent independent events of the WGN in separate channels. The multivariate signal, which consists of data and noise channels, is processed utilizing the MEMD method and the IMFs corresponding to the initial data are reconstituted to obtain the desired decomposition. In this way, unlike the EEMD, physically separate input and noise subspaces in the NA-MEMD prevent direct noise artifacts. In the multivariate data shifting process, when all reflected signals meet any stopping criteria accepted in the standard EMD, IMF can be stopped. The MEMD algorithm, when applied by adding WGN to a multidimensional signal, acts as a dyadic filter bank on each channel, showing a greatly improved rank order of IMFs corresponding to different channels over the same frequency range when compared to EMD as can be seen in results of this chapter. Using this feature of MEMD, the proposed NAMEMD technique is utilized to improve the mixing problem. This is accomplished by including a subspace with multivariate independent WGN and increasing the size of the data, and the resulting composite signal is processed using MEMD. In this way, the noise will remain in a different subspace and be used in raising the filter bank structure, never interfering with the useful data channels, thereby reducing the issue of mode-mixing and providing a better definition of the frequency bands. Only the IMFs corresponding to the initial input signal are preserved by subtracting the noise-dependent IMF subspace. Due to the noise subspace, the alignment of the IMFs adapts to the dyadic filter bank structure, thus providing an important tool for non-stationary analysis of narrow band gearbox signals and aligning the IMFs related to the original input signal. The performance of the NA-MEMD algorithm output depends on the power level of the attached noise channels. The algorithm performs like the standard EMD for infinite small noise amplitudes. Enhancing the power of noise will further strengthen the structure of the dyadic filter bank on the input data, but it overrides the data adaptive capability of the MEMD-based algorithms. Algorithm 5.2 outlines the details of the improved form of MEMD known as the NA-MEMD approach.

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Algorithm 5.2 Improved form of MEMD STEP 1: Create an uncorrelated m− channel white Gaussian noise of the same length as that of the input signal; STEP 2: Add m− channel noise created in step1 to the input multivariate signal n− channel, composing (m + n)− channel input signal; STEP 3: Decompose the (m + n)− channel multivariate signal using the MEMD Algorithm 5.1 to create multivariate IMFs; STEP 4: Obtain the resulting (m + n)− variate IMFs; STEP5: Discard the m− channel IMFs corresponding to the noise from the resulting (m +n)− channel and obtain the n− corresponding to the original multivariate signal.

5.2.5

Simulation Results

To demonstrate the comparison between the MEMD technique and the proposed NA-MEMD method in extracting the exact fault frequencies from the multivariate signal by minimizing mode mixing problems, and to verify the effectiveness of the proposed methodology, the collection of eq.(5.7) can be used to mimic a gearbox multivariate vibration signal as a six-channel multivariate synthetic signal. The frequencies of the multivariate signals are: f 1 = 2H z, f 2 = 8H z, f 3 = 16H z, f 4 = 32H z, f 5 = 50H z, where sampling point Ns = 1000 and sampling frequency f s = 1000H z. x1 (t) = 5cos(2π f 1 t) + 2.5cos(2π f 2 t) + 3cos(2π f 3 t) + 3sin(2π f 4 t) + 2sin(2π f 5 t) , x2 (t) = 2.5cos(2π f 2 t) + 3cos(2π f 3 t) + 3sin(2π f 4 t) + 2sin(2π f 5 t) , x3 (t) = 2.5cos(2π f 1 t) + 2.5cos(2π f 2 t) + 3cos(2π f 3 t) , x4 (t) = 5cos(2π f 1 t) + 2.5cos(2π f 2 t) + 3cos(2π f 4 t) + 2sin(2π f 5 t) , x5 (t) = 2.5cos(2π f 2 t) + 3cos(2π f 3 t) , x6 (t) = 3sin(2π f 1 t) + 2.5cos(2π f 2 t) .

(5.7) Standard EMD, MEMD, and the improved approach are employed to decompose the multivariate signals, as illustrated in Fig.5.3, 5.4 and 5.5 respectively. It can be seen that, the conventional univariate EMD, as shown in Fig.5.3, is applied to channel-wise multivariate signal x1 (t) to x6 (t) separately, which may produce a useful result for loosely coupled data. However, the typical univariate EMD applied to the multivariate data is suffered from many problems, including mode misalignment, mode mixing, and end effects, all of which result in mixed frequencies in the IMFs. It is noticed that different numbers of IMFs have been decomposed for each

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112 Development of Advanced Signal Processing Based Fault Diagnosis

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of these six channel signals. Furthermore, even with the same index of the IMFs, the mode mismatch is manifested. In addition, because the joint information is not taken into account, the univariate EMD cannot ensure the same number of IMFs in the decomposition results for each channel data in one hand, and the mode misalignment problem in the other hand. Applying the MEMD approach to multivariate signals can solve the mode misalignment problem, as shown in Fig.5.4. On the other hand, display aligned modes in the same scale and number across all six channels, overcoming the mode misalignment problem but still suffering from mode mixing between the different IMFs in the same channel. As demonstrated in Fig.5.4, the mode mixing between different IMFs in the same channel distorts the IMFs in the same channel. The proposed approach has been used to address this problem by minimizing the mode mixing problem where four white noise channel signals are added as an assisted signal in four independent (separate) channels. In order for the noise assisted channels to be efficient in decreasing mode mixing, the amplitude of the noise channels should be properly adjusted. Inspired by the univariate noise assisted EMD method, it is advised that the amplitude of the noise power should be (2 − 10)% of the amplitude of the input signal. Therefore, under this condition, the proposed method is effective and significantly reduce the mode mixing issue. It can be seen in Fig.5.5, the mode mixing problem between different IMFs in the same channel is greatly reduced, and all frequency modes for each channel are identified, resulting in accurate decomposition of the generated multivariate signals and accurate separation of each multicomponent frequency channel due to the reduced mode mixing problem. Furthermore, according to the proposed approach, when the number of assisted noise channels and the noise power of assisted noise channels are increased, the cross channel leakage is lessened. However, more assisted noise channel means more computational complexity, as a result, it is regarded as a trade-off between the computational complexity and noise leakage.

5.3

Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis

The decomposed IMFs from the NA-MEMD method still do not show up a very clear GMF and side-bands around the GMF due to the complexity of the vibration signal acquiring from different locations and a huge background noise. Motivated to this problem, and in order to extract the impulse features from the decomposed IMFs and achieve a clear indication of gear characteristic frequencies, a demodulation analysis with the aid of energy separation algorithm is combined with the improved MEMD to enhance the performance of the fault feature extraction. The decomposed

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IMFs from improved MEMD method could be separated into two parts, namely, AM and FM with the aid of improved TKEO. In this section, firstly the dyadic filter bank and mode alignment property of the MEMD method with their importance to measure the amount of mode mixing between different IMFs are explained. Then, to select the most effective multivariate IMFs, a new indication factor as an extension of the correlation coefficients in the univariate EMD will be introduced. After that, the energy separation algorithm based TKEO will be studied. Finally, an experimental case study of wind-turbine gearbox is then investigated and demonstrated to show the effectiveness of the combined multivariate signal processing framework in the detection of the faults in different stages of WTB gearbox.

5.3.1

Filter Bank Property of MEMD

Filter banks are a group of band pass filters used to separate different frequency bands in an input signal. [234] demonstrated that IMFs derived from the typical EMD method provide frequency responses similar to those of a dyadic filter bank. In this section, we investigated whether MEMD preserves the filter bank structure for multivariate input signals. It is worth noting that the concept of a filter bank for multivariate inputs is still confusing in a strict sense, because the concept of frequency for multivariate signals is not well defined. Even if we analyse the frequency response for the individual channels of a multivariate signal, the filter bank structure imposes a further constraint on the frequency output of each multivariate IMF including the overlapping of the filter bands associated with the corresponding (same-index) IMFs from multiple channels. This is critical for the physical meaning of the IMFs obtained via MEMD; any mismatch in the frequency contents of the related multichannel IMFs would render their matching and subsequent fusion applications meaningless.

5.3.2

Selection of Significant Multivariate IMFs

In order to know whether the frequency components achieved by EMD decomposition are meaningful to the analysis of the original signal, it is necessary to compute the correlation coefficients between each IMF and its original signal to select the most effective IMFs. The decomposed IMF can be considered as a noise or a redundant component if its correlation coefficient is infinitesimally small. In order to enhance the reliability of the signal analysis process using the correlation analysis

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[180], it is needed to determine both the most meaningful IMFs, which contain the fault feature frequencies and the noisy component, which are regarded as a redundant component. However, the information obtained by an individual determination of correlation coefficients between an IMF and one collected signal is not comprehensive and cannot describe the relationships between multiple sets of the IMFs, as there are multiple sets of IMFs in the MEMD decomposition process [234]. Motivated to this issue, in this thesis a novel method called correlation fault factor (CFF) analysis is proposed. In the MEMD decomposition of n−variate signal, all the m th IMFs corresponding to each signal create a matrix of n groups which all contain m IMFs. A constructed matrix is then used as a basic unit when proceeding the correlation analysis between n IMF groups and the collected multivariate signal. Then, the observation signal matrix is constructed by using the multivariate signal as  C(t) = c1k (t) c2k (t) · · · cnk (t) ,

(5.8)

where k is the number of different IMFs in each channel and n represents the number of channels. The correlation coefficient of a fault for the k th IMFs of the n th IMF groups can be defined as rnk . The Pearson correlation coefficient, in the signal processing applications, could be described as Ns  

ci (t) − c¯i xi (t) − x¯

rcx = r [ci (t), x(t)] = 

i=1 Ns

Ns   2

2 ci (t) − c¯i xi (t) − x¯

i=1

i=1

,

(5.9)

where xi denote the raw signal with i = 1, 2, . . . , Ns , t represents the time and Ns is the number of data points of the raw signal, x¯ and c¯i represent the sample mean of the raw signal and i th IMFs . First, Performing the correlation analysis on ith IMF of C(t) with each n-variate signal, respectively, after that averaging all the correlation coefficients, it will then provide the CFF of this order IMF, called rik . Then, this value cannot reveal the relationship between this order IMF and the same order IMF of other IMF groups achieved by multivariate EMD and only defines the correlation between this IMF and the collected-original signal. Theoretically, similar order of IMF contains the same fault feature frequency, meaning that the degree of correlation with the original collected signal should almost be the same. This value represents the average of all the vectors’ correlation in the correlation matrix analysis. Hence, taking the average computation of all the CFFs of the IMFs with the same order of n IMF groups yields

5.3 Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis

rk =

n 

rik /n ,

117

(5.10)

i=1

where i = 1, 2, ...., n. The bigger the value of r k in the correlation analysis the greater degree of fault feature frequency correlation between k th IMF of n IMF groups and the original collected signal. Based on the criterion of the Pearson correlation coefficients, when the value is bigger than 0.3, it can then be regarded as the quantities are relevant. Therefore, the orders of the most significant IMFs can be calculated using this approach in order to extract the defect feature frequency.

5.3.3

Demodulation Analysis Aided Fault Feature Extraction

The decomposed IMFs from the NA-MEMD approach still do not show up a very clear GMF and its side bands due to the complexity of the vibration signal acquiring from different locations and a huge background noise. Motivated to this problem, and in order to achieve a clear indication of gear characteristic frequencies, a demodulation analysis with the aid of energy separation algorithm is combined with the improved MEMD to enhance the performance of the fault feature extraction. The decomposed IMFs from improved MEMD method could be separated in to two parts, namely, AM and FM with the aid of improved TKEO. • TKEO and the energy separation algorithm The energy of a signal x(t) can be given by E=

T

−T

|x(t)|2 dt ,

(5.11)

this represents the energy of the signal over time 2T . Another method for estimating the energy of a signal is to utilize the squared absolute value of the distinct frequency bands of the Fourier transformed signal as a measure of the energy levels of the relevant bands. By investigating a second order differential equation, Kaiser discovered that the energy required to generate a basic sinusoidal signal varied with amplitude and frequency. Finally, an energy tracking operator is used to estimate the instantaneous energy of the signal. This is called TKEO, [.], which is a powerful nonlinear operator and can be defined as [109][146]: 2 c [x(t)] = [x(t)] ˙ − x(t)x(t) ¨ ,

(5.12)

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where x(t) denotes the signal and x(t) ˙ and x(t) ¨ are its first and second derivatives respectively. In the discrete time case, the time derivatives in eq.(5.12) can be approximated by time differences as d [x(n)] = x 2 (n) − x(n + 1)x(n − 1) .

(5.13)

The main goal of TKEO is to estimate the instantaneous amplitude and instantaneous frequency of the signal of interest. TKEO can improve fault-related transient features, making it ideally suited to detecting impact signals [136]. TKEO approach, in the field of fault diagnosis, is mainly used to extract the AM and FM signal of a faulty vibration signal, and then statistical features are derived from these AM and FM signals. Furthermore, because only three samples are required for the energy computation at each time instant, TKEO provides a good time resolution. The operators c and d were first developed by Teager during his research on speech signal modelling [227] where he studied the nonlinearities of speech signal and displayed a map of “the energy making sound,” but without revealing the algorithm for calculating this “energy”. Later, Kaiser proposed the algorithm developed by Teager in his work [94]. When c is applied to signals generated by a basic harmonic oscillator, such as a mass spring oscillator with a Newton’s law-derived equation of motion, consider a harmonic oscillation x(t) = Acos(2π f t + ϕ) ,

(5.14)

its first and second derivatives, i.e. velocity and acceleration, can be defined as x(t) ˙ = −2π A f sin(2π f t + ϕ) ,

(5.15)

x(t) ¨ = −4π 2 A f 2 cos(2π f t + ϕ) ,

(5.16)

and

where A denotes the amplitude, f is the frequency, and ϕ denotes an initial phase. By applying the Teager energy operator  to the harmonic oscillation x(t), and substituting for its first and second derivatives by the x(t) ˙ and x(t) ¨ from eqs.(5.15) and (5.16) yields

5.3 Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis

[x(t)] = [cos(2π f t + ϕ)] = 4π 2 A2 f 2 ,

119

(5.17)

[x(t)] ˙ = [−2π A f sin(2π f t + ϕ)] = 16π A f . 4

2

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The energy separation algorithm or in other word the absolute value for the instantaneous amplitude envelope a(t) and the instantaneous frequency f (t) can then be obtained as [145]  1 f (t) = 2π

[x(t)] ˙ , [x(t)]

[x(t)] , |A(t)| = √ [x(t)]

(5.19) (5.20)

both equations above can accurately estimate the instantaneous frequency and amplitude envelope of a sinusoidal signal, and the approximation errors are very small for the cases of AM and FM signals. Furthermore, if the instantaneous amplitude envelope and the instantaneous frequency do not vary too fast or too greatly compared to the signal carrier frequency, energy separation algorithm can also be generalized to signals with arbitrary time-varying amplitude and frequency. The energy separation algorithm could be extended to discrete-time signals by using difference to estimate derivatives. Various algorithms [145][146] have been proposed including the discrete time energy separation algorithms, namely, DESA1 and DESA-2. They both provide accurate estimations by determining the energy operator of the signal and its derivative. The DESA-2 technique has a slight computational advantage and approximates derivatives using a symmetric difference (between samples with time indices that differ by 2). The estimation of the instantaneous frequency and amplitude can be calculated using DESA-2 algorithm as 2d [xn ] , d [xn+1 − xn−1 ] 1 d [xn+1 − xn−1 ] i (n) ≈ ar cos(1 − ), 2 2d [xn ]

|A(t)| ≈ √

(5.21) (5.22)

where i (n) = ωi T . It can be seen from equation eq.(5.22) that the frequency of the signal can only be uniquely calculated up to π/2 (one quarter of the sampling frequency). The frequency i of a signal beyond π/2 is considered as its mirror frequency with

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respect to π/2. It has been demonstrated that the TKEO approach is much more sensitive to estimate the instantaneous features of the vibration signal than other energy separation methods such as the Hilbert transform technique.

5.3.4

Framework of Multivariate Signal Processing Based Gear Fault Detection

The faulty gear systems produce the non-stationary vibrations in the WTB drivetrain. Also, vibrations are transmitted from one stage to the next via the shafts in the WTB gearbox. This makes a detection of a WTB gearbox fault difficult. It has been proposed that a multivariate time-frequency analysis approach be used to generate accurate time-frequency localization, allowing good characteristics to be derived from the mono-component stationary signal to detect and isolate mechanical faults. Because of the load variation/speed fluctuations in the WTB systems, the fundamental GMF and associated harmonics smear across different frequency bins, limiting fault diagnosis performance. In addition, due to the complexity of the multistage WTB gearboxes, we propose an improved MEMD which decomposes multi-component non stationary signal into mono-component and combined with the TKEO to improve the fault feature extraction performance [181]. The methodology of the proposed work in this study is outlined as shown in Algorithm 5.3.

Algorithm 5.3 Proposed framework for multi-stage gearbox fault diagnosis STEP 1: Data acquisition system to collect the vibration signal; STEP 2: Multi-channel data is prepared from multiple sensors in different locations; STEP 3: The IMFs are extracted by applying improved MEMD technique as mentioned in Algorithm 5.2; STEP 4: Effective IMFs are selected using proposed CFF representing the basic oscillating modes in the multivariate data; STEP 5: Instantaneous amplitude and frequency are estimated by using TKEO (DESA2); STEP 6: Envelop spectrum is determined from the estimated instantaneous amplitude and frequency of the selected IMFs; STEP 7: The AM and FM signals at the GMF and its harmonics are indicators of failure modes.

5.3 Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis

5.3.5

121

Experimental Results

The MEMD decomposes the multivariate signal into several IMF groups, all of which have the same length, after collecting multivariate signal from various locations on the machine. The same frequency components of the signal, in these IMF groups, appear in the same order of each group which is the feature of MEMD technique and has been validated in Chapter 2. The correlations between each vector are usually used to determine the specific orders of effective IMFs, however in MEMD decomposition, the same order of each IMF group constitutes a matrix. In this work, by defining and calculating CFF, a correlation analysis between matrices is adopted to find the orders of the significant IMFs which contain fault information. Then, as an energy separation, TKEO will be used to improve the impulsiveness of the effective IMFs. Finally, fault characteristic frequencies are estimated using spectral analysis of the instantaneous amplitude and frequency. Fig.5.6 shows the main concept and framework of the proposed work. • Benchmark description The vibration datasets used in this work has been achieved from NREL Benchmark for vibration condition monitoring of the WTB. As shown in Fig.5.7, the

Simulated & Collected data

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Figure 5.6 Framework of development of advanced signal processing based multi-stage gear fault diagnosis

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Development of Advanced Signal Processing Based Fault Diagnosis B AN8

AN4

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Figure 5.7 A schematic diagram of NREL Dynamometer Test Rig and the sensor location [206]

observed vibration signals are monitored using eight accelerometer sensors mounted in various locations on the WTB drive train gearbox. The PXI -4472B National Instruments data acquisition system has been utilized to collect the data at sampling frequency 40 kHz per channel. The vibration datasets consist of ten one-minute data segments recorded at 1800 RPM on a normal high-speed shaft with 50% rated power and a main shaft speed of 22.1 RPM. In the experimental setup, the multistage gearboxes consist of a one planetary gearbox stage (Low speed stage) and two other parallel stages which are both the ISS and HSS. Table 5.1 shows the more details about the teeth number of gearbox [206]. The defect frequencies of each gear and the GMFs for all three WTB stages, as a prior knowledge of the rotational machines, should be first computed. For the planetary gear stage, based on the planetary gearbox configuration and running speed, as well as eq.(2.13)−(2.15), the characteristic frequency of each gear in planetary gearbox is determined and listed in Table 5.2. Furthermore, the characteristic frequency for the two other parallel stages, namely, the ISS and HSS gear, and related GMFs based on running speed and equations in Chapter 2 are calculated and listed in Table 5.2.

Table 5.1 The teeth number of multiple gears in WTB gearbox Zp

Zs

Zc

Z mi

Z mo

Z hi

Z ho

39

21

99

82

23

88

22

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Table 5.2 Characteristic frequencies and the GMFs Characteristic frequencies

Frequency (H z)

Ring gear fr Planet gear f p Sun gear f sun Intermediate gear pinion f I SS High speed gear pinion f H SS GMFs Planetary stage f mesh(L SS) Intermediate stage f mesh(I SS) High speed stage f mesh(H SS)

1.10 2.61 5.21 7.50 30 Frequency (H z) 36.45 172.50 660

• Result analysis and discussion Vibration data acquired from eight accelerometer sensors are formulated as 8-channel multivariate data. By applying NA-MEMD approach, this multivariate/multichannel signal has been decomposed into 18 IMFs/channels. Owing to the property of MEMD to align IMF frequency bands from different channels, the decomposed multivariate vibration signal is aligned in such a way that the high frequency vibration signals were present in the lower-index IMFs, while low frequency oscillation modes were contained in the higher index-IMFs. The first 12 IMFs were chosen and used to extract the fault characteristics in the proposed approach since the signatures of the faults are embedded in the high frequency modes. The multicomponent signal, in multivariate data, has been decomposed into amplitude and frequency modulated components. The extracted number of IMFs and their order are similar for all eight sensors. As illustrated in Fig.5.8, MEMD was applied to the multivariate/multichannel data and decomposed into a group of IMFs Ck (t). The presence of a mode mixing problem is not obvious from Fig.5.8. However, as illustrated in Fig.5.9, dyadic filter bank structure can measure the amount of mode mixing between IMFs. There is a leakage of higher index IMFs from multiple channels in Fig.5.9, and it is evident that the MEMD suffers from mode mixing problem from each channel. A proposed approach has been applied to the gearbox multivariate vibration signal to address this issue and to reduce mode mixing. The quasi-dyadic sub frequency bands in multivariate vibration signal are more appropriate so that in the proposed approach, the assisted noise in the separate channels aligns the same index oscillatory modes

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124 Development of Advanced Signal Processing Based Fault Diagnosis

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Figure 5.9 IMFs alignment using MEMD as a dyadic filter bank structure

from different channels without mode mixing. It can be noticed that in NA-MEMD, the IMF decomposition performance is improved. Fig.5.10 shows the IMFs that have been obtained using NA-MEMD. Only the first seven IMFs were shown due to the limited space. In this result, it is clear that the dyadic filter bank structure in the NA-MEMD approach is more superior to that of MEMD and the amount of overlapping of the frequency sub bands in dyadic filter bank structure is more in MEMD compared to NA-MEMD. Due to the use of NA-MEMD, the mode mixing phenomenon has significantly minimized, and mode alignment between same-index IMFs for separate channels has become more noticeable in a more fashion form as seen in IMF5 shown in black in Fig.5.9 and Fig.5.11. The lower index IMFs, on the other hand, are not adequately aligned in the NAMEMD, while the higher index IMFs retain frequency localisation in different bands with a large separation gap between the bands. When the MEMD is employed as shown in Fig.5.9, more mode-mixing problems occur, starting with IMF5 and IMF7, while a strict mode alignment is observed when NA-MEMD was used. Therefore, the mode mixing has been reduced as shown in Fig.5.11. Due to the addition of extra noise channels, the computational time of the NA-MEMD technique is increased compared to MEMD. Normally, the amount of mode mixing phenomenon in IMF1 is less significant in fault diagnosis, while the fault signatures are more prominent in the middle index IMFs, indicating that they are the best IMFs for visually inspecting the effects of damage (the periodic impulses are more visible) (IMF 2 -10). Therefore,

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they are selected as an important IMFs for analysing the faults. The instantaneous characteristics of these IMFs can then probably provide the significant features and physically meaningful information for fault diagnosis. A noise reduction is obtained by minimizing the mode mixing issues between different IMFs in each channels, hence the periodic impulses which refer to the fault information appear. Then, the AM-FM components can be clearly observed. This indicates that the independency rate between decomposed IMFs is higher in NA-MEMD, hence the mode mixing is reduced. In short, dyadic filter bank structure property of MEMD approach is used to measure the amount of mode mixing in such away that when the same index IMFs in different channels are aligned well, then it indicates that the mode mixing between the different IMFs in each channel is reduced. To make well alignment, noise channels in separate channels are added to assist the n-channel. This can be observed by comparing both Fig.5.9 and Fig.5.11. In addition, when the number of assisted noise channels and the noise power of assisted noise channels are increased, cross channel leakage is lessened. However, it is proven that the power(variance) of the additional noise should be appropriated adjusted between (2 − 10)% of the power of the input signal in order to get the optimal minimization of mode mixing. The concept of a noise-assisted approach is based on a study of the statistical properties of white noise. Further, because a larger assisted noise channel requires more computational complexity, there is always a trade-off between computational complexity and noise leakage.

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• Analysis of CFFs After applying proposed improved MEMD approach, the most effective orders of IMFs which contain fault feature frequency of the vibration signal can be calculated by using correlation analysis. For the multivariate signal, as mentioned, the CFF analysis method is utilized to analyse the correlation between the matrix created by the collected multivariate signal and the matrices created by employing the same order IMFs. The values of the CFF are calculated as shown in Table 5.3, these values demonstrate the correlation between collected multivariate signal and the specific order of the IMF groups. The greater the value of CFF is, the larger the correlation is. In the proposed work, only CFF is determined for the first seven orders of IMFs. It can be seen that the CFF values of three IMF orders, namely, IMF 2, IMF3 and IMF4 are larger than 0.3. Therefore, these three IMFs are used to exact fault feature frequencies. Among them, IMF3 has the maximum CCF value. Finally, by applying the TKEO to these three IMFs, different fault characteristic frequencies related to the different gear stages around the GMF can be identified through the envelop spectrum of the instantaneous frequency and amplitude which are all explained in this section. Table 5.3 Correlation Fault Factor (CFF) of each IMF for improved MEMD approach IMF order CFF

IMF1 0.1008

IMF2 0.4532

IMF3 0.5671

IMF4 0.3035

IMF5 0.2734

IMF6 0.1683

IMF7 0.1076

• Fault feature extraction for multi-stage WTB gearbox For the planetary gearbox stage, two channel sensors AN3 and AN4 have been used to analyse the faults occur in low speed stage due to the complexity of this stage. Only the results of sensor AN4 are shown here due to the limited page. The results of healthy and faulty signals obtained from sensor AN4 are shown in Fig.5.12 and Fig.5.13 respectively. When an improved MEMD is applied to a multivariate signal, numerous IMFs are generated, which are then utilized to calculate the appropriate instantaneous frequencies and amplitudes. Among these IMFs, only the most three relevant IMFs are plotted in this case. The AM and FM demodulations are predominant in faulty signals after applying TKEO to the corresponding IMFs as shown in Fig.5.12 and Fig.5.13. However, the healthy signal has no AM signal and minimum FM signal. TKEO gives a more accurate result for estimating the instantaneous envelopes compared to HT technique. Finally, envelop

5.3 Multi-stage Wind Turbine Gear Fault Diagnosis Aided Demodulation Analysis Healty IMFs

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0.5

1

1.5

0

2

0.5

1

1.5

600 400 200 0

2

Amplitude

0

-2 0

0

0.5

1

1.5

1

1.5

2

0

0.5

1 t [s]

1.5

2

0

0.5

1 t [s]

1.5

2

100

0

0.5

1 t [s]

1.5

600 400 200 0

2

Amplitude

Freq.(Hz)

IMF4

0.5

50

0

2

5

-5 0

0

100 Freq.(Hz)

IMF3

2

0

0.5

1 t [s]

1.5

50

0

2

Figure 5.12 Healthy signal’s IMF 2, 3, 4 of AN4-Low speed sensor (column 1), instantaneous frequency (column 2), and amplitude envelope (column 3) Faulty IMFs

Instantanous Frequency

0

Amplitude

600 Freq.(Hz)

IMF2

Amplitude envelope 20

2

400

10

200 -2 0

0.5

1

1.5

0

2

0.5

1

1.5

0

2

0

0.5

1

1.5

2

0

0.5

1 t [s]

1.5

2

0

0.5

1 t [s]

1.5

2

20 Freq.(Hz)

IMF3

0 -2 -4 0

Amplitude

600

2

400

10

200 0.5

1

1.5

0

2

0.5

1

1.5

0

2

20

10

0

-10 0

0.5

1 t [s]

1.5

2

Amplitude

Freq.(Hz)

IMF4

600 400 200 0

0

0.5

1 t [s]

1.5

2

10

0

Figure 5.13 Faulty signal’s IMF 2, 3, 4 of AN4-Low speed sensor (column 1), instantaneous frequency (column 2), and amplitude envelope (column 3)

spectrum is obtained for the amplitude envelop of IMF3. To further demonstrate the superiority of the proposed approach, the envelope spectrum of traditional EMD, MEMD, and the proposed approach combined with TKEO for the faulty case has been analysed and shown in Fig.5.14. It can be seen that, compared to both EMD and MEMD method, the faulty envelop spectrum based on the proposed approach exhibits a higher magnitude peaks and significant side-bands corresponding to the characteristic frequencies of sun/ring gear and their harmonics around the GMF. This indicates that these peaks and their side-bands, which are considerably related to the fault frequencies ( f mesh(L SS) ± n × fr /sun ) are dominant and a sure indication of

130

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Development of Advanced Signal Processing Based Fault Diagnosis

(

(

)

(

)

) _

(

)

(

± (×

/

)

) (

)

Sidebands

Figure 5.14 Comparison between EMD, MEMD and proposed approach for IMF3 (AN7HSS sensor)—faulty case

faults in ring /sun gears pinion in the planetary stage. While the side band magnitude peaks for both EMD and MEMD approaches are not significant compared to the proposed approach. Furthermore, the existence of such minor peaks in Fig.5.12 for the healthy scenario does not indicate gear fault because errors in manufacturing and assembling are inevitable, this might result insignificant peaks in the spectrum. In addition, the raw vibration signal from AN4 does not indicate any impulsive signal, while envelope of IMFs derived from AN4 sensor clearly exhibit the impulsive signature. When the damage in the planetary gear occurs, these periodic impulses are modulated. NREL post dynamometer retest result shows and confirms that the ring/sun pinion gears have a fault by scuffing and polishing failure modes. Furthermore, due to the transferring vibration effects across the shaft, certain extra characteristic frequency peaks corresponding to the ISS and HSS gear can be observed in the envelop spectra, demonstrating the benefit of applying multivariate signal processing. Similarly, for the ISS and HSS gear stages, the accelerometer sensor AN6 and AN7 are used to extract the feature faults in the intermediate and high stage gears. The high magnitude peaks and side-bands with their harmonics around GMF in both intermediate stage ( f mesh(I SS) ± n × f I SS ) and high speed stage ( f mesh(H SS) ± n × f H SS ) related to the all three stages could be observed in the envelop spectra indicating the effectiveness of applying multivariate signal processing methods to detect the fault features in different gear stages simultaneously.

5.4 Concluding Remarks

5.4

131

Concluding Remarks

The limitations of standard univariate EMD applied channel-wise on multivariate signals are defined in this chapter. These include both mode misalignment and mode mixing issues as a result of the interdependence between channels being ignored, making it difficult to extract common modes of oscillations within multi-channel signals. To address this issue, an improved MEMD called NA-MEMD is proposed to decompose an accurate IMF in multivariate signals by minimizing mode mixing problem with considering inter-channel correlation so as to detect fault feature in the multi-stage gearbox simultaneously. In this study, Gaussian white noise is added, under certain conditions, to the original multivariate signal in separate channels as an assisted channel which significantly reduces modal aliasing of MEMD and hence reduces mode mixing between different IMFs in each channel. By adding these separate assisted noise channels in NA-MEMD, the original MEMD could be improved to accurately estimate extrema of the multivariate data. In brief, the NA-MEMD adds noise in a separate subspace to the original multivariate data, resulting in minimal residual errors owing to infinitesimally small leakage of noise in the decomposition result. Then, a CFF is presented as an extension of correlation coefficients of univariate IMFs for selecting the most significant multivariate IMFs. A modified form of Energy separation algorithm TKEO based DESA-2 is then integrated with improved MEMD to enhance the accuracy of the instantaneous amplitude and instantaneous frequency estimation. A comparison among the NAMEMD, MEMD, and standard univariate EMD with the aid of TKEO has been studied to further demonstrate the superiority of the proposed technique. The ability to deal with multi-channel vibration signals with large speed/load fluctuations is a unique property and advantage of the proposed method. Therefore, NA-MEMD does not force any restrictions on the number of sensors and their installations (e.g. sensor directions and installation locations), hence overcomes the shortcoming of existing approaches, which can only deal with a single sensor signal. A simulation multivariate signals and experimental vibration data sets are utilized to show the effectiveness of the proposed combined multivariate signal processing algorithm for condition monitoring. The proposed approach is used to diagnose faults in a multistage WTB gearbox also it may has the ability to evaluate the degradation level of the gear mesh stiffness which will be quantitatively proved in future study. Visual inspection in the envelope spectrum of the instantaneous amplitude and frequency gives a pure indication of the failure modes and mechanical faults on the surface of the teeth gears such as severe scuffing, fretting corrosion and polishing wear.

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

In Chapter 4, in order to estimate the GFRFs for nonlinear frequency analysis, several analytical methods have been studied. The main goal of n th order GFRFs calculation is to accurately estimate the nonlinear output spectrum which can be utilized in the estimation of NOFRFs. However, these approaches for computation of GFRFs and NOS suffers from two main problems; first, the expensive computational cost, mathematical complexity especially for higher order nonlinear output spectrum and also the analytical relationship between model parameters and GFRFs and/or NOS are not clearly demonstrated. To circumvent these issues and address the optimal estimation of the NOS and NOFRFs with respect to the nonlinear model parameter, a concept of parametric characteristic analysis based separable function is proposed in this chapter which can be applied in crack evolution and detection in engineering structures. In this study, the estimation of the NOS with respect to the specific nonlinear model parameters together with the computation of NOFRFs are analysed to solve these issues.

6.1

Problem Formulation

The input−output relationship of nonlinear systems, under certain condition, can be represented by a Volterra series up to a sufficiently high order N as shown in eq.(4.53) and can be written as y(t) =

N  

∞

n=1 −∞

 ...

∞ −∞

h n (τ1 , ..., τn )

n 

u(t − τi )dτi .

i=1

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_6

133

134

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Estimation of NOFRFs Based Parametric Characteristic Analysis

The nonlinear system in the frequency domain has been described by the eq.(4.54). The result in eq.(4.54) can be utilized to achieve a more physically meaningful interpretation of the NOS or output frequency response of nonlinear system. In eq.(4.54), Hn ( jω1 , · · · , jωn ) is the n th order GFRF which can also be described as the multivariate Fourier transformation of h n (τ1 , · · · , τn ) as described in eq.(4.15). This result for computation of n th order GFRFs is known as non-parametric computation (i.e. time domain model is non parametric which is Volterra series model). There are some problems of this method including: • Large amount of data are required. • Only the frequency domain Volterra systems up to third order can be identified. • Direct computation of nonlinear system output spectrum involves very complicated integral and symbolic operation in a multidimensional complex space. To deal with these issues, an analytical solution called harmonic probing method was proposed for a recursive computation of the n th −order GFRFs of nonlinear systems [106][177]. This method is called parametric model-based approach (i.e. the parametric form of Volterra-type nonlinear model is utilized in order to map the nonlinear systems in time domain to frequency domain). Nonlinear systems can be represented by different parametric models. A parametric model of the nonlinear system under investigation can be used to compute the GFRFs. In this thesis, two different parametric models known as the NARX and NDE model are mainly used to describe the Volterra-type nonlinear systems in eq.(4.53). For the discrete time system, the Volterra series model in eq.(4.53) can be described in a parametric model which called NARX model as shown in eq.(4.24), and the n th order GFRFs computation of NARX models using the harmonic probing method can be recursively computed using eq.(4.26)−(4.30) [106][102]. For the continous time model, similarly, the Volterra series model in eq.(4.53) can be represented in a parametic model called NDE model as shown in eq.(4.25), and the n th order GFRFs computation of NDE model using the harmonic probing method can be recursively computed using eq.(4.38)−(4.42) [106][102]. Importantly, by comparing the n th order GFRF for the NDE model in eq.(4.42) to that for the NARX model in eq.(4.29), both eq.(4.42) and eq.(4.29) have the same notations and structure. This means that the parametric characteristics of the GFRFs for both the NDE model and the NARX model are the same. It can be seen from the recursive algorithm for the computation of the GFRFs in eq.(4.38) or eq.(4.42) amd eq.(4.26) or eq.(4.29) that the n th order GFRF is a parameter-separable polynomial function with respect to the nonlinear parameters in system models eq.(4.24) or eq.(4.25). For convenience, let

6.2 Parametric Characteristic Analysis

  p = 0 . . . m, p + q = m   C(n, K ) = c p,q (k1 . . . , k p+q )  2  m  n ,  ki = 0 . . . K , i = 1 . . . p + q

135



(6.1)

which contains all the nonlinear parameters from nonlinear degree 2 to n. The C(n, K ) basically includes all the nonlinear parameters involved in system model eq.(4.24) or eq.(4.25). Obviously, when p+q = 1, the corresponding parameters are then associated with the linear model terms. The coefficient extractor (CE) operator which will be explained in the next section as a part of parametric characteristic analysis will be applied to all the nonlinear parameters in C(n, K ) in this chapter. The analytical computation of n th order GFRFs (H n(·)) based on the harmonic probing algorithm [194] reveals the mapping between nonlinear parameters in time domain and the n th order GFRFs. However, this still suffers from some drawbacks including the complicated integral and symbolic operations in multi-dimensional complex space in one side and also the analytical relationship between model parameters and nonlinear output spectrum Y ( jω) is NOT clearly demonstrated. The effects of the nonlinear parameters in the time domain on both GFRFs (H n(·)) and Y (·) in nonlinear frequency analysis needs to be more clearly addressed. Motivated to this problem, in the next section a parametric characteristic analysis as a backbone in this thesis is first discussed as a mathematical tool to deal with this issue. Therefore, the main objective of this chapter is to compute the n th order GFRFs and nonlinear output frequency response with respect to the nonlinear parameters of interest using an alternative analytical solution known as parametric characteristic analysis and then use them in the application of fault detection.

6.2

Parametric Characteristic Analysis

The concept of PCA is defined in this section for a class of polynomial functions with parametrized coefficients. On the basis of this concept, a powerful mathematical operator known as the CE operator is demonstrated and defined, which plays a key role in parametric characteristic analysis approach for a class of parametrized polynomial functions with separable property. The main purpose of parametric characteristic analysis with the aid of separable function is to separate the nonlinear polynomial series model into two mathematically uncorrelated/independent functions and extract the interested nonlinear parameters in the function in order to deal with the effects of the nonlinear parameters in time domain on the frequency domain Hn (·), the NOS, Y ( jω), and the n th order nonlinear output frequency response, Yn ( jω). To this end, both mathematical tools known as separable function and CE

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Estimation of NOFRFs Based Parametric Characteristic Analysis

operator will be investigated to separate the polynomial series model function and mathematically extract the nonlinear parameters.

6.2.1

Preliminaries

In this section, the separable function and CE operator are introduced in more detail, which serve as the main mathematical tools in the area of parametric characteristic analysis. Besides, the interconnections between separable function and CE operator are further discussed, which builds the foundation for the study in the subsequent sections.

6.2.2

Separable Function

Separable functions can be described using the following definition. Definition 6.1 [103] ( Separable function ): If a function h(s; x) can be expressed as a polynomial function h(s; x) = g(x) · f 1 (s) + f 0 (s), then the function h(s; x) is said to be separable with respect to parameter x, where g(x) are functions related to all nonlinear parameters of interest x, and f i (·) are functions of variable s but independent of the parameter x for i = 0, 1. A function h(s; x) satisfying Definition 6.1 is referred to x-separable function or simply separable function, where x is referred to as the parameter of interest which may be a parameter to be analyzed or designed for a system, and s corresponds to other variables or parameters, which may be an independent variable or reference variable of a system such as time or frequency. Remark 6.1 [103] in the Definition 6.1 regarding an x-separable function h(s; x), x can be a vector including all the separable parameters or variables of interest, and s represents not only the independent variables of h(·), but also could include all the other non-interested/un-separable parameters in h(·). The values of the parameters x and s can be real or complex, but the detailed features of the function h(·) and its parameters are not necessarily taken into account here. Also note that, f 0 (s) and f 1 (s) are invariant/independent with respect to x and g(x) in definition 6.1. Therefore, h(s; x) can be considered as a pure function of x for any specific s. In this situation, if g(x) is known, and additionally the values of h(s; x) and g(x)

6.2 Parametric Characteristic Analysis

137

under some different values of x, for example x1 and x2 , can be achieved by specific approaches (simulations or experimental tests), then the values of f 0 (s) and f 0 (s) f 1 (s) can be obtained by using the least square approach, i.e. 

h(s; x1 ) = g(x1 ) · f 1 (s) + f 0 (s) h(s; x2 ) = g(x2 ) · f 1 (s) + f 0 (s)

=⇒



−1

f 0 (s) h(s; x1 ) 1 g(x1 ) = . f 1 (s) 1 g(x2 ) h(s; x2 ) (6.2)

As a result, at a given s, the function h(s; x), which is an analytical function of the parameter x, can be achieved. This provides a numerical method to find the relationship between the parameters of interest and the corresponding function. Furthermore, at a given point s, an x−separable function h(s; x) is denoted as  h(x)s or simply h(x)s . Consider a parameterized polynomial function series H (s; x) = g1 (x) f 1 (s) + g2 (x) f 2 (s) + · · · + gn (x) f n (s) = G · F T ,

(6.3)

where n > 1, gi (x)  and f i (x) for i = 1, . . . , n are all  scalar functions, Also assume the F = f 1 (s), f 2 (s), . . . , f n (s) and G = g1 (s), g2 (s), . . . , gn (s) , both x and s are parameterized vectors that include the interested parameters and other parameters(non-interested parameters), respectively. Because the series is x−separable, H (x)s is totally determined by the parameters in x or the values of g1 (x),g2 (x), and gn (x). Notice that the characteristics of the series H (s; x), at a given point s, is completely determined by G, and how the parameters in x are included in H (s; x) is totally demonstrated in G, too. Thus, the parametric characteristics of the series H (s; x) can be completely revealed by the function vector G. The vector G, therefore, is referred to as the parametric characteristic vector of the series. If the characteristic vector G is computed, then following the procedure described in Remark [6.1], the function H (x)s is obtained, which illustrates the analytical relationship between the interested parameter x and the series, allowing the effects of each parameter in x on the series to be analyzed. The function H (x)s is the term used to describe as parametric characteristic function of the series H (s; x). According to all the above discussions, the following result can then be obtained and concluded. Lemma 6.1 [103] If H (s; x) is a separable function with respect to the parameter x, then a parametric characteristic vector G and an appropriate function vector F must exist, such that the polynomial function H (s; x) = G.F T , where the compo-

138

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Estimation of NOFRFs Based Parametric Characteristic Analysis

nents of G are the functions of x but independent of s, and also the components of F are the functions of s and independent of x. Based on the discussion and definition above, it can be noticed that the n th order GFRF of the NARX model in eq.(4.24) and NDE model in eq.(4.25) are separable with respect to any nonlinear parameters of the corresponding models. As mentioned before, the parametric characteristic vector G should be achieved and extracted, in order to analyze the relationship between an interested function H (s; x) and its separable interested parameters x which is nonlinear parameters in this case. The parameterized vector G for a simple parameterized function may be simple and easy to achieve. However, this is not straightforward for a complicated function series involving recursive computations. In order to tackle this goal, and more importantly to achieve and study the parametric characteristic analysis of the n th order GFRF and output spectrum of Volterra-type nonlinear systems represented by eq.(4.24) or eq.(4.25), the CE operator will be introduced in the next section for extracting any parameters of interest mathematically included in a separable parameterized polynomial function.

6.2.3

Coefficient Extractor

The CE operator as a useful mathematical operator is defined in order to analyse the parametric characteristics of frequency response functions in nonlinear systems. Assume Cs is a set of parameters that takes values in C, Let P c denote a set ri  r1 r2 of monomial functions defined in C c , i.e. P = {c , c , . . . , c s c 1 2 I i ∈ C s , ri ∈     Z+ , I = Cs } where Cs  denotes the number of the parameters in Cs , and Z+ represents all the positive integers. Similar to Cs , let Ws be another param eter but Ws Cs = Ø and let P f be a function set defined in Ws , i.e. p f = { f (w1 , . . . , w I )wi ∈ Ws , I = Ws }. Let  represent all the finite order function series with the coefficients in Pc timing some functions in p f . A series in  is defined as HC F = s1 f 1 + s2 f 2 + · · · + sσ f σ ∈  ,

(6.4)  where si ∈ Pc , f i ∈ p f for i = 1, . . . , σ ∈ Z+ , C = s1 , s2 , . . . , sσ , and T  also F = f 1 , f 2 , . . . , f σ . This series is clearly separable with respect to the parameters in Cs and Ws .

6.2 Parametric Characteristic Analysis

139

A CE operator is defined as  → Pσc such that  C E(HC F ) = s1 , s2 , . . . , sσ = C ∈ Pcσ ,

(6.5)

where Pc = {[s1 , s2 , . . . , sσ ]|s1 . . . sσ ∈ Pc }. The CE operator has some mathematical properties [103]: 1) Reduced vectorized sum ⊕.   C E(HC1 F1 + HC2 F2 ) = C E(Hc1 F1 ) ⊕ C E(Hc2 F2 ) = C1 ⊕ C2 = C1 , C2 , 

also, C2 = V EC(C 2 − C 1 ∩ C 2 ) , where C 1 = {C1 (i)|1 ≤ i ≤ |C1 |}, and also C 2 = {C2 (i)|1 ≤ i ≤ |C2 |}, V EC(·) is a vector which includes all the elements in set (·).  C2 refers to a vector which includes all the elements in C2 except the similar elements as those in C1 . 2) Reduced Kronecker product ⊗. C E(HC1 F1 · HC2 F2 ) = C E(Hc1 F1 ) ⊗ C E(Hc2 F2 )      C = C1 (1)C2 , . . . , C1 (|C1 |)C2 = C1 ⊗ C2  V EC c  3 , c = C3 (i), 1 ≤ i ≤ |C3 |

which means that there will be no repetitive elements in C1 ⊗ C2 . (3) Invariant. i) C E(α · HC F ) = C E(HC F ), ∀α ∈ / Cs ; ii) C E(HC1 F1 + HC2 F2 ) = C E(HC(F1 +F2 ) ) = C. (4) Unitary. i) if ∂ H∂cC F = 0 for ∀c ∈ Cs ; then C E(HC F ) = 1; ii) if HC F = 0 for ∀c ∈ Cs , then C E(HC F ) = 0. When C E(HC F ) contains a unitary 1, the corresponding series of HC F contains a nonzero constant term that has no relation to the parameters in the Cs . (5) Inverse. C E −1 (C) = HC F . This implies any a vector C which consists of the Pc elements should relate to at least one series in .

140

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

(6) C E(Hc1 F1 ) ≈ C E(Hc2 F2 ) if the elements of both C1 and C2 are the same, where ≈ means equivalence. That is, both series actually yield the same result when the order of si f i in the series is considered since it has no influence on the HC F value of the series. Furthermore, this implies that the CE operator is also commutative and associative, for example, C E(HC1 F1 + HC2 F2 ) = C1 ⊕ C2 ≈ C E(HC2 F2 + HC1 F1 ) = C2 ⊕ C1 . (7) Separable with respect to only parameters of interest. A parameter can only be extracted in a series if the interested parameter and the series is separable with respect to this parameter. Hence, the result for different purposes is different. The CE operator establishes a mapping from  to Pcσ , as it can be seen from all the previous definition of the CE operator, and all operations are expressed in terms of the parameters in Cs [103]. For convenience, let ⊕ (·) and ⊗ (·) are the addition and (∗)

(∗)

multiplication by the reduced vectorized sum “⊕” and reduced Kronecker product k

“⊗” of the terms in (·) satisfying (∗), respectively; and ⊕ C p,q = C p,q ⊗. . .⊗C p,q i=1

can be written in a simple form as C kp,q . The ( p + q)−th degree nonlinear parameter vector for NDE model in eq.(4.25) can be defined as C p,q = [c p,q (0, · · · , 0), c p,q (0, · · · , 1), · · · , c p,q (K , · · · , K )] ,   

(6.6)

p+q=m

which contains all the nonlinear model parameters of the form c p,q (·) in model eq.(4.25). Similarly, for NARX model in eq.(4.24), the nonlinear parameters vector can be defined as C p,q = [c p,q (1, · · · , 1), c p,q (1, · · · , 2), · · · , c p,q (K , · · · , K )] .   

(6.7)

p+q=m

It should be noted that C p,q can alternatively be considered as a set of ( p + q)−th degree nonlinear model parameters with the form c p,q (·). Furthermore, if C E(HC F ) = 0, i.e. all the elements of C E(HC F ) are zero, then C E(HC F ) is regarded as empty as well. Thus, the CE operator provides an useful tool to study the analysis of the parametric characteristics of separable functions, i.e. analytically extracting the interested parameters in the series model. It can be demonstrated that the characteristics of the

6.2 Parametric Characteristic Analysis

141

nonlinear parametric of the GFRFs for eq.(4.24) or eq.(4.25) can be achieved by directly substituting the operations “+” and “·” by “⊕” and “⊗” in the corresponding recursive computation algorithms, respectively, and ignoring the corresponding multiplied frequency functions. To further demonstrate the process of mathematically extracting the interested nonlinear parameters using CE opertator, the following example is provided to demonstrate the use of the CE operator. Example 6.1 Apply the CE operator to the n th GFRFs in eq.(4.25), which is the GFRFs for NARX model, up to the 3rd order. When n = 1, eq.(4.26) becomes L(1)H1 ( jω1 ) =

K 

  c0,1 (k1 ) exp − j(ω1 k1 ,

k1 ,kn =1

when the CE operator is applied to the first order GFRF for the nonlinear model parameters for both sides of the above equation (using the properties of CE) yields K         C E H1 ( jω1 ) = C E L(1)H1 ( jω1 ) = C E c0,1 (k1 ) exp − j(ω1 k1 = 1 . k1 ,kn =1

When n = 2, it is known from eq.(4.26) to eq.(4.30) that, L(2)H2 ( jω1 , jω2 ) =

K 

  c0,2 (k1 , k2 ) exp − j(ω1 k1 + ω2 k2 )

k1 ,kn =1

+

K 

  c1,1 (k1 , k2 ) exp − j(ω2 k2 ) H1,1 ( jω1 )

k1 ,kn =1

+

K 

c2,0 (k1 , k2 )H2,2 ( jω1 )

k1 ,kn =1

=

K 

  c0,2 (k1 , k2 ) exp − j(ω1 k1 + ω2 k2 )

k1 ,kn =1

+

K 

    c1,1 (k1 , k2 ) exp − j(ω2 k2 ) H1 ( jω1 ) exp − j(ω1 k1 )

k1 ,kn =1

+

K  k1 ,kn =1

  c2,0 (k1 , k2 )H1 ( jω2 )H1 ( jω1 ) exp − j(ω1 k1 + ω2 k2 ) ,

142

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

applying the CE operator to the 2th order GFRF for nonlinear model parameters above (using the properties of CE) yields     C E H2 ( jω1 , jω2 ) = C E L(2)H2 ( jω1 , jω2 ) ⎧ K ⎫    ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c (k , k ) exp − j(ω k + ω k ) ⎪ ⎪ 0,2 1 2 1 1 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k1 ,kn =1 ⎪ ⎪ ⎪ ⎪ ⎪ K ⎨      ⎬ c1,1 (k1 , k2 ) exp − j(ω2 k2 ) H1 ( jω1 ) exp − j(ω1 k1 ) = CE + ⎪ ⎪ ⎪ ⎪ ⎪ k1 ,kn =1 ⎪ ⎪ ⎪ ⎪ ⎪ K ⎪ ⎪  ⎪ ⎪   ⎪ ⎪ ⎪ c2,0 (k1 , k2 )H1 ( jω2 )H1 ( jω1 ) exp − j(ω1 k1 + ω2 k2 ) ⎪ ⎪ ⎪ ⎩+ ⎭ k1 ,kn =1

 = C0,2 ⊕

2−1 2−q

⊕ ⊕

q=1 p=1

    2 C p,q ⊗ C E(H2−q, p (·)) ⊕ ⊕ C p,0 ⊗ C E(H2, p (·)) p=2

    = C0,2 ⊕ C1,1 ⊗ C E(H1,1 (·)) ⊕ C2,0 ⊗ C E(H2,2 (·)) .

It is noticed that H1 (·) has no relationship with nonlinear parameters, based on the definition of the CE operator, it can be achieved that C E(H1 (·)) = 1. Similarly, it can be also achieved that C E(H2,2 (·)) = 1. Thus, the parametric characteristic function vector of the second order GFRF is C E(H2 (·)) = C0,2 ⊕ C1,1 ⊕ C2,0 = C0,2 + C1,1 + C2,0 .

(6.8)

When n = 3, i.e. (3rd order), it can be known from eq.(4.26) to eq.(4.30) that, L(3).H3 ( jω1 , jω2 , jω3 ) K 

=

  c0,3 (k1 , k2 , k3 ) exp − j(ω1 k1 + ω2 k2 + ω3 k3 )

k1 ,k3 =1

+

3−q  3−1  K 

  q  c p,q (k1 , · · · , k p+q ) exp − j ω3−q+i k p+i

q=1 p=1 k1 ,k3 =1

i=1

× H3−q, p ( jω1 , jω2 , jω3−q ) +

K 3   p=2 k1 ,k p =1

c p,0 (k1 , · · · , k p )H3, p ( jω1 , jω2 , jω3 ) ,

6.2 Parametric Characteristic Analysis

143

similarly, applying the CE operator to the 3rd order GFRF for nonlinear model parameters above (using the properties of CE) yields     C E H3 (·) = C E L(3)H3 (·) ⎧ ⎫ K ⎪ ⎪    ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c0,3 (k1 , · · · , k3 ) exp − j(ω1 k1 + · · · + ω3 k3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ,k =1 1 3 ⎪ ⎪ ⎪ ⎪   ⎪  ⎪ q 3−q  2  K ⎪ ⎪  ⎪ ⎪ ⎨+ c p,q (k1 , · · · , k p+q ) exp − j ω3−q+i k p+i ⎬ = CE i=1 ⎪ q=1 p=1 k1 ,k3 =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H3−q, p ( jω1 , · · · , jω3−q ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 K ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + c (k , · · · , k )H ( jω , jω , jω ) ⎪ ⎪ p,0 1 p 3, p 1 2 3 ⎪ ⎪ ⎩ ⎭ p=2 k1 ,k p =1

 = C0,3 ⊕

    2 3−q 3 ⊕ ⊕ C p,q ⊗ C E(H3−q, p (·)) ⊕ C p,0 ⊗ C E(H3, p (·)) , q=1 p=1 p=2       middle term

last term

the middle and last terms of the above equation can be computed as: for the middle terms 

2 3−q

⊕ ⊕

q=1 p=1 2

= ⊕

p=1

  C p,q ⊗ C E(H3−q, p ( jω1 , · · · , jω3−q ))

     1 C p,1 ⊗ C E H2, p ( jω1 , · · · , jω2 ) ⊕ C p,2 ⊗ C E(H1, p ( jω1 )) p=1





    = C1,1 ⊗ C E H2,1 ( jω1 , jω2 ) ⊕ C2,1 ⊗ C E H2,2 ( jω1 , jω2 ) ⊕ C1,2 ⊗ C E H1,1 ( jω1 )       = C1,1 ⊗ C E H2,1 ( jω1 , jω2 ) ⊕ C2,1 ⊗ C E H1 ( jω1 )H1 ( jω2 ) ⊕ C1,2 ⊗ C E H1,1 ( jω1 )   = C1,1 ⊗ C E H2,1 ( jω1 , jω2 ) ⊕ C2,1 ⊗ 1 ⊕ C1,2 ⊗ 1   = C1,1 ⊗ C E H2,1 ( jω1 , jω2 ) ⊕ C2,1 ⊕ C1,2 ,

similarly, for the last terms 

 3 ⊕ C p,0 ⊗ C E(H3, p ( jω1 , · · · , jω3−q ))

p=2

    = C2,0 ⊗ C E H3,2 ( jω1 , · · · , jω3 ) ⊕ C3,0 ⊗ C E H3,3 ( jω1 , · · · , jω3 )

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6

Estimation of NOFRFs Based Parametric Characteristic Analysis

  = C2,0 ⊗ C E H1 ( jω1 )H2 ( jω2 , jω3 ) + H1 ( jω3 )H2 ( jω2 , jω1 )   ⊕ C3,0 ⊗ C E H1 ( jω1 )H1 ( jω2 )H1 ( jω3 )   = C2,0 ⊗ C E H2 ( jω1 , jω2 ) ⊕ C3,0 . Substituting the results of the middle and last terms to the main equation yields     C E H3 ( jω1 , · · · , jω3 ) = C E L(3)H3 ( jω1 , · · · , jω3 )         = C0,3 ⊕ C1,1 ⊗ C E H2 (·) ⊕ C2,1 ⊕ C1,2 ⊕ C2,0 ⊗ C E H2 (·) ⊕ C3,0 2 2 ⊕ C11 ⊗ C20 ⊕ C21 ⊕ C12 ⊕ C20 ⊗ C02 ⊕ C20 ⊗ C11 ⊕ C20 ⊕ C30 = C03 ⊕ C11 ⊗ C02 ⊕ C11 2 2 = C03 + C21 + C12 + C30 + C11 + C11 ◦ C20 + C20 ◦ C02 + C11 ◦ C02 + C20 .

(6.9)

It is obvious from both equations eq.(6.8) and eq.(6.9) that how nonlinear parameters of nonlinear degree 2 and 3 make a contribution in the construction of the 2nd and 3r d order GFRFs. Eq.(6.8) clearly demonstrates that different types and degrees of nonlinear parameters in C0,2 , C1,1 and C2,0 have independent influence on the second order GFRF with no interference, and that no other nonlinear parameters make any contribution on the 2nd order GFRF. This reveals an explicit insight into the analytical relationship between nonlinear parameters and the second order GFRF. Therefore, if H2 (·), for instance, is required to have a specific amplitude or phase, only the parameters in C0,2 , C1,1 and C2,0 may need to be considered and analyzed purposely. However, when cross multiplication terms between different types of nonlinear parameters appear in eq.(6.9), there may be some unusual nonlinear behaviour associated with these terms in the frequency response function. Consequently, eq.(6.9) obviously reveals how the different nonlinear parameters make a contribution in the generation of the 3rd order GFRF. Example 6.1 demonstrates that the CE operator is very effective tool for deriving the parametric characteristic vector of a separable function series around the parameters of interest. It provides a key technique for analysing parametric effects on parameter-separable function series in any system. With the demonstration of the parametric characteristic analysis of the polynomial separable function at hand, we are devoted to the compute the parametric characteristic of n th order GFRFs, i.e. C E(Hn (·)) and nonlinear output spectrum Y ( jω) with related to some specific nonlinear parameters in the subsequent section.

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

6.3

145

Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

In this section, a parametric characteristic of the n th order GFRFs C E(Hn (·)), nonlinear output spectrum i.e. NOS Y ( jω) and n th order system output response Yn ( jω) with related to the very specific nonlinear parameters are constructed based on the results given in [103]. This mainly consists of two parts: 1) the parametric characteristic of n th order GFRFs, under certain condition, for a specific nonlinear parameters C E(Hn (·)) is computed; 2) the n th order nonlinear output frequency response Yn ( jω) and the output spectrum of the nonlinear system or NOS, Y ( jω) for a specific nonlinear parameters based on the parametric characteristic of n th order GFRFs is analytically computed.

6.3.1

Computation of Parametric Characteristic of GFRFs with Respect to Model Nonlinear Parameters

A fundamental result firstly can be achieved for the parametric characteristic of the n th order GFRF in eq.(4.29) or eq.(4.42), which provides an essential foundation for the parametric characteristic analysis of the nonlinear frequency analysis in the next sections. Theorem 6.1 [103] Based on the parametric characteristic analysis and separable function in the previous section, the n th order GFRF in eq.(4.42) can be separated in to two different parts such that Hn ( jω1 , . . . , jωn ) = C E(Hn ( jω1 , . . . , jωn )) · f n ( jω1 , . . . , jωn ) ,

(6.10)

where f n ( jω1 , . . . , jωn ) denotes a complex valued function vector with an appropriate dimension. It is a function of jω1 , . . . , jωn and the linear parameters of the NDE model and also independent to the parametric characteristic C E(Hn ( jω1 , . . . , jωn )) in this study. Also, C E(Hn ( jω1 , . . . , jωn )) is the parametric characteristic vector of the n th order GFRF, and its elements include and only include all the nonlinear parameters in C0,n and all parameter monomials in C p,q ⊗ C p1 ,q1 ⊗ C p2 ,q2 ⊗ · · · ⊗ C pk ,qk for 0  k  n − 2 whose subscripts satisfy

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Estimation of NOFRFs Based Parametric Characteristic Analysis

⎧ k ⎪  ⎪ ⎨p + q + ( pi + qi ) = n + k i=1 ⎪ ⎪ ⎩1  p  n − k, 2  p + q  n − k, 2  p + q  n − k i i

.

(6.11)

Proof. Based on Lemma 6.1 and the corresponding discussions in Section 6.2, eq.(6.11) can be achieved based on the separable function. By applying the CE operator to eqs.(4.38)−(4.41), then eq.(6.11) is derived, firstly by applying CE operator to both sides of eq.(4.38) yields  C E(Hn (·)) = C0,n ⊕

   n ⊕ ⊕ C p,q ⊗ C E(Hn−q, p (·)) ⊕ ⊕ C p,0 ⊗ C E(Hn, p (·)) ,

n−1 n−q

q=1 p=1

p=2

(6.12) where   n− p+1 C E Hn, p (·) = ⊕ C E(Hi (·)) ⊗ C E(Hn−i, p−1 (·)) ,

(6.13)

i=1

or   C E Hn, p (·) =

n− p+1

⊕

r1 ,···  ,r p =1, ri =n

  p ⊗ C E Hri (·) ,

(6.14)

i=1

    and C E Hn,1 (·) = C E Hn (·) . For clarity, a simpler case can be considered that only output nonlinearities in eq.(6.12) is considered, then eq.(6.12) becomes

n

n

n− p+1

p=2

p=2

r1 ,···  ,r p =1, ri =n

⊕ C p,0 ⊗ C E(Hn, p (·)) = ⊕ C p,0 ⊗

note that

n− p+1

⊕

⊕

  p ⊗ C E Hri (·) ,

  p ⊗ C E Hri (·) contains all the combinations of (r1 , · · · , r p ) ,

r1 ,···  ,r p =1, i=1 ri =n

satisfying this condition p  i=0

(6.15)

i=1

ri = n, 1  ri  n − p + 1, and 2  p  n .

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

147

In addition, C E(H1 (·)) = 1 because there are no nonlinear parameters in it and related to linear parameters, also any repetitive combinations here have no any contribution. Hence,   n− p+1 p ⊕ ⊗ C E Hri (·) , r1 ,···  ,r p =1, ri =n

i=1

must include all the possible non-repetitive combinations of (r1 , · · · , rk ) satisfying this condition p 

ri = n − p + k, 1  ri  n − p + 1, and 1  k  p ,

i=0

so does C E(Hn ( jω1 , · · · , jωn )). The combinations of each subscript refers to a monomial of the contained nonlinear parameters. Therefore, by involving the term C p,0 and taken in to account the range of each variable (i.e. ri , p and k), C E(Hn ( jω1 , · · · , jωn )) must contain all the possible non-repetitive monomial functions of the nonlinear parameters in this form C p,0 ⊗ Cr1 ,0 ⊗ Cr2 ,0 ⊗ · · · ⊗ Crk ,0 , satisfying p+

p 

ri = n + k, 2  ri  n − k, and 0  k  n − 2 0  2  n − k .

i=0

The above case is only for the case when only pure output nonlinearity is considered. When considering all different types of nonlinearities, the results above can be extended to a more generic scenario in which the nonlinear parameters take the form as C p,q ⊗ C p1 ,q1 ⊗ C p2 ,q2 ⊗ · · · ⊗ C pk ,qk , which subscripts satisfying the condition

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Estimation of NOFRFs Based Parametric Characteristic Analysis

⎧ ⎪ 2  ( pi + qi )  n − k, ⎪ ⎪ ⎪ k ⎨2  ( p + q)  n − k,  p+q + ( pi + qi ) = n + k ⎪1  p  n − k, ⎪ i=1 ⎪ ⎪ ⎩0  k  n − 2. 

This completes the proof.

Remark 6.2 In Theorem 6.1, if the linear parameters of model eq.(4.38) are fixed, then f n ( jω1 , . . . , jωn ) is independent to C E(Hn ( jω1 , · · · , jωn )), i.e. it is not a function of C E(Hn ( jω1 , · · · , jωn )) and also f n ( jω1 , . . . , jωn ) is not varying at a specific frequency or point (ω1 , . . . , ωn ). Thus, Theorem 6.1 provides an explicit analytical expression for the n th order GFRF, revealing a straightforward relationship between the n th order GFRFs and nonlinear parameters of model eq.(4.38). Therefore, the n th order GFRFs Hn (·) is an explicit function of the nonlinear parameters C(n, k) at any specific frequency point (ω1 , . . . , ωn ). As a result, the function of the parametric characteristic of the n th order GFRF, defined by eq.(6.10), is denoted by Hn (C(n, K ))(ω1 ,...,ωn ) . Remark 6.3 Based on the definitions in previous section, the CE operator creates a mapping from  → Pcσ . In order to extract the nonlinear parameters from the n th order GFRFs of the NDE model (4.25), the CE operator can be applied to the GFRFs eq.(4.38), Cs = C(M, K ) , Ws = {ω1 , · · · , ω N } ∪ {c1,0 (k1 ), c0,1 (k1 )|0  k1  K },    Pc = {cr1 , cr1 , . . . , cr1 ci ∈ C(M, K ), ri ∈ Z 0 , I = C(M, K )} , 1

2

I

(6.16)

and  = {Hn (·)|1 ≤ n ≤ N }.

(6.17)

The condition expressed by eq.(6.11) provides a necessary and sufficient condition on how system model nonlinear parameters in eq.( 4.25) can appear in the frequency domain or in the n th order GFRF, and also how the n th order GFRFs is computed by using these parameters. Based on this result, further mathematical properties of C E(Hn ( jω1 , · · · , jωn )) for the NDE system model can be found in [103] for a deeper understanding of the parametric characteristic of GFRFs. Moreover, the following corollary is obtained

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

149

which is a simplified method to recursively compute parametric characteristic of the n th order GFRFs. Corollary 6.1 [103] The recursive computation of the n th order GFRFs of the model eq.(4.25) can be determined as     n−q−1 n−1 C E(Hn ( jω1 , · · · , jωn ) = C0,n ⊕ ⊕ Cn−q,q ⊕ ⊕ C p,q ⊗ χC n, p, q,  n−q  2 q=1 p=1     n−1 ⊕ Cn,0 ⊕ ⊕ C p,0 ⊗ χC n, p, 0,  n+1 , 2  p=2

(6.18) where · is to extract the integer part, and  χC (n, p, q, η) =

C E(Hn− p−q+1(·) ) C0,n− p−q+1

p≤η, p>η,

where η is a positive integer. The proof is straightforward using Theorem 6.1, the complete proof can be found in [103]. Remark 6.4 Corollary 6.1 provides an alternative recursive method to compute the parametric characteristic of the n th order GFRF with respect to all nonlinear model parameters. However, if there is a situation in which only some nonlinear parameters of interest in eq.(6.1) are considered, then eq.(6.18) and all the above conclusions and results can still be utilized by taking the rest parameters as 1 if they are nonzero, and/or as zero if they are equal to zero. Hence, whatever any nonlinear parameters or specific parameters (for example x) are interested, then by following the same approach established above, the function of parametric characteristic with respect to x expressed by Hn (x)(ω1 ,··· ,ωn ;C(n,k)x ) and the parametric characteristic C E(Hn ( jω1 , · · · , jωn ) can be all computed. In this section the parametric characteristic analysis can be used to illustrate how the interested nonlinear parameters affect the n th order GFRFs, as a result offers an important information for both the evaluation of GFRF and also system analysis in the frequency domain.

150

6.3.2

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

Effects of Nonlinearity on the GFRFs From Different Types of Nonlinearity

As previously described, the types of nonlinearities in system models eq.(4.24) and eq.(4.25) can be divided into three different categories: (1) Pure input nonlinearities: this corresponds to the nonlinear parameters term C0,n (·), which are the first part or tem in the parametric characteristics in eq.(6.18); (2) Pure output nonlinearities: this corresponds to the nonlinear parameters term C p,0 (·), which are the last part or term in eq.(6.18); (3) Input−output cross nonlinearities: this corresponds to the nonlinear parameters term C p,q (·), which are the second part or term in eq.(6.18). It is clear that different types of nonlinearity has a different influence on dynamics of systems. Different nonlinear parameters contributed in GFRFs refer to different types and degree of nonlinearities. As a result, the parametric characteristic analysis of the associated frequency response functions can reveal frequency features of frequency response functions as well as the effects of different nonlinear parameters on the behavior of the system output. The study of the nonlinear influence on the GFRFs from various types of nonlinearities can provide vital insight into the relationship between system frequency characteristics and physical model parameters since the GFRFs represent system frequency characteristics. The parametric characteristics based analysis, in this section, is discussed and investigated for the GFRFs in order to demonstrate how different model parameters have their effect on the nonlinear frequency response functions and also affect the characteristics of system frequency.

6.3.3

Computation of NOS Based on the Parametric Characteristic Analysis

Based on the theoretical result in Theorem 6.1 where n th order GFRFs are separated into two parts, similarly, the NOFRFs can also be expressed into two parts using the parametric characteristic analysis as described in definition 6.1, which can be described in the following result. Corollary 6.2 Based on the separable function definition, the nonlinear output spectrum Y ( jω) in eq.(4.54) is separable with respect to the nonlinear parame-

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

151

ters C(M, K ) in eq.(6.1), thus the nonlinear output spectrum based parametric characteristic function can be expressed as Y ( jω) = Y (C(M, K ))ω = C E(Y ( jω)) · Yn ( f n (·); jω)   N = ⊕ C E Hn (·) · Yn ( f n (·); jω) ,

(6.19)

n=1

where  C E(Y  ( jω)) represents the parametric characteristics of NOS, and C E Hn (·) which includes all interested nonlinear parameters is computed analytically using eq.(6.18). Proof. By substituting eq.(6.10) into eq.(4.54) yields N 

Y ( jω) =

n=1

Y ( jω) =

√ 1/ n (2π )n−1

N  n=1

 C E(Hn ( jω1 , . . . , jωn )) · f n ( jω1 , . . . , jωn )

n 

U ( jωi )dσωn ,

i=1

ω=ω1 +ω2 +···+ωn

√ 1/ n C E(Hn ( jω1 , . . . , jωn )) (2π )n−1

(6.20)

 f n ( jω1 , . . . , jωn )

n 

U ( jωi )dσωn ,

i=1

ω=ω1 +ω2 +···+ωn

(6.21) let √ 1/ n Yn ( f n (·); jω) = (2π)n−1

 f n ( jω1 , . . . , jωn ) ω=ω1 +ω2 +···+ωn

n 

U ( jωi )dσωn ,

i=1

(6.22) Giving Y ( jω) =

N 

C E(Hn ( jω1 , . . . , jωn ))Yn ( f n (·); jω) .

(6.23)

n=1

Taking CE operator for both sides of eq.(6.23) yields       N C E(Hn ( jω1 , . . . , jωn )) · C E Yn ( f n (·); jω) . C E Y ( jω) = C E n=1

(6.24)

152

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

 Based on the properties of the CE operator the summation can be written as a    reduced sum and C E Yn ( f n (·); jω) = 1, because it is constant , thus, eq.(6.24) can be written as N     C E(Hn ( jω1 , . . . , jωn )) , C E Y ( jω) = C E

(6.25)

n=1

therefore, Y ( jω) = Y (C(M, K ))ω = C E(Y ( jω)) · Yn ( f n (·); jω)   N = ⊕ C E Hn (·) · Yn ( f n (·); jω) .

(6.26)

n=1



The proof is complete.

Subsequently, using this result, the nonlinear output spectrum Y ( jω) can be computed using the parametric characteristic analysis function as an alternative theoretical solution. Comparing the eq.(4.54) and eq.(6.19), the n th order nonlinear output spectrum Yn ( jω) = Yn (C(n, K ))ω=ω1 +···+ωn with respect to the nonlinear parameters in eq.(6.1) can be achieved with respect to the nonlinear model parameters in terms of the parametric characteristic analysis as   Yn C(n, K )

ω=ω1 +···+ωn

  = C E Hn ( jω1 , . . . , jωn ) · Yn ( f n (·); jω) .

(6.27)

Using eq.(6.27), the nonlinear output frequency spectrum in each specific frequency point with respect to the nonlinear parameters of interest can be analytically computed. It gives an explicit representation of the output frequency response function of the NDE model in eq.(4.25) under general inputs, which is expressed as a polynomial form with respect to the model nonlinear parameters. Lang et al. [116] have proposed a different method to achieve a similar result. This expression is much more detailed in the current study, and the relationship between the model parameters and the output frequency response function is more clearly revealed.

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

153

According to above results, it is shown that the output spectrum of nonlinear systems can also be represented by a polynomial function of the nonlinear parameters in C(M, K ) using the results of the parametric characteristics of the GFRFs, and the detailed form of this polynomial function related to any parameters of interest is completely computed by its parametric characteristics. Remark 6.5 C E(Hn (·)) can be derived or analytically determined from the system model parameters based on the results in the Theorem 6.1 or using the proposed result. Given a specific time domain system model described by eq.(4.24) or eq.(4.25), then Y (C, K )ω can be achieved by the taking FFT of the time domain output data obtained from simulations or experiments at frequency ω. As a result, the least square method, as mentioned in Remark6.1, can be used to obtain as described in Yn ( f n (·); jω) for n = 1, . . . , N . Then Yn C(n, K ) ω=ω1 +···+ωn

eq.(6.27) can then be obtained which describes all frequency components at each frequency with the influence of the nonlinear parameters of interest. This provides a numerical way to compute the nonlinear output spectrum Y (C(M, K )) and its each order component n th order nonlinear output spectrum, Yn (C(n, K )). Both nonlinear frequency spectrum and n th order NOS, i.e. Y (C(M, K )) and Yn (C(n, K )), respectively, can be computed as analytical polynomial functions with respect to any interested nonlinear parameters. As a result, these model parameters can now be utilized to analyze and design the output performance of nonlinear systems in the frequency domain. The main goal of the proposed method is to greatly reduce the computational complexity in computing the n th order GFRFs and the NOS compared with the direct computation by using eq.(4.26)− (4.30) or eq.(4.38)− (4.42) and eq.(4.21)− (4.23). In addition, the parametric characteristic analysis and the related proposed result of this work, compared to the results in Lang et al. [114], provide an explicit and analytical expression for the relationship between the time domain model parameters and nonlinear output spectrum with detailed polynomial structure up to any order of interest and each order nonlinear output spectrum component can also be calculated. • General procedures for computation of the NOC based parametric characteristic analysis For the sake of clarity, the nonlinear output spectrum in eq.(6.19) can be simply written as

154

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

Y ( jω) = Y (C(M, K ))ω = χ · ( jω)T ,   N where χ = ⊕ C E Hn (·) and ( jω) = Yn ( f n (·); jω).

(6.28)

n=1

Therefore, it can be seen that the process of computing Y (C(M, K ))ω is NOT a straight forward process. Firstly, χ which is related to the parametric characteristics of n th order GFRFs or system model parameters should be determined. Then, the second part, ( jω)T , which is independence to the first part should also be estimated. Thus, the general procedure to compute the NOC can be summarized in the following two steps: Step 1: Computation of the χ This step is mainly to compute the parametric characteristic of the nonlinear output spectrum in eq.(6.28). In this step, two main points will be considered in order to compute χ as follows 1) The largest order N should be firstly determined, the first step is to find the maximum order of the Volterra series expansion for the nonlinear system i.e. N . It can be basically estimated by the magnitude of the truncation error in the finite order Volterra series expansion. Alternatively, the maximum value of N can also be determined by calculating the magnitude of the n th order output frequency response Yn ( jω). For instance, for the NARX model in eq.(4.24), the magnitude bound of Yn ( jω) can be evaluated by [95] |Yn ( jω)| ≤ αn bn h nT ,

(6.29)

where αn , h n denotes the complex valued functions, and bn represents a function vector of the system model parameters. In [95],the detailed definition regarding αn , bn and h n can be found. The largest order N can be achieved, if the magnitude bound of a specific order of Yn ( jω) is less than a predefined value (for instance 10−8 ). It should be noted that, for each nonlinear parameter, the largest ranges of interest should be considered in the estimation of |Yn ( jω)| Because the magnitude bound is a function of the model nonlinear parameters. 2) The parametric characteristic function of GFRFs should be determined after estimating the maximum order N to obtain the parametric characteristics of the n-th order GFRFs, i.e. C E(H n(.)) from n = 2 to N , which will be employed in the computation of “χ” which is a function of the nonlinear model parameters in eq.(6.1) and can be analytically computed. Basically, C E(H n(.)) can be analytically computed with respect to the nonlinear model parameters either by

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

155

recursive computation using eq.(6.18) or without recursive computation using the result of the Theorem 6.1. Consequently, based on the above steps the parametric characteristics of the n th order GFRFs with respect to any specific system model parameters of interest can be computed. Step 2: Estimation of ( jω) = Yn ( f n (·); jω) In this step, based on the Remark 6.1, the second part of eq.(6.28) denoted by ( jω), which is the invariant to the parametric characteristic vector or nonlinear parameters of interest, can be numerically estimated using Algorithm 6.1 [103]. Algorithm 6.1 Numerical estimation of ( jω) STEP 1: Choose a series of different values of the interested nonlinear parameters Sc , which are properly distributed in ∂ Sc , to form a series of vectors χ1 , · · · , χρ(N ) , where ρ(N ) = |χ| denotes the dimension of vector |χ|, such that  T T χ M = χ1T , · · · , χρ(N ) is not singular i.e. det(χ M )  = 0 . STEP 2: Given a frequency ω where the nonlinear output spectrum is to be analyzed. Excite the system using the same input under different values of the nonlinear parameters χ1 , · · · , χρ(N ) ; collect the time domain output y(t) for each case, and Finally evaluate the output frequency response [Y ( jω)1 , · · · , Y ( jω)ρ(N ) ] at the frequency ω by applying FFT technique. STEP 3: Following Step 2, ( jω) can be estimated ⎡

χM

⎡ ⎤ ⎤ χ1 Y ( jω)1 ⎢ χ2 ⎥ ⎢ Y ( jω)1 ⎥ ⎢ ⎢ ⎥ ⎥ · ( jω)T = ⎢ . ⎥ · ( jω)T = ⎢ ⎥ =: Y Y ( jω) , .. ⎣ .. ⎦ ⎣ ⎦ . χρ(N ) Y ( jω)ρ(N )  T T ( jω)T = φ1 φ2 · · · φρ(N ) = (χ M χ M )−1 χ M · Y Y ( jω) .

(6.30)

Remark 6.6 In Algorithm 6.1, at the first point, ρ(N ) different values of the parameter vector χ in the parameter space ∂ Sc are achieved such that det(χ M )  = 0 by selecting a grid of parameter values of the nonlinear parameters of interest properly spanned in ∂ Sc , or using a stochastic-based searching technique and/or other different optimization methods such as Genetic Algorithm to generate a non-singular matrix ∂ Sc with a proper inverse.

156

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Estimation of NOFRFs Based Parametric Characteristic Analysis

In the second point 2, given the maximum order N of the system output spectrum, it can be observed that this algorithm required ρ(N ) simulations to achieve ρ(N ) output responses under different parameter values. Note that from point 1 above   N ρ(N ) = |χ| = | ⊕ C E Hn (·) |, it is not only a function of the largest order N n=1

but also relevant to the number of interested parameters. This implies that, if the number of relevant parameters and the greatest order N increase, the simulation burden increases too. In point 3, the complex valued function vector ( jω) in eq.(6.30) is unique, if det(χ M )  = 0 , this means that since the truncation error incurred by the maximum order N of the Volterra series is sufficiently small, then the result in eq.(6.30) will sufficiently approximate to their real values. Therefore, by using the Algorithm 6.1, the complex valued function vector ( jω) for the specific input function at the given frequency ω can be accurately obtained. consequently, the NOS in eq.(6.28) subject to the specific input function at a specific frequency is explicitly computed by using point 1 and 2 in the above general procedure for computation of the nonlinear output spectrum for the nonlinear system of interest. Remark 6.7 However the function vector ( jω) is achieved using the numerical estimation based on the Algorithm 6.1 and consequently the computed NOS is not analytical function of the input and frequencies, however the achieved relationship between the output spectrum and model nonlinear parameters is analytical and explicit for the specific input function  at the given specific frequency ω. In addition, it can be seen that since C E Hn (·) is known, ( jω) can also be numerically estimated, then Yn ( jω, C(n, K )) = C E(Hn (·)) · ( jω)T is also computed, which represents the analytical expression for the n th order output frequency response of the nonlinear systems. In addition, it can also be noticed that, compared with the analytical computation based on the recursive algorithm for the GFRFs in [22] and the output spectrum for nonlinear systems provided in [111], the parametric characteristic analysis method demonstrated above enable the NOS to be achieved directly in a concise polynomial form as eq.(6.28) without the complex integration in the high-dimensional superplane ω = ω1 + · · · + ωn especially when the nonlinearity order n is high. By using the parametric characteristic method above, the NOS can be computed up to very high orders with respect to any specific interested model parameters and any specific input signal at any given frequency. The cost could be due to the fact that the new method requires ρ(N ) simulations.

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

6.3.4

157

Computation of NOS Based on Specific Nonlinear Parameters

As discussed in the previous section, the parametric characteristic vector C E(H n(·)) with respect to all model parameters with degree > 1 (related to nonlinear parameters) can be either non-recursively computed using Theorem 6.1 or recursively computed using eq.(6.18) in Corollary 6.1, and in eq.(6.18), if only a few or some parameters are of interest, the computation can be carried out by simply replacing the other nonzero parameters with 1. For the analysis of a specific nonlinear system, only some specific system model parameters, such as parameters in C p,q , are of interest in many circumstances. Consequently, the parametric characteristic vector computation in eq.(6.18) and eq.(6.19) can be considerably simplified. This section proposes some results for the computation of parametric characteristics with respect to one or more specific parameters in C p,q which can effectively simplify the computation of the NOS and the n th order nonlinear output spectrum. The main contribution of this section is to analytically compute C E(H n(·)) and C E(Y (·)) with respect to only some specific nonlinear parameters. This can significantly reduce the computation cost and the mathematical complexity in the computation of C E(H n(·)), C E(Y (·)) and then Y ( jω), Yn ( jω) especially for higher order nonlinear output spectrum which will be used in the computation of the NOFRFs. Lemma 6.2 Computation of the parametric characteristic vector C E(H n(·)) with respect to some specific nonlinear model parameters C p,q . Consider only the model nonlinear parameter C p,q = c. The parametric characteristic vector of the n th order GFRFs with respect to only some specific model parameters c is  n−1 

n−1  n−1 (1−δ( p)·u(n−q))·δ p+q−1 − p+q−1  , C E(Hn ( jω1 , . . . , jωn )) = 1, c, c2 , · · · , c p+q−1

(6.31) where · is to take the integer part of (·) ,  u(x) = ,

 δ( p) =

1 0

if x > 0, otherwise .

1 0

if p = 0, otherwise .

158

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

Proof. Consider all other model parameters as constants (i.e. no contribution of nonlinearity) or 1 (linear model parmeters) based on Theorem 6.1, and also some properties of CE operator which has been described in the previous sections, the following cases are concluded − Case 1: if p + q > n , which means maximum degree of nonlinearity is greater than the n th order in the frequency domain, based on the properties of CE operator, nonlinear model parameters have NO contribution to C E(Hn (·)) in this situation, i.e. C E(Hn ( jω1 , . . . , jωn )) = 1 . − Case 2: if p + q = n , then the model parameter has an independent contribution in the C E(H n(·)), i.e.  C E(Hn (·)) = 1

c .

− Case 3: if p + q < n, and p > 0 , then the independent contribution for this parameter in C E(Hn (·)) should be 

n−1



c p+q−1 , and the monomials c x are all NOT independent contributions in n−1 C E(Hn (·)), where 0 ≤ x <  p+q−1 , thus, in this case '

& C E(Hn ( jω1 , . . . , jωn )) = 1

c

c2 · · · · · · c

n−1 p+q−1

()

,

(6.32)

if p + q < n, and p = 0 , because in a complete monomial there should be at least one p > 0 (which means at least one output nonlinear parameter), Hence c x for any x in this latter case are not complete, therefore, in this case & C E(Hn ( jω1 , . . . , jωn )) = 1

' c

c2 · · · · · · c

n−1 p+q−1

(

−1

)

.

(6.33)

The parametric characteristic vector function with respect to the nonlinear model parameter c can be generalized for all the cases above and summarized into one general solution as  n−1 

n−1  n−1 (1−δ( p)·u(n−q))·δ p+q−1 − p+q−1  C E(Hn ( jω1 , . . . , jωn )) = 1, c, c2 , · · · , c p+q−1 .

This completes the proof.



6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

159

From the above result, the following can be concluded. Corollary 6.3 If all the other model nonlinear parameters are zero except C p,q = c  = 0.     n−1 − if: p + q < n, and p > 0 or p + q = n , and p+q−1 is an integer value, then ' n−1 ( C E(Hn (·)) = c p+q−1 , − else:

C E(Hn (·)) = 0 ,

this result can be generalized and summarized solution as C E(Hn ( jω1 , . . . , jωn )) = c

n−1  p+q−1 (1−δ( p)·u(n−q))·δ



n−1 n−1 p+q−1 − p+q−1 

 .

(6.34)

The proof can be directly achieved from both Theorem 6.1 and Lemma 6.2. Corollary 6.3 describes a more specific case of Volterra-type nonlinear systems in eq.(4.24) or eq.(4.25) represented by a parametric model. In the system under consideration, there are only some nonlinear parameters of the same nonlinear degree and type. The numerical studies in the next section will demonstrate this result. For the nonlinear output spectrum and its parametric characteristics, the following result can be achieved. Lemma 6.3 Consider only the specific nonlinear model parameter C p,q = c. The parametric characteristic vector function of the output spectrum in eq.(4.21) can also be separated using separable function with respect to the specific nonlinear parameter c can be written as   [N −1/ p+q−1]     N C E Y ( jω) = χ = ⊕ C E Hn (·) = ⊕ C E H( p+q−1)i+1 (·) n=1

 i=0  N −1 N −1 N −1  p+q−1 −δ( p)·δ p+q−1 − p+q−1  ·u(n−q)) 2 . = 1, c, c , · · · , c (6.35) Then a complex valued function vector which is independent to the parameters of interest F( jω1 , · · · , jωn ; C(M, K )/c) exists such that   Y (C)ω;C(M,K )/c = C E Y ( jω) · F( jω1 , · · · , jωn ; C(M, K )/c) .

(6.36)

160

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

If all the other model nonlinear parameters are zero except C p,q = c  = 0( p + q > 1). Subsequently, parametric characteristic vector function of the nonlinear output spectrum in eq.(4.21) with respect to the model parameter c can be defined as − if: p=0      χ = C E Y ( jω) = 1 ⊕ C E Hq (·) 1 − u(q − N ) & ) = 1 c · (1 − u(q − N )) ,

(6.37)

− else   [N −1/ p+q−1]     N ⊕ C E H( p+q−1)i+1 (·) C E Y ( jω) = χ = ⊕ C E Hn (·) = n=1

i=0



N −1

p+q−1 = 1 ⊕ C p,q ⊕ C 2p,q ⊕ C 3p,q ⊕ · · · · · · ⊕ C p,q ) &  N −1  = 1 c c2 · · · · · · c p+q−1 .



(6.38)

Proof. From Lemma 6.2 and Corollary 6.3, the results are straightforward and can be achieved as  n−1  n−1 n−1 ·δ p+q−1 − p+q−1  ·(1−δ( p)·u(n−q)) p+q−1 C E(Hn ( jω1 , . . . , jωn )) = c , (6.39)     N C E Y ( jω) = ⊕ C E Hn (·) .

(6.40)

n=1

Substitute eq.(6.39) in eq.(6.40 yields n−1   N ·δ C E Y ( jω) = ⊕ c p+q−1



n−1 n−1 p+q−1 − p+q−1 



·(1−δ( p)·u(n−q))

n=1

.

(6.41)

Let i = n − 1/ p + q − 1, i.e. n = ( p + q − 1)i + 1, then     [n−1/ p+q−1] ⊕ C E H( p+q−1)i+1 (·) C E Y ( jω) = i=0   =

[n−1/ p+q−1]

⊕

i=0

i·δ(i−i)· 1−δ( p)u ( p+q−1)i+1−q

c



 ,

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

161

this can be written as    [n−1/ p+q−1] ⊕ c C E Y ( jω) =



i· 1−δ( p)u ( p+q−1)i+1−q



 .

i=0

(6.42)

− Case 1: if p = 0, eq.(6.42) becomes    [n−1/q−1] ⊕ c C E Y ( jω) =



i· 1−u (q−1)i+1−q





i=0

   = 1 ⊕ C E Hq (·) (1 − u(q − n)) = 1

 c



1−u(q−n)

.

− Case 2: if p  = 0   [n−1/q−1] i C E Y ( jω) = ⊕ c 

i=0

= 1 c c2 · · · · · · c

n−1  p+q−1 ]

, 

i.e. the result is proved.

It can be clearly seen from Corollary 6.3 and Lemma 6.3 that different nonlinearities will produce a quite different polynomial function structure for the nonlinear output spectrum. As a result, the output frequency response of the system is affected in a different way. The influence of different nonlinearities on system output frequency characteristics can now be investigated by utilizing the established results above. Moreover, for the above results that, the nonlinear parameter, c, can be either one parameter or a vector of some nonlinear parameters of the same nonlinear degree and type in C p,q . Thus, the computation of cn = c ⊗ c · · · c ⊗ c, is required. Therefore, if c is an I −dimension vector, there will be several repetitive terms included in cn . The following lemma can be used to simplify the computation and to avoid the repetition terms. Lemma 6.4 [103] Let c = [c1 , c2 , · · · , c I ] which can also be described as c[1 : I ], and also cn = c ⊗ c · · · ⊗ c, n ≥ 1 and I ≥ 1. Then    n

162

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

) &   cn = cn−1 · c1 , · · · , cn−1 s(1)n − s(i)n + 1 : s(1)n · ci , · · · , cn−1 s(1n ) · c I , (6.43) I  s( j)n−1 , s(·)1 , and 1 ≤ i ≤ I . Moreover, D I M(cn ) = s(1)n+1 , where s(i)n = j=i

and the location of cin in cn is s(1)n+1 − s(i)n+1 + 1. The complete proof of Lemma 6.4 can be found in [103].

6.3.5

Numerical Example

In order to demonstrate the proposed results of Lemma 6.2, Corollary 6.3, and Lemma 6.3, and to illustrate the basic idea of computing the parametric characteristics of the n th order GFRFs and nonlinear output spectrum at each specific frequency points with respect to the only specific nonlinear parameters of interest in this case, consider the nonlinear system model described by a1 y¨ (t) + a2 y(t) + a3 y˙ (t) + c1 y˙ 3 (t) + c2 y˙ 2 (t)y(t) + c3 y˙ 3 (t) = bu(t) .

(6.44)

Eq.(6.44) represents a simple case of NDE model in eq.(4.25) with M = 3 (Maximum degree of nonlinearity) and K = 2 (Maximum order of derivatives) . Also, model parameters can be described based on the eq.(6.1) as − Linear parameters: c1,0 (2) = a1 , c1,0 (0) = a2 , c1,0 (1) = a3 , and c0,1 (0) = −b, − Nonlinear parameters: c3,0 (111) = c1 , c3,0 (110) = c2 , c3,0 (000) = c3 . − All other parameters are zero. Part 1: Computation of the C E(Hn (·)) with respect to only specific nonlinear parameters of interest using Corollary 6.3. The nonlinear parameters have the same degree and type of nonlinearity, thus all nonlinear parameters can be represented by a vector   c = C p,q = C3,0 = c3,0 (000) c3,0 (110) c3,0 (111) = c3 c2 c1 , based on the Corollary 6.3,

(6.45)

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

163

  [N −1/ p+q−1]     N C E Y ( jω) = χ = ⊕ C E Hn (·) = ⊕ C E H( p+q−1)i+1 (·) n=1

=

i=0

1 ⊕ C p,q ⊕ C 2p,q

& = 1

c

c2



⊕ C 3p,q

N −1

p+q−1 ⊕ · · · · · · ⊕ C p,q )  N −1  · · · · · · c p+q−1 ,



where p = 3, and q = 0. n−1 = n−1 n = 2i + 1, i = 1, 2, 3, ...., Assume i = p+q−1 2 , then     − if: p + q < n, and p > 0 or p + q = n , and if additionally integer value, then

n−1 p+q−1

is an

i  C E(Hn ( jω1 , . . . , jωn )) = ci = c3 c2 c1 for − else

n = 2i + 1, i = 1, 2, 3, . . . .

(6.46)

C E(Hn ( jω1 , . . . , jωn )) = 0 .

(6.47)

Following eq.(6.3.5) and eq.(6.47)is easier to compute C E(Hn ( jω1 , . . . , jωn )), − for n = 1 and 2, C E(Hn ( jω1 )) and C E(Hn ( jω1 , jω2 )) = 0 , − for all n = even numbers, i in this case will not be integer which means that not satisfy the condition, i.e. C E(Hn ( jω1 , . . . , jωn )) = 0 , where n is even numbers − for n = 3, i.e. i = 1,  C E(Hn ( jω1 , . . . , jωn )) = ci = c3

c2

c1

i

 = c3

c2

c1 ,

− for n = 5, i.e. i = 2,  C E(Hn ( jω1 , . . . , jωn )) = c3

c2

c1

2

  = c3 c2 c1 ⊗ c3 c2  2 2 2 = c3 , c 3 c 2 , c 3 c 1 , c 2 , c 2 c 1 , c 1 ,

c1



164

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

− for n = 7, i.e. i = 3, 3  2   C E(Hn ( jω1 , . . . , jωn )) = c3 c2 c1 = c3 c2 c1 ⊗ c3 c2 c1  = c33 , c32 c2 , c32 c1 , c3 c22 , c3 c2 c1 , c3 c12 , c23 , c22 c1 , c2 c12 , c13 .

Using the result of C E(Hn (·)), the n th order GFRFs can be computed based on the Theorem 6.1 with respect to a specific pure output nonlinear parameters of interest, − for n = 2i + 1, i.e. i = 1, 2, 3, . . ., Hn ( jω1 , . . . , jωn )|c1 ,c2 ,c3 = C E(Hn ( jω1 , . . . , jωn ))|c1 ,c2 ,c3 · f n ( jω1 , . . . , jωn ) , where f n ( jω1 , . . . , jωn ) is a complex valued function with appropriate dimension. − else Hn ( jω1 , . . . , jωn )|c1 ,c2 ,c3 = 0 . For the sake of simplicity in the mathematical analysis, suppose take the effect of only one nonlinear parameter c2 in c3,0 , i.e. − for n = 2i + 1, i.e. i = 1, 2, 3, . . .,  Hn ( jω1 , . . . , jωn ; c1 , c3 )|c2 = 1 c2 c22 · · · c2i · f n ( jω1 , . . . , jωn ; c1 , c3 ) , − else

Hn ( jω1 , . . . , jωn ; c1 , c3 )|c2 = 0 .

Part 2: Computation of parametric characteristic of nonlinear output spectrum C E(Y ( jω)) with respect to only specific nonlinear parameters of interest using Lemma 6.3 and using the result of Part 1 in this example, this will be a useful guide to compute Y ( jω) and Yn ( jω); Based on eq.(6.28), Y ( jω) = Y (C(M, K ))ω = C E(Y ( jω)) · Yn ( f n (·); jω)   N = ⊕ C E Hn (·) · Yn ( f n (·); jω) = χ · ( jω)T , n=1

where C E(Y ( jω)) = χ and Yn ( f n (·); jω) = ( jω)T .

6.3 Computation of GFRFs and NOS Aided Parametric Characteristic Analysis

165

Then for specific nonlinear parameters, the parametric characteristic function of the nonlinear output spectrum Y ( jω) with respect to nonlinear parameters c1 , c2 and c3 ,

Y (c1 , c2 , c3 )ω =

N 

C E(Hn (·)) · ( jω)T =

n=1

N −1/ p+q−1 

cip,q · Yi ( f i (·), jω)

i=0

  N −1/2  T = c3,0 · Y0 ( f 0 (·); jω) Y1 ( f 1 (·); jω)T · · · Yn ( f n (·); jω)T . i=0

For clarification, consider a much simpler case by only analyzing the effect of only one nonlinear parameter of interest, i.e. c1 = c3 = 0, which means only the effects of c2 is considered N −1/2

Y (c2 )ω =Y0 ( f 0 (·); jω) + c2 · Y1 ( f 1 (·); jω) + · · · + c2 · YN −1/2 ( f N −1/2 (·); jω) )  & T N −1/2 T · Y0 ( f 0 (·); jω) Y1 ( f 1 (·); jω) · · · YN −1/2 ( f N −1/2 (·); jω)T . = 1 c 2 · · · c2

(6.48) & ) Therefore, χ = 1 c2 · · · c2N −1/2 which are related to the nonlinear parameters of interest can be analytically computed using the result of Lemma 6.2. In addition, based on the Remark 6.1, the complex valued function vector Yn  T ( f n (·); jω) = Y0 ( f 0 (·); jω) Y1 ( f 1 (·); jω)T · · · YN −1/2 ( f N −1/2 (·); jω)T can be numerically obtained by exciting the system model for the specific input at the given frequency ω, then eq.(6.48) becomes ⎤ ⎡ ⎤ ··· c2 (0)N −1/2 1 c2 (0) Y ( jω)0 N −1/2 ⎥ ⎢ Y ( jω)1 ⎥ ⎢ 1 c2 (1) ··· c2 (1) ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ .. .. . . . . ⎦ ⎣ ⎦ ⎣ . 1 . . . N −1/2 Y ( jω)N −1/2 1 c2 (N − 1/2) · · · c2 (N − 1/2) ⎡ ⎤ Y0 ( f 0 (·); jω) ⎢ ⎥ Y1 ( f 1 (·); jω) ⎢ ⎥ ·⎢ ⎥ , .. ⎣ ⎦ . YN −1/2 ( f N −1/2 (·); jω) ⎡

166

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

⎡

⎤ ⎡ 1 c2 (0) Y0 ( f 0 (·); jω) ⎢ ⎥ ⎢1 Y1 ( f 1 (·); jω) c2 (1) ⎢ ⎥ ⎢ ⎢ ⎥=⎢ .. .. ⎣ ⎦ ⎣1 . . YN −1/2 ( f N −1/2 (·); jω) 1 c2 (N − 1/2) ⎤ ⎡ Y ( jω)0 ⎢ Y ( jω)1 ⎥ ⎥ ⎢ ·⎢ ⎥. .. ⎦ ⎣ . Y ( jω)N −1/2

⎤−1 ··· c2 (0)N −1/2 N −1/2 ⎥ ··· c2 (1) ⎥ ⎥ .. .. ⎦ . . · · · c2 (N − 1/2)N −1/2

(6.49)

For simplicity, assume that Yn ( f n (·); jω) = ( jω)T , then eq.(6.49) can be also written as T T ( jω)T = (χ M χ M )−1 χ M · Y Y ( jω) , ⎡ ⎤ Y0 ( f 0 (·); jω) ⎢ ⎥ Y1 ( f 1 (·); jω) ⎢ ⎥ where ( jω)T = ⎢ ⎥ .. ⎣ ⎦ . YN −1/2 ( f N −1/2 (·); jω)

(6.50)

⎤ 1 c2 (0) ··· c2 (0)N −1/2 N −1/2 ⎥ ⎢1 c2 (1) ··· c2 (1) ⎥ ⎢ χ =⎢ ⎥ , and .. . . . . ⎦ ⎣1 . . . N −1/2 1 c2 (N − 1/2) · · · c2 (N − 1/2) ⎡ ⎤ Y ( jω)0 ⎢ Y ( jω)1 ⎥ ⎢ ⎥ Y ( jω) = ⎢ ⎥. .. ⎣ ⎦ . ⎡

Y ( jω)N −1/2

Hence, eq.(6.50) can be computed using Algorithm 6.1. As a result, eq.(6.48), which is an analytical function with respect to the nonlinear parameter c2 , is explicitly determined. Therefore, the system output frequency response can be analyzed in terms of the nonlinear parameters. In addition, the NOFRFs in the next section can also be studied. A similar process, for more complicated cases, can be used to conduct a required analysis in terms of several nonlinear parameters for system model. Compared to the results in [114], this can considerably reduce the amount of simulation needed in the numerical method when several parameters are considered,

6.4 Estimation of NOFRFs Using Parametric Characteristic Analysis …

167

since for the output spectrum up to any order, the detailed polynomial structure can be explicitly computed.

6.4

Estimation of NOFRFs Using Parametric Characteristic Analysis for Damage Detection

In this section, a new method to analytically compute the NOFRFs based on the parametric characteristic analysis is proposed. The details of the computation of the each order frequency components of the NOFRFs with respect to the nonlinear parameters of interest is explained.

6.4.1

A Proposed Method for Computation of NOFRFs

Based on the proposed results for the computation of the n th order nonlinear output spectrum which demonstrated the analytical computation of the Y (M, K ) and Yn (c(n, K )) in terms of the interested nonlinear parameters, the NOFRFs, i.e. G n (C(M, K , n)), under certain condition, can also be analytically computed with respect to system model parameters as G n (c(n, K ))ω=ω1 +···+ωn =

Yn (c(n, K )) Un ( jω)

for ω ∈ f Yn ⊆ [a, b] .

(6.51)

Substitute eq.(6.27) to eq.(6.51), then eq.(6.51) becomes C E(Hn (·)) · Yn ( f n (·); jω) , √  n  1/ n U ( jω )dσ i n,ω (2π)n−1 ω=ω1 +ω2 +..+ωn i=1 (6.52) where G n (C(M, K , n))ω=ω1 +···+ωn is the n th order NOFRFs based parametric characteristic analysis in terms of the model nonlinear parameters and Un ( jω) is the Fourier transform for any input signal defined in Chapter 4. G n (C(M, K , n))ω=ω1 +···+ωn =

Remark 6.8 the n th order NOFRFs can also be represented by parametric characteristic functions with respect to the nonlinear parameters of interest as in eq.(6.51) or eq.(6.52). In which the numerator represents the nonlinear output spectrum of each order frequency components determined by using the proposed result in the previous section based on the parametric characteristic analysis. Therefore, eq.(6.52)

168

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

is an alternative analytical solution to the result defined by [115]. Here, the NOFRFs provide an explicit analytical expression with respect to the system model parameters. Thus, the effects of the system model parameters to the nonlinear output frequency response functions can be clearly noticed. The NOFRFs can be used for the fault diagnosis of structures and engineering systems. Algorithm 6.2 Estimation of NOFRFs based parametric characteristic analysis STEP1: Computation of the parametric characteristic function of n th order GFRFs   C E Hn (·) according to the Lemma 6.2; STEP2: Estimation of ( jω)T = Yn ( f n (·); jω) using the following steps: 1 : Computate of C E(Y ( jω)) based on the lemma 6.3; 2 : Determine of Y ( jω)) using the procedures described in Section(6.3.4); 3 : Estimate of ( jω)T = Yn ( f n (·); jω) using Algorithm 6.1; STEP3: Evaluation of NOFRFs as 1 : Determine the n th order of NOS, Yn ( jω) = C E(Hn (·)) · Yn ( f n (·); jω), using the results of Step 1 and 2; 2 : Compute Un ( jω) = F F T {u n (t)}; 3 : Evaluation of n th order NOFRFs, G n (c(n, K )), with respect to specific nonlinear parameters of interest G n (c(n, K )) =

Yn (c(n, K )) . Un ( jω)

Proposition 6.1 [174] Given a parametric model, input nonlinearity has NOT effect on the G n ( jω), i.e. only nonlinear parameters make contribution to the NOFRFs, under the assumption that the input is harmonic signal. Thus, NOFRFs is input independence under harmonic excitation



G n j(−n + 2k)ω F



k

n−k

      = Hn jω F , · · · , jω F , − jω F , · · · , − jω F .

(6.53)

Proof. Suppose that the nonlinear system in the frequency domain in eq.(4.23), is excited by a harmonic input excitation in eq.(6.39). Also, define the range of the output frequency components of Yn and Y as f Yn and f Y respectively, as explained in Section(4.3.2) as N * f Yn , (6.54) fY = n=1

6.4 Estimation of NOFRFs Using Parametric Characteristic Analysis …

169

where f Yn can be found by a set of frequencies {ω = ωk1 + · · · + ωkn |ωki = ±ω F ,

i = 1, · · · , n} ,

(6.55)

It is know from eq.(6.55) that if all ωk1 + · · · + ωkn are represented as −ω F , then ω = −nω F . If k of these are represented as ω F , then ω = (−n + 2k)ω F where the maximum value of k is n. Thus, the possible output generated frequencies of Y n( jω) can be represented as + f Yn = (−n + 2k)ω F

, k = 0, 1, · · · , n .

(6.56)

Furthermore, the output frequency components of Yn ( jω) can be described as fY =

N *

⎧ ⎨ −N ω F = 0 ⎩ N ωF

f Yn

n=1

k = −N , · · · , −1 k=0 k = 1, · · · , N .

(6.57)

Eq.(6.57) shows that when a nonlinear system is subject to a harmonic excitation, super-harmonic frequency components will be then generated. Since harmonic components are symmetrical about the frequency axis, hence, only positive frequencies also known as non-negative output frequency components will be taken into account. The NOFRFs subjected to a harmonic excitation can be derived by substituting eq.(4.23) in to the eq.(4.55) yields 

(1/2n ) G n ( jω) =

ω=ωk1 +ωk2 +···+ωkn

(1/2n )

Hn ( jωk1 , . . . , jωkn )A( jωk1 ) . . . A( jωkn ) 

A( jωk1 ) . . . A( jωkn )

,

ω=ωk1 +ωk2 +···+ωkn

(6.58) where n = 1, · · · , N , under the condition that An ( jω) =

1 2n



A( jωk1 ) . . . A( jωkn )  = 0 ,

(6.59)

ω=ωk1 +ωk2 +···+ωkn

where the G n ( jω) is only valid in the frequency range f Yn . As a result, the nonlinear output frequency response subjected to the harmonic excitation can be written as

170

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

Y ( jω) =

N 

Yn ( jω) =

n=1

N 

G n ( jω)An ( jω) .

(6.60)

n=1

If k of the n-frequencies of ωk1 , ωk2 , · · · , ωkn are represented as ω F and the reminder are represented as −ω F , substituting A( jω) as a part of eq.(6.40) into eq.(6.59) yields An ( j(−n + 2k)ω F ) =

1 k n j(−n+2k)β C |A| e , 2n n

(6.61)

therefore, G n ( jω) becomes k

  G n j(−n + 2k)ω F =

n−k

      (1/2n )Hn jω F , . . . , jω F , − jω F , . . . , − jω F Cnk |A|n e j(−n+2k)β (1/2n )Cnk |A|n e j(−n+2k)β k

n−k

      = Hn jω F , . . . , jω F , − jω F , . . . , − jω F ,

(6.62)

where Hn ( jω1 , · · · , jωn ) is considered as a symmetric function. + Therefore, G n ( jω) k= in this case,,over the range of the output frequency f Yn = (−n + 2k)ω F 0, 1, · · · , n is the same as the n th order GFRFs Hn ( jω1 , · · · , jωn ). This completes the proof.  This result demonstrates that the concept of NOFRFs can describe the nonlinear system dynamic characteristics, under harmonic excitation condition. Therefore, based on the difference between the structural dynamics behaviour in faulty and fault-free cases, it could be appropriate for the application of the detection of faults in mechanical or civil structures. Using the results of the Proposition 6.1, the NOFRFs based parametric characteristic analysis can also be represented by a dynamic behaviour of the system or nonlinear parameters of interest characteristic as follows. Corollary 6.4 Using the result of the Proposition 6.1, the input nonlinearity has no influence on the G n ( jω). In this case, the NOFRF is input independent and only the nonlinear parameters of interest has an effect on the NOFRFs based parametric characteristic analysis, under the condition that the input excitation is harmonic signal. This means that the NOFRFs are proportional to only the nonlinear parameter of interest

6.5 Concluding Remarks

171

  p = 0 . . . m, p + q = m   G n (C(M, K , n)) ∝ c p,q (k1 . . . , k p+q )  2  m  n .  ki = 0 . . . K , i = 1 . . . p + q (6.63) 

Proof. The proof is straight forward, according to the result of the Proposition (6.1),   (6.64) G n (C(M, K , n)) ≈ Hn (C(M, K , n)) ,   By substituting the n th order parametric characteristic GFRFs Hn (C(M, K , n)) , which is represented by eq.(6.10), into eq.(6.64) yields    G n (C(M, K , n)) ≈ C E Hn (C(M, K , n)) · f n ( jω1 , · · · , jωn ) .

(6.65)

As previously explained, C E(H n(·)) are related to the system behaviour with respect to the nonlinear parameters (C(M, K )). Furthermore, under the harmonic input with a specific frequency, the effects of the inputs and other non interested parameters are not considered. This means that the NOFRFs finally can be represented only with respect to the nonlinear parameters of interest and can be expressed as eq.(6.63) This completes the proof.  The result of eq.(6.63) indicates that, the NOFRFs function can be evaluated based only on the nonlinear parameters of interest (i.e. input independent) under the assumption that the input is sinusoidal function (which is mostly used in structural condition monitoring) with a specific frequency ω F . This result is suitable in the application of the structure damage detection and crack damage evaluation with the influence of the nonlinearity in the time domain model parameters on the frequency response.

6.5

Concluding Remarks

This chapter is dedicated to the study on the analytical computation of the NOFRF with respect to nonlinear parameter of interest based on the parametric characteristic analysis under the assumption that the nonlinear system model is known. To this end, the preliminaries on parametric characteristic analysis concept using separable

172

6

Estimation of NOFRFs Based Parametric Characteristic Analysis

function and CE operator as well as their interconnections are first introduced. Then, a parametric characteristics of GFRFs and nonlinear output spectrum with respect to all nonlinear model parameter is computed. After that, in order to reduce the computational cost, the parametric characteristic of GFRFs and nonlinear output spectrum with respect to only some specific nonlinear parameters are computed. The new computation of NOFRFs based PCA involves the determination of the NARX model of a system and computation of the NOFRFs with respect to the nonlinear model parameter for the purpose of the system analysis. The core method is a derivation of a novel algorithm in this study that can accurately estimate the NOFRFs of nonlinear systems up to an arbitrary order of interest with respsect to the nonlinear model parameter. The approach can be used in the nonlinear system frequency analysis for various purposes including condition monitoring and fault diagnosis.

7

Conclusions and Future Work

In this thesis, the advanced signal based fault diagnosis for gearbox in the first phase and nonlinear frequency analysis model-based fault detection in the second phase have been developed. In Chapter 1, the motivation and background of this thesis are presented. One of the most important parts of rotating machinery is the gearbox, whose faults can sometimes cause catastrophic accidents, expensive downtime, and equipment damage. Signal-based techniques, in the last decades, have achieved increasing attention by the researchers and become mainstream in the detection and diagnosis of faults in both rotational machines especially gears and structural systems. In addition, most of these methods are univariate based signal processing. Thus, they cannot be utilized to extract fault features in the multivariate signal. Therefore, when applying these univariate signal processing techniques to analyse multivariate signals, some problems between different channels appear. For example, when applying EMD technique as an advanced signal processing method to the multi-channel signal, it suffers from different issues including the mode mixing between different channels which cause incorrect frequency content extracting from the multivariate signals. Motivated to this issue, a development of the advanced EMD method in the form of improved MEMD is proposed in this thesis. Although several of these techniques are powerful for analysis non-stationary signals, but the effects of nonlinearity is ignored as they are typically model-free. In addition, they require a lot of knowledge and intervention of human to be successfully applied. Therefore, alternate techniques for diagnosing and detecting faults in the context of nonlinear frequency model based method should be developed. The thesis is therefore mainly focusing on three parts as a result of these observations:

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 H. Rafiq, Condition Monitoring and Nonlinear Frequency Analysis Based Fault Detection of Mechanical Vibration Systems, https://doi.org/10.1007/978-3-658-42480-0_7

173

174

7

Conclusions and Future Work

• Advanced signal based fault diagnosis using improved MEMD for detecting gearbox faults. • Estimation of NOFRFs using parametric charactestic analysis for damage detection. • Verification and analysis of the proposed advanced signal based method in the WTB gearbox fault detection. The basics of gearbox systems are addressed in Chapter 2 for a rudimentary understanding, which consists of the introductions on types, components and geometry of the gears. Furthermore, gear features and its characteristics during fault are presented to achieve a deeper insight into the mechanism of the fault and the characteristics of the frequency spectrum the dynamic response to the faults. In Chapter 3, various signal processing approaches are reviewed in terms of the time domain analysis, frequency domain analysis, Joint time-frequency domain analysis and some other special techniques, which play a predominant role in condition monitoring of rotational machinery including gearboxes. Chapter 4 are devoted to introduce the nonlinear frequency analysis in the framework of the frequency domain model-based fault diagnosis technique which is as an extension of signal based methods with respect to the nonlinearity effects in fault detection. In Chapter 4, different methods for the computation of GFRFs are discussed and then the generation of new frequency components in the output frequency response due to the influence of nonlinearity are investigated. In addition, a numerical estimation of the NOFRFs are demonstrated. In Chapter 5, an improved MEMD is proposed to reduce the problem of mode mixing in multivariate signals between different channels which provides a meaningful information from decomposed multivariate IMFs which related to the fault information. Furthermore, in order to extract accurate fault features in the multivariate IMFs decomposed from the multivariate signal, a demodulation analysis method called TKEO is combined to the improved MEMD approach to enhance the fault feature extraction in different locations of the machines simultaneously. The combined advanced signal based approach provides a development of the signal based fault detection. Finally, the effectiveness of the proposed combined approach is verified in the WTB mutli-stage gearbox for the purpose of fault detection in different stages simultaneously. The development of signal based fault diagnosis is achieved by analysing only the measured output, i.e. the systems behaviour and nonlinearity influence is not considered. To tackle the problem of ignorance of nonlinearity effects especially nonlinearity in the model parameters and their influence on the frequency domain, a nonlinear frequency analysis is developed in Chapter 6. To this end, a nonlinear output spectrum with respect to the nonlinear model parameter is computed based on the para-

7

Conclusions and Future Work

175

metric characteristic analysis in the first step. Then, the NOFRF is estimated with respect to the nonlinear model parameters. Compared to the numerical estimation of NOFRFs, the computation of NOFRFs based on the parametric characteristics provides an explicit analytical expression in terms of the nonlinear model parameters under certain condition. This work is devoted to investigate the WTB multi-stage gearboxes fault detection based on the development of advanced signal processing based fault diagnosis to analyse multi-channel signal in the first phase, and to study the nonlinear frequency analysis model based fault diagnosis in the second phase which can be used in damage detection of structure systems. Other issues that require further studies in the future are outlined and can be extended as: • Motivated to the lack of analytical solution in signal based fault diagnosis, a possible solution is suggested by using both signal- based and residual generation model-based methods. A powerful fault diagnosis framework for gearbox systems could be developed based on the combination of signal- and residual generation model-based methods. The combination of two techniques provides new opportunities and challenges. • The nonlinear frequency analysis model based fault detection method can be developed with respect to the nonlinear model parameters with uncertainty (noise and disturbance in the model parameter) which will be more complicated and challenging. In this case, the effects of nonlinear parameters with uncertainty on the frequency analysis could be formulated. Furthermore, the main limitation of the nonlinear frequency analysis model based fault diagnosis is that the fault transients are negligible which is the time-independence assumption, since faults often reveal some time dependent characteristics as systems can transit from a healthy to a faulty state over a period of time or at a particular instant. It will be assumed, in this case, that the fault transient is fast and has no an important role in characterizing the fault itself. In circumstances where the evolution of the defect is difficult to track, this is a reasonably practical assumption. For instance, the development of cracks over time, in structure health monitoring, is very difficult to observe, although the data collection before and after its appearance is manageable. This limitation implies a loss of information which could be beneficial for determining the time when the fault occurs, however, this issue can be partially alleviated by performing the fault diagnosis procedure periodically so that a reasonable detection window can be established. For this reason, the extension of this study to the analysis of the time-varying nonlinear system in the frequency domain for various engineering systems fault diagnosis applications is necessary and could be challenging.

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