Vibration and Structural Acoustics Analysis
C.M.A. Vasques J. Dias Rodrigues Editors
Vibration and Structural Acoustics Analysis Current Research and Related Technologies
Editors C.M.A. Vasques INEGI Universidade do Porto Campus da FEUP R. Dr. Roberto Frias 400 4200-465 Porto Portugal
[email protected] J. Dias Rodrigues Faculdade de Engenharia Universidade do Porto R. Dr Roberto Frias s/n 4200-465 Porto Portugal
[email protected] ISBN 978-94-007-1702-2 e-ISBN 978-94-007-1703-9 DOI 10.1007/978-94-007-1703-9 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011935121 © Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Vibration and structural acoustics analysis is nowadays an essential requirement for high-quality structural and mechanical design in order to assure acoustic comfort and the integrity, reliability and fail-safe behavior of structures and machines. In some conditions vibration and radiated sound in structures and machines is desirable, as is the case of the motion of a tuning fork, the enjoyable melody that a classical guitar may produce or the motion induced by vibration conveyors. More often, vibration and the underlying radiated noise are undesirable and inconvenient, as is the case of the vibrational motion of internal combustion engines, the noise generated by railway traffic, the imperfections in the milling and turning processes due to machine tool chatter or the vibration instability of light-weight aerospace structures. The underlying technologies of this field of multidisciplinary research are evolving very fast and their dissemination is usually scattered over different and complementary scientific and technical publication means. In order to make it easy for developers and technology end-users to follow the latest developments and news on the field, this book collects into a single volume selected, extended, updated and revised versions of the papers presented at the Symposium on Vibration and Structural Acoustics Analysis, coordinated by J. Dias Rodrigues and C.M.A. Vasques, of the 3rd International Conference on Integrity, Reliability & Failure (IRF’2009), co-chaired by J.F. Silva Gomes and Shaker A. Meguid, held at the Faculty of Engineering of the University of Porto, Portugal, 20–24 July 2009. The selected papers where chosen among the more than 60 papers presented at the conference symposium. Written by experienced practitioners and researchers in the field, this book brings together recent developments in the field, spanning across a broad range of themes: vibration analysis, analytical and computational structural acoustics and vibration, material systems and technologies for noise and vibration control, vibration-based structural health monitoring/evaluation, machinery noise/vibration and diagnostics, experimental testing in vibration and structural acoustics, applications and case studies in structural acoustics and vibration. Each chapter somewhat presents and describes the state of the art, presents current research results and discusses the need v
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for future developments in a particular aspect of vibration and structural acoustics analysis. The book is envisaged to be an appellative text for newcomers to the subject and a useful research study tool for advanced students and faculty members. Practitioners and researchers may also find this book an appellative reference that addresses current and future challenges in this field. The variety of case studies is expected to stimulate a holistic view of sound and vibration and related fields and to appeal to a broad spectrum of engineers such as the ones in the mechanical, aeronautical, aerospace, civil and electrical communities. With the synergistic combination of efforts of authors, editors and invited reviewers, this book brings together so many interrelated and yet diverse topics in a single volume. Hopefully, the editors expect it allows the readers to get an updated sense of the interest, technical diversity and applicability of this ever evolving research field, and that it may be used as a road-map to the required practical understanding and technical skills required to analyze and engineer new solutions for problems on vibration and structural acoustics fields. University of Porto Porto, Portugal
C.M.A. Vasques J. Dias Rodrigues
Acknowledgements
The editors strongly acknowledge the valuable efforts of all book contributors and authors. Putting forward this book was only possible due to their positive answer to the call for papers and to the number of contributions submitted and evaluated from where the best were selected. The contributions in this volume have undergone a full peer-review. In view of this, we are particular indebted to all peer reviewers that have kindly accepted the invitation to take part in the reviewing process and that provided valuable comments which certainly led to improvements in the chapters contained in this book. In particular, we would like to acknowledge the collaboration of: Aassif, E.H. (Ibn Zohr University, MA)
Collet, M. (University of Franche-Comté, FR)
Abdo, J. (Sultan Qaboos University, OM)
Devriendt, C. (Vrije Universiteit Brussel, BE)
Albarbar, A. (Manchester Metropolitan Univ., UK)
Ducarne, J. (Thales Alenia Space, FR)
Alfano, M. (Università della Calabria, IT)
Duffey, T. (Consulting Engineer, US)
Anthonis, J. (Katholieke Universiteit Leuven, BE)
Dutta, D. (Carnegie Mellon University, US)
Bard, D. (Lund University, SE)
Fidlin, A. (Karlsruhe Institute of Technology, DE)
Beck, B. (Georgia Institute of Technology, US)
Foltête, E. (University of Franche-Comté, FR)
Borza, D. (INSA de Rouen, FR)
Gil, L. (Univ. Politècnica de Catalunya, ES)
Chen, J.-T. (National Taiwan Ocean Univ., TW)
Gu, Y. (Queensland Univ. of Technology, AU) vii
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Han, J.-G. (Hainan University, CN)
Ravina, E. (University of Genova, IT)
Kessler, S.S. (Metis Design Corporation, US)
Ribeiro, A.M.R. (Technical University of Lisbon, PT)
Küchler, S. (University of Stuttgart, DE)
Ripamonti, F. (Politecnico di Milano, IT)
Latif, R. (Ibn Zohr University, MA)
Samandari, H. (University of Tabriz, IR)
Lee, D.-H. (Dong-Eui University, KR)
Sarigül, A.S. (Dokuz Eylül University, TR)
Li, H.J. (Ocean University of China, CN)
Seçgin, A. (Dokuz Eylül University, TR)
Loendersloot, R. (University of Twente, NL)
Sinha, J.K. (University of Manchester, UK)
Martarelli, M. (Univ. Politecnica delle Marche, IT)
Sohn, H. (KAIST, KR)
Matter, M. (EPF Lausanne, CH)
Stubbs, N. (Texas A&M University, US)
Montalvão, D. (Instituto Politécnico de Setúbal, PT)
Thomas, M. (Université du Québec, CA)
Neubauer, M. (Leibniz Universität Hannover, DE)
Trollé, A. (Université de Lyon, FR)
Oudjene, M. (ENSTIB, FR)
Tronchin, L. (Università di Bologna, IT)
Ouisse, M. (University of Franche-Comté, FR)
Varoto, P.S. (University of São Paulo, BR)
Pérez, M.A. (Univ. Politècnica de Catalunya, ES)
Woodhouse, J. (Cambridge University, UK)
Pierro, E. (Politecnico di Bari, IT)
Zhou, S. (Northeastern University, CN)
Qiao, P. (Washington State University, US)
Zhu, Q. (Oyama Nat. College of Technology, JP)
For helping putting forward this book, special thanks are also addressed to the Publishing Editor, Nathalie Jacobs, for the enthusiasm and support given to the editors in an early stage of the book planning and preparation. Last but not least, we wish also to express our gratitude to the IRF’2009 conference co-chairs, J.F. Silva Gomes and Shaker A. Meguid, for the consideration of a special symposium on Vibration and Structural Acoustics Analysis in IRF’2009 conference, from where the papers were selected.
Contents
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The Dynamic Analysis of Thin Structures Using a Radial Interpolator Meshless Method . . . . . . . . . . . . . . . . . L.M.J.S. Dinis, R.M. Natal Jorge, and J. Belinha 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of the State of the Art . . . . . . . . . . . . . . 1.3 The Natural Neighbour Radial Point Interpolation Method 1.4 Dynamic Discrete System of Equations . . . . . . . . . . 1.5 Dynamic Examples . . . . . . . . . . . . . . . . . . . . 1.5.1 Cantilever Beam . . . . . . . . . . . . . . . . . . 1.5.2 Variable Cross Section Beams . . . . . . . . . . . 1.5.3 Shear-Wall . . . . . . . . . . . . . . . . . . . . . 1.5.4 Square Plates . . . . . . . . . . . . . . . . . . . . 1.5.5 Shallow Shell . . . . . . . . . . . . . . . . . . . 1.6 Prospects for the Future . . . . . . . . . . . . . . . . . . 1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Selected Bibliography . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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Vibration Testing for the Evaluation of the Effects of Moisture Content on the In-Plane Elastic Constants of Wood Used in Musical Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.A. Pérez Martínez, P. Poletti, and L. Gil Espert 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . . . 2.3 Orthotropic Nature of Wood Properties . . . . . . . . . . . . . . . 2.4 Influence of Moisture Changes on Wood . . . . . . . . . . . . . . 2.5 Experimental Modal Analysis of Wooden Specimens . . . . . . . . 2.6 Numerical Model of Wooden Plate . . . . . . . . . . . . . . . . . 2.6.1 The Finite Element Method . . . . . . . . . . . . . . . . . 2.6.2 Free Vibrations of Kirchhoff Plates . . . . . . . . . . . . . 2.6.3 Perturbation of the Equation of Motion . . . . . . . . . . .
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Elastic Constants from Plate Vibration Measurements Results . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . Prospects for the Future . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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Short-Time Autoregressive (STAR) Modeling for Operational Modal Analysis of Non-stationary Vibration . . . . . . . . . . . . . V.-H. Vu, M. Thomas, A.A. Lakis, and L. Marcouiller 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . . 3.2.1 Operational Modal Analysis . . . . . . . . . . . . . . . . 3.2.2 Non-stationary Vibration . . . . . . . . . . . . . . . . . 3.2.3 Fluid-Structure Interaction . . . . . . . . . . . . . . . . . 3.2.4 Development of a New Method for Investigating Modal Parameters of Non-stationary Systems by Operational Modal Analysis . . . . . . . . . . . . . . . . . . . . . . 3.3 Vector Autoregressive (VAR) Modeling . . . . . . . . . . . . . . 3.4 The Short Time Autoregressive (STAR) Method . . . . . . . . . 3.4.1 Order Updating and a Criterion for Minimum Model Order Selection . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Working Procedure . . . . . . . . . . . . . . . . . . . . 3.5 Numerical Simulation on a Mechanical System . . . . . . . . . . 3.5.1 Discussion on Data Block Length . . . . . . . . . . . . . 3.5.2 Simulation on Mechanical System with Time-Dependent Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Experimental Application on an Emerging Steel Plate . . . . . . 3.7 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Numerical and Experimental Analysis for the Active Vibration Control of a Concrete Placing Boom . . . . . . . . . . . . . . . . G. Cazzulani, M. Ferrari, F. Resta, and F. Ripamonti 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . 4.3 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Test Rig . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Numerical Model . . . . . . . . . . . . . . . . . . . . 4.4 Active Modal Control . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Independent Modal Control . . . . . . . . . . . . . . . 4.4.2 The Modal Observer . . . . . . . . . . . . . . . . . . . 4.4.3 Numerical Analysis of Modal Control . . . . . . . . .
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4.5 Feed-Forward Control . . . . . . . . . . . . . . . . . . . 4.5.1 The Feed-Forward Control Logic . . . . . . . . . 4.5.2 Numerical Analysis of the Feed-Forward Control 4.6 Experimental Testing . . . . . . . . . . . . . . . . . . . 4.7 Prospects for the Future . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Selected Bibliography . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 5
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Modeling and Testing of a Concrete Pumping Group Control System C. Ghielmetti, H. Giberti, and F. Resta 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . . . 5.3 Description of the Entire System . . . . . . . . . . . . . . . . . . 5.4 Experimental Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Oil Continuity Equations . . . . . . . . . . . . . . . . . . 5.5.2 Concrete Continuity Equations . . . . . . . . . . . . . . . 5.5.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 5.6 Comparison Between Numerical and Experimental Results . . . . 5.7 Control System Design . . . . . . . . . . . . . . . . . . . . . . . 5.8 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . . . 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Vibration Based Structural Health Monitoring and the Modal Strain Energy Damage Index Algorithm Applied to a Composite T-Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 R. Loendersloot, T.H. Ooijevaar, L. Warnet, A. de Boer, and R. Akkerman 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . . . 124 6.2.1 Vibration Based Structural Health Monitoring . . . . . . . 124 6.2.2 Modal Strain Energy Damage Index Algorithm . . . . . . . 125 6.3 T-Beam with T-Joint Stiffener . . . . . . . . . . . . . . . . . . . . 126 6.4 Theory of the Modal Strain Energy Damage Index Algorithm . . . 126 6.5 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . 129 6.6 Experimental Analysis of the T-Beam . . . . . . . . . . . . . . . . 132 6.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 136 6.7.1 Validation of Numerical Model . . . . . . . . . . . . . . . 137 6.7.2 Length and Starting Point of Delamination . . . . . . . . . 140 6.7.3 Position of Evaluation Points . . . . . . . . . . . . . . . . 141 6.7.4 Number of Evaluation Points . . . . . . . . . . . . . . . . 142 6.7.5 Incorporation of Torsion Modes . . . . . . . . . . . . . . . 144 6.8 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . . . 145 6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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6.10 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7
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An Efficient Sound Source Localization Technique via Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Seçgin and A.S. Sarıgül 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . 7.3 Helmholtz Integral Equation and Boundary Element Method . 7.3.1 Full-Space Case . . . . . . . . . . . . . . . . . . . . . 7.3.2 Half-Space Case . . . . . . . . . . . . . . . . . . . . . 7.4 Theoretical Examples: Sound Field Determination . . . . . . . 7.5 Case Study: Sound Source Localization . . . . . . . . . . . . . 7.5.1 Surface Velocity Measurements . . . . . . . . . . . . . 7.5.2 Boundary Element Operations . . . . . . . . . . . . . . 7.5.3 Sound Source Identification and Characterization . . . 7.6 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion Analysis of Acoustic Circumferential Waves Using Time-Frequency Representations . . . . . . . . . . . . . . . . . . R. Latif, M. Laaboubi, E.H. Aassif, and G. Maze 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . 8.3 Time-Frequency Representations . . . . . . . . . . . . . . . . 8.3.1 Wigner-Ville Distribution . . . . . . . . . . . . . . . . 8.3.2 Spectrogram Distribution . . . . . . . . . . . . . . . . 8.3.3 Reassignment Spectrogram . . . . . . . . . . . . . . . 8.4 Acoustic Measured Signal Backscattered by an Elastic Tube . . 8.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . 8.4.2 Measured Acoustic Response . . . . . . . . . . . . . . 8.4.3 Resonance Spectrum . . . . . . . . . . . . . . . . . . . 8.5 Time-Frequency Images of Experimental Acoustic Signal . . . 8.5.1 Spectrogram and Wigner-Ville Images . . . . . . . . . 8.5.2 Reassigned Spectrogram Image . . . . . . . . . . . . . 8.6 Dispersion of the Circumferential Waves . . . . . . . . . . . . 8.6.1 Determination of Dispersion Curves of Circumferential Waves by the Theoretical Method . . . . . . . . . . . . 8.6.2 Determination of Dispersion Curves of Circumferential Waves by the Reassigned Spectrogram Image . . . . . 8.7 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . 8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.9 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 203 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9
Viscoelastic Damping Technologies: Finite Element Modeling and Application to Circular Saw Blades . . . . . . . . . . . . . . . . . . C.M.A. Vasques and L.C. Cardoso 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . . 9.3 Configurations of Viscoelastic Damping Treatments . . . . . . . 9.4 Viscoelastic Constitutive Behavior . . . . . . . . . . . . . . . . 9.5 Finite Element Modeling of Viscoelastic Structural Systems . . . 9.5.1 Some Comments on Deformation Theories . . . . . . . . 9.5.2 Spatial Modeling and Meshing . . . . . . . . . . . . . . 9.5.3 Damping Modeling and Solution Approaches . . . . . . 9.5.4 Frequency- and Time-Domain Implementations . . . . . 9.5.5 Commercial FE Software . . . . . . . . . . . . . . . . . 9.6 Vibroacoustic Simulation and Analysis . . . . . . . . . . . . . . 9.7 Circular Saw Blades Damping: Modeling, Analysis and Design . 9.7.1 Geometric and Material Properties of the “Saw” . . . . . 9.7.2 FE Modeling and Vibroacoustic Media Discretization . . 9.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . . 9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Vibroacoustic Energy Diffusion Optimization in Beams and Plates by Means of Distributed Shunted Piezoelectric Patches . . . . . . . M. Collet, M. Ouisse, K.A. Cunefare, M. Ruzzene, B. Beck, L. Airoldi, and F. Casadei 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Overview of the State of the Art . . . . . . . . . . . . . . . . . . 10.3 Classical Tools for Designing RL and RCneg Shunt Circuits . . . 10.3.1 Piezoelectric Modeling and Shunt Circuit Design . . . . . 10.4 Controlling the Dispersion in Beams and Plates . . . . . . . . . 10.4.1 Waves Dispersion Control by Using RL and Negative Capacitance Shunts on Periodically Distributed Piezoelectric Patches . . . . . . . . . . . . . . . . . . . 10.4.2 Periodically Distributed Shunted Piezoelectric Patches for Controlling Structure Borne Noise . . . . . . . . . . . . 10.5 Optimizing Wave’s Diffusion in Beam . . . . . . . . . . . . . . 10.5.1 Description and Modeling of a Periodic Beam System . . 10.5.2 Optimization of Power Flow Diffusion by Negative Capacitance Shunt Circuits . . . . . . . . . . . . . . . . 10.5.3 Optimization of Wave Reflection and Transmission . . . 10.6 Prospects for the Future . . . . . . . . . . . . . . . . . . . . . . 10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11 Identification of Reduced Models from Optimal Complex Eigenvectors in Structural Dynamics and Vibroacoustics . . . . M. Ouisse and E. Foltête 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview of the State of the Art . . . . . . . . . . . . . . . . 11.3 Properness Condition in Structural Dynamics . . . . . . . . . 11.3.1 Properness of Complex Modes . . . . . . . . . . . . 11.3.2 Illustration of Properness Impact on Inverse Procedure 11.3.3 Properness Enforcement . . . . . . . . . . . . . . . . 11.3.4 Experimental Illustration . . . . . . . . . . . . . . . 11.4 Extension of Properness to Vibroacoustics . . . . . . . . . . 11.4.1 Equations of Motion . . . . . . . . . . . . . . . . . . 11.4.2 Complex Modes for Vibroacoustics . . . . . . . . . . 11.4.3 Properness for Vibroacoustics . . . . . . . . . . . . . 11.4.4 Methodologies for Properness Enforcement . . . . . 11.4.5 Numerical Illustration . . . . . . . . . . . . . . . . . 11.4.6 Experimental Test-Case . . . . . . . . . . . . . . . . 11.5 Prospects for the Future . . . . . . . . . . . . . . . . . . . . 11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Selected Bibliography . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents
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303 304 305 306 307 308 311 315 315 316 317 318 320 321 324 324 325 325
Contributors
E.H. Aassif LMTI, Faculty of Science, Ibn Zohr University, Agadir, Morocco,
[email protected] L. Airoldi School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, USA,
[email protected] R. Akkerman Engineering Technology, Production Technology, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands,
[email protected] B. Beck G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA,
[email protected] J. Belinha Institute of Mechanical Engineering—IDMEC, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal,
[email protected] A. de Boer Engineering Technology, Applied Mechanics, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands,
[email protected] L.C. Cardoso INEGI, Universidade do Porto, Campus da FEUP, R. Dr. Roberto Frias 400, 4200-465 Porto, Portugal,
[email protected] F. Casadei School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, USA,
[email protected] G. Cazzulani Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy,
[email protected] M. Collet FEMTO-ST Institute, Applied Mechanics, University of Franche-Comté, 25000 Besançon, France,
[email protected] K.A. Cunefare G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA,
[email protected] L.M.J.S. Dinis Faculty of Engineering of the University of Porto—FEUP, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal,
[email protected] xv
xvi
Contributors
M. Ferrari Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy,
[email protected] E. Foltête FEMTO-ST Institute, Applied Mechanics, University of FrancheComté, 25000 Besançon, France,
[email protected] C. Ghielmetti Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy,
[email protected] H. Giberti Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy,
[email protected] L. Gil Espert Laboratori per a la Innovació Tecnològica d’Estructures i Materials, Universitat Politècnica de Catalunya, C/Colon, 11 TR45, 08225 Terrassa, Barcelona, Spain,
[email protected] M. Laaboubi LMTI, Faculty of Science, Ibn Zohr University, Agadir, Morocco,
[email protected] A.A. Lakis École Polytechnique, Montréal, QC, H3C 3A7, Canada,
[email protected] R. Latif ESSI, National School of Applied Science, Ibn Zohr University, Agadir, Morocco,
[email protected] R. Loendersloot Engineering Technology, Applied Mechanics, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands,
[email protected] L. Marcouiller Institut de Recherche Hydro Québec, Varennes, QC, J3X 1S1, Canada,
[email protected] G. Maze LOMC, Le Havre University, Le Havre, France,
[email protected] R.M. Natal Jorge Faculty of Engineering of the University of Porto—FEUP, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal,
[email protected] T.H. Ooijevaar Engineering Technology, Production Technology, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands,
[email protected] M. Ouisse FEMTO-ST Institute, Applied Mechanics, University of FrancheComté, 25000 Besançon, France,
[email protected] M.A. Pérez Martínez Department of Strength of Materials and Structures, Universitat Politècnica de Catalunya, C/Colon, 11 TR45, 08225 Terrassa, Barcelona, Spain,
[email protected] P. Poletti Department of Sonology, Escola Superior de Música de Catalunya, C/Padilla, 155, 08013 Barcelona, Spain,
[email protected] F. Resta Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy,
[email protected] Contributors
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F. Ripamonti Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy,
[email protected] M. Ruzzene School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, USA,
[email protected] A.S. Sarıgül Department of Mechanical Engineering, Dokuz Eylül University, 35100 Bornova, Izmir, Turkey,
[email protected] A. Seçgin Department of Mechanical Engineering, Dokuz Eylül University, 35100 Bornova, Izmir, Turkey,
[email protected] M. Thomas École de Technologie Supérieure, 1100 Notre Dame West, Montréal, QC, H3C 1K3, Canada,
[email protected] C.M.A. Vasques INEGI, Universidade do Porto, Campus da FEUP, R. Dr. Roberto Frias 400, 4200-465 Porto, Portugal,
[email protected] V.-H. Vu École de Technologie Supérieure, 1100 Notre Dame West, Montréal, QC, H3C 1K3, Canada,
[email protected] L. Warnet Engineering Technology, Production Technology, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands,
[email protected] List of Figures
Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4
Fig. 1.5 Fig. 1.6 Fig. 1.7
Fig. 1.8
Fig. 2.1
Fig. 2.2
Fig. 2.3
(a) Influence cell of interest point xI . (b) Construction process of integration points . . . . . . . . . . . . . . . . . . . . . . . . Shell-like nodal and integration mesh construction and distribution Cantilever beam. Examples of 2D and 3D, regular and irregular meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross section of the variable beams and respective nodal arrangement for the 2D and 3D modulation: (a) beam A and (b) beam B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear wall with four openings and respective mesh discretization for the 2D and the 3D modulation . . . . . . . . . . . . . . . . . Examples of used meshes . . . . . . . . . . . . . . . . . . . . . Thin clamped cylindrical shell panel geometric and material properties and regular and irregular meshes examples used in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Maximum deflection varying along time for a suddenly applied load (load case A) and for a load that suddenly vanishes (load case B) and (b) absolute value of the amplitude versus the load normalized frequency . . . . . . . . . . . . . . . . . . . . The principal axes useful for modelling wood as an orthotropic material. The longitudinal axis L is parallel to the cylindrical trunk and the tangential axis T is perpendicular to the long grain and tangential to the annual growth rings . . . . . . . . . . . . . Distortion caused by changes in humidity when a wooden plate is restrained on one side only. Both pieces of spruce (Picea Abies) were cut sequentially from the same plank. The width of the samples is approximately 14 cm . . . . . . . . . . . . . . . . Assembly made for modal testing. Both the applied excitation and the measured response were perpendicular to the plate. To reproduce free boundary conditions a vertical suspension was employed . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 5 10
11 13 14
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34 xix
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Fig. 2.4
Fig. 2.5 Fig. 2.6
Fig. 2.7
Fig. 2.8
Fig. 2.9
Fig. 2.10
Fig. 2.11
Fig. 2.12
Fig. 2.13
Fig. 2.14
Fig. 3.1
List of Figures
Experimental modes results with variations in moisture content. The maximum variation in moisture content which could be obtained was between 0% a 25%, the latter of which corresponds with saturation. The nomenclature (m, n) identifies the different mode types, where m and n refer to the number of nodal lines parallel to the y direction and x direction, respectively Kinematics of thin plate deformation under the Kirchhoff’s assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the first ten resonant frequencies as a function of the number of DoFs. Higher order modes converge slower than the lower order modes . . . . . . . . . . . . . . . . . . . . . . . The procedure, based on [28], is essentially an iterative process to minimize the difference between the numerical and the experimental response. An initial estimation of the elastic constants is required. The error function which defines the convergence criterion is based solely on frequency values. The updating of variables ensures a rapid convergence . . . . . . . . Percentage variations of the numerical frequencies of the ten first resonance modes caused by a 15% increase in the references values for the elastic moduli EL , ER , GLR and νLR . Convergence graph of the iterative process for the specimen with moisture content 6.01%. The error is completely minimized for the first three modes, since the update is done only using them . . Changes in longitudinal (left), radial and shear moduli (right) as a function of moisture content. The most significant variation is found in the radial modulus, and consequently in resonance modes associated therewith . . . . . . . . . . . . . . . . . . . . Comparison between the numerical and experimental modal analysis for the first ten mode shapes under 6.01% moisture content conditions. The subscripts E and S denote experimental and simulated results, respectively. Possible sources of error are wood’s heterogeneity, thickness irregularities, non-uniformity in the cross-grain direction and densities, as well as inaccuracies in geometrical or mass measurements . . . . . . . . . . . . . . . Percentage variations of the numerical frequencies of the ten first resonance modes caused by increasing the nominal values of length, width, thickness and mass by 5% . . . . . . . . . . . . Percentage errors in estimating the elastic properties EL , ER and GLR introduced by increasing the nominal values of length, width, thickness and mass by 5% . . . . . . . . . . . . . . . . . Curvature adopted by the plate under moisture conditions due to the imperfect perpendicularity between the earlywood and latewood. δ denotes deflection, and L, R and T are the longitudinal, radial and tangential directions, respectively . . . . Three DOF mechanical system . . . . . . . . . . . . . . . . . .
35 40
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54 66
List of Figures
Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7
Fig. 4.8 Fig. 4.9
Fig. 4.10
Fig. 4.11
Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12 Fig. 5.13
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Modal parameter identification with block size . . . . . . . . . . Optimal model order at different data block sizes . . . . . . . . . Simulated time-varying mass function . . . . . . . . . . . . . . Monitoring of minimum order on simulation . . . . . . . . . . . Monitoring of block size on simulation . . . . . . . . . . . . . . Monitoring of frequencies on simulation . . . . . . . . . . . . . Monitoring of damping ratios on simulation . . . . . . . . . . . Plate test configuration . . . . . . . . . . . . . . . . . . . . . . Plate temporal response . . . . . . . . . . . . . . . . . . . . . . Monitoring of plate minimum model order . . . . . . . . . . . . Monitoring of plate natural frequencies . . . . . . . . . . . . . . Short time Fourier transform . . . . . . . . . . . . . . . . . . . The test rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of the 2D model of the flexible boom . . . . . . . . . . . Reference systems for the numerical model of the flexible boom Block diagram of the active modal control logic . . . . . . . . . Block diagram of the modal observer . . . . . . . . . . . . . . . Large motion reference for the boom links; starting/final configuration (a) and rotation time history of the links (b) . . . . Numerical comparison of the acceleration of the end of the boom third link with and without modal control; (a) the large motion time history and (b) the spectrum . . . . . . . . . . . . . Block diagram of the feed-forward control logic . . . . . . . . . Numerical simulations without (initial part) and with (final part) the FF control logic application; the three link ends acceleration with one (case A) and three (case B) active actuators . . . . . . . Experimental comparison of the acceleration of the end of the boom third link with and without modal control; (a) the large motion time history and (b) the spectrum . . . . . . . . . . . . . Experimental data without (initial part) and with (final part) the FF control logic application; the third link end acceleration with one active actuator . . . . . . . . . . . . . . . . . . . . . . . . . Pumping group system, CAD 3D . . . . . . . . . . . . . . . . . Test rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test rig block diagram . . . . . . . . . . . . . . . . . . . . . . . Diagram of pumping group . . . . . . . . . . . . . . . . . . . . Oil chambers pressure profile, varying rpm of the pump . . . . . Pressure in oil chambers, varying the input oil flow . . . . . . . Oil chambers pressure profile, varying the reducing valves area . Pressure in the oil chambers . . . . . . . . . . . . . . . . . . . . Translational friction model . . . . . . . . . . . . . . . . . . . . Numerical vs experimental results (oil chambers pressure) . . . . Numerical vs experimental results (CLS back pressure) . . . . . Numerical vs experimental results (CLS back pressure) . . . . . Control system diagram . . . . . . . . . . . . . . . . . . . . . .
67 68 68 69 69 70 71 72 72 73 74 74 83 84 84 85 88 89
89 91
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94 100 102 102 103 105 106 106 107 110 111 111 112 113
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Fig. 5.14 Fig. 5.15 Fig. 5.16 Fig. 5.17 Fig. 5.18 Fig. 5.19 Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4 Fig. 6.5 Fig. 6.6
Fig. 6.7 Fig. 6.8
Fig. 6.9
Fig. 6.10
Fig. 6.11
List of Figures
Numerical vs experimental results (diff function) . . . . . . . . . Comparison between controlled and non-controlled numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical application of control . . . . . . . . . . . . . . . . . Oil chambers pressure, experimental comparison with and without control . . . . . . . . . . . . . . . . . . . . . . . . . . . CLS pumped concrete, numerical comparison with and without control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Percentage difference of pumped fluid, numerical comparison with and without control . . . . . . . . . . . . . . . . . . . . . . Position of the vibration based structural health monitoring (VB-SHM) techniques with respect to other damage identification methods . . . . . . . . . . . . . . . . . . . . . . . Traditional double L-shaped stiffener versus the new stiffener concept developed by Stork–Fokker AESP and the NLR. (a) Traditional stiffener, (b) new stiffener concept . . . . . . . . [0/90/0/90/0/90/0/90]s laminate lay-up and dimensions. The global coordinate system xyz is indicated as well as the material orientations 123 in the skin and the stiffeners . . . . . . . . . . . Cross-section of the T-beam, that is extruded to obtain the T-beam model . . . . . . . . . . . . . . . . . . . . . . . . . . . Bottom view of the T-beam in the xz-plane . . . . . . . . . . . . Cross-sectional view in the yz-plane at x = 0 of the T-beam showing the delamination. The open circles represent the nodes connected to both the skin (thick line at the bottom) and the stiffener, the filled circles represent two nodes, one connected to the skin, the other to the stiffener. Note that only nodes at the skin-stiffener interface are indicated by markers . . . . . . . . . Dynamic set-up and data acquisition for the experimental investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Location of the excitation point and the 3 × 30 measurement grid points at the T-beam (bottom view in the xz-plane). The 60 mm wide light gray area at the right-hand side of the T-beam is the clamped area . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the frequency response functions (FRF) of the intact and delaminated T-beam. A distinct shift in the higher natural bending frequencies can be observed, whereas the natural frequencies of the torsion modes remain relatively unaffected. (Single point FRF from point R1) . . . . . . . . . . Normalized mode shapes (amplitude A) of the 5th bending mode. The delamination between 500 and 600 mm from the clamping can be easily observed. Iso-view on top, yz-view at the bottom. (a) Intact, FN = 840 Hz, (b) delaminated, FN = 816 Hz General procedure to calculate the modal strain energy damage index, based on the nodal displacements of each natural mode shape of a structure . . . . . . . . . . . . . . . . . . . . . . . .
114 114 115 115 116 116
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126
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131 133
134
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137
List of Figures
Fig. 6.12
Fig. 6.13
Fig. 6.14
Fig. 6.15
Fig. 6.16
Fig. 6.17
Fig. 6.18
Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7
xxiii
Comparison between calculated (n) and measured (e) natural bending and torsion frequencies. The light gray bars refer to the intact T-beam, whereas the dark gray bars refer to the delaminated T-beam. (a) Natural bending frequencies, (b) natural torsion frequencies . . . . . . . . . . . . . . . . . . . Damage indices for delamination lengths varying between 10 and 100 mm. A peak in the damage index indicates damage. Delamination starting at (a) 300 mm from the clamping, (b) 500 mm from the clamping, (c) 700 mm from the clamping . . . . . Damage indices for delamination lengths varying between 10 and 100 mm, measured at 0.041 m from the center line of the T-beam. The tip of the delamination is located at 500 mm from the clamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . The sum of the damage index values exceeding a threshold value of 2 for delamination lengths of 10–100 mm plot as a function of the perpendicular distance x to the delamination. The gray area covers the cases in which the probability of detection is low . . . . . . . . . . . . . . . . . . . . . . . . . . Damage indices for delamination lengths varying between 10 and 100 mm, using equidistant data points. A peak in the damage index indicates damage. (a) Damage index based on 33 equidistant data points. (b) Damage index based on 17 equidistant data points . . . . . . . . . . . . . . . . . . . . . . . Damage indices for delamination lengths varying between 10 and 100 mm, using clustered sets of data points. A peak in the damage index indicates damage. (a) Damage index based on 25 clustered data points (3 per cluster). (b) Damage index based on 13 clustered data points (3 per cluster) . . . . . . . . . . . . . . Damage indices for delamination lengths varying between 10 and 100 mm, using the first 12 torsion modes. Note the different scales for the axis of β. (a) Damage index using the torsion strain energy formulation (6.9). (b) Damage index using the bending strain energy formulation (6.8) . . . . . . . . . . . . . . Representation of (a) full-space, (b) half-space, (c) quarter-space and (d) half-space-contact geometries . . . . . . . . . . . . . . . (a) A sphere in half space; (b) two spheres in full space . . . . . Boundary element discretization of the sphere . . . . . . . . . . Near field of a dilating sphere near a rigid surface (z = 0, b = 3a, ka = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . Near field of a dilating sphere near a rigid surface (z = 0, b = 2a, ka = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . Near field of a dilating sphere near a rigid surface (z = a/3, b = 2a, ka = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . Near field of a dilating sphere near a rigid surface (z = 2a/3, b = 2a, ka = 1) . . . . . . . . . . . . . . . . . . . . . . . . . .
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145 156 163 163 164 164 165 165
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Fig. 7.8 Fig. 7.9 Fig. 7.10 Fig. 7.11 Fig. 7.12 Fig. 7.13 Fig. 7.14 Fig. 7.15 Fig. 7.16
Fig. 7.17
Fig. 7.A1 Fig. 7.B1
Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4
Fig. 8.5 Fig. 8.6 Fig. 8.7
Fig. 8.8
List of Figures
Near field of a dilating sphere near a rigid surface (z = 0, b = 2a, ka = 0.1) . . . . . . . . . . . . . . . . . . . . . . . . Near field of a dilating sphere near a rigid surface (z = 0, b = 2a, ka = 2) . . . . . . . . . . . . . . . . . . . . . . . . . The tested refrigerator . . . . . . . . . . . . . . . . . . . . . . Measurement system . . . . . . . . . . . . . . . . . . . . . . Variation of internal temperature of the refrigerator with time . Layout of the boundary element model . . . . . . . . . . . . . Measured surface velocity distributions at (a) 50 Hz and (b) 100 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicted surface pressure distributions at (a) 50 Hz and (b) 100 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . Near field sound radiation patterns at the horizontal crosssection through the compressor at (a) 50 Hz and (b) 100 Hz; zoomed view at (c) 50 Hz and (d) 100 Hz. (a = 0.71 m, b = 0.69 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . Near field sound radiation patterns at the horizontal crosssection through the fan at (a) 50 Hz and (b) 100 Hz; zoomed view at (c) 50 Hz and (d) 100 Hz (a = 0.71 m, b = 0.69 m) . . A boundary element . . . . . . . . . . . . . . . . . . . . . . . An interface of ‘in-house BEM code’ conforming the co-ordinates and incidence matrix defining the geometry of the refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . The time-frequency plan decomposed into small rectangular windows, called the Heisenberg boxes . . . . . . . . . . . . . Field form of the reassignment vector orientations of the energy distribution in the time-frequency plan . . . . . . . . . . . . . Experimental setup . . . . . . . . . . . . . . . . . . . . . . . Experimental signal backscattered by an air-filled aluminium tube immersed in water with radius ratio 0.9; (b) is a zoom of the selection part of the signal in (a) . . . . . . . . . . . . . . . Mechanism of echoes showing the specular reflection (1), circumferential waves (2) and Scholte wave (3) . . . . . . . . . Resonance spectrum for the signal given in Fig. 8.4 . . . . . . Time-frequency spectrogram images (Gabor transform) of an experimental signal backscattered by an aluminium tube b/a = 0.9 (the window used is Gaussian with α = 0.01 and the window size is 65 points for (a) and (c) and is 200 points for (b) and (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-frequency Wigner-Ville image of the experimental signal backscattered by an aluminium tube with radius ratio 0.9 (the h window size is 256 points and the g window size is 3 points) .
. 166 . . . . .
166 167 167 168 169
. 170 . 170
. 171
. 172 . 174
. 179 . 188 . 191 . 191
. 192 . 193 . 193
. 195
. 195
List of Figures
Fig. 8.9
Fig. 8.10
Fig. 8.11
Fig. 8.12
Fig. 8.13 Fig. 8.14 Fig. 8.15 Fig. 9.1
Fig. 9.2 Fig. 9.3
Fig. 9.4
Fig. 9.5 Fig. 9.6 Fig. 9.7
Fig. 9.8
xxv
The Blackman analysis windows employed in the three Short Time Fourier Transforms used to compute reassigned times and frequencies. Waveform (a) is the original window function, waveform (b) is time weighted window function, and waveform (c) is the frequency-weighted window function . . . . . . . . . . Time-frequency spectrogram images of the signal backscattered by an aluminium tube; (a) is the image with vector field of displacement and (b) is the reassigned spectrogram image . . . . Time-frequency images of the signal backscattered by a copper tube with radius ratio 0.9; (a) is the spectrogram and (b) is the reassigned spectrogram images . . . . . . . . . . . . . . . . . . Time-frequency Wigner-Ville images of the signal backscattered by a copper tube with radius ratio 0.9 (the h window size is 256 points and the window g size is 20 points) . . . . . . . . . . . . The group velocity dispersion curve of the symmetric circumferential wave S0 . . . . . . . . . . . . . . . . . . . . . . The group velocity dispersion curve of the anti-symmetric circumferential wave A1 . . . . . . . . . . . . . . . . . . . . . . The group velocity dispersion curve of the symmetric circumferential wave S1 . . . . . . . . . . . . . . . . . . . . . . Viscoelastic damping treatments configurations: (a) unconstrained layer damping (ULD); (b) passive constrained layer damping (PCLD); (c) PCLD with the strain magnifying effect of a spacer-layer; (d) active constrained layer damping (ACLD) . . . . . . . . . . . . . . . . . . . . . . . . . Generic viscoelastically damped sandwich or constrained plate and elemental volume . . . . . . . . . . . . . . . . . . . . . . . “Composite” FE plate (or shell) models of viscoelastically damped structures using existing commercial or in-house FE codes/elements . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete-layer (layerwise) 2D FE plate (or shell) models of viscoelastically damped structures using new dedicated FE codes/elements . . . . . . . . . . . . . . . . . . . . . . . . . . . FE-based viscoelastic time and frequency domain solution alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curve fitted ADF curves with 1 and 3 series of parameters at 27 °C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FE mesh and zoom of the discretized annular plate (light gray), viscoelastic damping layer (green) and constraining layer (dark gray), and clamped inner cylindrical surface (red) . . . . . . . . Undamped (flexural) mode shapes and natural frequencies of the bare annular plate (saw); ∗ , duplicated modes; cold color (dark blue) meaning zero transverse displacement, as is the case for example in the clamped inner circle, and hot color (red) meaning relevant transverse flexural displacement . . . . . . . .
197
198
198
199 200 200 201
214 221
221
226 230 243
244
245
xxvi
Fig. 9.9
Fig. 10.1
Fig. 10.2 Fig. 10.3 Fig. 10.4 Fig. 10.5 Fig. 10.6 Fig. 10.7
Fig. 10.8
Fig. 10.9 Fig. 10.10
Fig. 10.11
Fig. 10.12 Fig. 10.13
Fig. 10.14
Fig. 10.15 Fig. 10.16
List of Figures
Driving-point receptance, measured at the outer radius, mean-square (MS) velocity and radiated sound power per unit force applied at the outer radius of the undamped (dotted line) and viscoelastically damped (solid line) annular plate (saw) configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of a used piezo composite beam: (a) Shunted with a resistive (R) circuit; (b) shunt with R and negative Capacitance (RCneg) circuit . . . . . . . . . . . . . . . . . . . . . . . . . . Root locus induced by a resistive shunt circuit . . . . . . . . . . The effective shunted piezoelectric composite stiffness [35] as a function of the connected negative capacitance term . . . . . . . Different R induced mono-modal root loci with different negative capacitance part in the connected shunt . . . . . . . . . Plate with RL piezo-shunted device . . . . . . . . . . . . . . . . (a) Piezo-shunted RLC effect on resonance; (b) example of multimodal electric circuit . . . . . . . . . . . . . . . . . . . . . (a) Example of periodic structure; (b) associated band gaps diagram on which the blue domains correspond to evanescent waves propagation . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of periodically distributed shunted piezoelectric patches with RL and RC (negative) circuits for controlling a plate. The upper part shows the band gaps estimation δ and the lower part the corresponding collocated FRFs: (a) Two bimodal RL circuits ruled according two sets of different blocking circuits; (b) RC (negative) circuits with two different resistors . . Periodic piezoelectric plate and associated RL shunt circuits . . . Spatial average of the plate velocity frequency response function with RL circuits tuned at ftun = 1720 Hz with two different resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualization of the velocity distribution [m/(s V)] over the plate when vibrating at 1720 Hz in the short (a) and closed (b) RL circuit cases tuned at the same frequency . . . . . . . . . . . . . Experimental FRF of the hybrid configuration with RL circuits tuned at 1150 Hz and R = 100 . . . . . . . . . . . . . . . . . Panel vibration and corresponding cavity pressure at 650 Hz with (a) short circuited patches and (b) RL shunted patches tuned at the same frequency . . . . . . . . . . . . . . . . . . . . (a) Experimental set-up for structural acoustic control and comparison between (b) measured sound pressure levels without any connected circuit and with (c) RL shunt circuit tuned at 1600 Hz connected to the periodically distributed piezoelectric patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept overview and view of one cell of the periodic piezoelectric patch distribution . . . . . . . . . . . . . . . . . . Ratio of the transmitted flexural power flux as a function of negative capacitance shunt at 30 Hz, 1500 Hz and 3000 Hz . . .
248
268 271 272 273 273 274
275
277 278
278
279 279
280
281 282 290
List of Figures
xxvii
Fig. 10.17 Optimal shunt capacitance for reflection optimization (C-shunt) . 292 Fig. 10.18 Optimal shunt resistance for reflection optimization (RC-shunt) . 292 Fig. 10.19 Criterion value vs. freq. for transmission optimization with RC shunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Fig. 10.20 10 cells damped power function for D10 . . . . . . . . . . . . . 294 Fig. 10.21 Criterion value vs. freq. for transmission optimization with RC shunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Fig. 10.22 10 cells damped power function: D10 . . . . . . . . . . . . . . . 295 Fig. 11.1 Impact of noise on eigenvectors on properness norm . . . . . . . 308 Fig. 11.2 Impact of noise on eigenvectors on error on identified matrices . 308 Fig. 11.3 Eigenvectors of the first mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line) . . . . . . . . . . . . . . . . . . . . . . . 309 Fig. 11.4 Eigenvectors of the second mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line) . . . . . . . . . . . . . . . . . . . . 310 Fig. 11.5 Eigenvectors of the third mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line) . . . . . . . . . . . . . . . . . . . . . . . 310 Fig. 11.6 Eigenvectors of the fourth mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line) . . . . . . . . . . . . . . . . . . . . . . . 311 Fig. 11.7 Experimental test-case: two bending beams coupled by common clamping device . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Fig. 11.8 Comparison of measured and synthesized FRF11 . . . . . . . . 312 Fig. 11.9 Comparison of measured and synthesized FRF12 . . . . . . . . 313 Fig. 11.10 Comparison of measured and synthesized FRF22 . . . . . . . . 314 Fig. 11.11 Methodologies for properness enforcement on numerical test-case 322 Fig. 11.12 Methodologies for properness enforcement on guitar measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
List of Tables
Table 1.1 Table 1.2 Table 1.3 Table 1.4 Table 1.5 Table 1.6 Table 1.7
Table 2.1 Table 2.2
Table 2.3
Table 3.1 Table 4.1 Table 6.1 Table 6.2
Convergence of the first natural frequency ω (rad/s) varying the number of nodes dis-cretizing the cantilever beam domain . . . 10 First ten natural frequencies ω (rad/s) obtained for the cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Obtained natural frequencies ω (rad/s) with the meshless solutions and FEM for beam A . . . . . . . . . . . . . . . . . . 12 Obtained natural frequencies ω (rad/s) with the meshless solutions and FEM for beam B . . . . . . . . . . . . . . . . . . 12 First five natural frequencies ω (rad/s) obtained for the shear-wall 13 Convergence of the first natural frequency ω (rad/s) varying the number of nodes dis-cretizing the square plate domain . . . . . 14 Convergence of the first natural frequency ω (rad/s) varying the number of nodes dis-cretizing the thin clamped cylindrical shell domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Results of the influence of moisture content β on the geometric parameters and mass. Units in mm, g and kg/m3 , respectively . . 31 Initial and final estimated Young’s and shear moduli (EL , ER and GLR , respectively, in MPa) percentage error and elastic ratios under different states of moisture content β . . . . . . . . 48 Experimental (Exp) and numerical (FEM) resonant modes (in Hz), corresponding mode shapes and percentage deviation between frequencies under different moisture conditions β . . . 51 Modal identification of the emerging plate . . . . . . . . . . . . 73 Decrease in RMS of acceleration due to FF control with one and three active actuators . . . . . . . . . . . . . . . . . . . . . 93 Homogenized material properties of the uni-directional composite, based on the measured material data . . . . . . . . . 130 Dimensions and number of elements in the different sections of the T-beam. Note that the width to the right and left are equal for a symmetric T-beam . . . . . . . . . . . . . . . . . . . . . . 131 xxix
xxx
Table 6.3
Table 6.4
Table 7.1 Table 7.B1 Table 7.B2 Table 9.1 Table 9.2 Table 9.3 Table 9.4 Table 9.5
List of Tables
Experimentally determined natural bending and torsion frequencies FN [Hz] and viscous damping coefficients ζ [%] for the intact and delaminated T-beam . . . . . . . . . . . . . . Natural frequencies [Hz] calculated by the numerical model, including the relative error [%] with respect to the experimentally determined natural frequency [30] (the natural frequency is not measured in case no error value is given). The absolute maximum error is indicated by the bold-face numbers, the mean value is based on the absolute error values . . . . . . . Benefits and limitations of some sound source determination procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computed coordinates of front surface of the refrigerator . . . . Incidence matrix for the front surface of refrigerator . . . . . . . Main features of the “composite” and discrete-layer elemental models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damping modeling capabilities and general features available in some commercial FE softwares . . . . . . . . . . . . . . . . Material and geometric properties of the vibroacoustic system . Identified GHM and ADF parameters for 3M ISD112 at 27 °C using three series (n = 3) . . . . . . . . . . . . . . . . . . . . . Resonant frequencies, modal loss factors and response reductions for the plate with and without the PCLD treatment . .
135
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Chapter 1
The Dynamic Analysis of Thin Structures Using a Radial Interpolator Meshless Method L.M.J.S. Dinis, R.M. Natal Jorge, and J. Belinha
Abstract In this chapter an improvement of the Natural Neighbour Radial Point Interpolation Method (NNRPIM), a recently developed meshless method, is presented. A new approach of the NNRPIM is proposed, the NNRPIM 3D Shell-Like formulation, in order to analyse dynamically thin three-dimensional structures. The NNRPIM uses the Natural Neighbour concept to enforce the nodal connectivity and to construct the integration background mesh (totally node-dependent), which is used in the numerical integration of the NNRPIM interpolation functions. The essential and natural boundaries are imposed directly once the NNRPIM interpolation functions possess the delta Kronecker property. Several dynamic plate and shell problems are studied to demonstrate the effectiveness of the method.
1.1 Introduction In this chapter it is presented a recently developed meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM), applied to the dynamic analysis of thin three-dimensional structures. The scope of this chapter is to show the flexibility and the accuracy of this meshless method. The main motivation in the development of the NNRPIM was, without doubt, to create a meshless method: (a) easy to implement; (b) with interpolation functions (to simplify the essential and natural boundary imposition); (c) accurate; (d) with a low computational cost. All these purposed goals were successfully achieved, however the authors felt that the efficiency of the method could be improved, particularly in the analysis of L.M.J.S. Dinis () · R.M. Natal Jorge Faculty of Engineering of the University of Porto—FEUP, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal e-mail:
[email protected] R.M. Natal Jorge e-mail:
[email protected] J. Belinha Institute of Mechanical Engineering—IDMEC, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_1, © Springer Science+Business Media B.V. 2011
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thin three-dimensional structures. As so, for this kind of structures, a new NNRPIM approach was developed, the NNRPIM 3D Shell-Like formulation. The outline of this chapter is as follow: In Sect. 1.3 the NNRPIM 3D Shell-Like approach is presented, the creation of the influence-cells and the used integration scheme are summarized, as well as the construction of the interpolation functions. In Sect. 1.4 the dynamic discrete system of equations is presented and developed. In Sect. 1.5 benchmark dynamic examples of plates and shells in free and force vibration are solved. The article ends with the prospects for the future in Sect. 1.6 and the conclusions and remarks in Sect. 1.7.
1.2 Overview of the State of the Art The assemblage of spatial thin shells structures permits the construction of several engineering structures, such as roof structures, boat hulls and aeroplane fuselages, among many others. Nowadays, shell structures are design to be light, being the shells themselves the load main supporting structure, reducing the number of structure stiffeners. On the other hand this structural material optimization has a design disadvantage, it leads to lower fundamental frequencies, increasing the risk of collapse by resonance. Thus the dynamic analysis became an important part in shell structures design. Numerical methods are an important tool in the modulation of such complex structures. For many years the Finite Element Method (FEM) was the most widespread numerical method used [1]. However in the last fifteen years meshless methods [2] enlarge their application field, and are today a competitive and alternative approach in structural analysis. Numerous shell structures present elaborated curvatures and several holes or discontinuous essential boundaries, and for these conditions meshless methods are efficient. As it was in the beginning with the FEM, in this work the analysed thin structures are solved as three-dimensional (3D) problems, with some awareness in the integration along the smallest dimension in order to obtain the most reliable results. In meshless methods [3], generally, the nodes discretizing the problem domain can be randomly distributed, since the field functions are approximated within a flexible influence domain rather an element. In meshless methods the influence domains may and must overlap each other, in opposition to the no-overlap rule between elements in the FEM. Meshless methods that use the weak form solution can be divided in two categories, the ones that use approximation functions [4–9] and others that use interpolation functions. Meshless methods based in approximation functions have been successfully applied in computational mechanics and even its difficulty on imposing the essential and natural boundary conditions, due to the lack of the delta Kronecker property, has been overcome with the use of efficient numerical methods [10]. At the time, to solve the mentioned difficulty of the approximation functions, several meshless methods, using interpolation functions, were developed [11–17]. More recently meshless methods [2] were extended to numerous engineering fields. In the biomechanical field [18] meshless methods were applied from bone
1 The Dynamic Analysis of Thin Structures Using the NNRPIM
3
fracture simulation [19] to blood flow simulation [20]. Meshless have been also extended from multiscale analysis [21] to impact problems [22]. Advances in the theoretical study of meshless methods have also been made [23, 24] showing that meshless methods are continuously improving and growing. In this chapter a radial interpolator meshless method is used, the Natural Neighbour Radial Point Interpolation Method (NNRPIM) [25, 26], where the nodal connectivity and the background integration mesh, totally dependent on the nodal mesh, are achieved using mathematic concepts, such as Voronoï Diagrams [27] and the Delaunay tessellation [28]. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed with the Radial Point Interpolators (RPI) [17] and possess the delta Kronecker property. Although being the NNRPIM a recent meshless method, it was already extended to several computational mechanical problems. In statics it was extended to linear elasticity for two-dimensional and three-dimensional problems [25, 29] and for plate and laminate bending problems [26]. It was also extended to non-linear problems, such as elasto-plasticity [30] and large deformation analysis [31]. In dynamics it was extended to the analysis of twodimensional and three-dimensional problems [32] and to the analysis of plates and composites [33]. Within the NNRPIM a new approach is proposed, the NNRPIM 3D Shell-Like formulation [29]. The construction procedure is simple, first a two-dimensional nodal mesh, coincident with the middle surface of the plate or of the shell, is created. Then, based in this two-dimensional mesh, the background integration mesh is constructed. Afterwards the middle surface nodal mesh is projected to the top surface and bottom surface of the plate or shell, and the two-dimensional integration mesh is distributed along the plate or shell thickness respecting a Gauss quadrature scheme.
1.3 The Natural Neighbour Radial Point Interpolation Method The natural neighbours [34] determination of each node belonging to the global nodal set N = {n1 n2 . . . nN } ∈ R3 is achieved in the NNRPIM using the Voronoï diagrams and the Delaunay triangulation. This theory is applicable to a NDimensional space. The Voronoï diagram of N is the partition of the domain defined by N in sub-regions VI , closed and convex. Each sub-region VI is associated with the node I , nI , in a way that any point in the interior of the VI is closer to nI than any other node nJ , where nJ ∈ N(J = I ). The sub-regions Vk are defined as ‘Voronoï cells’ which form the Voronoï diagram, k = 1, . . . , N . In mathematical terms the Voronoï cell is defined by VI = {x ∈ R3 : En (x, xI ) < En (x, xJ ) ∀J = I }, considering En (xJ , xI ) as the Euclidean metric norm. The construction of such geometric structures is well described in [17]. The nodal connectivity is imposed by the overlap of the influence-cells [25, 26], a recent concept in meshless methods similar to the influence domain concept. The influence cells are obtained from the Voronoï cells. The cell formed by n nodes that contributes to the interpolation of the interest point xI is called ‘influence-cell’. In order to obtain the influence-cell a point of interest, xI searches for its neighbour nodes following the Natural Neighbour Voronoï construction. Then the natural
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Fig. 1.1 (a) Influence cell of interest point xI . (b) Construction process of integration points
neighbours of these first natural neighbours found are added to the set. In Fig. 1.1(a) it is presented the obtained cell of an interest point xI . After an initial phase where the Voronoï cells of each node are constructed, these cells are considered for the construction of a background mesh for integration purpose. In the NNRPIM an integration scheme based on the Voronoï tessellation and the Delaunay triangulation is used. This work focuses in thin 3D structures, in which one of the directions is much smaller than the others. Thus, firstly the problem is reduced to a two-dimensional (2D) problem using the two larger dimensions of the structure, and then this fictitious 2D domain is discretized. With the 2D nodal mesh obtained it is possible to construct a support 2D background integration mesh. Using the previous referred geometrical constructions small areas are established. Figure 1.1 shows the process. Firstly an original Voronoï cell, VI , is divided in m quadrilateral sub-cells, SI i , and it is known that AVI =
m
ASI i ,
∀ASI i ≥ 0,
(1.1)
i=1
being AVI the Voronoï cell area and ASI i the sub-cell area. If the set of Voronoï cells are a partition, without gaps, of the global domain then, the set of sub-cells are also a partition, without gaps, of the global domain. The obtained m quadrilateral are then filled with integration points [25]. This work only uses one integration point per quadrilateral, xI = 4−1 4i=1 xi , and the weight of this integration point it is the area of its own sub-cell, ASI i . After this stage the 2D nodal mesh constructed is projected to the upper and bottom face of the solid and the obtained surface integration mesh is reproduced along the thickness of the plate or the shell respecting Gauss quadrature distribution, being this procedure shown in Fig. 1.2. The 3DSL is to be used only with two node layers: an up nodal layer, a bottom nodal layer and in the middle a core of integration points (totally nodal dependent). There is no need to fill the core of the shell with nodes. This is the major advantage of this formulation. Less nodes and less integration points, combined with the higher accuracy obtained, lead to higher computational efficiency.
1 The Dynamic Analysis of Thin Structures Using the NNRPIM
5
Fig. 1.2 Shell-like nodal and integration mesh construction and distribution
The present 3D shell-like (3DSL) formulation was firstly presented for static problems [29]. In this work it is extend to the dynamic analysis. It was shown in [29] that the 3DSL formulation leads to more accurate results and reduces the computational cost of the meshless method (it is three times faster than the NNRPIM classical approach), therefore increasing the computational efficiency of the NNRPIM. In the same article the efficiency of the NNRPIM using the 3DSL formulation was compare with the FEM analysis. The results show clearly that the NNRPIM 3DSL is computational more efficient than the FEM. Previous works on the NNRPIM [25, 26] prove that the polynomial basis of the classic Radial Point Interpolators (RPIs) is unnecessary if the shape parameters of the radial basis function (RBF) are chosen carefully. Such innovation in the RPIs conducts to an increase in the computational efficiency of the method. Consider a function u(x) defined in the domain, , which is discretized by a set of N nodes. In the NNRPIM the function u(x) passes through all nodes using a radial basis function. It is assumed that only the nodes within the influence-cell of the point of interest xI have effect on u(x). The value of function u(x) at the point of interest xI is obtained by u(xI ) =
n
Ri (xI )ai (xI ) = R(xI )a(xI ),
(1.2)
i=1
where Ri (xI ) is the RBF, n is the number of nodes inside the influence-cell of xI . The coefficients ai (xI ) are non constant coefficients of Ri (xI ). In the RBF the variable isthe distance RI i between the relevant node xI and the neighbour node xi , rI i = (xI − xi )2 + (yI − yi )2 + (zI − zi )2 . Several known RBFs are well studied and developed in [17]. The present work uses the Multiquadric (MQ) function proposed initially by Hardy [35], p (1.3) R(rI i ) = rI2i + c2 . In Eq. (1.3) c and p are two shape parameters. The optimized values of c and p can be found in [25]. Equation (1.2) can be written as a matrix, us = RG a,
(1.4)
and solved by substitution on Eq. (1.2), u(xI ) = R(xI )R−1 G us = ϕ(xI )us . In order to clarify Eq. (1.5) vector R(xI ) is defined as R(xI ) = R(rI 1 ) R(rI 2 ) . . . R(rI n ) ,
(1.5)
(1.6)
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L.M.J.S. Dinis et al.
being matrix RG defined as ⎡
R(r11 ) ⎢ ⎢ R(r21 ) RG = ⎢ ⎢ .. ⎣ . R(rn1 )
R(r12 ) R(r22 ) .. . R(rn2 )
... ... .. . ...
⎤ R(r1n ) ⎥ R(r2n ) ⎥ ⎥ .. ⎥ . . ⎦ R(rnn )
The interpolation function ϕ(xI ) obtained in Eq. (1.5) is defined as ϕ(xI ) = R(xI )R−1 G = ϕ1 (xI ) ϕ2 (xI ) . . . ϕn (xI ) .
(1.7)
(1.8)
The partial derivative of ϕ(xI ) in order to a variable ξ is defined as ϕ,ξ (xI ) = R,ξ (xI )R−1 G ,
(1.9)
and the partial derivatives of the MQ-RBF in order to a variable ξ by p−1 R,ξ (rij ) = 2p rij2 + c2 (ζj − ζi ).
(1.10)
Early works on the RPI [17] show that these interpolation functions possess the delta Kronecker property and also that the partition of unity is satisfied. The lack of compatibility is an inconvenient property of the RPI functions. Using the conforming RPI [36] the compatibility is achieved, however it was concluded as well [36] that the RPI is much more simple and efficient than the conforming RPI. The lack of consistency in the RPI functions, constructed without a polynomial basis, is the reason why this formulation cannot pass the standard patch test, although it is proved that it can approach polynomials in the desired accuracy and the convergence is guaranteed when the nodal mesh is refined [37].
1.4 Dynamic Discrete System of Equations Consider the solid with a domain bounded by . In the absence of damping effects, the dynamic equilibrium based on the principle of virtual work can be written as T T T δε σ d + δu ρ u¨ d − δu b d − δuT t d = 0, (1.11)
t
where u and u¨ are respectively the displacement and the acceleration field, b is the body force vector and t the traction on the natural boundary. The strain vector ε is defined as ε = Lu, where L is the differential operator defined in Eq. (1.13), defined as ⎤T ⎡ ∂ ∂ ∂ 0 0 ∂y 0 ∂z ∂x ⎥ ⎢ ∂ ∂ ∂ ⎥ L=⎢ ⎣ 0 ∂y 0 ∂x ∂z 0 ⎦ . ∂ ∂ ∂ 0 0 ∂z 0 ∂y ∂x
(1.12)
(1.13)
1 The Dynamic Analysis of Thin Structures Using the NNRPIM
7
The linear constitutive relations can be given by σ = cε,
(1.14)
where σ is the stress tensor and c the material matrix defined as ⎡ ⎤ 1 ν ν 0 0 0 ⎢ν 1 ν 0 0 0 ⎥ ⎢ ⎥ ⎢ν ν 1 0 0 0 ⎥ ⎢ ⎥, c = μ1 ⎢ 0 ⎥ ⎢ 0 0 0 μ2 0 ⎥ ⎣ 0 0 0 0 μ2 0 ⎦ 0 0 0 0 0 μ2
(1.15)
where μ1 = E/(1 − ν 2 ) and μ2 = (1 − ν)/2. The Young Modulus is defined as E and ν is the Poisson ratio. Generally the first term of Eq. (1.11) can be presented as δε σ d = T
n n
I
δuI
J
BTI cBJ duJ ,
(1.16)
where KI J = BTI cBJ d is the stiffness matrix. The second term of Eq. (1.11) can be developed as
δu ρ u¨ d =
¨ d = δ(Hu)T ρ(Hu)
T
n n I
J
δuI
¨ (1.17) HTI ρHJ du,
from where the mass matrix is obtained, MI J = HTI ρHJ d. B is the deformation matrix, defined in Eq. (1.18), H is the interpolation function diagonal matrix, HJ = ϕJ I, and ρ = ρ ∗ I, being ρ ∗ the mass density of the material and I the identity matrix with size 3 × 3, ⎡ ∂ϕI ∂ϕI ∂ϕI ⎤ 0 0 0 ∂x ∂y ∂z ⎢ ⎥ ∂ϕI ∂ϕI ∂ϕI T ⎢ 0 0 ⎥ BI = ⎣ 0 (1.18) ∂y ∂x ∂z ⎦. ∂ϕI ∂ϕI ∂ϕI 0 0 0 ∂z ∂y ∂x The force vector, neglecting the body force vector of Eq. (1.11), is defined by FI =
n I
t
HTI f dt ,
(1.19)
being f the external force vector applied in the natural boundary t . The essential boundary conditions can be directly imposed in the mass matrix and in the stiffness matrix as in the FEM, since the interpolation function possesses the delta Kronecker property. Thus, the equilibrium equations governing the linear dynamic response, neglecting the damping effect, can be represented in the matrix form ¨ + KU = F, MU
(1.20)
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¨ = u. ¨ The fundamental mathematical method used to solve where U = u and U Eq. (1.20) is the separation of variables. This approach [38] assumes that the solution can be expressed in the form U(t) = X(t),
(1.21)
where is a m × m square matrix containing m spatial vectors independent of the time variable t, X(t) is a time dependent vector and m = 3N on the 3D formulation, being N the total number of nodes in the problem domain. The components of ¨ X(t) are called generalized displacements. From Eq. (1.21) it follows that U(t) = ¨ X(t). It is required that the space functions satisfy the following stiffness and mass orthogonality conditions T K =
and T M = I,
(1.22)
where is the diagonal matrix which contains the free vibration frequencies, ωi2 . After substituting Eq. (1.21) and its derivatives in order to time into Eq. (1.22) and pre-multiplying it by T , the equilibrium equation that corresponds to the modal generalized displacement is obtained, ¨ + X(t) = T F(t). X(t)
(1.23)
The initial conditions on X(t) are obtained as follows, X0 = T MU0 , ˙ 0 = T MU ˙ 0. X
(1.24)
Equation (1.23) can be represented as m individual equations of the form x¨i (t) + ωi2 xi (t) = fi (t), fi (t) = φ Ti F(t),
(1.25)
with the initial conditions xit=0 = φ Ti MU0 , ˙ 0, x˙ t=0 = φ T MU
(1.26)
i
i
being the vector φ i called the ith mode shape vector, = [φ 1 φ 2 . . . φ m ]. For the complete response, the solution to all m equations in Eq. (1.25) must be calculated and then the modal point displacements are obtained by superposition of the response in each mode, U(t) =
m
φ i xi (t).
(1.27)
i=1
In this work when forced vibrations are imposed three different time-dependent loading conditions are considered, f (t) = F × g(t). A time constant load—load case A, gA (t) = 1,
(1.28)
1 The Dynamic Analysis of Thin Structures Using the NNRPIM
9
a transient load—load case B,
gB (t) = 1 if t ti , gB (t) = 0 if t > ti and a harmonic load—load case C, gC (t) = sin(λt).
(1.29)
(1.30)
The solution of each equation in Eq. (1.25) can be calculated using the Duhamel integral, 1 t fi (τ ) sin ωi (t − τ ) dτ + αi sin(ωi t) + βi cos(ωi t), (1.31) xi (t) = ωi 0 where αi and βi are determined from the initial conditions, Eq. (1.26) and fi (t) = φ Ti F(t). For load case A and load case B the obtained solution is xi (t) =
x˙ t=0 fi (t) 1 − cos(ωi t) + i sin(ωi t) + xit=0 cos(ωi t). 2 ωi ωi
For load case C the obtained solution is λ fi (t) sin(λt) − xi (t) = 2 sin(ωi t) . ωi ωi − λ2
(1.32)
(1.33)
1.5 Dynamic Examples In this section in order to show the accuracy of the NNRPIM 3D Shell-Like (3DSL) approach, in the context of dynamic analysis, several examples are presented and the numerical results are compared with other numerical methods solutions, analytical solutions and FEM solutions, which are available in the literature. All results are obtained using the consistent mass matrix. It was found in a previous work [29] on NNRPIM 3D Shell-Like that the use of two or three integration layers along the thickness, Fig. 1.2, is sufficient in order to achieve the solution stabilization. As so, in the next examples, for the NNRPIM 3D Shell-Like formulation, it will be used three integration layers along the thickness.
1.5.1 Cantilever Beam The NNRPIM is applied to analyze the free vibration of a cantilever beam, Fig. 1.3. The geometrical parameters of the beam are, L = 48.0 m, D = 12.0 m and thickness h = 1.0 m. The material properties are Young’s modulus E = 3 × 107 N/m2 , Poisson ratio ν = 0.3 and mass density ρ = 1.0 kg/m3 . A convergence study was performed. The problem was analyzed in 2D and 3D, with several numerical methods. Considering the problem as a two-dimensional problem (2D plain stress deformation theory), it was used the FEM, 9 node element, the Radial Point Interpolation
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Fig. 1.3 Cantilever beam. Examples of 2D and 3D, regular and irregular meshes
Table 1.1 Convergence of the first natural frequency ω (rad/s) varying the number of nodes discretizing the cantilever beam domain Regular mesh 2D nodes RPIM FEM 9 p = 1.03; nodes c=1
3D NNRPIM nodes classic
Regular mesh
Irregular mesh
NNRPIM NNRPIM NNRPIM NNRPIM classic shell-like classic shell-like
27
29.725
167.830 29.599
27 × 2 29.501
29.635
28.294
85
28.054
83.301 28.050
85 × 2 28.085
28.109
28.097
28.211
175
27.845
55.478 27.840
175 × 2 27.892
27.904
27.868
27.968
28.422
297
27.785
41.594 27.775
297 × 2 27.833
27.840
27.735
27.825
451
27.759
33.270 27.747
451 × 2 27.807
27.812
27.719
27.808
637
27.746
27.722 27.732
637 × 2 27.793
27.797
27.701
27.787
Method [17] and the NNRPIM. Considering the problem as a three-dimensional problem it was used in the analysis the NNRPIM 3D classic formulation [25] and the NNRPIM 3D Shell-Like formulation. The problem domain was discretized in two different meshes, regular meshes and irregular meshes, Fig. 1.3. In Table 1.1 are presented the obtained results regarding the first natural frequency. It is perceptible that the NNRPIM 3D Shell-Like formulation produces results very close to the others formulations and present a fast convergence to the final solution. The use of irregular meshes does not seam to disturb the method performance. In Table 1.2 are presented the first ten natural frequencies obtained for the finest mesh indicated in Table 1.1. In order to determine the NNRPIM 3D Shell-Like formulation efficiency, the same problem was analysed. The time spend in the analysis was registered and compared with the time spend with the NNRPIM classic formulation. The result was categorical, the NNRPIM 3D Shell-Like formulation is three times faster than the NNRPIM classic formulation.
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Table 1.2 First ten natural frequencies ω (rad/s) obtained for the cantilever beam Vibration mode
Regular mesh RPIM FEM p = 1.03; 9 nodes c=1 2D 2D
NNRPIM classic 2D
Regular mesh NNRPIM NNRPIM classic shell-like 3D 3D
Irregular mesh NNRPIM NNRPIM classic shell-like 3D 3D
1 2 3 4 5 6 7 8 9 10
27.746 141.285 179.746 325.371 526.705 536.799 735.818 881.841 905.421 1006.030
27.732 141.257 179.737 325.214 526.269 536.741 734.954 881.650 904.427 1005.051
27.793 141.429 179.961 325.538 526.807 537.396 735.873 882.648 905.415 1006.148
27.701 140.933 179.246 324.455 524.940 535.459 732.426 879.683 901.604 1002.522
27.722 140.838 179.714 323.724 522.970 536.534 729.172 881.058 898.567 999.162
27.797 141.478 179.967 325.722 527.238 537.438 736.666 882.776 906.243 1006.969
27.787 141.373 179.806 325.469 526.581 537.133 734.716 882.432 904.423 1005.656
Fig. 1.4 Cross section of the variable beams and respective nodal arrangement for the 2D and 3D modulation: (a) beam A and (b) beam B
1.5.2 Variable Cross Section Beams In this example two different cantilever beams with a variable cross section are presented, Figs. 1.4(a) and 1.4(b). The geometrical parameters of beam A are, L = 10.0 m, D1 = 5.0 m and D2 = 3.0 m. For the beam B, L1 = 8.0 m, L2 = 2.0 m, D1 = 3.0 m and D2 = 1.0 m. For both beams the thickness is h = 1.0 m and the material properties are Young’s modulus E = 3 × 107 N/m2 , Poisson ratio ν = 0.3 and mass density ρ = 1.0 kg/m3 . The problem is analysed considering a 2D and a 3D approach. The nodal arrangements for both beams are presented in Fig. 1.4.
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L.M.J.S. Dinis et al.
Table 1.3 Obtained natural frequencies ω (rad/s) with the meshless solutions and FEM for beam A Vibration mode
MLPG 2D
LRPIM p = 1.03; c=1 2D
NNRPIM classic 2D
NNRPIM classic 3D
NNRPIM shell-like 3D
FEM Abaqus 2D
1
263.21
262.13
262.58
262.20
262.39
262.09
2
923.03
920.81
925.21
921.32
921.65
918.93
3
953.45
952.06
952.43
953.42
953.39
951.86
4
1855.14
1854.32
1874.89
1863.06
1863.43
1850.92
5
2589.7S
2589.87
2589.76
2590.23
2590.37
2578.63
Table 1.4 Obtained natural frequencies ω (rad/s) with the meshless solutions and FEM for beam B FEM 9 nodes 2D
NNRPIM classic 2D
NNRPIM classic 2L 3D
NNRPIM classic 3D
NNRPIM shell-like 3D
1
132.21
134.01
146.61
138.14
131.59
2
484.63
476.90
521.85
518.53
469.88
3
878.63
891.45
981.41
943.94
878.31
4
1117.20
1075.85
1158.75
1168.69
1062.42
5
1942.27
1883.92
1981.18
2002.34
1863.21
Vibration mode
For the 2D analysis the beam A is discretized with 231 nodes and beam B with 287 nodes. In the 3D analysis 1122 nodes are used for beam A and 918 nodes for beam B. The first five natural frequencies (rad/s) obtained for beam A with the three 2D-NNRPIM formulations, with the classical 3D-NNRPIM formulation, with the MLPG [39], with the LRPIM [40] and with the FEM-ABAQUS are presented in Table 1.3. The results obtained for beam B are presented in Table 1.4. The results of the distinct analyses are obtained for the same nodal discretization. It is visible that the results obtained with the NNRPIM are in a good agreement with both meshless methods and with the FEM.
1.5.3 Shear-Wall A shear wall with four openings is studied in this example. The geometrical parameters of the shear wall are presented in Fig. 1.5, as well as the 2D nodal mesh of 559 nodes and 3D nodal mesh of 1118 nodes. The material properties are Young’s modulus E = 1000 N/m2 , Poisson ratio ν = 0.2 and mass density ρ = 1.0 kg/m3 . The problem is analysed with the three NNRPIM formulations and the obtained results are compared with the MLPG [39], with the LRPIM [40], with the Boundary Element Method (BEM) [41], and with the FEM-ABAQUS for the same nodal arrangement. The obtained results of the first five frequencies are presented in Table 1.5.
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Fig. 1.5 Shear wall with four openings and respective mesh discretization for the 2D and the 3D modulation Table 1.5 First five natural frequencies ω (rad/s) obtained for the shear-wall Vibration mode
MLPG 2D
BEM 2D
LRPIM p = 1.03; c=1 2D
NNRPIM classic 2D
NNRPIM classic 1L 3D
NNRPIM shell-like 3D
FEM Abaqus 3D
1
2.069
2.079
2.086
2.098
2.276
2.280
2.073
2
7.154
7.181
7.152
7.110
7.107
7.106
7.096
3
7.742
7.644
7.647
7.647
7.520
7.532
7.625
4
12.163
11.833
12.019
12.353
12.297
12.351
11.938
5
15.587
15.947
15.628
15.418
15.790
15.789
15.341
1.5.4 Square Plates Several plates with various shapes are studied in this sub-section. In all examples the material properties are the same: Young modulus E = 1000 N/m2 , Poisson ratio ν = 0.3 and mass density ρ = 100 kg/m3 . A square plate with the dimensions L = D = 10.0 m and h = 0.1 m was considered in first place. Several boundary conditions are considered in the analysis. The boundary x = 0 is identify as bc1 and x = L corresponds to bc3. The bc2 and the bc4 are respectively the boundaries y = 0 and y = D. Respecting the order bc1-bc2-bc3-bc4 four types were considered: SSSS, CCCC, SCSC and SCSS, where S stands for simply supported and C for clamped edge. In the convergence study of the square plate two types of meshes are used in the analysis, nodal meshes with regular distribution and nodal meshes with irregular distribution. Examples of both are shown in Fig. 1.6. As it was explained in Sect. 1.3 there is no nodes along the plate or shell thickness, there is only one up-layer and bottom-layer of nodes. In Table 1.6 the convergence studies for the SSSS, CCCC, SCSC and SCSS square plates are presented. The NNRPIM 3DSL
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L.M.J.S. Dinis et al.
Fig. 1.6 Examples of used meshes Table 1.6 Convergence of the first natural frequency ω (rad/s) varying the number of nodes discretizing the square plate domain Nodes
SSSS
CCCC
SCSC
SCSS
Regular Irregular Regular Irregular Regular Irregular Regular Irregular 5×5×2
50 6.612
6.853
19.866 22.331
9.549
11.022
7×7×2
98 3.598
4.466
8.828 10.640
6.842
8.114
4.789
6.041 4.301
15.789 17.788
9×9×2
162 2.762
3.367
6.079
7.306
4.799
5.662
3.536
11 × 11 × 2
242 2.418
2.973
5.061
6.341
4.005
4.900
3.037
3.810
13 × 13 × 2
338 2.255
2.701
4.583
5.532
3.626
4.368
2.800
3.366
17 × 17 × 2
578 2.119
2.403
4.159
4.607
3.297
3.666
2.599
2.897
21 × 21 × 2
882 2.068
2.224
3.976
4.224
3.161
3.366
2.519
2.699
25 × 25 × 2 1250 2.044
2.136
3.879
4.026
3.091
3.193
2.480
2.578
RMAS
2.000
3.642
3.642
2.930
2.930
2.394
2.394
2.000
formulation is compared with the Reissner-Mindlin analytical solution (RMAS), [42]. The Table shows the first natural frequencies obtained with the 3DSL formulation for increasing nodal meshes. The natural frequencies are normalised with the expression ωnorm = ω L2 π 2 ρh/κ, (1.34) where κ = Eh3 /(12(1 − υ 2 )).
1.5.5 Shallow Shell In this section a thin clamped cylindrical shell panel is considered. This example was experimented in Southampton University [43] and then it was verified numerically with the FEM [44]. The geometric and material properties of the problem are presented in Fig. 1.7. Taking advantage of the symmetry, only one quarter of the cylindrical shell panel was considered and the necessary symmetric or antisymmetric boundary conditions were introduced along the lines of symmetry. The problem domain was discretized in several regular and irregular meshes in order to
1 The Dynamic Analysis of Thin Structures Using the NNRPIM
15
Fig. 1.7 Thin clamped cylindrical shell panel geometric and material properties and regular and irregular meshes examples used in the analysis Table 1.7 Convergence of the first natural frequency ω (rad/s) varying the number of nodes discretizing the thin clamped cylindrical shell domain Mesh nodes
6 × 5 × 2 11 × 9 × 2 16 × 13 × 2 21 × 17 × 2 25 × 21 × 2 60 198 416 714 1092
NNRPIM
3DSL regular
1190.19
874.38
822.01
813.77
811.19
NNRPIM
3DSL irregular
1457.50
904.09
891.24
674.73
825.53
FEM
Shell 132 nodes
Experimental results
869.00 814.00
study the convergence of the numerical method. Examples of used meshes in the analysis are presented as well in Fig. 1.7. The first vibration frequencies (Hz) with increasing number of nodes discretizing the problem domain are presented in Table 1.7. The results are compared with the experimental results [43] and the FEM solution [44]. For future comparison the final converged first vibration frequencies obtained were f3DSL-RM = 811.19 Hz and f3DSL-IM = 825.53 Hz. The reference solutions are fexp = 814 Hz and fFEM = 869 Hz. Notice that the NNRPIM solution, in comparison with the FEM solution, is much closer to the experimental solution. The same shells is now considered for forced vibration analysis. The thin clamped cylindrical shell panel is subjected to a uniform distributed load, f (t) = g(t), where g(t) is the time-dependent load presented in Sect. 1.3, Eqs. (1.28), (1.29) and (1.30). In the load case B, ti = 0.003 s. In this analysis the domain discretization respect the regular nodal arrangement present in Fig. 1.7, it was used a regular mesh of 13 × 16 × 2 = 416 nodes. In Fig. 1.8 the responses to the dynamic load cases A, B and C for the analysed shell are presented. Notice that, in Fig. 1.8(a), the shell transient response to the load case B, where the shell is subjected to a suddenly vanishing load at t = ti , is equal to the shell transient response to the load case A until the instant t = ti , and after t = ti the shell vibrates freely around the initial position, due the absence of applied loads. To obtain Fig. 1.8(b) the frequency λ of the dynamic load, load case C, is varied and the maximum amplitudes of the shell
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L.M.J.S. Dinis et al.
Fig. 1.8 (a) Maximum deflection varying along time for a suddenly applied load (load case A) and for a load that suddenly vanishes (load case B) and (b) absolute value of the amplitude versus the load normalized frequency
transverse deflection are obtained for each one of the load frequencies applied. In Fig. 1.8(b) it is presented the maximum value of the transverse displacement versus the normalized frequency of the dynamic load, λnorm = λ S 2 /π 2 ρh/κ, where S = R sin(β).
(1.35)
1 The Dynamic Analysis of Thin Structures Using the NNRPIM
17
1.6 Prospects for the Future When first meshless methods were developed the main purpose was to give a suitable response to the problems cause by remeshing or ill-conditioned meshes. Meanwhile researchers found that meshless methods are more than numerical methods with a good response to irregular meshes or complex boundary edges. Meshless methods prove to be more accurate and more stable than the Finite Element Method (FEM). Today meshless methods pass beyond FEM applications, and are entering a world where the FEM cannot penetrate, such as flexible domains, changing with time and load [45], or the fluid particle dynamics [4]. The NNRPIM, and particularly the formulation presented in this work, is a recent meshless method, which proved to be accurate and efficient. The NNRPIM 3D Shell-Like formulation does not deplete in solid mechanical problems. The biomechanical applications are the next natural step for this formulation. The study of the dynamic behaviour of the tympanic membrane is one example of a future application, or the dynamic behaviour of the blood flow [20]. The unique characteristics of this numerical method place it in the first line of numerical methods suitable to deal with such promising applications.
1.7 Summary In this chapter a three-dimensional shell-like approach for the dynamic analysis of thin plates and shells using the Natural Neighbour Radial Point Interpolation Method (NNRPIM 3DSL) was proposed. Several well-known three-dimensional benchmark examples were solved. The obtained results, and the experience acquired along the development of this work, permit to conclude: (a) The NNRPIM is a stable and accurate interpolator meshless method. (b) Generally the convergence rate is high and the final converged solution is always very close to the considered problem analytical solution. (c) The interpolation functions of the NNRPIM, which are very simple to construct, permit an easy imposition of the boundary conditions. (d) The 3DSL prove to be more efficient than the classical NNRPIM approach, presenting lower computational times, higher convergence rates and more accurate results. (e) The variation of the boundary conditions and the irregularity of the mesh do not disturb significantly the method performance. The NNRPIM 3DSL approach proved to be an efficient and alternative method in the three-dimensional dynamic analysis of thin plates and shells.
1.8 Selected Bibliography The work of Belytschko [3] is a good starting point for those how want to initiate them selves in the meshless methods world. Afterwards the reading of Liu’s book it is advised [37] here one can find the description and the mathematical development
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L.M.J.S. Dinis et al.
of the most important meshless methods. Besides the construction description of the approximation and interpolation functions of several meshless methods, this book contains the results of many benchmark examples (static and dynamic analysis). In parallel it is recommended the reading of Dolbow work [46] and Nguyen work [2], in order to begin, from a safe starting point, the meshless programming. There are many meshless methods, each one with its own variations, and the NNRPIM is just one of them. However the NNRPIM have a huge advantage to the others. The NNRPIM is a recent meshless method, giving the firsts steps in the computational mechanics. Numerous engineering fields are waiting to be experimented with the NNRPIM. In order to start the study of the NNRPIM, the first step is to get familiar with the work of Wang [17]. Then, the first NNRPIM paper is mandatory [25]. In this paper the interpolation functions are presented, as well as the integration scheme (complete nodal-dependent) and the original studies regarding the shape parameters optimization. Concerning the dynamic analysis two other NNRPIM papers are suggested [26] and [32]. In this article a 3D shell-like formulation was presented, in order to understand it better the reading of [29] is advised. Acknowledgements The authors truly acknowledge the funding provided by Ministério da Ciência, Tecnologia e Ensino Superior—Fundação para a Ciência e a Tecnologia (Portugal), under grant SFRH/BD/31121/2006, and by FEDER/FSE, under grant PTDC/EME-PME/81229/2006.
References 1. Yang, H.T.Y., Saigal, S., Liaw, D.G.: Advances of thin shell finite elements and some applications-version-I. Compos. Struct. 35, 481–504 (1990) 2. Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008) 3. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(1), 3–47 (1996) 4. Monaghan, J.J.: Smoothed particle hydrodynamics: theory and applications to non-spherical stars. Mon. Not. Astron. Soc. 181, 375–389 (1977) 5. Lancaster, P., Salkauskas, K.: Surfaces generation by moving least squares methods. Math. Comput. 37, 141–158 (1981) 6. Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992) 7. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin method. Int. J. Numer. Methods Eng. 37, 229–256 (1994) 8. Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20(6), 1081–1106 (1995) 9. Atluri, S.N., Zhu, T.: A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22(2), 117–127 (1998) 10. Liu, G.R., Yan, L.A.: A modified meshless local Petrov-Galerkin method for solid mechanics. Adv. Comput. Eng. Sci. 39, 1374–1379 (2000) 11. Traversoni, L.: Natural neighbour finite elements. In: Int. Conf. on Hydraulic Engineering Software. Hydrosoft Proc. Computational Mechanics Publications, vol. 2, pp. 291–297 (1994) 12. Sukumar, N., Moran, B., Belytschko, T.: The natural element method in solid mechanics. Int. J. Numer. Methods Eng. 43(5), 839–887 (1998)
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13. Sukumar, N., Moran, B., Semenov, A.Y., Belikov, V.V.: Natural neighbour Galerkin methods. Int. J. Numer. Methods Eng. 50(1), 1–27 (2001) 14. Cueto, E., Sukumar, N., Calvo, B., Cegoñino, J., Doblaré, M.: Overview and recent advances in the natural neighbour Galerkin method. Arch. Comput. Methods Eng. 10(4), 307–387 (2003) 15. Liu, G.R., Gu, Y.T.: A point interpolation method for two-dimensional solids. Int. J. Numer. Methods Eng. 50, 937–951 (2001) 16. Liu, G.R.: A point assembly method for stress analysis for two-dimensional solids. Int. J. Solids Struct. 39, 261–276 (2002) 17. Wang, J.G., Liu, G.R.: A point interpolation meshless method based on radial basis functions. Int. J. Solids Struct. 54, 1623–1648 (2002) 18. Doblare, M., Cueto, E., Calvo, B., Martínez, M.A., Garcia, J.M., Cegonino, J.: On the employ of meshless methods in biomechanics. Comput. Methods Appl. Mech. Eng. 194(6–8), 801– 821 (2005) 19. Lee, J.D., Chen, Y., Zeng, X., Eskandarian, A., Oskard, M.: Modeling and simulation of osteoporosis and fracture of trabecular bone by meshless method. Int. J. Eng. Sci. 45(2–8), 329–338 (2007) 20. Tsubota, K., Wada, S., Yamaguchi, T.: Particle method for computer simulation of red blood cell motion in blood flow. Comput. Methods Programs Biomed. 83(2), 139–146 (2006) 21. Venkataraman, P., Ng, T.Y., Li, H.: Development of a novel multi-scale numerical technique. Comput. Mater. Sci. (2010). doi:10.1016/j.commatsci.2009.12.039 22. Zhang, Y.Y., Chen, L.: Impact simulation using simplified meshless method. Int. J. Impact Eng. 36(5), 651–658 (2009) 23. Fili, A., Naji, A., Duan, Y.: Coupling three-field formulation and meshless mixed Galerkin methods using radial basis functions. J. Comput. Appl. Math. 234(8), 2456–2468 (2010) 24. Zeze, D.S., Potier-Ferry, M., Damil, N.: A boundary meshless method with shape functions computed from the PDE. Eng. Anal. Bound. Elem. 34(8), 747–754 (2010) 25. Dinis, L., Jorge, R.N., Belinha, J.: Analysis of 3D solids using the natural neighbour radial point interpolation method. Comput. Methods Appl. Mech. Eng. 196, 2009–2028 (2008) 26. Dinis, L., Jorge, R.N., Belinha, J.: Analysis of plates and laminates using the natural neighbour radial point interpolation method. Eng. Anal. Bound. Elem. 32, 267–279 (2008) 27. Voronoï, G.F.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième Mémoire: Recherches sur les parallélloèdres primitifs. J. Reine Angew. Math. 134, 198–287 (1908) 28. Delaunay, B.: Sur la sphére vide. A la memoire de Georges Voronoï. Izv. Akad. Nauk SSSR Otd. Mat. Est. Nauk 7, 793–800 (1934) 29. Dinis, L., Jorge, R.N., Belinha, J.: A 3D shell-like approach using a natural neighbour meshless method: isotropic and orthotropic thin structures. Compos. Struct. 92, 1132–1142 (2010) 30. Dinis, L., Jorge, R.N., Belinha, J.: Radial natural neighbours interpolators: 2D and 3D elastic and elastoplastic applications. In: Ferreira, A.J.M., Kansa, E.J., Fasshauer, G.E., Leitão, V.M.A. (eds.) Computational Methods in Applied Sciences – Progress on Meshless Methods. Springer, Berlin (2008) 31. Dinis, L., Jorge, R.N., Belinha, J.: Large deformation applications with the radial natural neighbours interpolators. Comput. Mod. Eng. Sci. 44(1), 1–34 (2009) 32. Dinis, L., Jorge, R.N., Belinha, J.: The natural neighbour radial point interpolation method: dynamic applications. Eng. Comput. 26(8), 911–949 (2009) 33. Dinis, L., Jorge, R.N., Belinha, J.: Composite laminated plates: a 3D natural neighbour radial point interpolation method approach. J. Sandw. Struct. Mater. 12(2), 119–138 (2009) 34. Sibson, R.: A vector identity for the Dirichlet tessellation. Math. Proc. Camb. Philos. Soc. 87, 151–155 (1980) 35. Hardy, R.L.: Theory and applications of the multiquadrics – biharmonic method (20 years of discovery 1968–1988). Comput. Math. Appl. 19, 127–161 (1990) 36. Liu, G.R., Gu, Y.T., Dai, K.Y.: Assessment and applications of interpolation methods for computational mechanics. Int. J. Numer. Methods Eng. 59, 1373–1379 (2004)
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37. Liu, G.R.: Mesh Free Methods – Moving Beyond the Finite Element Method. CRC Press, Bota Raton (2002) 38. Bathe, K.J.: Finite Element Procedures. Prentice-Hall, Englewood Cliffs (1996) 39. Gu, Y.T., Liu, G.R.: A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput. Mech. 27, 188–198 (2001) 40. Gu, Y.T., Liu, G.R.: A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids. J. Sound Vib. 246(1), 29–46 (2001) 41. Brebbia, C.A., Telles, J.C., Wrobel, L.C.: Boundary Element Techniques. Springer, Berlin (1984) 42. Dawe, D.J., Roufaeil, O.L.: Rayleigh-Ritz vibration analysis of Mindlin plates. J. Sound Vib. 69, 345–359 (1980) 43. Nath, D.: Dynamics of rectangular curved plate. Ph.D. thesis, Southampton University, UK (1969) 44. Au, F.T.K., Cheung, Y.K.: Free vibration and stability analysis of shells by the isoparametric spline finite strip method. Thin-Walled Struct. 24, 53–82 (1996) 45. García, J.A., Gascón, L., Cueto, E., Ordeig, I., Chinesta, F.: Meshless methods with application to liquid composite molding simulation. Comput. Methods Appl. Mech. Eng. 198(33–36), 2700–2709 (2009) 46. Dolbow, J., Belytschko, T.: An introduction to programming the meshless element free Galerkin method. Arch. Comput. Mech. 5(3), 207–241 (1998)
Chapter 2
Vibration Testing for the Evaluation of the Effects of Moisture Content on the In-Plane Elastic Constants of Wood Used in Musical Instruments M.A. Pérez Martínez, P. Poletti, and L. Gil Espert Abstract The present work provides an experimental-numerical investigation into the effects of moisture content on the in-plane elastic constants of wood for the specific use of the construction of soundboards of musical instruments. The vibrational behavior of a rectangular plate of spruce has been observed using vibration testing under different humidity conditions. The use of a nondestructive test method permits direct examination of a material sample which will eventually become part of a real instrument. The proposed approach is intended to minimize the difference between the numerical and experimental dynamic response through an iterative process, which allows identifying the elastic characteristics of wood specimens. It has been demonstrated that the vibrational behavior of timber varies considerably with variations in humidity. However, not all elastic properties are equally affected by such changes. The most significant variation is found in the transverse elastic modulus, and consequently in resonance modes associated therewith.
2.1 Introduction Wood is one of the oldest and best-known materials which, given its unique mechanical and acoustical properties, is by far the most widely used material in the making of stringed instruments [46]. Despite the variety of construction materials currently M.A. Pérez Martínez () Department of Strength of Materials and Structures, Universitat Politècnica de Catalunya, C/Colon, 11 TR45, 08225 Terrassa, Barcelona, Spain e-mail:
[email protected] P. Poletti Department of Sonology, Escola Superior de Música de Catalunya, C/Padilla, 155, 08013 Barcelona, Spain e-mail:
[email protected] L. Gil Espert Laboratori per a la Innovació Tecnològica d’Estructures i Materials, Universitat Politècnica de Catalunya, C/Colon, 11 TR45, 08225 Terrassa, Barcelona, Spain e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_2, © Springer Science+Business Media B.V. 2011
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available, such as synthetic polymers and carbon fibre composites, luthiers have for the large part continued using wood. Wood has the capacity to react and adapt to environmental conditions, thus the effects of changing moisture levels are important from a practical point of view and are often a decisive factor for the material’s mechanical performance. Generally, the speed of sound increases and the internal damping decrease with a reduction of moisture content and, as a consequence, the musical instrument often sounds noticeably brighter [22]. At the other end of the spectrum, under extremely moist conditions, the expansion experienced by the material due to its hygroscopicity can become large enough that permanent damage results. Other material properties which are critical for the acoustical performance, such as elastic moduli or density, are also affected [18, 44]. Therefore, one of the most important considerations in the process of the construction of a wooden musical instrument is to understand the alterations caused by changes in humidity. Independent of the basic nature of any material, its elastic properties play a fundamental role in science and technology, permitting a description of the mechanical behavior of a material which is essential to design and experimental stress analysis. There are a wide number of methods to characterize the elasticity of materials. Several static methods have been codified due their simplicity despite the fact that they introduce serious difficulties, primarily their destructive nature, but also including other problems, such as boundary effects, sample size dependencies, difficulties in obtaining homogeneous stress strain fields and localized data, the drawbacks which make these methods less attractive. In the present work, an initial attempt was made to determine the elastic characteristics of the specimens using extensometric techniques. Aside from the disadvantages of static tests enumerated, this method introduces a series of additional difficulties due to the fact that the reading returned by the sensor represents localized conditions within the material. When examining a material which is not completely homogeneous, the location of the sensor is critical. In the case of wood, different readings are obtained when the sensor is placed over the earlywood or latewood, a problem which might be resolved by using large strain gauges such that the reading corresponds to the integration of the field deformation of the covered surface. Another problem is that the adhesive can interfere with the reading by producing a local increase in rigidity of the material. Finally, when the specimen is placed under extreme conditions of temperature and humidity, it is difficult to guarantee the perfect adherence of the gauge to the surface of the specimen. Due to all of these drawbacks, the technique of extensometry does not seem to be the best option for the determination of the elastic properties of wooden specimens under different moisture contents. Given the problem of direct measurement of the elastic properties, an indirect method that allows the measurement of experimental related parameters is useful to derive the unknown properties. A large number of techniques have been developed for the experimental study in the elasticity of wood [46], among them being resonance methods, which are attractive from both the experimental and numerical point of view. These methods are based on a relationship between an analytical
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expression describing the vibration behavior of the specimens and experimental results [28]. However, due to the complex nature of the material in combination with problem of boundary conditions, the applicability of this technique is limited. As discussed in the following sections, since no closed form analytical solution exists for the partial differential equation governing the free transverse vibration of a rectangular plate [30], the problem must be addressed by numerical approximations. Finally, the inverse problem of estimating the in-plane elastic constants is reduced to minimizing the difference between the numerical and the experimental response through an iterative process. The present work provides an experimental-numerical investigation on the effects of moisture content on the in-plane elastic constants of wood for the specific use of the construction of soundboards of musical instruments. This chapter has been organized with the intent of giving the reader a structured and comprehensive knowledge of the problem treated. After a brief overview of the state of the art, Sect. 2.3 discusses wood’s orthotropic nature and exposes the constitutive model used in the study. Section 2.4 describes the effects of moisture content in wood and the specimens used, as well as the procedures used to induce humidity changes in the samples and the experimental results of the influence of such changes on the geometry and mass parameters. In Sect. 2.5, the vibrational behavior of a rectangular plate of spruce has been examined using vibration testing under different moisture conditions. Section 2.6 is devoted to the description of the adopted numerical model. In Sect. 2.7, the methodology which allows identifying the elastic characteristics from vibration measurements through an iterative process is presented and discussed. The results of numerical simulations and the comparison with experimental results are then discussed in Sect. 2.8. Finally, Sects. 2.9 and 2.10 contain concluding remarks and suggestions for further lines of inquiry.
2.2 Overview of the State of the Art In 1680, the experimental philosopher R. Hooke unveiled for the very first time to the eyes of the world the astonishing patterns of movement traced by a vibrating glass plate. In 1787, the scientist E.F.F. Chladni repeated the pioneering experiments of Hooke, providing one of the major 18th century experimental stimuli to 19th century continuum mechanics [6], the results of which are commonly referred to today by his name: Chladni Patterns [30]. Several centuries after this discovery, although the methods employed are rather more sophisticated than the bow and flour used by Hooke, the same phenomenon is still used today by both the instrument maker and the scientist to unravel the complex vibration behavior of plates. Even though 18th century scientists such as J.B. Biot in 1816 [6] were already using dynamic experiments to calculate the elastic moduli of isotropic materials, new approaches continue to be developed today, and in the past decades, a large number of dynamic methodologies for the elastic characterization of rectangular plates have been presented in the literature.
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In essence, the modern procedure is based on an optimization process that minimizes the difference between the experimental and the numerical dynamic response. The material’s parameters in the numerical model are iteratively updated until the numerical response approximates as closely as possible the experimental response [37]. The values which produce convergence in the response of the system are the unknown elastic characteristics. In other words, the method is basically an optimal curve fitting of vibration test data to equations expressing the dynamics of the test specimen [4]. These methods have in common the requirement of the measurement of a limited number of natural frequencies of the plate, usually with free boundary conditions; the problem is then to relate the mode shapes, natural frequencies and deflection fields to the unknown parameters, since unfortunately there is no closed form analytical solution for the eigenvalue problem in the case of a rectangular plate with free boundary conditions, a difficulty already mentioned in Rayleigh’s classical book Theory of Sound [39]. In the first approximation, classical analytical methods have been used to determine exact analytical solutions of the governing differential equations for a rectangular plate simply supported or having two opposite edges simply supported with free conditions at the other edges. However, for other combinations of boundary conditions, the solutions are much more complex [23, 30]. In order to provide alternative solutions, many researchers have resorted to approximate analytical methods. Rayleigh’s method [3, 12, 33], the Rayleigh-Ritz technique [9, 43] and numerical approaches, e.g. Finite Element Method [1, 2, 15, 16, 25, 35], are the three principal methods1 usually employed. At the beginning of the 1980’s, Sol [43] developed a method for the identification of the elastic moduli of thin orthotropic plates, based on the comparison of numerically calculated resonant frequencies of a thin plate specimen with corresponding experimental data which allows for the simultaneous identification of the four independent in-plane elastic constants. Deobald and Gibson [9] used a Rayleigh-Ritz technique in order to model the vibrational behavior of an orthotropic rectangular plate with clamped and free boundary conditions. Starting from the natural frequencies of the plates so obtained, they determined the four in-plane elastic properties. Ayorinde and Gibson [3] developed a method using the classical lamination theory and an optimized three-mode Rayleigh’s formulation to determine elastic coefficients of orthotropic plates with free boundary conditions. McIntyre and Woodhouse [33] presented a similar approach including the determination of both damping and elastic constants. Similar results were obtained for the determination of stiffness and damping properties of orthotropic composites plates by De Visscher et al. [10] using a numerical model of the specimen in combination with the modal strain energy method. Mota Soares et al. [35] obtained elastic characteristics by solving an optimization problem of an error functional expressing the difference 1 In
[24], the reader can find a comparative overview and a discussion of the advantages and disadvantages of these principal methods for calculating the eigenfrequencies and eigenmodes based on the classical thin plate theory.
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between measured higher frequencies of a plate specimen and the corresponding numerical ones. Grédiac et al. [15, 16] proposed a method allowing the direct determination of the flexural stiffness from natural frequencies which does not require an initial estimate parameter, based on a set of relevant weighting functions associated with different natural modes of vibration. More recently, Lauwagie [29] has developed a vibration-based identification technique to determine the elastic properties of the constituent layers of layered materials. The majority of cited references focus on the study of samples of aluminium or composite materials, thereby extending the methodology to anisotropic materials. However, an extensive research regarding wood’s elastic characterization has also been carried out. From a perspective in which wood is treated as a structural material, other authors [28, 36] have offered previous solutions to the problem of the elastic characterization of wood by vibration testing. However they have primarily dealt with samples of large physical dimensions under constant humidity, although moisture content has a significant influence on elastic properties, as shown in [20, 21]. From a musical instrument maker’s perspective, in which wood takes precedence over other materials due to its acoustic properties [46, 47], there have also been several discussions of the orthotropic properties of wood, the possible ways to characterize them and their importance in instrument making [22, 34]. For instance, Rodgers [41] concluded, based solely on a finite elements analysis, that only three (longitudinal and radial Young’s Moduli and shear modulus) of nine elastic properties were of the first order importance in violin wood selection and analysis. Haines [17] presented an extensive and complete characterization of the musically important properties of wood for instruments, concluding that the ratio between longitudinal and radial Young’s Moduli strongly determines the vibrational mode shapes of plates. However, his tests were conducted under closely controlled temperature and constant humidity conditions. In the second part of his work [18], the changes in density and stiffness due to the effects of moisture change were quantified, concluding that both elastic moduli are reduced while the density increases with increasing moisture content. However, the shear modulus was not measured and only two moisture contents were tested. Thompson [44] measured the frequency response of back and front violin plates under six different relative humidity levels, between 15% and 79%, concluding that changes in frequency are significant in a way that decreases as the relative humidity increases; however, in this study, only the second and fifth modes were measured. According to the authors’ knowledge, this topic has hitherto not been fully studied. Advances in experimental techniques and the versatility and accessibility of numerical methods now makes it possible to perform further study.
2.3 Orthotropic Nature of Wood Properties Wood is a complicated composite of hard-celled cellulose microfibrils (organic cells known as tracheids) embedded in a lignin and hemicellulose resin matrix. The seasonal variation in the cell wall density of a tree is evident when looking at the end
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Fig. 2.1 The principal axes useful for modelling wood as an orthotropic material. The longitudinal axis L is parallel to the cylindrical trunk and the tangential axis T is perpendicular to the long grain and tangential to the annual growth rings
of the cut trunk, where a concentric ring structure formed by the walls of the long slender tracheids can be observed. Commonly referred to as growth rings, this architecture composed of alternating layers of earlywood (formed in the spring and summer) and latewood (formed at the end of the growing season) is responsible for wood’s high anisotropic and viscoelastic behavior [46]. Wood may be described as an orthotropic material because its mechanical properties are independent and can be defined in three perpendicular axes (see Fig. 2.1). The longitudinal axis L is parallel to the cylindrical trunk of the tree and therefore to the long axis of the wood fibres as well (parallel to the grain). The tangential axis T is perpendicular to the long grain and tangential to the annual growth rings. Both the tangential and radial directions are referred to as being perpendicular to the grain. Taking the tree trunk as a series of concentric cylindrical shells and cutting thin radial slices, the growth ring curvature is negligible and occurs in straight parallel lines orthogonal to both the longitudinal and the tangential axis. In the case where the long axis is parallel to the grain fibre orientation and the width is in the radial direction, the piece is said to be quarter-sawn. The wood used in soundboards is almost always of quarter-sawn timber, which causes the speed of sound to be higher and the values of damping to be lower than for wood cut at an angle to the grain [22, 46]. In general, the mechanical properties vary the most between the longitudinal grain and the other two radial and tangential directions. In spite of the fact that wood is a viscoelastic material, viscoelastic phenomena do not appear relevant in the present work because of the small strains and the nature of dynamic problem. Therefore, in the current study, linear elasticity can be assumed as an hypothesis. When linear elastic behavior is assumed, the generalized Hooke’s law gives the stress-strain relation, which can be written in matrix form as {σ } = [D]{ε},
(2.1)
{ε} = [S]{σ },
(2.2)
or where [S] = [D]−1 , {σ } are the stress components, [D] the elastic matrix, [S] the compliances matrix, and {ε} the strain components [40]. Since {σ } and {ε} are elements of R6 , there are 36 components in both the stiffness and compliance matrices, but these reduce to 21 because of the symmetry of the stresses and small strains tensor [26]. When three mutually orthogonal planes of material symmetry exist and
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the coordinate system employed is aligned with the principal material directions, the number of elastic coefficients is reduced to 9, and such material is called orthotropic. Then, the stress-strain relation takes the form ⎧ ⎫ ⎡ ⎫ ⎤⎧ ε1 ⎪ 0 0 0 σ1 ⎪ D11 D12 D13 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ D12 D22 D23 σ2 ⎪ 0 0 0 ⎥ ε2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎢ ⎨ ⎨ ⎬ ⎢ ⎬ σ3 0 0 0 ⎥ ε3 D13 D23 D33 ⎥ ⎢ =⎢ . (2.3) 0 0 D44 τ23 ⎪ 0 0 ⎥ γ23 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎢ 0 ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 0 D55 τ13 ⎪ 0 ⎦⎪ γ13 ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎭ ⎭ τ12 γ12 0 0 0 0 0 D66 An important feature of orthotropic materials that can be observed in Eq. (2.3) is the uncoupled behavior between the normal stresses and the shear strains, the shear stresses and normal strains, and the shear stresses and shear strains. For plate-like structures a state of generalized plane stress can be considered, which implies that all three transverse stress components are negligible, σ3 = 0,
τ23 = 0,
τ13 = 0.
(2.4)
Furthermore the transverse normal strain, ε3 , is small enough to be ignored. Therefore, for an orthotropic plate material with principal materials axes (x1 , x2 , x3 ) coinciding with the plate coordinates (x, y, z), i.e. the local and global axes coincide, the plane stress-strain reduced constitutive equations can be expressed as ⎫ ⎡ ⎫ ⎧ ⎤⎧ Q11 Q12 0 ⎨ εx ⎬ ⎨ σx ⎬ σy = ⎣ Q12 Q22 0 ⎦ εy , (2.5) ⎭ ⎭ ⎩ ⎩ τxy γxy 0 0 Q66 where Qij are the plane strain-reduced stiffnesses. Engineering constants, such as Young’s, shear moduli and Poisson’s ratio, are used instead of the stiffness coefficients due to its direct and obvious physical meaning. Hence the stiffness coefficients are related to the engineering constants as Ex , 1 − νxy νyx Ey Q22 = , 1 − νxy νyx Q11 =
Q12 =
νxy Ey νyx Ex = , 1 − νxy νyx 1 − νxy νyx
Q66 = Gxy .
(2.6)
The so-called thermodynamic constraints are based on the principle that the sum of the work done by all stresses must be positive in order to avoid the creation of energy. Formally, it can be proven that the matrices relating stresses to strains must be positive-definite [26], i.e., the diagonal elements must be positive. This mathematical condition applied to Eq. (2.5) implies that Q11 ,
Q22 ,
Q66 > 0,
(2.7)
Gxy > 0
(2.8)
or in terms of the engineering constants, Ex , whereupon from Eq. (2.6)
Ey ,
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(1 − νxy νyx ) > 0.
(2.9)
Using the compliance symmetry condition νyx νxy = , Ex Ey the inequality in Eq. (2.9) can be rewritten as Ey Ex |νxy | < or |νyx | < . Ey Ex
(2.10)
(2.11)
In the specific case of wood, the axes are given a special nomenclature, as has been mentioned above, defining a longitudinal and radial direction in reference to the manner in which the specimen has been cut from the original tree trunk. In this manner, the engineering constants are related to the stiffness coefficients in the following way: ⎤⎧ νLR ER ⎧ ⎫ ⎡ EL ⎫ 0 1−νLR νRL 1−νLR νRL ⎨ εL ⎬ ⎨ σL ⎬ ⎥ ⎢ νRL ER ER σR = ⎣ 1−ν ν (2.12) 0 ⎦ εR . 1−νLR νRL LR RL ⎩ ⎩ ⎭ ⎭ τLR γLR 0 0 GLR As can be seen, the elastic behavior of an orthotropic plate-like structures can be described by only four independent engineering constants, i.e. EL , ER , νLR and GLR .
2.4 Influence of Moisture Changes on Wood Wood is a hygroscopic material which shows coupling between the moisture transfer phenomena and mechanical behavior at two levels. Moisture changes induce swelling or shrinkage in the material, the so-called hygroexpansive effect; after the initial drying process or seasoning, wood continues to interact with the moisture in the ambient atmosphere in order to reach a state of equilibrium, a process which significantly alters the proportion of its total mass which consists of water [46]. Its geometry and elastic characteristics are also altered by these changes in moisture content, and, as shall be demonstrated, its vibrational behavior as well. Furthermore, with a mechanical load acting under moist conditions, an additional deformation is induced [21], known as mechano-sorptive creep. Most constitutive models consider that the total strain, ε, is assumed to consist of additive strain terms, ε = εe + εu + εve + εms ,
(2.13)
where ε is the total strain, εe the elastic, εu the hygroexpansion (swelling and shrinkage), εve the viscoelastic and εms the mechano-sorptive. The strain terms are treated
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by associating a separate differential equation to each one.2 Due to the application being studied in this work, the effects of the combined action of simultaneous moisture changes with mechanical loading has not been examined. Furthermore, as has been mentioned above, strain rates are so low that the linear elastic hypothesis can be taken as valid, and therefore the viscoelastic effects that occur in higher strain rates can be neglected. The moisture content β is defined as the mass fraction of free water in the wood,3 m0 − mdry β= (2.14) mdry where m0 is the initial mass and mdry is the mass of the specimen with all free water removed. While varnishes and other finishes may retard the movement of water vapour between the atmosphere and the wood, they cannot stop it completely. In any event, the interior surfaces of musical instruments are almost never finished in any way whatsoever, and the ubiquitous presence of acoustic venting holes in the resonating cavity assures that a rapid equalization of internal and external atmospheric humidity is inevitable. For the constructor of a musical instrument (or for that matter, any large complex wooden object), the more subtle variations of mass and elasticity are far outweighed by the dimensional variation in the radial and tangential directions (wood being essentially stable longitudinally). The larger plate-like objects of many types of musical instruments, including not only the soundboard but also the walls and bottom of resonating cavities, are almost inevitably attached to other structural elements whose longitudinal grain orientation runs in a perpendicular or near-perpendicular orientation relative to the longitudinal grain of the plate itself, elements such as ribbing or barring of soundboards and major structural supports of case walls and bottoms. The fact that the plate-like elements are constrained precisely in the orientation which is dimensionally variable is an inescapable reality which the builder ignores at his own peril. In the best of cases, whenever the ambient humidity is other than that at which the instrument was assembled, the plate will be deformed by the dimensional differences between the restrained and free sides, the free side becoming convex under wet conditions and concave under dry conditions. In the worst scenarios, under extremes of humidity, the tension or pressure applied to the material can become large enough that permanent damage results, either by splitting the wood apart or causing the cell structure to become permanently crushed, a phenomenon known as compression set. Wood which has suffered compression set will thereafter be even more susceptible to splitting under dry conditions, as its effective dimension has already been reduced at a cellular level before any excessive drying reduces it further. 2 More
background information on moisture induced eigen-stresses in timber can be found in, e.g., [20] and an informative and comprehensive review of constitutive models is presented by Hanhijärvi [21].
3 Although there an explicit relation between relative humidity and moisture content [45], these parameters should not be confused.
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Fig. 2.2 Distortion caused by changes in humidity when a wooden plate is restrained on one side only. Both pieces of spruce (Picea Abies) were cut sequentially from the same plank. The width of the samples is approximately 14 cm
Figure 2.2 illustrates the distortion which can be caused by changes in humidity when a wooden plate is restrained on one side only. Both pieces of spruce (Picea Abies) were cut sequentially from the same plank. The single rib was attached to the upper sample after it had reached saturation content using the methods described in Sect. 2.5 below, and to the lower sample after it had been dried to 0% moisture content. The photo was taken after leaving the samples in a stable ambient humidity of 55% for several days. A soundboard is usually considered to be the structural element which most critically affects the acoustic behavior of the instrument as a whole. In the present work, the behavior of a rectangular plate of spruce (Picea Abies), which is the type of wood most commonly used for the soundboards of guitars, violins, pianos, and other string instruments, has been addressed using modal analysis under different conditions of humidity. The wood specimens examined in this work are of quarter-sawn timber, and the dimensions in the growth ring plane are small enough (≤2 mm) so that the properties are essentially orthotropic. The wooden specimens had a size of 432 × 189 × 5.1 mm and a density of 470 kg/m3 under normal conditions.4 At this point it is important to justify why only two specimens have been used for this study. No two pieces of wood are exactly identical, and while it is hypothetically possible that two soundboards made from different trees of the same species might demonstrate subtly-different mechanical/acoustical behaviors, the actual limits of any such variations are greatly constrained by the traditional process of material selection, dictated by a set of requirements which include not only the type of wood, the orientation of the cut and the general width of the annular rings, but also a certain raw acoustic characteristic which is empirically determined by tapping the plank and listening to its response. Furthermore, the readily observable dimensional changes caused by moisture variations are quite consistent for a vast sampling of planks of the same species, as is demonstrated by the high degree of correlation between the values which can be found in tables of expansion/contraction rates presented in any 4 As
shown below, uncertainty in the thickness measurement may have major effects on the estimation of elastic properties. To measure the thickness across the whole surface, the authors recommend the use of an accurate caliper thickness with resolution of 0.01 mm. Moreover, the length and width of the specimen should not be the same, in order to avoid the degeneration of the modes (2 0) and (0 2) into ring and x-modes [33], which would complicate the initial estimation of the elastic properties according to the methodology described below.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
31
Table 2.1 Results of the influence of moisture content β on the geometric parameters and mass. Units in mm, g and kg/m3 , respectively β (%)
0.11
1.43
6.01
9.38
15.73
17.22
18.82
20.37
21.80
24.71
Lengtha
431.5
431.5
431.5
431.5
431.5
432.0
432.0
432.0
432.0
432.0
Widthb
186.5
187.0
188.5
189.0
191.0
191.6
191.5
192.0
192.6
192.5
Thicknessc
4.90
4.91
4.97
5.02
5.13
5.15
5.18
5.20
5.25
5.32
Massd
175.0
177.3
185.3
191.2
202.3
204.9
207.7
210.4
212.9
218.0
Density
443.8
447.5
458.4
467.0
478.5
481.6
484.7
487.8
488.3
492.8
All test were performed at room temperature 23 ± 2 °C at 38 ± 2% relative humidity a Tolerance: ±0.5 mm; b ±0.1 mm; c ±0.01 mm; d ±0.1 g
number of common technical manuals dealing with wood [45]. Therefore, while the natural inconsistencies of the material might seem to imply a significant factor of uncertainty to those with little first-hand experience with the material and/or little or no knowledge of the craft of instrument making, a level of uncertainty which under other circumstances might require a much larger data set, in regards to the specific application under consideration it is highly unlikely that a study involving a greater number of relevant samples would produce results markedly different from those obtained from the two upon which present work is based. To measure the dry mass, the plates were subjecting to a standard oven-drying process, in which they were kept at a temperature of 100 ± 2 °C, the mass of each specimen being checked periodically until no further reduction of mass was detected. The results obtained were 174.7 and 174.9 g. Care was taken to perform the weighing of the sample in ambient conditions as quickly as possible, since reabsorption of moisture can occur very rapidly at low content levels. In order to increase the moisture content, the plates were suspended just above a tray filled with water without allowing them to come into actual contact with the fluid, the tray and plate then being covered with plastic film and left for a period of ten days, after which no further increase of mass was detected. To simulate the behavior of the specimen it was necessary to estimate which characteristics had changed with respect to the normal conditions of temperature and humidity. Table 2.1 shows the data on the geometry, temperature, relative humidity and mass of the specimen taken during the experiments. There was a significant variation in density with values ranging between 443.8 and 492.8 kg/m3 . The length of the plate along the grain direction varied by less than 0.5 mm, which supports the hypothesis that wood is essentially stable longitudinally. However, in thickness and across the grain, differences reached 0.42 and 6 mm respectively under extreme conditions, which corresponds to a variation of ±4% and ±1.6% of the nominal value, respectively. The reader may be surprised that the tests under the driest conditions were performed with some moisture content. This was caused by the fact that the recommended reversible adhesive to ensure proper coupling between the accelerometer and the specimen is common beeswax. However, the temperature of the specimen remains high enough to melt the wax adhesive for some time after removal from
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M.A. Pérez Martínez et al.
the drying oven, and therefore it is necessary to allow the specimen to cool for a short period of time, during which some reabsorption of moisture is unavoidable. Alternative solutions might be the use of adhesives whose properties are maintained at higher temperatures such as cyanoacrylate, but such adhesives are not reversible; perhaps the best solution would be to use a laser-based vibration transducer. Unlike traditional contact vibration transducers, laser vibrometers require no physical contact with the test object.
2.5 Experimental Modal Analysis of Wooden Specimens Experimental modal analysis is the process of determining the modal parameters (frequency, damping and mode shapes) of a linear time-invariant system.5 It is generally based upon a theoretical relationship between measured quantities, the socalled Frequency Response Function (FRF), and classical vibration theory represented as matrix differential equations of motion [19]. A system can be described by its equation of motion, which general mathematical representation is expressed by [M] x(t) ¨ + [C] x(t) ˙ + [K] x(t) = f (t) , (2.15) where [M] is the mass matrix, [C] the damping matrix and [K] the stiffness matrix, describing the spatial properties of the system, {x(t)}, ¨ {x(t)} ˙ and {x(t)} are vectors of time-varying acceleration, velocity and displacement response, respectively, and {f (t)} is a vector of the time-varying external excitation forces. By taking the Fourier transform, the differential equation of motion, Eq. (2.15), can be written as (2.16) −[M]ω2 + j[C]ω + [K] X(ω) = F (ω) , where ω is the frequency variable, j the imaginary j-operator, {X(ω)} the output vibration vector, and {F (ω)} the input force vector. Grouping the terms on the left, one obtains B(ω) X(ω) = F (ω) , (2.17) where [B(ω)] is the system response matrix. This equivalent representation has the advantage of converting a differential equation to an algebraic equation in the frequency domain (magnitude and phase). The system transfer function [H (ω)], or FRF, is the inverse of the system response matrix. The FRF relates the Fourier transform of the system input to the Fourier transform of the system response. That is, −1 {X(ω)} B(ω) = H (ω) = . {F (ω)} 5 The
(2.18)
theory of experimental modal analysis is presented in a shortened, comprehensive form. The interested reader is referred to [11, 19, 32] where more details are given.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
33
The modal parameters such as frequency, damping and mode shapes are finally extracted from measured input-output data [H (ω)], using numerical techniques, e.g. curve-fitting algorithms, which separate the contributions of individual modes of vibration in measurements [11, 19, 32]. An important feature is that [B(ω)] and [H (ω)] are symmetric since [M], [C] and [K] are symmetric. This implies that Hpq = Hqp , which means that a force applied at degree of freedom (DoF) p cause a response at DoF q that is the same as the response at DoF p caused by the same force applied at DoF q. Is said that the structure obeys Maxwell’s reciprocity, one of the basic assumptions which underlies experimental modal analysis [19], whose practical results are obvious. The test environment involves several factors which must be taken into consideration, as well as the appropriate boundary conditions of the structure, especially if the vibration behavior is to be subsequently compared with numerical results. There are many options ranging from completely free to completely clamped edges, simply supported or combinations of them all. From an experimental point of view, arriving at a completely clamped boundary condition is extremely difficult. This detail was also stated in [1, 9], who concluded that it is better to perform the tests with free boundary conditions. Another option is to perform the test with simple supported boundaries. The problem found is that when the specimen is small, as force is applied with the hammer, the specimen itself rebounds and thereby looses contact with the supporting surfaces. To reproduce free boundary conditions in the laboratory several options exist: either the plate can be situated on an undulated foam surface, or it can be suspended either horizontally or vertically. For the present work, the various situations were tested and the results indicated a minimal difference; therefore, a vertical suspension was ultimately employed. Among different testing configurations, tests were performed with a singlereference testing employing a roving hammer test. To measure a FRF of a structure basically two channels are needed: one channel is used to measure the excitation force and the other one to measure acceleration response of the plate. A stationary monoaxial accelerometer has been attached to a single DoF reference point whereas the hammer roved around exciting the specimen at well distributed measurement DoFs.6 The experiments were conducted using a Brüel & Kjaer type 4518-003 uniaxial accelerometer, whose mass is less than 1.5 g, along with a miniature transducer hammer Brüel & Kjaer model 8204 for the excitation of the system. Due to the accelerometer being attached to the test specimen, there is a certain amount of mass which is added to the structure. It is important to assure that the mass of the transducer is negligible when compared with the effective modal mass of a mode of vibration so as not to interfere with the vibrational behavior.7 To determine the mass threshold above which the presence of the transducer interferes with the vibration 6 This 7 It
is also called Single-Input Single-Output (SISO) testing.
is generally accepted that a ratio of less than 1:10 is required [11], in this study the ratio was approximately 1:130.
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M.A. Pérez Martínez et al.
Fig. 2.3 Assembly made for modal testing. Both the applied excitation and the measured response were perpendicular to the plate. To reproduce free boundary conditions a vertical suspension was employed
behavior of the plate, a simple experiment was conducted by gradually adding portions of plasticine modeling clay to a point on the other side of the plate directly opposite the accelerometer, examining the frequencies and vibration modes at each stage. The results showed that there are no variations in the behavior until the added mass reaches 14.3 g, which corresponds with the mass of the plate by a ratio of 1:13. The problem may come when smaller specimens need to be tested and the total added mass becomes significant. In these circumstances, it is advisable to perform an operation known as mass cancellation [32] or, as already discussed above, to use laser vibrometers. Both the applied excitation and the measured response were perpendicular to the plate. The range of frequencies studied was between 0 and 1000 Hz with a resolution of 0.25 Hz. The data acquisition system was a Brüel & Kjaer model 3050-B-6/0. An analysis of the measurements was performed using Brüel & Kjaer PULSE v.13, and for the estimation parameters, the modal and structural analysis software ME’Scope VES™ 7754 was employed. Figure 2.3 shows the measurement set-up for modal testing. In order to have sufficient spatial distribution for comparison to a finite element model it is necessary to have a reasonable number of well distributed measurement DoFs. While a larger number of finite matrix measurement points results in a higher description for the mode shape, because of time and cost constraints, the number of measured points was restricted to one hundred FRF’s on each plate.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
35
Fig. 2.4 Experimental modes results with variations in moisture content. The maximum variation in moisture content which could be obtained was between 0% a 25%, the latter of which corresponds with saturation. The nomenclature (m, n) identifies the different mode types, where m and n refer to the number of nodal lines parallel to the y direction and x direction, respectively
The selection of the reference location is a critical step of performing an experimental modal test. If the accelerometer location is in a nodal position of a mode, FRF’s may not contain strong response for this mode and consequently would be poorly represented or, in the worst case, completely absent. In regular geometries, as is the case of the specimens studied, it is possible to deduce that the nodal lines of the first vibration modes correspond to the symmetry axes or regular subdivision surface. However, with irregular geometry, this task is not trivial. Prior knowledge obtained from pre-testing or a finite element model is very beneficial to accomplish this.8 Figure 2.4 shows the evolution of the frequencies of the first ten modes. Both graphics illustrate the same data, though the right graphic uses the proportional unit of cents in order to better illustrate the musical significance of the variations,9 since luthiers often tune the resonances of plates to certain notes or musical intervals according to traditional precepts. Clearly the frequencies do not decrease linearly with 8 If
obtaining a large number of modes is desired, the option of taking the test in several stages should be considered, with the position of the accelerometer varied for each test. The final result is obtained by the superposition of all FRF curves.
9 The
cent is a logarithmic unit of measure used for musical intervals which correspond to a onehundredth of an equal-tempered semitone. The formula n = 1200 log2 (f1 /f2 ) returns the number of cents measuring the frequency interval between f1 and f2 .
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M.A. Pérez Martínez et al.
increasing moisture content. This fact is empirically supported by luthiers who note that the timbre of a completed instrument undergoes subtle changes with variations of relative humidity, often sounding noticeably brighter with low percentages of moisture. From a practical point of view, the behavior of wood is moisture-content dependent, and thus the measurements made at the two different times are inevitably inconsistent, an unfortunately unavoidable violation of another of the basic assumptions of time invariance which underlies experimental modal analysis [19]. Considering the rapid reabsorption rate at such low content levels, it is very important to assure that a constant moisture content is maintained during the tests. Given the number of discrete points used to describe the specimen, it is practically impossible to realize the test in only one step without the moisture content changing. One possible solution would be to perform the test in chambers under controlled humidity conditions, which unfortunately would increase the costs prohibitively. As an alternative solution, the test were performed in sequential steps; after drying the specimen in an oven, its mass was determined and the procedure was begun of acquiring the data at the initial points of the mesh, controlling the mass at regular intervals. When variations of mass were detected, which often occurred within less than a minute, the specimen was returned to the oven to reduce the moisture content yet again, a process which was repeated until data from all points of the mesh was acquired. The authors’ experience has demonstrated that for the range of moisture content from completely dry to saturation, the order of the modal shapes did not appear to vary10 ; therefore, the numerical and experimental modes can be compared sequentially, the number of points can be reduced drastically and, as a consequence, the amount of time required for the test. Unlike of the finite element approach, where adding additional nodes and elements generally has a direct effect on the results, in a vibration test, a particular frequency response measurement has no relationship to other measurements. However this practice presents a clear drawback: the drastic reduction in the number of DoFs results in poor spatial distribution and consequently the mode shapes would be poorly represented. This fact makes it difficult to make a comparison between the numerical and experimental mode shapes using correlation tools such as the Modal Assurance Criterion (MAC) [32]. Even so, MAC is only an indicator of the vector correlation, i.e. mode shapes, and can indicate only consistency, not validity, but in this application the vector correlation may not be as critical and the frequency correlation is more important.
2.6 Numerical Model of Wooden Plate Among the wide number of theories that govern the deformation of plates, the wellknown classical plate theory (CPT), or Kirchhoff’s plate theory, has been chosen 10 Except
modes (3 1) and (0 3) which swap positions at a point close to saturation (see Fig. 2.4 and Table 2.3). However this does not become a critical issue since these modes are not strictly involved in the elastic properties estimation.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
37
since the wood specimens studied satisfy the thin plate condition11 in the sense that the thickness h is small compared to the characteristic lengths and thickness is either uniform or varies slowly so that three-dimensional stress effects are ignored [13]; therefore, CPT is a more suitable, straightforward and useful model. The partial differential equation governing the free transverse vibration of a rectangular plate is given by ∂ 2w = 0, (2.19) ∂t 2 where w is transverse deflection, ∇ 4 = ∇ 2 ∇ 2 is the biharmonic differential operator, D the flexural rigidity which is a function of Young’s modulus, Poisson’s ratio and plate thickness, ρ is mass density per unit area and t represents time [30]. Although a closed form analytical solution exists for the eigenvalue problem of predicting the natural frequencies of a rectangular plate, this tend to be restricted to regular geometries and simple boundary conditions such a simply supported, clamped edges or any combinations of them. Even the free edges can also be taken into account when the other pair of opposite edges is simply supported or clamped [30]. Unfortunately there is no closed form analytical solution for the case of completely free edges. When the problems do not yield to analytical treatment an alternative practice is needed, e.g. numerical simulation.12 D∇ 4 w + ρ
2.6.1 The Finite Element Method The Finite Element Method (FEM) is a procedure for the numerical solution to the mathematical equations governing a problem, and recently has become the dominant discretization technique in structural mechanics.13 The basic concept in the physical FEM is the subdivision of the mathematical model into non-overlapping domains of simple geometry termed finite elements. The response of the mathematical model is then considered to be approximated by that of the discrete model obtained by connecting or assembling the collection of all elements [13]. The matrix equations of motion, Eq. (2.15), describing both individual element and general system models can be derived from the minimum principles, i.e. Hamilton’s principle, stated as t1 δ (T + W ) dt = 0, (2.20) t0 11 The
CPT overpredicts frequencies of plates with side-to-thickness ratios of the order of 20 or less, i.e. thick plates [31, 40]. The side-to-thickness ratio for the specimens studied, as can be seen in Sect. 2.4, is about 40. 12 References [14, 27] may be helpful in order to implement a finite element model to solve the problem of simulation of vibration of a plate. Also, most available FEM analysis packages, e.g. ANSYS, ABAQUS or NASTRAN, among others, are appropriate and useful. 13 The basic concepts, procedures and whole formulations can also be found in many existing textbooks. The interested reader is referred to, e.g. [5, 13], where more details are given.
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M.A. Pérez Martínez et al.
where T is the system kinetic energy, W the work done by internal and external forces, δ the variational operator and [t0 , t1 ] the interval of time. For a free undamped14 vibration system, dissipative interior work is neglected and the work performed by externally applied forces is zero; thus Eq. (2.20) becomes t1 (T − U ) dt = 0, (2.21) δ t0
where U is the conservative interior elastic potential energy, also called strain energy. Using an assumed displacement field u(x, y, z, t) written in terms of the nodal DoFs {ue } by means of the so-called element shape functions [N], the kinetic and the strain energy of each element can be expressed in the form of the element mass and stiffness matrices. The strain field {ε} within the element volume is related to the assumed displacement by {ε} = ∂u(x, y, z, t) = {∂}[N] ue = [B] ue , (2.22) where [N] is the shape function matrix of the element e, {ue } the vector of the nodal DoFs, {∂} the differential operator matrix and [B] the element strain matrix. The stress field {σ } within the element volume is expressed as {σ } = [D]{ε} = [D][B] ue , (2.23) where [D] is the elastic matrix used in Eq. (2.1). Using the above expressions, the strain and kinetic energies in the element e are given by15 e T T 1 1 Ue = {ε}T {σ } dV = u [B] [D][B] ue dV 2 Ve 2 Ve 1 e T e e K u , (2.24) = u 2 and e T 1 1 Te = ρ{u} ˙ T {u} ˙ dV = u˙ ρ[N ]T [N] u˙ e dV 2 Ve 2 Ve 1 e T e e = u˙ M u˙ , (2.25) 2 respectively, where V denotes volume, [K e ] is the element stiffness matrix, ρ the ˙ y, z, t) = [N ]{u˙ e }, at a point in the material density, {u˙ e } the velocity, so that u(x, e element e and [M ] the element mass matrix. In order to obtain the complete finite element system model, an assembly process has to be carried out, which consists in the allocation of the individual element’s 14 Since the numerical differences between undamped and damped natural frequencies are not significant, the damping is neglected as it causes problems in eigenvalue calculations. 15 The
superscript { }T and [ ]T denotes the transpose of a vector and matrix, respectively.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
39
contribution to the system stiffness and mass matrices [5, 13, 14]. Substituting the expressions for the complete system energies into Hamilton’s principle, and conducting the routine variation operation [40], results in [M] x(t) ¨ + [K] x(t) = {0}, (2.26) which is the system differential vibration equation from the finite element discretization, called free undamped equation of motion, where [K] is the global stiffness matrix, [M] the global mass matrix and {x(t)} ¨ and {x(t)} are vectors of time-varying acceleration and displacement response, respectively [19]. Assuming a harmonic motion, the natural vibration frequencies and the associated mode shapes of a homogeneous linear undamped system can be obtained by solving the generalized matrix eigenvalue problem,16 defined as [ ][M][ ] = [K][ ] ⇒ [K] − [ ][M] [ ] = 0, (2.27) where [K] and [M] are the global stiffness and mass matrices, respectively, [ ] is a diagonal matrix listing the corresponding eigenvalues (natural vibration frequencies) and [ ] is the eigenvector matrix (mode shapes).
2.6.2 Free Vibrations of Kirchhoff Plates Kirchhoff’s plate theory provides a theoretical model of plate behavior by assuming that normals to the undeformed midplane remains straight and perpendicular to the deformed midplane (implying that the transverse shear strain is neglected), that only the transverse displacement w is considered and that transverse normal stress σz is negligible. The first and second hypothesis allow the definition of the displacement field through the thickness of the plate and the third hypothesis affects the stress-strain relationship. From the above assumptions, the displacement field for time-dependent deformations can be derived: ∂w(x, y, t) u(x, y, z, t) = −zθx (x, y, t) = −z , ∂x ∂w(x, y, t) , (2.28) v(x, y, z, t) = −zθy (x, y, t) = −z ∂y w(x, y, z, t) = w(x, y, t), 16 The
eigenvalue problem usually requires a huge computational effort to be solved, especially if the number of elements and consequently the number of DoFs are high. When the number of eigenvalues to be determined is around 30 (as in the present study where the number is limited by the experimental results) there are numerous subroutines readily available based on a vector iteration, i.e. subspace iteration method among others, which are well-documented in [5]. It is also highly recommended to use sparse matrices and diagonally lumped mass matrix, given its storage and computational advantages.
40
M.A. Pérez Martínez et al.
Fig. 2.5 Kinematics of thin plate deformation under the Kirchhoff’s assumptions
where t is time, u and v are the in xy-plane displacements due to the rotation of the cross section, w denotes the transverse deflection and θx and θy are the bending rotations of a transverse normal about the xz and yz planes, respectively, as shown in Fig. 2.5. The so-called motion vector {u} which contains the transverse displacement and rotations is defined by ⎧ ⎫ ⎧w⎫ ⎬ ⎨ ⎪ ⎨w⎬ ⎪ , (2.29) {u} = θx = ∂w ∂x ⎩ ⎭ ⎪ ⎭ ⎩ ∂w ⎪ θy ∂y
whose components will correspond to the DoFs of each of the nodes of the discretization. The strain field of the thin plate associated with the displacement field in Eq. (2.28) can be expressed by εx =
∂u ∂θx ∂ 2w = −z = −z 2 , ∂x ∂x ∂x
(2.30a)
εy =
∂θy ∂v ∂ 2w = −z = −z 2 , ∂y ∂y ∂y
(2.30b)
∂w = 0, ∂z
(2.30c)
∂θy ∂v ∂u ∂θx ∂ 2w + = −z + , = −2z ∂x ∂y ∂x ∂y ∂x∂y
(2.30d)
∂w ∂u + = 0, ∂x ∂z
(2.30e)
εz = γxy =
γxz =
∂w ∂v + = 0, (2.30f) ∂y ∂z where the value of the transverse shear strains γxz and γyz are due to the first hypothesis. Collecting non-zero components, one obtains ⎧ 2 ⎫ ⎫ ⎧ ∂ w ⎪ ∂θx ⎧ ⎪ ⎫ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ∂x 2 ⎪ ⎪ ⎬ ⎨ εx ⎬ ⎬ ⎨ ⎨ ∂θy ∂ 2w , (2.31) {ε} = εy = −z = −z ∂y ∂y 2 ⎪ ⎩ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ γxy ⎪ ⎭ ⎪ ⎩ ∂θy ∂θx ⎪ 2 ⎩2 ∂ w ⎭ ∂x + ∂y ∂x∂y γyz =
where {ε} is the strain vector.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
41
The stresses in a linear and orthotropic plate are computed from Hooke’s law,17 ⎤ ⎡ E νxy Ey x ⎧ ⎧ ⎫ ⎫ 0 1−ν ν 1−ν ν xy yx xy yx ⎨ σx ⎬ ⎢ ⎥ ⎨ εx ⎬ νyx Ey Ey σy = ⎢ (2.32) 0 ⎥ ⎦ ⎩ εy ⎭ = [D]{ε}. ⎩ ⎭ ⎣ 1−νxy νyx 1−νxy νyx τxy γxy 0 0 Gxy As shown, this relation coincides with Eq. (2.12) for plane stress problems.18 This coincidence is due to the hypothesis of null normal stress (σz = 0) across the thickness, common to both problems. The orthogonality hypothesis on which CPT is based only holds for thin plate thickness. As the thickness of the plate increases, normals to the undeformed midplane remain straight and unstretched in length but not necessarily normal to the deformed plane.19 This hypothesis amounts to including transverse shear strains in the classical plate theory. In these cases, the Reissner-Mindlin theory is an improved approximation of the plate’s strain [31, 40]. However, since the components of the transverse shear strains γxz and γyz are not zero, in the case of an orthotropic material, two more engineering constants appear on the stress-strain relation (Gxz and Gyz shear moduli). The aim of the work is to analyze the influence of moisture content on the in-plane elastic constants, hence the determination of two transverse shear moduli is beyond the scope of this study, due to the fact that in the thin plate condition the effects of transverse shear do not become significant. As mentioned above, the problem of determining the natural vibration frequencies and the associated mode shapes of a plate always leads to solving the eigenproblem of the linear undamped system. In order to obtain the complete finite element model of the whole plate, it is first necessary to determine the expressions of strain and kinetic energy of each element and then to place them into Hamilton’s principle, resulting in the equation of motion. The plate strain energy is computed by integrating the strain energy density over the volume, 1 1 T {ε} {σ } dV = {ε}T [D]{ε} dV , (2.33) U= 2 V 2 V and the kinetic energy due to the three velocities is computed through integration of the kinetic energy of the volume element, 1 T= 2 17 The
V
1 ρ u˙ 2 + v˙ 2 + w˙ 2 dV = 2
h 2
− h2
ρ u˙ 2 + v˙ 2 + w˙ 2 dS dz, (2.34)
S
reader is reminded that the stress-strains relation is exposed in Sect. 2.3.
18 In
Eq. (2.12), the axes are given in a wooden nomenclature, in which the subscripts L and R are equivalent to x and y, respectively.
19 In
addition, if one tries to analyze thicker specimens, it should be taken into account that the growth ring curvature may not necessarily be negligible, and the effects of such assumptions may become significant.
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M.A. Pérez Martínez et al.
Fig. 2.6 Evolution of the first ten resonant frequencies as a function of the number of DoFs. Higher order modes converge slower than the lower order modes
where ρ is the mass density per unit volume, h the thickness, S denotes the integration area and the superposed dot on a variable indicates time derivative [40]. By substituting the displacement field in Eq. (2.28) and integrating through the thickness dimension, the kinetic energy can be expressed as T=
ρh 2
h2 2 w˙ 2 + θ˙x + θ˙y2 dS, 12 S
(2.35)
where it has been preferred to express T as a function of the nodes DoFs (w, θx , θy ) instead of the transverse deflection w. Among the numerous readily available finite elements for plate bending, in this work a 4-node quadrilateral element has been implemented in a Matlab code, whose DoFs are the transverse displacement and the bending rotations of a transverse normal about the xz and yz planes. This results in a total of 12 DoFs per element. Since with a numerical approach the number of nodes generally has a direct effect on the results, before carrying out the simulations it is necessary to conduct a mesh convergence study in order to determine its effect and estimate the computational effort. Due to the fact that there is no closed form analytical solution for the case studied, it is not possible to estimate the accuracy of the numerical approach in absolute terms. In spite of this, a study of the evolution of the first ten resonant frequencies for an increasing mesh density was performed. Figure 2.6 shows the evolution of vibration modes as functions of the number of DoFs. The aspect ratio of an element has been tried to be close to one. As can be seen, higher order modes converge slower than the lower order modes. Accordingly, the mesh density was chosen so that a further mesh refinement results in a changes in frequencies, that is, of an order less than the tolerance defined in the convergence criterion (see Sect. 2.7), which corresponds to a discretization of 70 × 30 elements, equivalent to 6603 DoFs.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
43
2.6.3 Perturbation of the Equation of Motion As discussed above, modal parameters (frequency and mode shapes) are characteristics of a structure’s mass and stiffness. Considering the effect of wood moisture content as a widespread damage across the whole volume whose influence on the stiffness and density of the material is established, it can be expected that such deterioration affects the natural frequencies and associated mode shapes. Quantitative relations between damage and the resulting changes in modal parameters can be developed from a perturbation of the equation of motion in Eq. (2.26). The eigenvalue problem for free undamped vibration of a structure is governed by the generalized matrix eigenvalue problem, previously defined as [ ][M][ ] = [K][ ] ⇒ [K] − [ ][M] [ ] = 0, (2.36) where [K] and [M] are the global stiffness and mass matrices, respectively, [ ] is a diagonal matrix listing the corresponding eigenvalues (natural vibration frequencies) and [ ] is the eigenvector matrix (mode shapes). For the perturbed system, Eq. (2.36) becomes [K] + [K] − [ ] + [ ] [M] + [M] [ ] + [ ] = 0, (2.37) where [ ] and [ ] are the variation of the natural frequencies and mode shapes, respectively, induced by [K] and [M]. Given a non-zero eigenvector {φ}, from the generalized Rayleigh quotient, defined as {φ}T [K]{φ} = ω2 , R [K], [M], {φ} = {φ}T [M]{φ}
(2.38)
where ω2 is the natural vibration frequency, the effect of [K] and [M] on a particular mode can be determined [42]. Particularizing Eq. (2.38) for the ith natural frequency and associated ith mode shape, the generalized Rayleigh quotient can be expressed as ωi2 + ωi2 =
[{φi }T + {φi }T ][[K] + [K]][{φi } + {φi }] . [{φi }T + {φi }T ][[M] + [M]][{φi } + {φi }]
(2.39)
Because of the fact that R([K], [M], {φ}) reaches the stationary value when {φ} is equal to the eigenvector {φi } [42], Eq. (2.39) can be simplified as ωi2 = {φi }T [K] − [M] {φi }, (2.40) which indicates that an increase in mass induced by the increase in moisture content causes a decrease in natural frequencies and vice versa, as shown in Fig. 2.4 and represented in Sect. 2.5. On the other hand, a decrease in moisture content leads to a rise in natural frequencies, and therefore intuitively an induced increase in stiffness can be expected, as shown in the following section.
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M.A. Pérez Martínez et al.
2.7 Elastic Constants from Plate Vibration Measurements The above experiments have shown that the vibration behavior of a wooden plate is heavily dependent on moisture content (see Fig. 2.4), a fact previously noted by several researchers and commonly observed by musicians and luthiers [18, 22, 44, 46, 47]. In addition to density variations, significant changes in geometry have also been observed (see Table 2.1). When attempts were made to simulate modal behavior under different moisture contents by employing the dimensional and density variations observed in the specimens, the results did not correspond with observed reality, proving that there must be other variables which intervene in the vibration problem. From Eq. (2.36), it can be derived that the dynamic behavior is governed by the mass and stiffness matrices, hence a perturbation in these matrices will significantly influence the vibration behavior; thus, it is evident that elastic characteristics must also be affected by moisture content. Given the problem of direct measurement of the elastic properties, an indirect method that allows the measurement of experimental related parameters is useful to derive the unknown properties. Resonance methods20 are based on a relationship between an analytical expression, describing the vibration behavior of the specimens, and experimental results. The inverse problem of estimating the in-plane elastic constants is reduced to minimizing the difference between the numerical and the experimental response through an iterative process, in which the resonant frequencies obtained numerically for the initially estimated parameters are compared to the experimental resonant frequencies, followed by an update of the parameters in order to minimize the difference in the responses until convergence is achieved for a predefined error criteria. The flowchart of the procedure followed in this work is shown schematically in Fig. 2.7. Prior to beginning the iterative process, it is necessary to introduce initial estimations of the in-plane elastic moduli to the numerical model. The success of the convergence depends in part upon the correct choice of these initial constants. These variables have been estimated using the simple methodology proposed in [28], which is based principally on the assumption that the resonance modes (2 0), (0 2) and (1 1)21 are influenced principally by elastic constants EL , ER and GLR , respectively. This assumption may be demonstrated by conducting a sensitivity analysis of 20 Nevertheless, the main drawback of the resonant technique applied to samples of wood is related to the type of cut and shape of the specimen. For solid wood, it is easy to obtain plates in quartersawn, i.e. the long axis is parallel to the grain fibre orientation and the width is in the radial direction (LR) (see Fig. 2.1), or plain-sawn, i.e. the long axis is parallel to the grain fibre orientation and the width is in the tangential direction (LT). However, it is very difficult in the radial tangential plane (RT), especially when it comes to thin slices, as well as maintain the integrity of samples and ensure orthotropic behavior. 21 As
mentioned above, the nomenclature (m, n) identifies the different mode types, where m and n refer to the number of nodal lines parallel to the y direction and x direction, respectively.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
45
Fig. 2.7 The procedure, based on [28], is essentially an iterative process to minimize the difference between the numerical and the experimental response. An initial estimation of the elastic constants is required. The error function which defines the convergence criterion is based solely on frequency values. The updating of variables ensures a rapid convergence
numerical model response from the variation of input parameters, as explained here below. In Fig. 2.8, the effect upon the numerical frequencies of the ten first resonance modes caused by an increase of 15% in the reference values for the elastic moduli EL , ER , GLR and νLR has been represented. It is evident that some elastic moduli influence some mode types more predominantly than others. That is, a specific eigenmode can be more sensitive to the variation of some elastic constants than others. For example, the first mode, corresponding to torsional mode (1 1), is clearly dependent on the shear modulus GLR . Similarly the second and sixth mode, mode shapes (2 0) and (3 0) respectively, or the third and ninth mode, mode shapes (0 2)
46
M.A. Pérez Martínez et al.
Fig. 2.8 Percentage variations of the numerical frequencies of the ten first resonance modes caused by a 15% increase in the references values for the elastic moduli EL , ER , GLR and νLR
and (0 3) respectively, are essentially sensitive to the elastic moduli EL and ER , respectively. In modes in which m = 0 or n = 0, the influence is shared. Another important conclusion derived from sensitivity analysis, also discussed in [28, 41], is that the influence of the Poisson’s ratio on the numerical frequencies is minimal. Since it does not seem feasible to use the bending modes for the estimation of Poisson’s ratio, consequently this work has been limited to the identification of three of the four in-plane elastic moduli, even though the results show that this does not become a critical issue.22 The in-plane elastic moduli may be initially estimated by using the closed form eigenvalue solution for free and torsional beam, assuming B h, that is BL 2 19L4 19B 4 2 2 2 ER ≈ f(0 2) ρ 2 2 , GLR ≈ f(1 1) ρ , EL ≈ f(2 0) ρ 2 2 , 2π h 2π h h (2.41) where ρ is the density, L the plate length, B the plate width, h the thickness and the subscript (m n) denotes the mode shape. One might think that the numerical approximation, and consequently the iterative process described, is entirely dispensable if a beam model is assumed to define the behavior of the plate. However, as discussed below, a refinement in the solution is required in order to minimize the differences between the numerical and experimental response. Moreover, the model allows the determination of the influence and sensitivity of different parameters involved in the problem of plate behavior, as well as identifying any deficiencies or errors in the experimental data. 22 As
described in [28], the Poisson’s ratio can be estimated from an in-plane compression mode. To perform a vibration test in which one wishes to acquire bending and compression modes simultaneously, the use of a biaxial accelerometer and a biaxial excitation are required.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
47
Fig. 2.9 Convergence graph of the iterative process for the specimen with moisture content 6.01%. The error is completely minimized for the first three modes, since the update is done only using them
Once the numerical response for the initially estimated parameters is obtained, the eigenfrequencies associated to the mode shapes (1 1), (2 0) and (0 2) are compared with the corresponding experimental frequencies. If the convergence criterion is not achieved, parameters are updated as follows: FEM Experimental Experimental 2 f f(m, n) − f(m, n) > 0.025% ⇒ PN = PN −1 (m, n) if , FEM Experimental f(m,n) f(m,n) (2.42) where f is the numerical or experimental frequency associated to the mode (m, n), P the parameter to be update, i.e. EL , ER and GLR , N is the iteration number and the percentage corresponds to the maximum error defined to achieve convergence. It must be observed that in both the case of the error function as well as that of the update parameters function, the calculations are done for each independent mode, which means that each elastic constant depends upon and is updated based upon a single mode. Among the wide variety of error functions, most of which calculated the error by comparing several modes simultaneously, it is preferable in this case perform the modal correlation and updating the elastic constants individually, as proven by the results. This is due to the independent and nearly uncoupled behavior of the modes analyzed, as illustrated in Fig. 2.8. In Fig. 2.9 the convergence graph of one of the cases studied (moisture content 6.01%) has been represented. As can be seen, convergence is achieved after only a few iterations, as was also true in all the other cases. It should be pointed out that the error is completely minimized for the first three modes, since the update is done using them only. Despite this, as shown in the following section, the errors that occur in other frequencies for any specimen are less than 5%, which prove that the implemented routine is appropriate.
2.8 Results In the previous sections, the influence of moisture content upon wooden plate vibration behavior has been experimentally demonstrated. Moreover, procedures and
48
M.A. Pérez Martínez et al.
Table 2.2 Initial and final estimated Young’s and shear moduli (EL , ER and GLR , respectively, in MPa) percentage error and elastic ratios under different states of moisture content β β (%)
0.11
1.43
6.01
9.38
15.73
17.22
18.82
20.37
21.80
24.71
ELi
13274 14049 13254
12604
12199
10816
10857 10458 10021 9563
ELf
13102 13871 13080
12439
12065
10685
10736 10347 9919
9491
Error
1.31
1.28
1.33
1.32
1.11
1.22
1.12
1.07
1.04
0.76
ERi
920.5
947.1
963.5
913.8
745.4
711.1
671.8
635.5
595.6
512.3
ERf
887.8
923.0
938.6
890.1
725.6
692.3
653.9
618.3
579.4
497.6
Error
3.69
2.61
2.65
2.66
2.74
2.72
2.74
2.77
2.79
2.94
GLRi
927.4
996.5
1088.3 1054.5 1015.5 1024.1 969.3
908.6
859.5
721.1
GLRf
845.0
914.0
1004.8 974.9
942.7
956.1
903.7
846.1
800.4
668.1
Error
9.75
9.03
8.31
8.16
7.72
7.11
7.26
7.39
7.39
7.94
ERf /ELf
0.068
0.067
0.072
0.072
0.060
0.065
0.061
0.060
0.058
0.052
GLRf /ELf
0.064
0.066
0.077
0.078
0.078
0.089
0.084
0.082
0.081
0.070
GLRf /ERf
0.952
0.990
1.070
1.095
1.299
1.381
1.382
1.368
1.381
1.343
Subscripts i and f indicate initial and final estimation, respectively
methodologies to identify which variables are affected and quantitatively assess the magnitude of the variation due to such moisture changes have been discussed. The value of elastic moduli were determined using the exposed iterative method, solving the inverse problem. It should be noted that the rapid convergence is due in part to the proper estimation of the initial variables, involving not only the validation of assumptions, but also requiring that density variations, changes in the geometry and overall experimental frequencies have been properly determined. In Table 2.2, the initial and final estimated elastic properties, the percentage error between them and the elastic ratios for each of the states of moisture content are presented. In spite of the difference between specimens of the same species (spruce), good agreement between Ref. [45] and obtained elastic ratios have been observed. It is noteworthy that in all cases the initial estimate tends to stiffen the plate. Although errors are relatively small for the elastic moduli, the shear modulus error is almost 10%,23 demonstrating the need for refinement from the numerical model. Figure 2.10 shows the variation of the final estimated elastic characteristics EL , ER and GLR as a function of moisture content for Picea Abies specimens. Although the species studied are different, there is a clear trend that roughly corresponds with that published in [46] and a good correlation has been observed with the recently 23 To
determine the consequences that would cause the error in a modal simulation, the influence of shear modulus in the first ten resonance modes represented in the graph in Fig. 2.8 must be analyzed.
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
49
Fig. 2.10 Changes in longitudinal (left), radial and shear moduli (right) as a function of moisture content. The most significant variation is found in the radial modulus, and consequently in resonance modes associated therewith
published results in [38] for a proposed computational micromechanical model for the influence of moisture on softwood. As anticipated in Sect. 2.6, Eq. (2.40) indicated that an increase in mass induced by the increase in moisture content caused a decrease in natural frequencies and vice versa. On the other hand, a decrease in moisture content led to a rise in natural frequencies, and therefore intuitively an induced increase in stiffness could be expected as a result of the alleged elastic moduli dependence on moisture content. While this hypothesis has been shown to be true, it should be noted that the behavior is clearly not linear. Fitting a third order polynomial curve is excellent for the radial elastic modulus, but for the longitudinal and shear moduli, major discrepancies occur. Not all elastic properties are equally affected by moisture content changes, as can also be derived from the elastic ratios shown in Table 2.2; the most significant variation is found in the radial modulus, in which the difference between the maximum and minimum modulus reached approximately 88%, while for the shear and longitudinal moduli, variation is about 51% and 47%, respectively. From the illustrated results, one can conclude that the elastic moduli is reduced with increasing moisture content and vice versa; however, while the minimum value corresponds to the saturation point, the maximum value does not correspond to the driest condition. This fact can be analyzed from a micromechanical approach. The properties of the wood’s cell walls are responsible for its macromechanical behavior. As mentioned earlier, its highly anisotropic behavior is due to alternating layers of earlywood and latewood. In terms of the microscopic structure, wood is essentially composed of cellulose, hemicelluloses and lignin. Latewood has a higher
50
M.A. Pérez Martínez et al.
cellulose content, which is the main contributor to the stiffness, and a lower lignin content than earlywood. Unlike cellulose, hemicellulose is strongly influenced by moisture changes, and along with lignin, it is the primary cause of moisture sorption. At lower moisture content, the hard-celled cellulose microfibrils, known as tracheids, are stiffer and conversely they become softer at higher moisture content. The dependence of stiffness constants of hemicellulose and lignin on moisture content can be found in [7, 8, 38], in which it is observed the lignin stiffness has a clear trend that roughly corresponds with the curves in Fig. 2.10. The methodology results in numerical eigenfrequencies that fit closely with the experimental frequencies. Based on the estimated elastic properties24 displayed in Table 2.2 and taking into account the geometric and mass variations experienced by the specimen displayed in Table 2.1, wooden plate vibration behavior has been simulated under different moisture conditions. Experimental and numerical modes, the corresponding mode shapes and the relative deviation between frequencies under different moisture conditions have been presented in Table 2.3. Figure 2.11 shows a comparison between the numerical and experimental modal analysis under 6.01% moisture content conditions for the first ten mode shapes. As expected, the minimum error corresponds to the first three modes. This fact is due to the independent use of the first three eigenfrequencies for the updating of the elastic characteristics at each iteration; even so, the maximum error for the rest of the modes is below 5%. The main source of error is due to the inevitable heterogeneity of wood’s properties, such as a local dispersion and orientation of the grain, irregularities in the thickness, non-uniformity in the cross-grain direction and densities. Inaccuracies in geometrical or mass measurements are other possible sources of error. In order to quantify the influence of these inaccuracies in the eigenfrequencies obtained in the numerical model, a sensitivity analysis has been carried out by increasing the nominal values of length, width, thickness and mass by 5%. Variations in relation to a reference value of each of the resonance modes of the wooden plate analyzed are illustrated in Fig. 2.12. A decoupled behavior has been observed between the x-bending and y-bending modes with a strong relation to the length and width, respectively, as well as constant errors of 5% and −2.5% in all resonance modes, corresponding to variations in thickness and mass, respectively. It must be emphasized that these increases would result with an error in the nominal measure of approximately 10 g, 21.6 mm, 9.6 mm and 0.26 mm, respectively, all of which are well above tolerance levels (see Table 2.1). Errors in the estimation of elastic properties as a result of inaccuracies in the geometrical measurements and mass are also quantifiable. The percentage errors in estimating the elastic properties for a 5% increase in length, width, thickness and mass have been represented in Fig. 2.13. Although errors illustrated are much higher than tolerance, it is demonstrated that geometric variation is not necessarily negligible and therefore should be taken into account in order to obtain results which are as accurate as possible. 24 Since the Poisson’s ratio does not have a significant effect on bending modes as shown in the graph of sensitivity (see in Fig. 2.8), a reference value of 0.35 has taken to perform simulations [45].
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
51
Table 2.3 Experimental (Exp) and numerical (FEM) resonant modes (in Hz), corresponding mode shapes and percentage deviation between frequencies under different moisture conditions β β
Modes (1 1)
0.11%
Exp
88.0
146.7
206.8 224.8 272.8
397.1
442.3
470.4
549.2 611.5
FEM
88.0
146.7
206.8 230.2 270.3
405.7
436.6
481.5
569.3 625.7
Error
0.02a
0.00
0.00
1.43%
6.01%
9.38%
(0 2)
(2 1)
2.40
(1 2)
(3 0)
−0.92 2.16
(2 2)
(3 1)
−1.29 2.36
(0 3)
3.65
(1 3)
2.32
Exp
90.8
150.6
208.2 228.3 275.7
399.9
443.0
474.9
558.9 615.0
FEM
90.8
150.6
208.2 236.9 275.0
416.5
447.6
495.0
573.1 632.6
Error
0.02
0.00
0.00
1.03
4.22
2.54
Exp
94.2
146.3
206.7 232.9 280.6
395.6
450.2
480.1
552.5 618.3
FEM
94.2
146.3
206.7 239.5 278.2
404.9
456.1
491.0
569.0 633.1
Error
0.00
0.00
0.00
1.31
2.26
2.99
3.78
2.83
−0.27 4.15
−0.86 2.35
2.86
2.40
Exp
92.5
142.8
200.4 232.4 275.3
397.1
443.7
481.9
533.0 604.4
FEM
92.5
142.8
200.4 234.7 271.3
395.1
446.4
480.1
551.5 615.3
Error
0.00
0.00
0.00
15.73%
17.22%
18.82%
20.37%
21.80%
24.71%
(2 0)
0.98
−1.44 −0.51 0.61
−0.38 3.47
1.79
Exp
90.7
141.8
178.9 220.5 255.7
378.4
411.4
458.5
475.5 550.0
FEM
90.7
141.8
178.9 230.9 253.4
392.5
429.9
474.4
492.1 560.1
Error
0.00
0.00
0.00
4.50
3.47
3.50
4.73
−0.91 3.72
1.83
Exp
90.9
133.3
174.4 218.6 252.6
371.3
407.4
451.0
458.7 539.8
FEM
90.9
133.3
174.4 226.1 250.6
369.0
425.7
456.0
479.9 549.6
Error
−0.03 0.00
0.00
1.10
4.62
Exp
88.6
133.9
169.5 214.8 246.5
368.7
438.9
450.0 520.0
FEM
88.5
133.9
169.5 222.7 243.8
370.7
Error
−0.02 0.00
0.00
Exp
85.6
131.5
164.1 210.5 238.7
FEM
85.5
131.5
164.1 216.5 235.8
Error
−0.02 0.00
0.01
3.44
3.67
2.83
−0.81 −0.61 4.49 398.1
1.82
415.8
453.0
466.3 534.4
4.46
3.22
3.62
364.7
388.3
428.5
440.0 515.0
364.0
402.7
−1.10 0.54
−1.22 −0.18 3.70
2.77
442.4
451.4 517.1
3.25
2.58
0.40
Exp
83.9
129.9
159.9 207.4 233.6
361.8
381.4
419.3
436.8 496.5
FEM
83.9
129.9
159.9 212.8 230.4
359.6
394.6
435.8
439.7 504.5
Error
−0.02 0.00
3.94
0.66
0.00
2.58
−1.35 −0.61 3.46
1.61
Exp
77.4
128.0
149.2 196.9 217.7
353.8
363.6
412.7
393.1 461.7
FEM
77.4
128.0
149.2 201.4 213.7
354.4
367.7
420.4
409.8 469.3
Error
0.00
−0.01 0.01
1.12
1.86
4.25
a Non-zero
2.30
−1.83 0.17
errors when equal frequencies are due to rounding off decimals
1.65
52
M.A. Pérez Martínez et al.
Fig. 2.11 Comparison between the numerical and experimental modal analysis for the first ten mode shapes under 6.01% moisture content conditions. The subscripts E and S denote experimental and simulated results, respectively. Possible sources of error are wood’s heterogeneity, thickness irregularities, non-uniformity in the cross-grain direction and densities, as well as inaccuracies in geometrical or mass measurements
Finally, another source of error may be due to the curvature in the transverse direction which the specimen assumes under different moisture conditions. Although none of the sides of the plate are attached to any structural element, as mentioned in Sect. 2.3 (see Fig. 2.2), it has been observed during the tests that the plate becomes slightly curved across the grain under different moisture conditions, with less than 1 mm deflection in the radial-tangential plane, as shown schematically in Fig. 2.14. This distortion is due to the imperfect perpendicularity between the earlywood and latewood. This curved geometry has an influence on the modes in that it tends to soften the bending modes in the y-plane and stiffen them in the x-plane;
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
53
Fig. 2.12 Percentage variations of the numerical frequencies of the ten first resonance modes caused by increasing the nominal values of length, width, thickness and mass by 5% Fig. 2.13 Percentage errors in estimating the elastic properties EL , ER and GLR introduced by increasing the nominal values of length, width, thickness and mass by 5%
consequently, the experimental resonance would be slightly lower and higher, respectively. Since the deflection is minimal, the curvature of the plate mid-surface in the reference configuration can be considered zero, hence Kirchhoff’s plate theory can be used. However, it may become a critical issue for thicker specimens, or for grain orientations other than quarter-sawn.
2.9 Concluding Remarks Tradition is very strong in stringed musical instruments making. Despite scientific effort to introduce new materials such as synthetic polymers and carbon fibre composites, wood remains by far the most widely used material, due to its aesthetic appeal and unique mechanical and acoustical properties. However, its susceptibility to moisture changes and the irreversible consequences that may result therefrom
54
M.A. Pérez Martínez et al.
Fig. 2.14 Curvature adopted by the plate under moisture conditions due to the imperfect perpendicularity between the earlywood and latewood. δ denotes deflection, and L, R and T are the longitudinal, radial and tangential directions, respectively
forces the builder to focus on this aspect, becoming one of the most important considerations in the process of construction. The aim of this contribution is to illustrate the dynamic effects caused by moisture in the wood, to provide a practical tool for quantifying them, and finally to reaffirm the benefits of applying the techniques of simulation and modal analysis to the problem of furthering the understanding of the complex behavior of musical instrument. It has been shown that wood’s dynamic behavior is heavily dependent on moisture content. Wood changes dimension differently in the longitudinal, radial and tangential directions; in the latter two orientations, the increments have a near-linear relationship to moisture content until saturation point. The most significant effects were observed in the radial direction due to the major contribution of earlywood, and the strong influence of humidity on cell wall components. One can conclude that the lower the percentage of earlywood, that is, the denser or more fine-grained the wood, the less sensitive to moisture changes. The combination of numerical with experimental modal analysis is a potential alternative approach for the elastic characterization of materials. Its nondestructive nature permits direct examination of a material sample which will eventually become part of a real instrument. The use of numerical approximation to solve the plate’s vibration problem has allowed sensitivity studies of the influence on the dynamic response of the parameters involved in the problem, showing, for example, the influence of a isolated parameter in global behavior, the errors in the estimation due to geometric inaccuracies, or which of the elastic properties are of first order importance in bending modes. In addition, numerical approximation allows one to work with irregular shapes such as a real instrument’s soundboard. In this case, vector correlation tools, e.g. MAC, are useful, since the comparison between the numerical and experimental mode shapes may not be trivial. In general, the results of the study agree well with the empirical knowledge of many luthiers regarding changes in timbre associated with variations in ambient humidity. A high degree of correlation between the numerical and experimental results and cited references shows that the global dynamic behavior of the wooden specimen is accurately described assuming a homogeneous, orthotropic and linear elastic material. Furthermore, given the geometric characteristics of the specimen
2 Effects of Moisture Content on the In-Plane Elastic Constants of Wood
55
and in accordance with the results, plate theory can properly be applied. It must nonetheless be pointed that if one wishes to analyze specimens other than quartersawn, samples which are thicker or curved, or samples with varying thicknesses, these assumptions must be reconsidered.
2.10 Prospects for the Future While the general lines to follow in the evaluation of the effects of moisture content on the in-plane elastic constants of wooden specimens have been roughly sketched out in the present work, it is certainly necessary to refine the numerical model by including other parameters such as Poisson’s ratio, the consideration of more modes for the error functions and examining aspects other than just modal frequencies, incorporating a theory that provides a solution for thicker plates by using a shear deformation theory, as well as dealing with plates which vary in thickness, in which case three-dimensional stress effects could not be neglected. Future lines of enquiry might be to compare the results with the estimation of elastic properties using ultrasonic inspection techniques, and the effects upon dynamical behavior due to different types of glue or varnish used in the construction of instruments. Finally, since the most important aspect in the making and preservation of a musical instrument is its sound quality, future lines of enquiry might include the preservation of samples of the wood used in the construction of soundboards of specific instruments in order to allow the correlation of observed variations in the elastic characteristics or states of stress of the material with any acoustic alterations which might be detected in the instrument’s sound, a study which could be carried out without having to subject the instrument itself to the sort of testing needed to establish mechanical characteristics of its component parts. In addition, further work is required on the influence of wood’s moisture content on the internal friction, i.e. damping, given its significance for the final musical result.
2.11 Summary The present work illustrates the effects on the vibration behavior caused by moisture content in wood and provides a mixed experimental-numerical tool to quantify these effects on the in-plane elastic constants. The vibrational behavior of a specimen of quarter-sawn spruce has been observed using vibration testing, from dry conditions to the saturation point. It has been shown that wood’s dynamic behavior is heavily dependent on moisture content, and a significant variations in geometry and mass measurements have been observed. The methodology results in numerical eigenfrequencies that fit closely with the experimental frequencies. In general, the results show that an increase in moisture content causes a nonlinear decrease in natural frequencies and, consequently, in elastic moduli. Good agreement between
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reference solutions and the results obtained have been observed, proving that the global dynamic behavior of the wooden specimen have been accurately described under the assumptions considered.
References 1. Alfano, M., Pagnotta, L.: Determining the elastic constants of isotropic materials by modal vibration testing of rectangular thin plates. J. Sound Vib. (2006). doi:10.1016/j.jsv.2005.10.021 2. Araújo, A.L., Mota Soares, C.M., Moreira de Freitas, M.J.: Characterization of material parameters of composite plate specimens using optimization and experimental vibrational data. Compos. Part B Eng. (1996). doi:10.1016/1359-8368(95)00050-X 3. Ayorinde, E.O., Gibson, R.F.: Elastic constants of orthotropic composite materials using plate resonance frequencies, classical lamination theory and an optimized three-mode Rayleigh formulation. Compos. Eng. (1993). doi:10.1016/0961-9526(93)90077-W 4. Ayorinde, E.O., Yu, L.: On the elastic characterization of composite plates with vibration data. J. Sound Vib. (2005). doi:10.1016/j.jsv.2004.04.026 5. Bathe, K.J.: Finite Element Procedures. Prentice Hall, New York (1996) 6. Bell, J.F.: Mechanics of Solids: Volume 1—The Experimental Foundations of Solid Mechanics. Springer, Berlin (1973) 7. Cousins, W.J.: Elastic modulus of lignin as related to moisture content. Wood Sci. Technol. (1976). doi:10.1007/BF00376380 8. Cousins, W.J.: Young’s modulus of hemicellulose as related to moisture content. Wood Sci. Technol. (1978). doi:10.1007/BF00372862 9. Deobald, R.L., Gibson, R.F.: Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh-Ritz technique. J. Sound Vib. (1988). doi:10.1016/ S0022-460X(88)80187-1 10. De Visscher, J., Sol, H., De Wilde, W.P., Vantomme, J.: Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method. Appl. Compos. Mat. (1997). doi:10.1007/BF02481386 11. Ewins, D.J.: Modal Testing: Theory and Practice. Research Studies Press, Baldock (1986) 12. Fällström, K.E., Jonsson, M.A.: Determining material properties in anisotropic plates using Rayleigh’s method. Polym. Compos. (1991). doi:10.1002/pc.750120503 13. Felippa, C.: Introduction to finite element methods. http://www.colorado.edu/engi-neering/ cas/courses.d/IFEM.d/ (2009). Cited Jun 2010 14. Ferreira, A.J.M.: MATLAB Codes for Finite Element Analysis. Springer, Berlin (2009). doi:10.1007/978-1-4020-9200-8 15. Grédiac, M., Paris, P.A.: Direct identification of elastic constants of anisotropic plates by modal analysis: theoretical and numerical aspects. J. Sound Vib. (1996). doi:10.1006/ jsvi.1996.0434 16. Grédiac, M., Fournier, N., Paris, P.A., Surrel, Y.: Direct identification of elastic constants of anisotropic plates by modal analysis: experimental results. J. Sound Vib. (1998). doi:10.1006/jsvi.1997.1304 17. Haines, D.W.: On musical instrument wood, Part I. J. Catgut. Acoust. Soc. (1979). Published in [22], paper 88 18. Haines, D.W.: On musical instrument wood, Part II: surface finishes, plywood, light and water exposure. J. Catgut. Acoust. Soc. (1980). Published in [22], paper 89 19. Harris, C.M., Piersol, A.G.: Harris’ Shock and Vibration Handbook. McGraw-Hill, New York (2002) 20. Häglund, M.: Parameter influence on moisture induced eigen-stresses in timber. Eur. J. Wood Prod. (2009). doi:10.1007/s00107-009-0377-2
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21. Hanhijärvi, A.: Advances in the knowledge of the influence of moisture changes on the long-term mechanical performance of timber structures. Mater. Struct. (2000). doi:10.1007/BF02481695 22. Hutchins, C.M., Benade, V.: Research Papers in Violin Acoustics, 1973–1995. Acoust. Soc. Amer., USA (1996) 23. Hurlebaus, S., Gaul, L., Wang, J.T.S.: An exact series solution for calculating the eigenfrequencies of orthotropic plates with completely free boundary. J. Sound Vib. (2001). doi:10.1006/jsvi.2000.3541 24. Hurlebaus, S.: Calculation of eigenfrequencies for rectangular free orthotropic plates—an overview. J. Appl. Math. Mech. (2007). doi:10.1002/zamm.200710349 25. Hwang, S.F., Chang, C.F.: Determination of elastic constants of materials by vibration testing. Compos. Struct. (2000). doi:10.1016/S0263-8223(99)00132-4 26. Jones, R.M.: Mechanics of Composite Materials. Taylor & Francis, London (1999) 27. Kwon, Y.W., Bang, H.: The Finite Element Method Using MATLAB. CRC Press, Boca Raton (1997) 28. Larsson, D.: Using modal analysis for estimation of anisotropic material constants. J. Eng. Mech. (1999). doi:10.1061/(ASCE)0733-9399(1997)123:3(222) 29. Lauwagie, T.: Vibration-based methods for the identification of the elastic properties of layered materials. Ph.D. thesis, Katholieke Universiteit Leuven, Belgium (2005) 30. Leissa, A.W.: Vibration of Plates. Acoust. Soc. Amer., USA (1993) 31. Liew, K.M., Wang, C.M., Xiang, Y., Kitipornchai, S.: Vibration of Mindlin Plates: Programming the p-Version Ritz Method. Elsevier, Amsterdam (1998) 32. Maia, N.M.M., Silva, J.M.M.: Theoretical and Experimental Modal Analysis. Research Studies Press, Baldock (1997) 33. McIntyre, M.E., Woodhouse, J.: On measuring the elastic and damping constants of orthotropic sheet materials. Acta Metall. (1988). doi:10.1016/0001-6160(88)90209-X 34. Molin, N.E., Lindgren, L.E.: Parameters of violin plates and their influence on the plate modes. J. Acoust. Soc. Am. (1988). doi:10.1121/1.396430 35. Mota Soares, C.M., Moreira de Freitas, M., Araújo, A.L., Pedersen, P.: Identification of material properties of composite plate specimens. Compos. Struct. (1993). doi:10.1016/0263-8223(93)90174-O 36. Ohlsson, S., Perstorper, M.: Elastic wood properties from dynamic tests and computer modeling. J. Struct. Eng. (1992). doi:10.1061/(ASCE)0733-9445(1992)118:10(2677) 37. Pagnotta, L., Stigliano, G.: Elastic characterization of isotropic plates of any shape via dynamic tests: theoretical aspects and numerical simulations. Mech. Res. Commun. (2008). doi:10.1016/j.mechrescom.2008.03.008 38. Qing, H., Mishnaevsky, L.: Moisture-related mechanical properties of softwood: 3D micromechanical modeling. Comput. Mater. Sci. (2009). doi:10.1016/j.commatsci.2009.03.008 39. Rayleigh, J.W.S.: The Theory of Sound, vol. 1. Macmillan Co., New York (1877) (Reprinted by Dover Publications, New York (1945)) 40. Reddy, J.N.: Theory and Analysis of Elastic Plates. Taylor & Francis, London (1999) 41. Rodgers, O.E.: The effect of the elements of wood stiffness on violin plate vibration. J. Catgut. Acoust. Soc. (1988). Published in [22], paper 48 42. Silva, C.W.: Vibration and Shock Handbook. CRC Press, Boca Raton (2005) 43. Sol, H.: Identification of anisotropic plate rigidities using free vibration data. Ph.D. thesis, University of Brussels, Belgium (1986) 44. Thompson, R.: The effect of variations in relative humidity on the frequency response of free violin plates. J. Catgut. Acoust. Soc. (1979). Published in [22], paper 53 45. United States Department of Agriculture: The Encyclopedia of Wood. Skyhorse Publishing, New York (2007) 46. Voichita, B.: Acoustics of Wood. CRC Press, Boca Raton (1995) 47. Wegst, U.G.K.: Wood for sound. Am. J. Bot. 93(10), 1439–1448 (2006)
Chapter 3
Short-Time Autoregressive (STAR) Modeling for Operational Modal Analysis of Non-stationary Vibration V.-H. Vu, M. Thomas, A.A. Lakis, and L. Marcouiller
Abstract In this chapter, a method based on an autoregressive model in a shorttime scheme is developed for the modal analysis of vibrating structures whose properties may vary with time and is called Short-Time AutoRegressive (STAR) method. This new method allows for the successful modeling and identification of an outputonly modal analysis. The originality of the proposed method lies in its specific handling of non-stationary vibrations, which allows the tracking of modal parameter changes in time. This chapter presents an update of the model with respect to model order and a noise-to-signal based criterion for the selection of the minimum model order. A length equal to four times the period of the lowest natural frequency has been numerically found to be efficient for the data block size and may be recommended for experimental applications. To validate the method, a system with three degrees of freedom is first simulated under a random excitation, and both stationary and non-stationary vibrations are considered. The method is finally applied on the real multichannel data measured on an experimental steel plate emerging from water, and is compared to the conventional Short-Time Fourier Transform (STFT) method. It is shown that the proposed method outperforms in terms of frequency identification, whatever the non-stationary behaviour (either slow or abrupt change) due to the added mass effect of the fluid.
V.-H. Vu () · M. Thomas École de Technologie Supérieure, 1100 Notre Dame West, Montréal, QC, H3C 1K3, Canada e-mail:
[email protected] M. Thomas e-mail:
[email protected] A.A. Lakis École Polytechnique, Montréal, QC, H3C 3A7, Canada e-mail:
[email protected] L. Marcouiller Institut de Recherche Hydro Québec, Varennes, QC, J3X 1S1, Canada e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_3, © Springer Science+Business Media B.V. 2011
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3.1 Introduction This chapter presents the modal monitoring of a non stationary system by operational modal analysis. An emerging steel plate is investigated in order to identify the added mass and damping due to the interaction effect of the fluid. The fluid has an inertial effect on the mass of the structure and hence significantly affects its vibration behaviour. Furthermore, the modal damping ratios need to be investigated since no analytical model is available and hence an experimental modal analysis is the unique technique to consider [28, 32, 35]. The chapter outline starts with a brief overview of the art followed by a presentation of the autoregressive model and the STAR method. Several discussions are given on numerical simulations and an application on the emerging steel plate exhibits the performance of the method. Important conclusions can be found in the summary.
3.2 Overview of the State of the Art The classical modal analysis, usually conducted in the frequency domain, has been in decades a companion to Modal Testing experimentalists [5]. An overview of modal analysis methods can be found in [20]. However, this technique is not reliable when the tested system is working in operating conditions and in the last decades, modal analysis has migrated to the Operational Modal Analysis (OMA), called also Output-only Modal Analysis or Natural Excitation Modal Analysis.
3.2.1 Operational Modal Analysis This novel technique processes the identification of modal parameters (natural frequencies, damping ratios and structural modes) directly from only the output responses of the system without having to know the excitation forces [15, 34]. The last two decades have witnessed a trend toward the use of the time series models. Time series models are parametric models which are able to evaluate a time dependent phenomena. The most applicable models for mechanical and structural systems are the Autoregressive model (AR), Autoregressive Moving Average (ARMA) and their variants [9, 23, 38]. Industrial applications of OMA can be found for analysing an offshore structure excited by natural excitations such as sea wind and waves [14], but difficulties in frequency and damping identification can appear when harmonic excitations are considered [21].
3.2.2 Non-stationary Vibration Non-stationary vibration [24, 25, 33, 37] is a common phenomena in real life systems where the modal properties vary with respect to the time [4]. Such problems
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can be found in various mechanical and structural systems like in robotic [17] where the modal parameters vary with the manipulator extension, or in civil engineering like a bridge vibration under traffic loads [22, 36]. In damage monitoring or crack detection, it is important to monitor the changes in the modal properties of the structures over time [1, 2, 30]. Modal and vibration analysis of such time-dependent systems may be analyzed through non parametric time-frequency methods [3, 7, 11] included the Short-Time Fourier Transform (STFT), Wavelet and Wigner-Ville transforms [27]. However, parametric models offer a number of advantages such as improving accuracy and resolution which explain why the time domain methods are generally preferred [6].
3.2.3 Fluid-Structure Interaction In fluid-structure interaction, such as for ship structures analysis [16] or hydraulic turbines [8], the analysis of the added mass and especially of the added damping is necessary to compute the dynamic stresses. Since no analytical method can be applied for estimating the damping under a turbulent flow, an operational modal analysis is the only technique that can be applied. Reference [32] presents a recent experimental research on modal analysis of a submerged plate excited by a turbulent flow to experimentally evaluate the added mass and damping in stationary conditions.
3.2.4 Development of a New Method for Investigating Modal Parameters of Non-stationary Systems by Operational Modal Analysis In this chapter, a linear multivariate autoregressive model [13, 15, 18] is used and is sequentially computed in a short-time scheme [26] by using a sliding window. The model parameters are estimated by least squares via the fast and stable computation of the QR factorization. A model order is selected from a minimum value which robustly gives a convergence of the noise and signal separation. Rather than varying the parameters on a sample by sample basis, a sliding window [19, 31] is used to keep the parameters constant inside each window and adjusts the parameters, window by window. The window length size is automatically adjusted based on the greatest period. The proposed method may thus be considered as the time domain counterpart of the Short-Time Fourier Transform and can be used with multichannel measurements. It possesses efficiency in tracking the modal parameters and monitoring their evolution on both stationary and non-stationary vibrations.
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3.3 Vector Autoregressive (VAR) Modeling Assuming a random measurement environment, the excitation may be ignored. Since the modal analysis normally requires a measurement at multiple locations supposed d sensors, the response data can be represented by an order p vector autoregressive model of dimension d, y(t) + A1 y(t − 1) + A2 y(t − 2) + · · · + Ap y(t − p) = e(t).
(3.1)
The model can be rewritten as a multiple regression convenient form [37], y(t) = z(t) + e(t),
(3.2)
where • = [−A1 −A2 . . . −Ap ], of size d × dp, is the model parameters matrix, • Ai (i = 1, 2, . . . , p), of size d × d, is the matrix of autoregressive parameters relating the output y(t − i) to y(t), • z(t), with z(t)T = [y(t − 1)T y(t − 2)T . . . y(t − p)T ], is the regressor for the output y(t), • y(t − i), of size d × 1, is the output vector with delayed time iT , • T is the sampling period (s), and • e(t), of size d × 1, is the residual vector of all output channels, considered as the model error. If the data are assumed to be measured in a white noise environment, the least squares estimation can be applied. Consider N successive vectors of the output responses from y(t) to y(t + N − 1), the model parameters matrix and the estimated ˆ and also of the error part Eˆ (both of size covariance matrices of the unnoised part D d × d; the “hat” denotes the estimated value) can be given via the computation of the QR factorization [37], −1 −1 T = R11 R12 , (3.3) = RT12 R11 RT11 R11 ˆ = 1 RT R12 , D N 12
(3.4)
1 Eˆ = RT22 R22 . N
(3.5)
In these formulas R11 , of size dp × dp, R12 , of size dp × d, and R22 , of size d × d, are sub-matrices of the upper triangular factor R (size N × dp + d) derived from the QR factorization of the data matrix K = Q × R,
(3.6)
where Q, of size N × N , is an orthogonal matrix (that is Q × QT = I), R has the form
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⎡
R11 R=⎣ 0 0
⎤ R12 R22 ⎦ , 0
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(3.7)
and data matrix K, of size N × (dp + d), is constructed from N successive samples, ⎤ ⎡ y(t)T z(t)T ⎢ z(t + 1)T y(t + 1)T ⎥ ⎥ ⎢ (3.8) K=⎢ ⎥. ⎦ ⎣ ... ... z(t + N − 1)T
y(t + N − 1)T
Once the model parameters matrix has been estimated, the modal parameters, such as natural frequencies, damping ratios and mode shapes, can be directly identified from the eigen-decomposition of the dp × dp state matrix S [23] given by ⎡ ⎤ −A1 −A2 . . . −Ap−1 −Ap ⎢ I 0 ... 0 0 ⎥ ⎢ ⎥ ⎢ I ... 0 0 ⎥ S=⎢ 0 (3.9) ⎥. ⎣ ... ... ... ... ... ⎦ 0 0 ... I 0
3.4 The Short Time Autoregressive (STAR) Method In operational modal analysis, the dynamic parameters of the system are unknown, and thus, a priori knowledge about the model order is not available. Since we are concerned with short-time modeling, we propose that the data is processed in blockwise Gabor expansion [7]. From the above modeling, it is found that the number of samples in each block N must satisfy N > dp + d, where p is the computing model order, and thus can be variable in non-stationary vibration. It is also clear that the block size must be long enough to allow an exhibition of the vibratory features of the system and to cover the largest period in the signal. That is why the block length of the sliding window must be adjusted from the greatest period. The optimal model must be selected from the order 2 to the maximum available order which fits data of the whole block size. Since it is time-consuming to repeat the computation for each order value, this procedure should be avoided. Below, we present an algorithm allowing for an effective updating of the solution with respect to model order, which requires only the triangularization on a sub-matrix of the data matrix.
3.4.1 Order Updating and a Criterion for Minimum Model Order Selection The data matrix K(p) of order p can be rewritten as
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⎡ ⎢ ⎢ K(p) = ⎢ ⎣
z(t)T
y(t)T
z(t + 1)T ...
y(t + 1)T ...
z(t + N − 1)T
y(t + N − 1)T
⎤ ⎥
⎥ ⎥ = K(p) 1 ⎦
K2 ,
(3.10)
(p)
where K1 and K2 are of size N × dp and N × d, respectively. If the model is updated to p + 1, the data matrix has the form
∗ K , K(p+1) = K(p) K 2 1 where K∗ of size N × d are the added d columns, ⎡ y(k − (p + 1))T ⎢ y(k + 1 − (p + 1))T ⎢ K∗ = ⎢ ⎣ ...
(3.11)
⎤ ⎥ ⎥ ⎥. ⎦
(3.12)
y(k + N − 1 − (p + 1))T We can then compute the matrix
T T Q(p) K(p+1) = Q(p) T K(p) Q(p) K∗ 1 (p)
(p) R11 T1 R12 = , (p) 0 T2 R22
T
Q(p) K2
(3.13)
where T1 , of size dp × d, and T2 , of size (N − dp) × d, are extracted from T1 (p) T ∗ . (3.14) K = Q T2 We must now triangularize the right term matrix in Eq. (3.13). This can be done with a set of Householder transformations or Givens rotations [10]. If we decompose only the small sub-matrix T2 , it easily yields RT T 2 = QT , (3.15) 0 where RT , of size d × d, is an upper diagonal matrix and QT , of size (N − dp) × (N − dp), is the product of the Householder transformations or Givens rotations. Equation (3.13) then becomes ⎡ (p) ⎤ R(p) T1 R12 11 T I 0 ⎣ ⎦ (3.16) Q(p) K(p+1) = 0 RT (p) , 0 QT QTT R22 0 0
I
0
0 QTT
⎡
(p)
R11 ⎢ (p) T (p+1) Q K =⎣ 0 0
T1 RT 0
⎤ (p) R12 ⎥ R∗22 ⎦ , R∗∗ 22
(3.17)
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where R∗22 , of size d × d, and R∗∗ 22 , of size (N − dp − d) × d, are obtained from multiplication,
R∗22 = QTT R22 . (3.18) R∗∗ 22 It can be seen that the first d × p rows of the right hand side in Eq. (3.17) are not affected by above transformations, and the factor matrix R(p+1) , of order p + 1, is thus updated to
(p) (p) R11 T1 R12 (p+1) (p+1) (p+1) , R12 , R22 R11 = = = R∗∗ 22 . (3.19) 0 RT R∗22 Matrix Q(p) is also updated to Q
(p+1)
=Q
(p)
I
0
0 QT
,
(3.20)
as well as the two covariance matrices from Eqs. (3.4) and (3.5), T
T
ˆ (p) + R∗ T R∗ , (3.21) ˆ (p+1) = R(p+1) R12 (p+1) = R(p) R(p) + R∗ T R∗ = D D 22 22 22 22 12 12 12 T
(p+1) (p+1) (p) (p) Eˆ (p+1) = R22 R22 = R22 R22 − R∗22 T R∗22 = Eˆ (p) − R∗22 T R∗22 .
(3.22)
Finding the optimal model order pmin is crucial in parametric model-based methods [12, 29]. From a statistical point of view, AIC (Akaike Information Criterion) and MDL (Minimum Description Length) criteria can be used to select the optimal model order [18]. It is seen from Eqs. (3.21) and (3.22) that as the model order increases, the norm of the deterministic covariance matrix increases while the one of the error parts decreases with the same amount. The global Noise-to-Signal Ratio (NSR) is therefore monotonically decreased in terms of the model order and is given by NSR =
ˆ tr(E) . ˆ tr(D)
(3.23)
The Noise-rate Order Factor (NOF) defines the change in the NSR within two successive model order values, NOF(p) = NSR(p) − NSR(p+1) .
(3.24)
It is seen that the convergence of the NSR can be served as a criterion for the selection of optimal model order, which has been inspired in AIC or MDL in combination with a linear penalty function. In this chapter, only the convergence of the NSR is utilized in term of the order-wise NOF since this latter is insured to be always positive and keeps the convergence of the NSR to zero. Since the NSR decreases significantly at low orders and quickly converges, the convergence of NOF is obviously observable and pick-able. It is evident that the convergence of the NOF may not give the optimal model order as do the AIC and MDL, but the minimum
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Fig. 3.1 Three DOF mechanical system
required order for the modal analysis. This minimum model order should therefore be chosen after a significant change in the NOF before stably converging (Fig. 3.3). Since the model solution is effectively updated with respect to model order and the window is moving, the selection of minimum order can be applied for time-varying systems.
3.4.2 Working Procedure Since there is no leakage present in parametric model-based modal analysis, it is therefore not necessary to apply a window function on the data, so we propose that the data is processed in combination with a progressive search for the model order, as follows. Firstly, the above VAR model is initially applied to a block of data with a reasonable low order value. The length of the first block size could thus be specified from the smallest natural frequency of the structure, as discussed in the next section. The modal parameters are identified and the natural frequencies are estimated by using the signal-to-noise ratio of each eigenvalue (MSN) [23] in order to find the smallest frequency to use to specify the length of the next block data. Once the block size is chosen, the minimum model order is selected by the NOF and an order equal to or higher than this minimum value is used to get the modal parameters. An overlapping process can also be employed by changing the sliding step, which can vary from only one sample to the whole length of the rectangular block window.
3.5 Numerical Simulation on a Mechanical System A numerical simulation of the proposed method was applied on a system with 3 degrees of freedom (DOF), as shown in Fig. 3.1, under a unmeasured random force excitation.
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Fig. 3.2 Modal parameter identification with block size
3.5.1 Discussion on Data Block Length The mechanical properties of the system are first kept constant and possess three natural frequencies 6.4 Hz, 12.6 Hz and 25.0 Hz with damping rates 2.0%, 4.0% and 7.8% respectively. Frequencies and damping rates identified by the VAR method are shown in Fig. 3.2 with varying block data lengths. It is observed that the block size must be larger than 3 times the longest period Tmax in order to produce the smallest natural frequency value. For that reason, the block size was chosen to be equal to 4 times the period of this frequency. The same result was also obtained from others simulations with different numbers of degree of freedom. In conclusion, when a moderate damped system is subjected to a random excitation, its modal parameters can be monitored at block sizes equal to four times the period of the smallest natural frequency. The NOF curves of the 3 DOF system at various data length sizes are plotted in Fig. 3.3. It is seen that the minimum model order is found accurately at 3 regardless the data length showing the stability of the
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Fig. 3.3 Optimal model order at different data block sizes
Fig. 3.4 Simulated time-varying mass function
NOF criterion with respect to data length. It also confirms that if the properties of the structure are subjected to change, the optimal order can still be tracked, and does not depend on the block size once this latter is long enough to reveal the smallest natural frequency.
3.5.2 Simulation on Mechanical System with Time-Dependent Parameters The above 3 DOF system has been modified to vary its mechanical properties in the time domain, and is always subjected to a random excitation. The mass M2 is now a time variant factor which changes following the function shown in Fig. 3.4. Since the data is non-stationary, the minimum model and modal parameters can vary with time, and therefore, the block size should be changed. The initial block
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Fig. 3.5 Monitoring of minimum order on simulation
Fig. 3.6 Monitoring of block size on simulation
size is chosen to be four times of the fundamental period. When more data are acquired, the block size is adjusted based on the smallest frequency identified in the previous step. The minimum model and the computing block size used to track the change in the system properties are given in Figs. 3.5 and 3.6. It is shown that the minimum order is primarily monitored at 3, except for some outer values. We can observe an adjustment on the block length when the change appears at the second half of the monitoring time. The changes in frequencies and damping rates are plotted on Figs. 3.7 and 3.8, respectively. As the mass is increasing, all natural frequencies decrease and can be well tracked. However, we observe a high variance on the damping rates ranging from 0% to 5% for mode 1 and from 0% to 10% for modes 2 and 3. This could be due to the necessity to use a higher computing order when the system is continuously varying, and thus the monitoring of the damping ratios requires further researches.
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Fig. 3.7 Monitoring of frequencies on simulation
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Fig. 3.8 Monitoring of damping ratios on simulation
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Fig. 3.9 Plate test configuration
Fig. 3.10 Plate temporal response
3.6 Experimental Application on an Emerging Steel Plate The method is applied to the monitoring of vibration occurred in a submerging steel plate. The plate measures 500 mm × 200 mm × 2 mm, and emerges from water while it is always excited by a random turbulent flow. The configuration of the test is presented in Fig. 3.9. In Fig. 3.10, the time response data is given where the low amplitude portion corresponds to the submerging period and the high amplitude portion is attributed to the emergence of the plate from the water to the air. Before the plate rises, its modal parameters are both calculated and identified using analytical and experimental methods [35], as shown in Table 3.1. The minimum model order applied to the data over measuring time is plotted in Fig. 3.11. It is seen that the minimum order is found primarily from 3 to 6. Since the minimum model orders are tracked from 3 to 6 and any higher order can be used for the model fitting, Fig. 3.12 shows the monitoring of frequencies where the variations are clearly revealed. The changes in frequencies correspond
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Table 3.1 Modal identification of the emerging plate Mode
Frequency (Hz) in submerging conditions (Depth/Plate length) 0.6 (totally submerged)
0.4
0.2
0.1
0 (totally in air)
1st
11.9
12.0
12.2
12.7
39.4
2nd
34.1
34.1
34.2
35.0
75.0
3rd
77.7
77.9
78.1
79.5
108.6
4th
135.3
135.4
135.6
137.5
164.0
5th
151.3
151.3
151.4
152.6
210.0
Fig. 3.11 Monitoring of plate minimum model order
to the emergence of the plate. Both the slow change when the plate was still in water and the abrupt change when it appears on the surface are monitored. The natural frequencies highly match the calculated values in Table 1 to show that the effect of added mass on the plate is accurately monitored [35]. Compared to the STFT computed on the first channel with the same configuration in Fig. 3.13, it is seen that the proposed STAR method outperforms in terms of revealing the natural frequencies.
3.7 Prospects for the Future Non-stationary vibration is a topic of interest. It can be seen that modal analysis and monitoring of such vibration is a new trend for the research. Parametric models with time depending characteristics reserve always a prospective future. Despite its specific application in this chapter on the short time manner with the VAR model, several research directions and improvements can be pointed out. It is seen first that the least squares is the basic estimate for the model parameters. In this chapter, the least squares are implemented via the QR factorization. The stability of this numerical factorization has been addressed in [10] but it should be extensively evaluated for
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Fig. 3.12 Monitoring of plate natural frequencies
Fig. 3.13 Short time Fourier transform
the non-stationary data since this stability influence all the result of the modal identification. A recursive computation method is also of interest in order to accelerate the implementation. The model order selection is the most important aspect for the parametric modeling. This chapter has presented the introduction of a minimum model order which is
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very effective for identification of natural frequencies. Further development should be taken on the selection of the computing model order for identifying the damping ratios. It is found that the variance of damping ratios is higher than of the natural frequencies. The effect of computing model order to the damping identification may be investigated in the future by taken from a threshold of the damping variance with respect to model order. The performance of the proposed method has successfully been investigated on a structure dynamically emerging from water. Various applications could be prospectively applied in mechanical and civil engineering such as the machinery start-up or shutting-down, the detection of cracks in structural health monitoring applied to rotors [30] or bridges, the dynamic behaviour of robots, the on line identification of lobe stabilities in high speed machining monitoring, etc. However, the results for the monitoring of damping changes are actually not satisfying depending on the system variation type and extended researches should be carried out. Ongoing research actually focuses on the evaluation of the uncertainty of the damping ratios with respect to model order, noises and type of excitation.
3.8 Summary An application of the multivariate autoregressive model in the short-time scheme (STAR) was presented, and covered the monitoring of changes in the modal parameters of non-stationary structures under unknown excitations. In order to track the frequency variations, it is not suitable to use a stabilization diagram and it is preferred to select a minimum order for computing the modal parameters. This model order is effectively selected by the convergence of a newly introduced Noise-rate Order Factor. The model was fast and stably updated with respect to the order by the QR factorization. It is found that the minimum model order value does not depend on the block size if this latter is long enough to identify the first natural frequency. The block size was minimally found from numerical simulations to be equal to four times the period of the first natural frequency and it has been successfully used for real structures. Consequently, the block sizes vary with variations in the first natural frequency. Numerical simulations and experiments show that the proposed method can be used to track a slow change as well as a sudden change in the frequency of the structure, and that it outperforms the STFT method. While the monitoring of natural frequencies has been successfully dealt with, those of damping rates are however not enough precise if the excitation is random or the system is continuously varying. Research is thus still ongoing on damping identification of time-varying system.
3.9 Selected Bibliography This chapter addresses the monitoring of modal parameters in non-stationary vibrations by using operational modal analysis. The fundamental model is the autoregressive (AR). Readers are recommended to back to [4] for the introduction on time
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series modeling and detail of vector autoregressive model (VAR) model for operational modal analysis. This multivariate model has been used in modal analysis in some works such as [13, 15, 16, 22, 38]. A very good book on modal analysis using the AR model is [23] where detail on modal identification and selection of modes can be interesting found. A critical survey of the applications of non-stationary vibrations can be found in [25]. Especially, damage monitoring can be found in [1, 2, 36]. Finally, fluid-structure interaction is experimentally described in [32, 35]. Acknowledgements The support of NSERC (Natural Sciences and Engineering Research Council of Canada) through Research Cooperative grants is gratefully acknowledged. The authors would like to thank Hydro-Quebec’s Research Institute for the collaboration.
References 1. Basseville, M.: Detecting changes in signals and systems—A survey. Automatica 24(3), 309– 326 (1988) 2. Basseville, M., Benveniste, A., Gach-Devauchelle, B., Goursat, M., Bonnecase, D., Dorey, P., Prevosto, M., Olagnon, M.: Damage monitoring in vibration mechanics: Issues in diagnostics and predictive maintenance. Mech. Syst. Signal Process. 7(5), 401–423 (1993) 3. Bellizzi, S., Guillemain, P., Kronland-Martinet, R.: Identification of coupled nonlinear modes from free vibrations using time-frequency representations. J. Sound Vib. 243(2), 191–213 (2001) 4. Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. San Francisco, Holden-Day (1970) 575p 5. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd edn. Wiley, New York (2000) 562p 6. Fassois, S.D.: Parametric identification of vibrating structures. In: Braun, S.G., Ewins, D.J., Rao, S.S. (eds.) Encyclopedia of Vibration, pp. 673–685. Academic Press, New York (2001) 7. Gabor, D.: Theory of communication. J. IEEE (London) 93, 429–457 (1946) 8. Gagnon, M., Tahan, S.A., Coutu, A., Thomas, M.: Operational modal analysis with harmonic excitations: Application to a hydraulic turbine. In: Proceedings of the 24th Seminar on Machinery Vibration, pp. 320–329. Canadian Machinery Vibration Association, Montreal, ISBN 2-921145-61-8 (2006) 9. Gang, L., Wilkes, D.M., Cadzow, J.A.: ARMA model order estimation based on the eigenvalues of covariance matrix. Trans. Signal Process. 41(10), 3003–3009 (1993) 10. Golub, G., Van Loan, C.: Matrix Computations. London. Johns Hopkins University Press, Baltimore (1996) 732p 11. Hammond, J.K., White, P.R.: The analysis of non-stationary signals using time-frequency methods. J. Sound Vib. 190, 419–447 (1996) 12. Hannan, E.J.: The estimation of the order of an ARMA process. Ann. Stat. 8(5), 1071–1081 (1980) 13. He, X., De Roeck, G.: System identification of mechanical structures by a high-order multivariate autoregressive model. Comput. Struct. 64(1–4), 341–351 (1997) 14. Hermans, L., Van der Auweraer, H.: Modal testing and analysis of structures under operational conditions: Industrial applications. Mech. Syst. Signal Process. 13(2), 193–216 (1999) 15. Huang, C.S.: Structural identification from ambient vibration measurement using the multivariate AR model. J. Sound Vib. 241(3), 337–359 (2001) 16. Li, C.S., Ko, W.J., Lin, H.T., Shyu, R.J.: Vector autoregressive modal analysis with application to ship structures. J. Sound Vib. 167(1), 1–15 (1993)
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17. Li, J., Liu, Z., Thomas, M., Fihey, J.L.: Dynamic analysis of a planar manipulator with flexible joints and links. In: Fifth International Conference on Industrial Automation, Montréal, ROB06 (2007) 4p 18. Lutkepohl, H.: Introduction to Multiple Time Series Analysis, 2nd edn. Springer, Berlin, (1993) 545p 19. Mahon, M., Sibul, L., Valenzuela, H.: A sliding window update for the basis matrix of the QR-decomposition. IEEE Trans. Signal Process. 41, 1951–1953 (1993) 20. Maia, N.M.M., Silva, J.M.M.: Modal analysis identification techniques. Philos. Trans. R. Soc. Lond. A 359, 29–40 (2001) 21. Mohanty, P., Rixen, D.J.: Operational modal analysis in the presence of harmonic excitation. J. Sound Vib. 270, 93–109 (2004) 22. Owen, J.S., Eccles, B.J., Choo, B.S., Woodings, M.A.: The application of auto-regressive time series modelling for the time-frequency analysis of civil engineering structures. Eng. Struct. 23, 521–536 (2001) 23. Pandit, S.M.: Modal and Spectrum Analysis: Data Dependent Systems in State Space. John Wiley and Sons, New York (1991) 415p 24. Poulimenos, A.G., Fassois, S.D.: Non-stationary vibration modelling and analysis via functional series TARMA models. In: 5th International Conference on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques (Surveillance 5), Senlis, France (2004) 10p 25. Poulimenos, A.G., Fassois, S.D.: Parametric time-domain methods for non-stationary random vibration modelling and analysis—A critical survey and comparison. Mech. Syst. Signal Process. 20(4), 763–816 (2006) 26. Rissanen, J.: Modeling by shortest data description. Automatica 14, 465–471 (1978) 27. Safizadeh, M.S., Lakis, A.A., Thomas, M.: Using short time Fourier transform in machinery fault diagnosis. Int. J. Cond. Monitor. Diagn. Eng. Manag. 3(1), 5–16 (2000) 28. Sinha, J.K., Singh, S., Rama, Rao A.: Added mass and damping of submerged perforated plates. J. Sound Vib. 260(3), 549–564 (2003) 29. Smail, M., Thomas, M., Lakis, A.A.: Assessment of optimal ARMA model orders for modal analysis. Mech. Syst. Signal Process. 13(5), 803–819 (1999) 30. Smail, M., Thomas, M., Lakis, A.A.: Use of ARMA model for detecting cracks in rotors. In: Proceedings of the 3rd Industrial Automation International Conference AIAI, Montreal, pp. 21.1–21.4 (1999) (in French) 31. Strobach, P., Goryn, D.: A computation of the sliding window recursive QR decomposition. In: Proc. ICASSP, pp. 29–32 (1993) 32. Thomas, M., Abassi, K., Lakis, A.A., Marcouiller, L.: Operational modal analysis of a structure subjected to a turbulent flow. In: Proceedings of the 23rd Seminar on Machinery Vibration. Canadian Machinery Vibration Association, Edmonton (2005) 10p 33. Uhl, T.: Identification of modal parameters for non-stationary mechanical systems. Arch. Appl. Mech. 74, 878–889 (2005) 34. Vu, H.V., Thomas, M., Lakis, A.A.: Operational modal analysis in time domain. In: Proceedings of the 24th Seminar on Machinery Vibration, pp. 330–343. Canadian Machinery Vibration Association, Montreal, ISBN 2-921145-61-8 (2006) 35. Vu, H.V., Thomas, M., Lakis, A.A., Marcouiller, L.: Identification of added mass on submerged vibrated plates. In: Proceedings of the 25th Seminar on Machinery Vibration, pp. 40.1–40.15. Canadian Machinery Vibration Association, St John (2007) 36. Vu, H.V., Thomas, M., Lakis, A.A., Marcouiller, L.: Identification of modal parameters by experimental operational modal analysis for the assessment of bridge rehabilitation. In: Proceedings of International Operational Modal Analysis Conference (IOMAC 2007), Copenhagen, Denmark, pp. 133–142 (2007) 37. Vu, H.V., Thomas, M., Lakis, A.A., Marcouiller, L.: Online monitoring of varying modal parameters by operating modal analysis and model updating. In: CIRI2009, Reims, France (2009) 18p 38. Wahab, M.M.A., De Roeck, G.: An effective method for selecting physical modes by vector autoregressive models. Mech. Syst. Signal Process. 13(3), 449–474 (1999)
Chapter 4
A Numerical and Experimental Analysis for the Active Vibration Control of a Concrete Placing Boom G. Cazzulani, M. Ferrari, F. Resta, and F. Ripamonti
Abstract Concrete placing booms are subjected to vibrations causing an increase of mechanical stress and a reduction of the boom lifetime. The aim of this paper is the development of an active control methodology in order to suppress boom vibrations using the same actuators performing the boom large motion. The control logic is based on two contributions, a feed-forward and a feed-back one. In this work, a nonlinear flexible multibody model has been created in order to simulate the dynamic behavior of the boom. Moreover, this model has been used in order to develop and test the control logic. Finally the control methodology has been validated on an experimental test rig.
4.1 Introduction The present work deals with booms for the concrete placing. These booms consist of a series of several flexible elements (called “links”). There are different typologies, differing in dimension and number of links. A pipeline, through which the concrete flows, is connected along the boom. The movement of these booms is performed by hydraulic actuators, driven by a human operator, who provides the boom correct positioning. The so-given actuators force is able to guarantee the correct positioning of the boom, but it cannot avoid vibrations due to structure flexibility. These vibrations cause an increase in mechanical stress, adding a dynamic contribution to the static (or quasi-static) one due to the boom weight and introducing fatigue problems with a reduction in boom lifetime. G. Cazzulani () · M. Ferrari · F. Resta · F. Ripamonti Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy e-mail:
[email protected] M. Ferrari e-mail:
[email protected] F. Resta e-mail:
[email protected] F. Ripamonti e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_4, © Springer Science+Business Media B.V. 2011
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During the last years, this problem became much more critical because of a continuous development in terms of size increasing and lightening. Vibrations are mainly due to two contributions: • Concrete flow along the pipeline: the concrete is pumped into the pipeline by a volumetric pump. As a consequence, concrete flow has an irregularity which implies a periodic excitation of the boom. The phenomenon is particularly significant when the pump frequency is close to the first natural one of the boom; • Motion law given by the human operator. The aim of the present work is the definition of a control logic to reduce vibrations due to both the boom large motion and the concrete pumping. For this reason, at first, a nonlinear numerical model describing the boom behavior has been implemented. This model could be used to define the parameters of the control law and to simulate the behavior of the real booms with and without the vibration control. This model has been obtained using finite element method (FEM) [17] to describe the links flexibility, while the large motion of the boom is modeled with the multibody (MB) theory [16]. Then an experimental test rig has been designed and realized. It consists in a reduced-scale boom reproducing the dynamic behavior of the commercial ones. Both the test rig and the numerical model are described in the third section of this chapter. Section 4.4 shows the feedback control logic proposed for this application. The first part of the paragraph describes the feedback control, based on the independent modal space control algorithm. The second part describes the modal observer, estimating the contribution of each modal shape to the boom motion. In the last part some numerical results on a boom typical movement are presented and discussed. Section 4.5 describes, in the first part, the feed-forward control logic that can be used to suppress vibrations during the concrete pumping. In the second part some numerical results, showing the capability of the feed-forward control to suppress vibrations due to an external excitation, are presented. Finally, Sect. 4.6 shows some significant experimental results validating the proposed control logic.
4.2 Overview of the State of the Art In mechanical and aerospace fields, the last years were characterized by a wide development of large and flexible structures. These structures are typically characterized by an high flexibility and a low material damping ratio, which lead to high-level vibrations and, as a consequence, to fatigue problems. For this reason, the need to develop methodologies, acting on the system to reduce its vibratory state, grew. The target of most of these logics is to increase the system damping to dissipate energy and reduce vibration amplitude and decay time. The first and simplest solution is represented by the use of tuned passive mass dampers [7, 21]. It consists in mounting on the structure a mass-spring-damper system. The parameters of this added system can be tuned in order to dissipate energy
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at a given natural frequency of the structure. The main disadvantage of this methodology is that it requires to add a mass on the structure, increasing the material static stress. Besides, it becomes ineffective far from the frequency at which it was designed. The great development of computer and electronic systems led to a lot of studies about semi-active and active control techniques. The first one is based on the same idea of passive systems, but some parameters can be actively modified through an actuator. This solution, with respect to a simple passive system, is able to adapt itself to the operating conditions or to the modifications of the structure properties that can occur during its lifetime [1, 2]. Active control methods, on the contrary, provide a feedback force, calculated from one or more measurements of the system vibratory state. In case of flexible systems it must be considered that, even if they are theoretically characterized by an infinite number of modal shapes, infinite number of Dofs (degrees of freedom), only the first ones significantly contribute to their dynamics. In fact, higher-frequency modes, associated to higher damping ratios, are excitable with difficulty and, as a consequence, their dynamic contribution can be neglected. For this reason, it becomes suitable to describe the dynamic behavior of the system in modal coordinates, considering only the first ones in the definition of the active control law. The calculation of the control gain matrix can be performed using different formulations, like Pole Placement or Optimal Control [12]. These formulations couple the modal shapes of the controlled system. For this reason, Meirovitch and Balas [3, 14] introduced the so-called Independent Modal Space Control (IMSC) which, under certain assumptions [3, 14], allows to control each mode, setting its frequency and damping independently from the others. In the following years some modifications to IMSC were proposed. For example, Fang et al. [5] proposed a Modified Independent Modal Space Control (MIMSC), that reduces the so-called control spillover, a deterioration of the non-controlled modes behavior under the application of the control force. In the same years, other authors [18] proposed the Efficient Modal Control (EMC), that allows to reduce the amplitude of the required control forces with respect to IMSC and MIMSC. All these modal control formulations require the knowledge of the system modal coordinates. In flexible structures, these coordinates can be directly measured only with distributed piezoelectric sensors [11, 19]. If they are not available, it becomes necessary to calculate them from the available measurements. Mainly two methods can be applied: • Modal filters [13, 20]: they use the kinematic relationship between the modal coordinates and the measures and they need a number of measures equal to the number of considered modes; • Modal observers [9]: they consist in a dynamic system, numerically integrated on the control board, that returns an estimation of the modal coordinates from the available measurements. In this case, there are no limits about the number of sensors. In fact, even if a higher numbers of sensors allows to estimate modal coordinates in a better way, it is possible to achieve very good performances even with a single sensor (as shown in the present work). Another advantage of the observer is that it works as a filter and, for this reason, it is more robust to measurement noise with respect to modal filters. Besides, thanks to their predictor/corrector
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layout, they are less sensible to model errors with respect to other model-based methods like modal filters. Both modal filters and observers may be subjected to observation spillover [4], due to the influence of the non-considered modes on the estimation of the considered ones. It occurs when a mode which plays an important role in the system dynamics is neglected and it can lead the system to instability. Some solution was proposed to reduce the spillover problem. For example Kim and Inman [10] proposed to use a sliding-mode observer, while other authors [6] introduced another control method, called “Positive position feedback”. The spillover problem can also be avoided by observing an increased number of modes. As previously said, modal control can modify the frequency and damping associated to each mode. Increasing the damping, it becomes possible to reduce vibrations when the structure is excited near to its natural frequencies. On the contrary, if the excitation frequency is far from the natural frequencies of the structure, modal control is not effective. For this reason a second control law, dedicated to non-resonant disturbance rejection, can be introduced in addition to the modal control. For example, some author proposed to use a disturbance estimator in order to estimate and balance the disturbance force, giving to the system another force opposing to it [20]. The work proposed in this chapter, starting from the studies found in literature, wants to integrate the performances of IMSC feedback with a feed-forward contribution, to achieve a robust control logic in the whole medium-low frequency range.
4.3 The System 4.3.1 Test Rig The booms under investigation are characterized by an high flexibility and a low damping factor. In order to reproduce in laboratory the dynamic behavior of these systems a test rig has been created (Fig. 4.1). This test rig (a three link boom 1:3 scaled with respect to a commercial one) allows to develop and test control strategies and to validate the control law for the vibration suppression anticipating the testing step on a real boom. The boom length and the link sections have been chosen in order to obtain the same dynamic behavior of a commercial boom, in terms of first natural frequencies and modal shapes. The natural frequencies can be modified, in a limited range, by means of a load that can be mounted in different positions on the last link. In order to reduce the weight of the entire structure, each link is made of highstrength steel and is connected to the others by revolution joints. An hydraulic actuator allows the large motion of each link through a kinematic chain similar to the one used on real booms: the actuator, connected to a triangular element mounted at the end of a link, pushes a rod connected to the following link. The actuators are driven by a servo-valve controlled by a computer-based controller which allows a great repeatability of the large motion laws. The test rig has been instrumented with:
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Fig. 4.1 The test rig
• 3 load cells, in order to know the total forces applied by the actuators; • 3 LVDT sensors, that measure the length of the actuators and allow to obtain the boom configuration; • 3 accelerometers, located at the end of each link, having measurement direction normal to the link and in the vertical plane. Capacitive accelerometers are used in order to measure vibrations at very low frequencies. From the input sensors mounted on the boom, the control logic (performed by a dSpace board) calculates the control action.
4.3.2 Numerical Model A numerical model of the described system has been developed as an instrument to define the control logic and to simulate the boom behavior. Due to the planar motion of the boom, the model can simulate only the 2D dynamics. The other assumptions introduced are small deformations, in order to linearly relate strains and deformations and to decouple axial and bending deformations. Moreover, only small link rotation speed are considered so that all the centrifugal and Coriolis’ contributions can be neglected. This model is able to describe the large motion of the links and their deformations due to flexibility. The scheme of the system is shown in Fig. 4.2. The boom kinematic have been solved using the floating frame of reference formulation [16, 17]. As described in [15], to describe the motion of each link, the following reference systems have been defined (Fig. 4.3): • the global reference system (O − x, y); • the i-th link reference system (O − xi , yi ), positioned at the beginning of the link and rotated of θi with respect to the global reference system; • the reference system of the k-th element of the i-th link (O − xi,k , yi,k ).
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Fig. 4.2 Sketch of the 2D model of the flexible boom
Fig. 4.3 Reference systems for the numerical model of the flexible boom
This formulation describes the motion of a generic link as a sum of two contributions, the absolute position and rotation and the deformation. The absolute position is described as (Oi − O) and it depends on the degrees of freedom of the previous links, while the link rotation is described by the independent variable θi . The link deformation is described by using the FEM formulation, considering 2 nodes beam elements. Due to the assumption of planar motion, the motion of each node is described by a vector d i,k of three independent variables: the axial deformation, the transversal deformation and the rotation. The total independent variables vector, containing all the degrees of freedom, is defined as θ z= . (4.1) d Using the Lagrange formulation it’s possible to obtain a nonlinear equation of motion of the boom, because all the terms depend on the motion variables and, in particular, on the links configuration. In order to study the vibration problem and obtain system eigenvalues, the nonlinear equation can be evaluated and linearized in each links configuration. In a generic configuration of the system “z = zj ” the equation of motion becomes Mj δ¨zj + Rj δ˙zj + Kj δzj = fg zj + Tj Fact ,
(4.2)
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Fig. 4.4 Block diagram of the active modal control logic
where • Mj and Kj respectively represent the inertial and stiffness matrices; • the damping term Rj is assumed to be proportional to stiffness and inertial ones (a typical assumption for mechanical systems); the proportional coefficients are estimated starting from experimental data, considering the first three modes of the system (the same modes considered for the active control); • fg zj represents the constant contribution of the gravitational term; • Fact represents the actuator forces and j represents the kinematic relationship between the actuator length ρ and the independent coordinates vector z. In the next paragraphs two different control logics are presented and investigated. The two coefficients have been estimated through an experimental campaign, considering the first three modes of the system (the same modes considered for the active control).
4.4 Active Modal Control In this section a control logic able to reduce the structure vibrations is presented. In operating conditions the system, as previously introduced, is subject to a generic large motion causing, due to the links high flexibility, significant vibration levels. In order to suppress these vibrations (δz), the idea proposed in this paper is to apply a control action using the same actuators in charge of moving each link. As shown in Fig. 4.4, control forces uc work in parallel to the large motion forces Fact . The vibration control logic adopted is based on an independent modal approach [13]. Following this approach the control action is calculated from the system vibratory state and a gain matrix opportunely defined. In particular, the vibratory state is described by a set of modal coordinates representing the system dynamics in the frequency range of interest. The modal coordinates cannot be measured directly for a generic application (unless using distributed sensors [11]) and they have to be estimated by a modal observer. In the following subsection, the theoretical background about modal approach and modal control for a linear system is reviewed (the generic boom configuration is indicated with the subscript “j ”. In this application, as the system is nonlinear, natural frequencies and modal shapes change during the boom motion. For this reason all the matrices and gains of the control and the observer change with time.
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4.4.1 Independent Modal Control The control synthesis can be done starting from Eq. (4.2), not considering the gravitational and external forces terms because these contributions do not modify the poles value. Considering a generic second order mechanical system and defining the state-space vector δ˙z x= , (4.3) δz Equation (4.2) can be written in state-space form as −1 T M−1 −M−1 j Rj −Mj Kj j j x+ uc = Aj x + Bj uc . x˙ = I 0 0
(4.4)
The aim of the control force uc is to increase the damping ratio of the system in order to reduce the vibrations through the imposition of the system eigenvalues, so that the original eigenvalues λ become λc by increasing the only real part absolute value. The imaginary one has not been changed, in order to avoid relevant control forces and, as a consequence, additional mechanical fatigue stress of the boom material. This aim could be obtained through a state-space control design method, like the pole placement. As previously described, in the present work the equation of motion has been obtained through a FEM discretization of the links. It means that the variable vector z contains all the nodal displacements of the structure and its dimension is very large. Since few modes take part in the structure dynamics (high-frequency modes usually have higher damping ratio and they can hardly be excited), only the first ones can be considered for the modal control definition. Particularly, as can be seen later, it is necessary to consider the first three modes, thus equaling the number of the actuators available on the boom. Defining qz,j as the vector of the first m modal coordinates of the entire system, it is possible to perform the coordinate change δz = φ j qz,j ,
(4.5)
where φ j represents the eigenvectors of the considered modes extracted by the complete eigenvector matrix. Inserting Eq. (4.5) into (4.2), the equation of motion becomes a set of independent modal equations mj q¨ z,j + rj q˙ z,j + kj qz,j = φ Tj Tj uc ,
(4.6)
where mj , rj and kj are m × m diagonal matrices, obtained as mj = φ Tj MTj φ j rj = φ Tj RTj φ j kj = φ Tj KTj φ j .
(4.7)
In order to apply the state-space control approach to the reduced modal system, Eq. (4.6) can be written in state-space form as −1 −1 T T −m−1 m r −m k φ j j j j j j j q˙ j = qj + uc = Aq,j qj + Bq,j uc , (4.8) I 0 0
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q˙ z,j
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(4.9)
uc = −Gq qj ,
(4.10)
q˙ j = [Aq,j − Bq,j Gq ]qj = Ac qj ,
(4.11)
qz,j
Defining the feedback control law as
Eq. (4.8) becomes
where the eigenvalues λc of the matrix Ac represent the poles of the controlled system. Defining Uq,j as the right eigenvector matrix of the system (controlled and not controlled systems have the same eigenvectors for an independent modal controller), the gain matrix of the control can be calculated by −1
(4.12) Gq = Bq,j 0 Aq,j − Uq,j λc U−1 q,j , where Bq,j represents the upper half of Bq,j . This formulation can be applied to any second order system that has as many inputs as degrees of freedom (Bq,j must be square). Besides, the system must be controllable, so any row and column of φ Tj Tj must be non-zero. It means that at least one control force must have a non-zero contribution on every mode and each control force must take action on one mode at least. For a generic state-space equation the Moore-Penrose pseudo-inverse of Bq,j should be used.
4.4.2 The Modal Observer Under the assumption that the exact modal coordinates are known, the proposed method is able to guarantee a completely decoupled modal control and to avoid spillover problems [8], even if the number of controlled modes is smaller than the number of the structure modes. Nevertheless, in most flexible structure control applications, the modal coordinates are unknown and so they have to be estimated through a modal observer or filter [14]. In this case the control system is not immune to spillover problems. The implemented modal observer scheme is shown in Fig. 4.5. The observer equation, in a state-space form, can be written as ˆ q˙ˆ = Aq qˆ + Bq uc + Go (μ − μ), where • Go is the observer gain matrix, that will be defined later; • qˆ represents the estimated modal coordinates;
(4.13)
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Fig. 4.5 Block diagram of the modal observer
ˆ respectively represent the vector of accelerometers measurements and • μ and μ ˆ depends on the estimated measurements; μ is an input of the observer, while μ observer state by the relation ˆ = Co qˆ + Do uc , μ
(4.14)
ˆ on the estimated modal coordinates where Co and Do represent the dependence of μ and the control forces. It is possible to define the gain matrix Go by using pole placement method. In this case, a single-input observer (one accelerometer) has been implemented. The gain matrix Go has been calculated as Go = T−1 (co − c),
(4.15)
where the column vectors c and co contain the characteristic polynomial coefficients of the reduced modal state-space matrix and observer state-space matrix (Aq and Aq − Go Co ), while T−1 is known as Toeplitz matrix.
4.4.3 Numerical Analysis of Modal Control In this section, some numerical results are presented. All the simulations refer to the numerical model of the test rig then used for the experimental campaign. All the results (with and without vibration control) refer to the same large movement of the boom (Fig. 4.6). As said in Sects. 4.4.1 and 4.4.2, control and observer gains have been calculated imposing the poles of the controlled system. In this simulations, the poles of the controlled system have been chosen in order to keep the imaginary parts equal to those of the uncontrolled system and to set the controlled system damping ratios to 10% for the controlled modes in each boom configuration. The poles of the observer, instead, have been chosen in order to make the observer dynamics damped and faster than the system dynamics. In this way the observer is able to follow the
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Fig. 4.6 Large motion reference for the boom links; starting/final configuration (a) and rotation time history of the links (b)
Fig. 4.7 Numerical comparison of the acceleration of the end of the boom third link with and without modal control; (a) the large motion time history and (b) the spectrum
system vibrations without introducing significant delays. The imaginary parts of the observer poles have been set at twice the system ones, while the real parts have been set equal to 30% of the imaginary ones. As previously said, the boom modes and poles change continuously with the configuration. In theory, with the independent modal control, for every configuration a gain matrix as in Eq. (4.12) and a modal observer should be defined, resulting in an infinite number of gain matrices and observers. Practically only a finite number of gain matrices and modal observers can be computed. In this application a gain-scheduling method is used, calculating all the control and observer matrices in a discrete set of configurations and pre-allocating them in the control board memory. During the boom motions, the boom configuration is calculated through the actuator length measurements and, at each time step, one combination of matrices is extracted from the pre-allocated set. This solution have been used for both the numerical and experimental analysis. Figure 4.7 shows the damping effect of the vibration modal control. The acceleration of the third link end is considered as the indicator of the control performance.
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On the left, Fig. 4.7a shows the acceleration time history during the large motion described in Fig. 4.6. The time history, also including the static acceleration due to gravity, highlight the performances of the proposed method. On the right, Fig. 4.7b shows, on a logarithmic scale, the acceleration spectrum of the final part of the time history shown in Fig. 4.6 (from 15 s to the end). The figure shows that modal control provides an high increase in damping. The consequence is the reduction in vibrations near the controlled natural frequencies, while the system response can slightly increase between them.
4.5 Feed-Forward Control The active control logic presented is able to suppress the vibrations of the boom and the transient motion through the increase in the structure damping factor. However it doesn’t manage to reduce vibrations due to an external force when the force has harmonic components far from the natural frequencies of the boom. Focusing for example on the case of a concrete displacing boom, the flow of the concrete in the pipe rigidly connected to each link produces a distributed force along the structure. This causes a boom waving, clearly visible in the tip motion, heavily affecting the material fatigue. In what follows a feed-forward action has been studied and presented in order to generate an action opposing to the pumping forcing.
4.5.1 The Feed-Forward Control Logic The feed-forward (FF) controller has to suppress the effects of a deterministic force applied to the structure. Since the only predictable force present on a boom is that due to the concrete flow, the FF logic is studied for this specific application. Investigations on the field highlight that the concrete pump produces a multi-harmonic flow/force, but only the first component has a non-negligible effect and has to be considered. Moreover, the pumping process is stationary since the variations of speed and intensity are negligible. Each piston of the concrete pump provides a pulse signal when it changes its movement direction. The delay between two consecutive signals gives the information about the period (and consequently the frequency) of the pumping force. It is important to underline that, in normal application, the frequency of the pumping force is always lower than the first resonant one and typically between 0.4 and 0.6 Hz depending on the operating conditions. The level of the vibratory state is provided by the measurement of acceleration in different points, including the most important one at the tip of the last link. Furthermore, in this case, the control force is provided by the same actuators working with the active modal control logic. The FF algorithm consists in several steps (Fig. 4.8): 1. First of all the s acceleration measurements are collected, where s is the number of considered sensors;
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Fig. 4.8 Block diagram of the feed-forward control logic
2. Each signal aj is analyzed on an integer number of periods with a FFT algorithm, giving the information about amplitude and phase at the known frequency, without applying the feed-forward force; 3. Each i-th active actuator applies, one at a time, an harmonic force Fi with small amplitude and random phase, and the response is analyzed as previously described; 4. The effect of each actuator on each sensor (ai,j ) is evaluated by comparing the responses with and without the Fi force; 5. The frequency response function (FRF i,j ) between the Fi and the ai,j at the known frequency is evaluated for each actuator; 6. The FF control force is calculated by pre-multiplying the a vector, measured at point 1, by the matrix (FRF)−1 and taking the opposite sign. The matrix can be inverted under the assumption that the number of measures (s) is equal to the number of active actuators (a). The proposed approach guarantees the vibration reduction for all the considered points but no information is available about other points of the structure. However, for frequencies lower than the first resonant one, it is possible to show that a reduction in vibrations for a few points results in a reduction for all the points of the structure.
4.5.2 Numerical Analysis of the Feed-Forward Control The FF logic is tested on the previously described numerical model of the boom. The concrete flow effect is simulated with a number of forces applied along the structure at the same frequency. Two different conditions have been investigated. The first simple case (case A) considers only one accelerometer at the end of the
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Fig. 4.9 Numerical simulations without (initial part) and with (final part) the FF control logic application; the three link ends acceleration with one (case A) and three (case B) active actuators
last link and only one active actuator (the first). The second one (case B) takes into account three measurements at the end of each link and it is possible to use all the three actuators for the FF control logic. For both cases, a pumping frequency of 0.5 Hz is considered and the following conditions have been simulated: • 20 s with only the pumping force applied (FF inactive); • 20 s for each active actuator, for the relative FRF evaluation; • 40 s with the FF active. Figure 4.9 shows the acceleration at the end of each link in case A and case B, while Table 4.1 shows the reduction in RMS of acceleration due to feed-forward
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Table 4.1 Decrease in RMS of acceleration due to FF control with one and three active actuators Reduction in acceleration RMS [%] 1st link
2nd link
3rd link
FF with one actuator
88%
82%
99%
FF with 3 actuators
98%
96%
99%
control. It is possible to notice that, even if the system’s response can slightly increase during the identification interval, after the application of the feed-forward control force, the acceleration at the end of the third link drops to zero in both cases. On the other hand, the accelerations at the end of the second and the first links are completely canceled only in case B. Anyway, in case A, there is a significant reduction in vibrations for the non-measured points, even if they do not drop to zero.
4.6 Experimental Testing In order to validate the numerical results and the proposed approaches, in this section an experimental campaign on the test rig is presented. As in the numerical simulation, the acceleration of the end of the third link is considered in order to estimate the modal control performances. The experimental testing, considering the same control settings of the numerical analysis, highlights spillover problems and the application of the control forces lead the higher uncontrolled modes of the structure to instability. It means that those modes cannot be neglected in the boom dynamics and they must be included in the definition of the reduced system. This phenomenon is due to the differences between the numerical model and the experimental test rig, that lead to a non-rigorous independent modal control law. Spillover problems are strictly related to the system damping. Damping factors have been estimated experimentally and, for mechanical systems, they are always subjected to high uncertainty, especially for the high-frequency modes. In order to solve the problem, a five-mode observer has been implemented while only the first three modes feed-back has been considered for the definition of uc . Figure 4.10 shows the experimental results using the improved control settings. Comparing it with Fig. 4.7 it is possible to observe a good agreement with the numerical data. At the same time the FF logic is also tested. Because of its small size, it is impossible to mount a concrete pipe and pump. Therefore an external actuator has been used to replicate the concrete flow force. It had been placed on the ground and connected to the end of the first link by means of a decoupling spring, in order not to modify the natural frequencies and modal shapes of the boom. Only case A is investigated. The accelerometer at the end of the third link is considered and the FF control force is applied by means of the first actuator. Figure 4.11 shows the benefit of the proposed logic; a considerable reduction in vibrations due to the external force is clearly visible when comparing the two portions of the time history, with a decrement of 85%.
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Fig. 4.10 Experimental comparison of the acceleration of the end of the boom third link with and without modal control; (a) the large motion time history and (b) the spectrum Fig. 4.11 Experimental data without (initial part) and with (final part) the FF control logic application; the third link end acceleration with one active actuator
4.7 Prospects for the Future The presented research activity is still in progress. In particular further developments are under investigation both in terms of system dynamic model and control actuation methodologies. About the first aspect, the numerical model of the boom could be linked to a second algorithm for the simulation of the pumping group. The concrete fluid dynamics will be modeled and a co-simulation of the two subsystems will be carried out. This improvement will allow to numerically reproduce the multi-harmonic distributed pumping force and to validate the feed-forward logic in a more realistic situation. Moreover, a deeper insight into the modal control actuation is due. In particular, for the practical application on a commercial boom, a very important role is played by the hydraulic actuators and their transfer function between the command and the generated force. In fact, for these actuators the hypothesis of ideal actuators, which has been considered in this laboratory application, cannot be assumed anymore. The main reason of that is the use of electro-proportional valves instead of servo-valves. This choice is due to many reason, like for example cost, filtering requirements of the oil, conditions in which the boom works. For this reason, a numerical model of
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the actuator will be developed and included in the boom one. In this way, the simulations can consider both the systems, since they mutually affect their dynamics. In any case, this problem could be solved also by considering a different point of view. In fact, about the control logic, the possibility to introduce embedded sensors and actuators can be analyzed. This choice, adopting actuators dedicated only to the actuation of the vibration control logic, could improve the control performance, accepting the cost increase of the improved design. Finally, the application of different control laws is under investigation. In particular, the introduction of a disturbance estimator could allow to identify the pumping force contribution and to feed-back a force opposite to it. This solution will be studied as an alternative to the feed-forward contribution, while the modal control will be maintained.
4.8 Summary An active control logic to suppress vibrations of high flexible booms has been implemented both numerically and experimentally on a test rig. The numericalexperimental comparison showed a good agreement, demonstrating the efficiency of the proposed methodology. Issues related with the logic implementation, for example the spillover problem, have been investigated and solved. This control strategy can find practical applications on industrial concrete placing booms in order to reduce the mechanical fatigue of the material and increase the system safety and performance.
4.9 Selected Bibliography For a deeper insight into the topics of this chapter, the following books are suggested. Shabana, A.A., Dynamics of Multibody Systems, John Wiley & Sons (1998), provides a very good overview to multibody dynamics, considering, in particular, flexible structures. The first part covers the basic ideas of kinematics and dynamics of rigid and deformable bodies. Then the book describes some topics, including finite element formulation, that have been widely used for the numerical model of the boom. Finally the book provides some example and applications of multibody theory. About control of structures, there are many books providing a detailed state-ofthe-art. Probably the most complete is Levine, W.S., The Control Handbook, CRC Press (1996), that covers many topics related with control, both from the pure mathematical and practical points of view. This book is very appellative for expert researchers in the field, but it could be very difficult for newcomer readers. Other interesting books are: • Friedland, B., Control System Design, McGraw-Hill (1986);
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• Gawronski, W.K., Dynamics and Control of Structures: A Modal Approach, Springer (1998); • Meirovitch, L., Dynamics and Control of Structures, John Wiley & Sons (1990); • Preumont, A., Vibration Control of Active Structures: An Introduction, Kluwer (2002). All these books provide an introduction to active control of structures, focusing, in particular, on state-space methods (modal control, pole placement, optimal control, etc.) and on state observers and Kalman filter. Friedland (1986) focuses in a particular way on the control theories, while the other authors also provide some interesting practical examples about flexible aerospace structures, robotics, vibration suppression, etc. Lastly, the book Fuller C.R, Elliott S.J., Nelson, P.A., Active Control of Vibration, Academic Press (1996), deals with active control of vibrations and noise, providing information about actuators and transducers typically used for these applications and introducing feed-forward control for deterministic disturbances. Acknowledgements
The research has been developed with the financial support of CIFA S.p.A.
References 1. Abdel-Rohman, M.: Optimal design of active TMD for buildings control. Build. Environ. 19(3), 191–195 (1984) 2. Abdel-Rohman, M., John, M.J.: Control of wind-induced nonlinear oscillations in suspension bridges using multiple semi-active tuned mass damper. J. Vib. Control 12(9), 1011–1046 (2006) 3. Balas, M.J.: Active control of flexible systems. J. Optim. Theory Appl. 25(3), 415–436 (1978) 4. Balas, M.J.: Feedback control of flexible systems. IEEE Trans. Autom. Control AC-23(4), 673–679 (1978) 5. Fang, J.Q., Li, Q.S., Jeary, A.P.: Modified independent modal space control of m.d.o.f. systems. J. Sound Vib. 261(3), 421–441 (2003) 6. Goh, C.J., Caughey, T.K.: On the stability problem caused by finite actuator dynamics in the collocated control of large space structures. Int. J. Control 41(3), 787–802 (1985) 7. Igusa, T., Xu, K.: Vibration control using multiple tuned mass dampers. J. Sound Vib. 175(4), 491–503 (1994) 8. Inman, D.J.: Active modal control for smart structures. Philos. Trans. R. Soc. 359, 205–219 (2001) 9. Juloski, A.L., Heemels, W.P.M.H., Weiland, S.: Observer design for a class of piecewise linear systems. Int. J. Robust Nonlinear Control 17(15), 1387–1404 (2007) 10. Kim, M.H., Inman, D.J.: Reduction of observation spillover in vibration suppression using a sliding mode observer. J. Vib. Control 7(7), 1087–1105 (2001) 11. Lee, C.K., Moon, F.C.: Modal sensors/actuators. J. Appl. Mech. 57(2), 434–441 (1990) 12. Meirovitch, L., Baruh, H.: Optimal control of damped flexible gyroscopic systems. J. Guid. Control Dyn. 4(21), 157–163 (1981) 13. Meirovitch, L., Baruh, H.: Robustness of the independent modal-space control method. J. Guid. Control Dyn. 6(1), 20–25 (1983) 14. Meirovitch, L., Baruh, H.: Nonlinear control of an experimental beam by IMSC. J. Guid. Control Dyn. 7(4), 437–442 (1984) 15. Resta, F., Ripamonti, F., Cazzulani, G., Ferrari, M.: Independent modal control for nonlinear flexible structures: An experimental test rig. J. Sound Vib. 329(8), 961–972 (2010)
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16. Shabana, A.A.: Flexible multibody dynamics: Review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997) 17. Shabana, A.A., Schwertassek, R.: Equivalence of the floating frame of reference approach and finite element formulations. Int. J. Non-Linear Mech. 33(3), 417–432 (1998) 18. Singh, S.P., Pruthi, H.S., Agarwal, V.P.: Efficient modal control strategies for active control of vibrations. J. Sound Vib. 262(3), 563–575 (2003) 19. Tzou, H.S., Tseng, C.I.: Distributed modal identification and vibration control of continua: Piezoelectric finite element formulation and analysis. J. Dyn. Syst. Meas. Contr. Trans. ASME 113(3), 500–505 (1991) 20. Wang, D.A., Huang, Y.M.: Modal space vibration control of a beam by using the feedforward and feedback control loops. Int. J. Mech. Sci. 44(1), 1–19 (2002) 21. Warburton, G.B., Ayorinde, E.O.: Optimum absorber parameters for simple systems. Earthquake Eng. Struct. Dyn. 8, 197–217 (1980)
Chapter 5
Modeling and Testing of a Concrete Pumping Group Control System C. Ghielmetti, H. Giberti, and F. Resta
Abstract This paper describes a study of the behavior of a “truck-mounted concrete pump” used industrially to pump large quantities of concrete. The characteristics of the pump, its functioning and its numerical model will be explained. Numerical model results will be compared with experimental ones. The paper demonstrates, using the mathematical model, how the pumping group system is improved by introducing an active control system. In particular, the aim of the study is to optimize the mechanical layout in order to satisfy certain concrete flow and pressure requirements and to develop and complete an active control system application. The control system described in this paper is particularly simple, easy to put in practice and therefore suitable for use in a hostile environment. This control system was designed to reduce dangerous vibrations induced by the operation of the pumping system on another large structure which the pump is constrained.
5.1 Introduction This paper focuses mainly on the concrete pumping system. The aim of this system is to pump large amounts of concrete a long way from the cockpit. The concrete is pumped along a pipeline rigidly connected to a long mechanical boom. The pump develops variable flow and pressure on the basis of the type of work required. Usually it is used in dirty field conditions and it is in contact with extremely aggressive materials, so the design solution chosen had to be strong and easy to clean and repair. The pumping group (Fig. 5.1) is an alternate volumetric pump with two pistons. One of these sucks up the concrete from a container while, at the same time, the other piston pumps the material along the pipeline. The resultant motion of C. Ghielmetti () · H. Giberti · F. Resta Mechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156 Milan, Italy e-mail:
[email protected] H. Giberti e-mail:
[email protected] F. Resta e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_5, © Springer Science+Business Media B.V. 2011
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Fig. 5.1 Pumping group system, CAD 3D
the concrete is periodic and discontinuous and induces a force on the pipeline due to the friction between the concrete and the metal pipe. These forces induce dangerous mechanical oscillations on the boom and compromise its fatigue lifetime. Moreover, the simple layout and functioning of the pumping system make it difficult to control the flow provided. But control of the concrete flow, in terms of continuity and regularity, is fundamental for limiting oscillations on the entire system and especially on the boom [15]. A numerical model is developed to facilitate an understanding of the way the system reacted when certain characteristic parameters were varied. The model was used to identify parameters and compare them with experimental ones. We must underline the fundamental importance of the test rig (shown in Fig. 5.2) on which many tests were carried out. The use of a test rig was very important because in this way it was possible to investigate several different working configurations and it can also be easily used to test different kinds of concrete. The numerical model includes a study of the pumping group in terms of its mechanical and hydraulic behavior. Moreover, this model helps us to understand the control of the pump flow, the diminution of the forces on the mechanical boom and the optimization of the design of the pumping group in terms of back-pressure, noise, duration and pumping capacity. This chapter compares some tests, underlining the good matches obtained and introducing an initial study of a complete active control system for the motion of the pistons in terms of numerical and experimental analysis. The chapter is structured as follows: Sect. 5.1 is the introduction; in Sect. 5.2, there is a brief overview of the state of the art; Sect. 5.3, describes the entire system; Sect. 5.4, the experimental tests; and Sect. 5.5, presents the mathematical model. Moreover, Sect. 5.6 compares the numerical and experimental results while Sect. 5.7 shows the control system design. Finally, Sects. 5.8 and 5.9 present the prospects for the future and summary.
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5.2 Overview of the State of the Art Retracing the history of the pumped concrete, it can be said that this technique has been known in the world for almost 70 years. Pumps in which the concrete circulates in metal pipes were used in the U.S. as early as 1933. Freshly-mixed concrete can be transported to where it is needed without using buckets or conveyor belts. The first pumping units were based on a mechanism designed to push the concrete along a pipeline. Subsequently these systems have been replaced by hydraulic cylinders and the first two piston pump were introduced [14]. Now the pumping technique is widely used in industrial construction, and has seen many improvements, particularly on the pumps (for example, high-span pumps and pumps with high pressure). A number of different studies presented in literature are about fresh concrete pumpability. These studies consist, for example, of measuring the pressure required to maintain the concrete in continuous movement [1, 2, 4, 7]. Many efforts of researchers are undertaken to find a link between different concrete types and its pumpability. These theoretical studies are aimed at modeling the behavior of concrete. For this reason other authors deal with regard to the rheological properties of concrete and a large amount of literature is available [3, 6, 10]. Their efforts demonstrate it is very difficult to reach a predictive model to evaluate the oil pressure during a concrete pumping task. The other research frontier on concrete pumping is to improve the concrete pump truck boom performance regarding pumping capacity, maximum pumping distance attainable and safety, analyzing the boom behavior. In fact the entire system, called a “truck mounted concrete boom pump”, is made up of two subsystems: the “concrete pumping group” and the “boom”. The control design of pump truck boom impact directly the overall work performance and safety. The difficulty of the control is caused by the remarkable elasticities and nonlinearities both of the slim shape and of the hydraulic actuators. Concrete placing booms are subjected to vibrations causing an increase of mechanical stress and a reduction of the boom lifetime. In literature some contributions can be found. As an example [8] implemented a FEM model of a flexible boom in order to define a desired control logic for the vibration suppression. Other authors created boom models for the tip trajectory synthesis [13] or the pouring process automatization [4] and pump models for enhanced components design. The vibration problems relating to the boom have been studied by a research group of the department of mechanical engineering of the Politecnico di Milano and some results are presented in [11]. Currently are not available studies where it analyzes the mechanical behavior of the concrete pump in relation to vibration problems. In literature, this kind of application has not yet been studied, probably due to the particular and unusual research field.
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Fig. 5.2 Test rig
Fig. 5.3 Test rig block diagram
5.3 Description of the Entire System In Figs. 5.2 and 5.3 the pumping group test rig and its block diagram are shown; the position of the group, its fixing to the ground, its inclination and all circuits provide a perfect simulation of the real group mounted on the truck. The pump was positioned on a steel ground constrained structure which has the same connections of the truck-mounted concrete boom pump so that the simulation is as close as possible to reality. The test rig is composed of the pumping group (the physical model in which we are interested) and of a pair of auxiliary circuits; the hydraulic circuit and the concrete circuit (called the CLS circuit). Both the hydraulic circuit, connected to diesel engines, and the CLS circuit are similar to those in the real system. The work
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Fig. 5.4 Diagram of pumping group
fluid (CLS in Fig. 5.3), flows in a dedicated circuit that is connected to a steel tank almost three meters high. It flows out alternately from the cylinders on the concrete side and returns to the hopper of the pumping group after passing through a closed circuit consisting of a pipeline, a series of regulating valves and a steel holding tank. A series of valves are mounted in the CLS circuit to pilot and to adjust the fluid pressure, or to be more precise the load to which the pistons are subjected from the outside. This complete test rig makes it possible to carry out many kinds of experiments. The oil pumps are powered by diesel engines whose rpm can be varied to regulate the oil input flow. Figure 5.4 explains the behavior of the system, giving a simple description of the components of the pumping group and how they interact with each other to pump the concrete along the metal pipeline. The pumping group is a volumetric pump composed of two pistons (E) that, alternately, suck up concrete from the concrete tank (H) and, at the same time, pump it out along the pipeline (I). The pistons are driven by two hydraulic actuators (B and C). The pipes containing the pistons are immersed in the concrete vessel (H). To direct the fluid into the metal pipeline a device is used to orientate the pumped flow toward the outlet pipeline; this is the S Valve (G), and we will explain below how it works. We can divide the pumping group system into two completely separated subsystems: the “oil side” and “concrete side”. The oil side represents the power unit; the other is directly in contact with the concrete. A series of key words describe some important components and their states. The term “active” refers to the piston (C) that, in the operational phase, as it moves pushes out and compresses the concrete accumulated in the pipes. The second piston (B) is said to be “passive” because it is driven by the active one through a hydraulic circuit (the Slave, D) and is currently refilling its cylinder. It is not, in this phase of the cycle, directly involved in pumping the concrete. The two pistons are described as active or passive according to the phase of the cycle in which they find themselves. In addition to these important components, there are a number of hydraulic, electrical and mechanical elements in the machine. The oil side elements are the oil pump (P), the ducts (A), the proximity sensors (X) and the slave. The oil is pumped into the oil chambers (C1 or C4 depending on the phase of the work
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cycle) and leaves them (C4 or C1) through two pipes in which the oil pumps create a flow. These pumps send the flow in one side and, at the same time, receive another flow in the other. The alternating movement of the pistons is produced both by the changes in pressure in the chambers adjacent to them and by a system of proximity sensors. Figure 5.4 shows also a series of pipes called ducts, with retaining valves, which form a closed circuit on the oil side cylinder. The ducts function as brakes, as oil conduits (because of oil leakage) and as a piston interlock during the operating cycles. These components are very important because they permit the passage of oil from chamber C1 to C2 (or from C4 to C3) creating a balanced distribution of oil inside the circuit. The oil pump is put into reverse by the proximity sensors at the end of chambers 1 and 4 (C1 and C4). These are intercepted by the passive piston when it is almost at the end of its stroke, and give an electric signal to the oil pumps and to the S valve (G) in order to invert their sense of operation and position respectively. The concrete side elements, shown in Fig. 5.4, are the S valve and the concrete vessel (H). The S valve connects the end of the cylinder in the concrete side subsystem to the beginning of the metal pipeline. The switching (rotation) of this valve is, as mentioned earlier, triggered by a signal from the proximity sensors. The concrete vessel is continuously refilled by a concrete mixer and holds the fluid used during the cycles.
5.4 Experimental Tests Many experimental tests were performed with different values for the rpm of the pump and for the area of the reducing valve. The tests investigated how the system worked with three different pump speeds, several different values for the oil input flow and sensitivity analysis values of the area of the reducing valve. This allowed the behavior of the pumping group to be defined for a large range of operational values. Data were acquired using sensors that measured the stroke frequency of the pistons, their position (also velocity and acceleration as a consequence of the numerical calculation), pressure in all the chambers of the pumping group (the complete acquisition system included thirteen pressure sensors) and the flows of oil and concrete, as well as other less important measurements. The tests produced interesting results on the principal phenomena. Following are some experimental test results. Considering only one piston, the single working cycle is composed of two phases: • in the first (passive phase) there is an outflow of work fluid from the concrete vessel, • in the second (active phase) the work fluid is pumped along the metal pipeline through the S-valve.
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Fig. 5.5 Oil chambers pressure profile, varying rpm of the pump
The passage from one phase to the other is signaled by the position of the passive piston; this determines the direction of the pump and the position of S-valve. The oil pumps in the circuit are driven by a diesel engine, and by varying its rpm the oil flow can be regulated. Figure 5.5 shows the pressure profile of the fluid in the test circuit (oil side) as a function of diesel engine speed. Oil mass flow can also be changed by adjusting the plates of the oil pump that feeds the chambers of the cylinders. Figures 5.6 and 5.7 show the oil pressure profile in the pump chamber X when the oil input flow and the reducing valves area change: in the firs case it is noted the cycle time reduction and the consequent increasing of concrete flow while in the second case it is note an increasing of the pressure. In our tests water was used as the work fluid. The presence of work fluid in the concrete vessel represents a force to be overcome by the active piston and the load to which the piston is subjected is determined by the regulation of the area of proportional choking valve. The load conditions, that in the real situation are due to the pressure of concrete, are simulated by adjustable choking on the circuit. In this way the desired back pressure is obtained. Figure 5.8 shows the pressure profile inside the pump during some working cycles. The main phenomena highlighted in the experimental tests, are the high pressure peaks and dynamics of the pressure in the oil chambers and the pressure regime values during the working cycles and the flow of work fluid. An understanding of these phenomena helps us to conceive a system that can pump a lot of concrete (m3 /h) quickly, without creating external forces.
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Fig. 5.6 Pressure in oil chambers, varying the input oil flow
Fig. 5.7 Oil chambers pressure profile, varying the reducing valves area
To understand and solve the problem it was necessary to understand how the pumping group executed its working cycle. After analyzing the physical model and understanding the working cycle of the system, it was necessary to study all the phenomena described earlier using a mathematical and numerical model because was not easy to explain these phenomena by experimental means.
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Fig. 5.8 Pressure in the oil chambers
5.5 Mathematical Model The system was modeled with differential equations that described both the motion of pistons and the continuity of working fluids (oil and concrete). In general we can identify three sets of equations used to describe the system: the dynamics of the piston, the dynamic of pressures in the oil side and the dynamic of pressures in the concrete side (back pressure). All these equations define the mathematical model of the entire system.
5.5.1 Oil Continuity Equations Starting from the mass continuity equation for a low compressibility fluid it is explained the term that described the derivative of pressure. For example in oil chamber 1 (C1 in Fig. 5.4) the following formula is used: β(Q − Ay˙1 − Ci P − Ce P1 ) P˙1 = , V1
(5.1)
where Q = Qin − Qout gives the difference between the incoming and outgoing flow in chamber 1 while P = P1 − P2 gives the difference in pressure between two separate chambers (separated by the piston head). The values Qin and Qout are the incoming and outgoing flows, y˙1 is the speed of the piston, V1 is the volume of the chamber (variable), Ci and Ce are the coefficients of internal and external leakage of oil and β is the coefficient of compressibility.
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In order to verify such parameters as the commutation time of the hydraulic pump, the commutation time of the S valve, the oil compressibility coefficient β and outflow coefficients Ci and Ce , tests were carried out. The state variable P˙1 describes the variation of pressure within the chamber. Similar equations are written for all the other oil side chambers (C2, C3 and C4 in Fig. 5.4), whose volume is variable and is defined by position of pistons at specific points in time. The pipes that connect the oil pump to the cylinders are represented as constant volume chambers. For this type of element the following equation is used: βQ . P˙t = Vt
(5.2)
The flow expression in Eq. (5.1) is described by the equation between the oil pipes and chambers, or in the chambers through the ducts or slave. The expression of the flow is given by 2|Pi − Pi − 1| sign(Pi − Pi−1 ), (5.3) Qi = Ai Cqi ρoil where Qi is the flow that depends on the pressure difference between the chambers considered. The term Ai shows the area of transition between the chambers and with Cqi the coefficient that takes into account the various drop is represented. The initial and final flow of the oil pump [12] are assumed to be known at all times. The dynamic of the pump in terms of oil flow was implemented taking into account its data sheet and the experimental data during the trial. The coefficients are assumed to be constant and depend on the geometry of the cylinders.
5.5.2 Concrete Continuity Equations The equations on the CLS side are similar to those on the oil side. To calculate the variation of pressure, Eq. (5.1) is used. To define the flow of incoming and outgoing water, we have to take into account the S Valve dynamic. This element introduces a discontinuity; it connects the active chamber (concrete side) to the concrete pipeline and leaves the passive chamber free to get the concrete from the concrete vessel (passive chambers). To model this component two equations were used: one of these relating the metal concrete transport pipeline pressure with the concrete chamber pressure and the other the concrete vessel pressure with the CLS chamber pressure. These equations are 2|Pcls − Ptube | sign(Ptube − Pcls ), (5.4a) Qcls-tube = Acls Cq-tube ρcls 2|Pcls − Ptank | Qcls-tank = Acls Cq-tank sign(Ptank − Pcls ), (5.4b) ρcls
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where Qcls-tube is the flow between the chamber and the transportation pipeline, Qcls-tank represents the flow between the chamber and the concrete vessel. The pressure in the chamber, in the vessel and in the transportation pipeline are indicated respectively as Pcls , Ptank , Ptube . The value of the outflow coefficients Cq-tube and Cq-tank depends on the position of the S Valve. In Eqs. (5.4a) and (5.4b), Cq-tank can be adjusted from its maximum value (passive chamber) to zero (active chamber). When the chamber is active there will be a flow from the chamber to transport pipeline while that from the concrete tank to the chamber will be invalid. On the other side, in the other equation, it is possible to change the value of Cq-tube . The reversal follows a ramp trend. Another continuity equation was used to describe the concrete transport pipeline. Here the flows used were the chambers flow and the ambient flow.
5.5.3 Equations of Motion The system has 2 DOF and the equations describing the motion of the two pistons are P1 A1 − P2 A2 + Pcls1 Acls1 − Fatt1 + mg sin(α) y¨1 = , (5.5a) m P4 A4 − P3 A3 + Pcls4 Acls1 − Fatt2 + mg sin(α) , (5.5b) y¨2 = m where y¨1 and y¨2 are the accelerations of the pistons, P1 , P2 , P3 , P4 , Pcls1 , Pcls4 , are the pressure values in the chambers, Acls1 and Acls2 are the different piston head areas, m is the mass of the single piston, Fatt1 and Fatt2 are the friction forces and α is the slope of the pumping group. The force on the pistons thus depends both on the pressures in the chambers and on the piston head area, on the gravity force and on the friction force. The LuGre friction force model [5] for translational friction was used; generally this model is used for hydraulic actuators and is given in the literature as Fatt = (Fc + Fs ) sign(ν) + rν,
(5.6)
where Fatt is the friction between moving bodies. The friction force is simulated as a function of the relative velocity v and is assumed to be the sum of Stribeck, Coulomb and viscous components. Stribeck friction, Fs = (Fbrk − Fc )e−cν|ν| , is decreasing friction at low velocities the negatively sloped characteristics taking place at low velocities (see Fig. 5.9). Coulomb friction, Fc , results in a constant force at any velocity. Viscous friction, rν, gives a force directly proportional and opposite to the relative velocity. The sum of Coulomb and Stribeck friction close to zero velocity is often referred to as breakaway friction, Fbrk . The system of equations that describes the complete model pumping group is in the matrix form x˙ = Ax + B,
(5.7)
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Fig. 5.9 Translational friction model
where the state vector, x, is x = [y˙1 , y1 , y˙2 , y2 , Pt1 , Pt2 , P1 , P2 , P3 , P4 , P6 , P7 , PT ]T .
(5.8)
5.6 Comparison Between Numerical and Experimental Results In order to study the state variables and the resolution of the differential equations system, a numerical approach was implemented; in particular, the mathematical model that describes the problem was implemented in a Matlab program. In this approach both the non-linear equations of the mechanical part and the fluid dynamic part are considered. Through this program it is possible to plot a series of variables that are comparable with the experimental measurements. In this case, in order to show the validation of the model, there are some comparison between numerical and experimental results of a not controlled model. The numerical model takes into account the initial conditions and, particularly, the position and speed of pistons, the initial pressures in the oil side chambers and the pressure in the concrete side chambers due to the presence of the concrete. The output variables are the pressures and flows in all chambers in the system (transitional phase and regime phase), as well as the positions, speeds and acceleration of the pistons and the flows and pressures of the concrete. The dynamics of the oil pumps and S Valve were also taken into account. Figures 5.10, 5.11 and 5.12 compare the numerical and experimental values of some important variables. A comparison of the oil chambers pressures in Fig. 5.10 shows a good agreement both in the transitional zone (beginning and end of piston strokes) and in the regime zone; there is also an excellent agreement in terms of cycle time. Note how the pressure evaluated through experiments is well approximated by the mathematical model even if some significant differences in terms of absolute values are present. These differences are hard to be minimized improving the mathematical model since they depends on the effects of elasticity of structure, oil and
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Fig. 5.10 Numerical vs experimental results (oil chambers pressure)
Fig. 5.11 Numerical vs experimental results (CLS back pressure)
working fluid. While a more detailed model could give more precise results, it would be unnecessarily complex for the aim of the work. Figure 5.11 shows the agreement also in the CLS side, considering the backpressure given by concrete. Figure 5.12 shows the comparison between experimental and numerical results about the output flow of concrete. Also in this case the
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Fig. 5.12 Numerical vs experimental results (CLS back pressure)
comparison is good both in terms of shape and in time approximation. The little difference on the shape is due to the part of the real system not deeply modeled.
5.7 Control System Design The high pressure peaks in Figs. 5.5, 5.6, 5.7 and 5.8 are worth investigating because it seems that physically they represent the impact of the piston at the end of its stroke; this fact creates problems of induced forces, wear on the system, noise and induced vibration on the boom. For this reason it is important to check both the pressure difference between chambers 1 and 2 (or between 3 and 4, see Fig. 5.4 C1, C2, C3 and C4) and the flow of oil passing through the ducts. Owing to the presence of the ducts, the pumping group becomes a 2 DOF system in which the movement of the pistons becomes independent and phase displacement is possible. Concerning the piston’s position, the impact phenomenon simply depends on the phase displacement between the pistons. If this distance is so great that the active piston arrives at the end of its stroke before the passive one does not again intercepts the proximity sensor, the pump proceeds (for a few ms) to push the active chamber’s oil against the stopped actuator (C), immediately increasing the pressure. In order to eliminate this phenomenon of high pressure peak a piston displacement control system was introduced [9]. This system allows us to control the instant. In Fig. 5.13 there are all the geometric parameters used for the definition of the control system. In terms of pressure, there are important improvements with the reduction of pressure peaks and, as a result, with a decrease of the force induced on the boom.
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Fig. 5.13 Control system diagram
To avoid the impact of the pistons against the head of the cylinder a simple control system was devised based on the position of the pistons combined with a function called the “diff function”. The function diff is zero when yactive + ypassive = K and, in this case, the pistons are said to be in phase. The pistons are not in phase and the impact occurred if, defining the distance ys as the proximity sensor position and x as the length of the piston rod, the following conditions are satisfied: yactive = K, ypassive + ys > x.
(5.9a) (5.9b)
From Eqs. (5.9a) and (5.9b) the theoretical and ideal limit condition impact value can be defined as diff ≥ 0.
(5.10)
Alongside the definition of the ideal limit condition impact value it is important to consider the real condition, which takes into account the inertia of the pistons, the back pressure value due to the presence of concrete in the CLS circuit and the participation of some components of the hydraulic circuits. In terms of diff function and how was confirmed both by the numerical model and by experimental tests, the real limit condition impact value is diff ≥ 4 mm.
(5.11)
Figure 5.14 shows the diff function obtained by experimental tests compared with numerical ones. The agreement between model and real tests and confirms the importance of this variable. In some cases, the value of numerical results in the transitional part is different from the experimental one. This is due to the asymmetry of the test rig, not modeled in the numerical model. The concrete in the vessel is most important because it creates a back-pressure that, over certain values, produces a force contrasts the inertia and, at the same time,
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Fig. 5.14 Numerical vs experimental results (diff function)
Fig. 5.15 Comparison between controlled and non-controlled numerical results
eliminates the pressure peaks in chambers 1 and 4. All these physical phenomena can be checked and controlled through the diff function, defined by the pistons position. So the idea of the control system is to measure the function in a certain position of the active piston and to decide whether the control system should be activated or
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Fig. 5.16 Numerical application of control
Fig. 5.17 Oil chambers pressure, experimental comparison with and without control
not. When the active piston is 8 mm from the end of its stroke, if the diff function is diff = 4 or diff < 4, the pump maintain its default working mode; otherwise, if diff > 4, the signal is given in order to invert the motion of the pump. From Figs. 5.15–5.19 some of the results obtained with the numerical model and, the experimental test are shown. Figure 5.15 shows a purely numerical comparison of the pressure in the oil chambers. It is clear the exclusion of the pressure peaks maintaining the same work cycle time and the same shape. As confirmation
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Fig. 5.18 CLS pumped concrete, numerical comparison with and without control
Fig. 5.19 Percentage difference of pumped fluid, numerical comparison with and without control
of the numerical estimation, an experimental result of the same variables previously described, is shown in Fig. 5.17. The comparison is good and there is a total cancellation of the pressure peak, improving the functioning of the entire pumping group system.
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In fact the presence of the control system improves the synchronization of the pistons and eliminates the phenomenon of impact. Figure 5.16 shows the application of control during a work cycle, underling the expected value of real impact value of 4 mm from which the high pressure peak disappear. The control was applied at second 24. The lack of the high pressure peak makes it certain that there will be lower induced forces on the entire system and fewer problems in terms of noise, wear and vibrations. Moreover, as Figs. 5.18 and 5.19 show, the difference— in terms of CLS flow—between the controlled and not controlled system operating and the percentage difference of concrete pumped with or without control application are shown. With the application of the active control system, the percentage of losses concrete is minimal, and this is good because it means that the control does not affect the main parameter for the industry application, the pumping flow.
5.8 Prospects for the Future The mathematical and numerical models allow us to make a deeper study on the pumping group system. Moreover, the simple control logic brings an improvement on the machine without loss of performance. The good results of this study allow us to continue the research developing a complete system for pumping concrete, tacking into account the pumping group and the boom models, coupled one to each other. With regard on the future developments, the numerical models will be coupled; to obtain the model of a truck mounted concrete boom pump. This model will consider all the geometrical characteristics of the complete system and it will use the work fluid in order to define the forces exchanged between the pumping group and the boom. Moreover, the coupling between the mechanical system and the hydraulic system will permit to modify the model in its minimum detail. As for the single models, also in this case we will proceed with a validation of the numerical model and some tests will be done in order to have more sensibility on the main parameter: the oscillation of the boom.
5.9 Summary This paper presents and describes the results obtained by a comparison of numerical and experimental tests on a concrete pumping group system in various operating conditions. The mathematical model developed was implemented in a calculation code that allowed us to simulate the behavior of the pumping group. With the test rig was possible to verify the preliminary characterization of the fundamental parameters of the mathematical model. The numerical model, as demonstrated, can simulate the behavior of the pumping group in terms of pressure of the actuator chambers, the pressure and flow of the pumped fluid, and the position of the pistons.
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Using this model was possible to test the influence of the control system and verify what advantages it offered, and also to find optimal values for certain important parameters. The control system enables to reduce the pressure in the oil chambers (1 and 4) and the induced forces on the boom when the work cycle is commutated. By the way, the diff function gives a simple manner to verify the position of the pistons in order to reverse the S-valve working and the oil pump functioning. This approach improves the entire system working in terms of oscillations induced on the boom, noise and wear. As latest consideration, the quantity of work fluid pumped with controlled machine, is almost the same as in the system without control. In conclusion, the implementation of the control system makes the entire machine more manageable and suitable for construction site work.
5.10 Selected Bibliography • Anthony Esposito, Fluid Power with Applications, Pearson Prentice Hall, 2008, ISBN 0-1351-3690-3. Updated to reflect current fluid power technology and industrial applications, this book focuses on the design, analysis, operation, and maintenance of fluid power systems. Provide readers with realistic ways to obtain desired speeds of hydraulic cylinders and motors. Enhances understanding of the operation of hydraulic pumps and motors. • Raymond Mulley, Flow of Industrial Fluids: Theory and Equations, ISA-The Instrumentation, Systems, and Automation Society, ISBN 0-8493-2767-9. To describe the flow of industrial fluids, the technical literature generally takes either a highly theoretical, specialized approach that can make extracting practical information difficult, or highly practical one that is too simplified and focused on equipment to impart a thorough understanding. This book takes an approach that bridges the gap between theory and practice. • Igor J. Karassik, Pump Handbook, ISBN 0-07-034032-3. Handbook that provides practical data and know-how on the design, application, specification, purchase, operation, troubleshooting, and maintenance of pumps of every type. It is an essential working tool for engineers in a wide variety of industries all those who are pump specialists, in addition to those who need to acquaint themselves with pump technology. • J.M. Illston, Construction Materials: Their Nature and Behaviour, ISBN 0-41915470-1. This book provides a comprehensive coverage of the main construction materials for civil engineering and the construction. It gives an understanding of materials and a knowledge of their chemical and physical structure, leading to an ability to judge their behavior in service and construction. • A.K. Tamimi, Workability and Rheology of Fresh Concrete: Compendium of Tests, RILEM Technical Committee 145-WSM, Workability of Special Concrete Mixes, ISBN 2-912143-32-2. This book brought out by special tests, the main chemical— physical characteristics of fresh concrete and describes in detail the workability of building materials.
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Thanks to CIFA S.p.a. for financial support and opportunities.
References 1. ACI Committee 304: Placing concrete by pumping methods. ACI Mater. J. 92(4), 441–464 (1995) 2. Austin, S.A., Goodier, C.I., Robins, P.J.: Low-volume wet-process sprayed concrete: pumping and spraying. Mater. Struct. 38(276), 229–237 (2005) 3. Banfill, P.F.G.: The rheology of fresh cement and concrete. International Cement Chemistry Congress, Durban (2003) 4. Bartos, P.: Fresh Concrete: Properties and Tests. Elsevier, Amsterdam (1992) 5. Canudas de Wit, C., Olsson, H., Aström, K.J., Lischinsky, P.: A new model for control of systems with friction. IEEE Trans. Autom. Control 40(3), 419–425 (1995) 6. Ferraris, C.F., de Larrard, F.: Testing and modelling of fresh concrete rheology. National Institute of Standards and Technology, Internal Report 6094 (1998) 7. Jolin, M., Chapdelaine, F., Gagnon, F., Beaupré, D.: Pumping concrete: A fundamental and practical approach. In: ASCE, 10th International Conference on Shotcrete for Underground Support, Whistler, British Columbia, Canada, pp. 334–347 (September 2006) 8. Khulief, Y.A.: Vibration suppression in rotating beams using active modal control. J. Sound Vib. 242(4), 681–699 (2001) 9. Mei, T.X., Goodall, R.M.: Position control for a subsea pump system driven by a linear motor. Control Eng. Pract. 3(3), 301–311 (1995) 10. Mori, H., Tanigawa, Y.: Simulation methods for fluidity of fresh concrete. Mem. School Eng. Nagoya Univ. 44, 71–134 (1992) 11. Resta, F., Ripamonti, F., Cazzulani, G., Ferrari, M.: Independent modal control for nonlinear flexible structures: An experimental test rig. J. Sound Vib. 329(8), 961–972 (2009) 12. Tam, C.M., Tong, T.K.L., Wong, Y.W.: Selection of concrete pump using the superiority and inferiority ranking method. J. Constr. Eng. Manage. 130(6), 827–834 (2004) 13. Wang, T., Wang, G., Liu, K.: Simulation control of concrete pump truck boom based on PSO and adaptive robust PD. In: Chinese Control and Decision Conference CCDC (5192810), pp. 960–963 (2009) 14. Weber, R.: The transport of concrete by pipeline. Cement and Concrete Association (C & CA library translation No. 129), Londres (1963) 85 p 15. Zhou, S., Zhang, S.: Co-simulation on automatic pouring of truck-mounted concrete boom pump. In: Proceedings of the IEEE, International Conference on Automation and Logistics, pp. 928–932 (2007)
Chapter 6
Vibration Based Structural Health Monitoring and the Modal Strain Energy Damage Index Algorithm Applied to a Composite T-Beam R. Loendersloot, T.H. Ooijevaar, L. Warnet, A. de Boer, and R. Akkerman
Abstract A Finite Element based numerical model for a vibration based damage identification method for a thin-walled slender composite structure is discussed in this chapter. The linear dynamic response of an intact and a locally delaminated 16-layer unidirectional carbon fibre PEKK reinforced T-beam is analysed. The capabilities of the modal strain energy damage index algorithm to detect and localize a delamination is assessed. Both bending and torsion modes of the structure are used in the algorithm. Both an experimental set-up and a numerical model are discussed. Measurements are performed on an intact and an artificially delaminated structure, using a laser-vibro measuring system to determine the response to a force excitation. A commercially available Finite Element package is employed for the numerical model. The aim of the numerical model is to perform a parametric study. The study is preceded by an experimental verification of the numerical model. Subsequently, it is used to analyse the effect of the size and location of a delamination, as well as the number of data points employed, on the damage index.
R. Loendersloot () · A. de Boer Engineering Technology, Applied Mechanics, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands e-mail:
[email protected] A. de Boer e-mail:
[email protected] T.H. Ooijevaar · L. Warnet · R. Akkerman Engineering Technology, Production Technology, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands T.H. Ooijevaar e-mail:
[email protected] L. Warnet e-mail:
[email protected] R. Akkerman e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_6, © Springer Science+Business Media B.V. 2011
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Nomenclature Roman E f F FN G H i, j I J l Ld Ls LT M n N s T u U W1 W2 x, y, z
modulus of elasticity [N m−2 ] fraction [–] fractional strain energy [–] natural frequency [Hz] shear modulus [N m−2 ] height of stiffener [m] indices 2nd moment of inertia [m4 ] rotational moment of inertia [m4 ] length [m] length of delamination [m] start of delamination [m] total length [m] momentum [N m] natural mode index [–] total number [–] location of data lines [m] torque [N m] displacement [m] strain energy [N m] skin flange width [m] distance to data lines [m] cartesian coordinates
Subscripts B bending mode related value T torsion mode related value thrs threshold value Greek α β ε εmax ζ θ ν ρ
damage severity [–] damage index [–] relative error [%] maximum relative error [%] damping [N s m−1 ] angle [rad] Poisson’s ratio [–] volumetric density [kg m−3 ]
Mathematical || absolute value ¯ mean value ˜ damaged variant of parameter/variable ∂ partial derivative
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derivative/infinitesimal part summation
Abbreviations EMA Experimental modal analysis FBG Fibre bragg grating FRF Frequency response function MSE-DI Modal strain energy damage identification PEKK PolyEhterKatoneKatone SHM Structural health monitoring TRL Technology readiness level VB Vibration based
6.1 Introduction Development of structural health monitoring technologies for composite based structural components for aircrafts is one of the objectives of the European research program Clean Sky/Eco-design. The skin-stiffener based structures, of which for example wing sections of an airplane are built, are of special interest. The connection between the skin and the stiffener is vulnerable for delamination damage and must be frequently tested or the part is replaced prior to its end of life, which is cost and resource inefficient. Vibration based damage identification methods are promising as an alternative for the time consuming and costly non-destructive testing (NDT) methods currently available. The change of the dynamic properties is employed to identify damage such as delaminations. Damage detection based on a shift of the natural frequencies [19] allows for a detection, but localization of the damage is difficult, in particular if multiple damages are present. The modal strain energy damage index (MSEDI) algorithm [36] is therefore adopted. This method is one of the vibration based damage identification methods currently available and is based on the observation that local changes in the modal strain energy are a sensitive indicator of damage. The MSE-DI algorithm is mainly applied to flexural bending. Duffey et al. [13] are the only ones who included torsion modes and vibrations in their research. Including the effect of damage on the torsion modes of (complex) thin-walled, slender structures may enhance the sensitivity of the MSE-DI algorithm, similarly to the enhancements achieved by separating the axial and bending strain energies [25] and is therefore investigated. This chapter discusses the numerical model used to explore the capabilities of the MSE-DI algorithm. The numerical model is validated based on experiments discussed in detail in [30]. It is not the intention to assess the validity of the numerical model in depth here. The main focus is on the application of a numerical model in the analysis of a damaged structure. To this end, a parametric study is performed by varying the size and location of the delamination, as well as the number of data points used in the analysis. The numerical model developed can be used to support
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Fig. 6.1 Position of the vibration based structural health monitoring (VB-SHM) techniques with respect to other damage identification methods
the implementation of a damage identification method in an application, without an extensive and costly experimental program. An overview of the current state of the art is presented in Sect. 6.2 prior to a detailed description of the structure in Sect. 6.3. The theory of the modal strain energy method is subsequently explained in Sect. 6.4. In Sect. 6.5, the finite element model that is developed to perform the parametric study is discussed. The experimental work, comprising laser-vibro based dynamic measurements are the subject of Sect. 6.6. The comparison between the numerical model and the experimental results, as well as the results of the parametric study are presented and discussed in Sect. 6.7. Then, the readers is presented with a prospect to the future, Sect. 6.8: what can be expected from the future research on vibration based structural health monitoring technologies given the results discussed here? The chapter ends with a summary and a selection of interesting literature for those interested to explore this subject further (Sect. 6.9).
6.2 Overview of the State of the Art 6.2.1 Vibration Based Structural Health Monitoring Structural health monitoring (SHM) is the general process of implementing a strategy for damage identification by non-destructive testing (NDT). This process involves the definition of potential damage scenarios for a structure, the observation of the structure over a period of time using periodically spaced measurements, the extraction of damage sensitive features from these measurements and the analysis of these features to determine the current state of health of the structure. The output of this process is periodically updated information regarding the ability of the structure to perform its intended function in consideration of the applied loadings and aging degradation resulting from the operational environments. An extensive number of NDT technologies can be applied to achieve this formally formulated objective, see for example [41]. Vibration based SHM (VB-SHM) technologies are only a subset of this wide variety, as indicated by Fig. 6.1. VBSHM can offer some specific advantages compared to other NDT technologies, in
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particular regarding the ability to assess the health of the structure online, i.e. during operation of the structure.
6.2.2 Modal Strain Energy Damage Index Algorithm In the early 90s, Stubbs et al. [21, 36] presented the modal strain energy damage index (MSE-DI) algorithm and applied it to a steel bridge. The core of the MSE-DI algorithm consists of the phenomenon that the modal strain energy of a vibrating structure exhibits local discontinuities due to a loss in stiffness. The modal strain energy is determined from a modal analysis. In general, only a limited number of natural mode shapes is required to determine the damage index and does the algorithm not rely on a normalization of the modes shapes [15, 16]. The magnitude of the damage index is used to estimate the severity of the damage [22]. Farrar and Jauregui [15, 16] compared the MSE-DI algorithm to methods based on the mode shape curvature [32], flexibility coefficients [31], stiffness coefficients [43] and the curvature of the uniform load surface [38]. They used both experimental data and a numerical model of the steel bridge discussed in [36]. Their final conclusion is that the MSE-DI algorithm is the most powerful method. Alvandi and Cremona [2] also assessed different vibration based damage identification techniques. They focussed on the performance of different methods considering experimental data including a certain level of noise. Strain energy based methods are concluded to be the most stable methods. The 1D MSE-DI algorithm is applied to 3D off-shore frame structures by Li et al. [25, 26]. Failure is assessed at truss level. The results are promising, but the stiffness reductions are relatively large (5% stiffness reduction for a complete truss element) compared to the stiffness reductions caused by delaminations as discussed here. The main advantage is that the structural safety can be monitored during operation despite the difficult accessibility of the structure or visibility of the damage. This advantage also applies for (skin-stiffener) composite components, which are subject of this research, or for composite sandwich structures, as studied by Kumar et al. [24]. Here, the 1D algorithm is applied to a thin-walled, slender composite structure. It could be argued to implement 2D or 3D theory. However, it was shown by [11] that the extension from 1D to 2D theory for the analysis of plates does not increase the sensitivity of the MSE-DI algorithm to detect and localize damage. Choi et al. [9, 10] and Kim et al. [20] concluded from their research to damage identification on plate structures that a compliance based damage index outperforms the strain energy based damage index algorithm, but they do not provide a comparison with a 1D approach. The MSE-DI algorithm relies for practical applications on dynamic measurements. Inevitably, these dynamic measurements suffer from noise resulting in a probability of false damage identifications. Statistical methods are therefore employed to assess the performance of the damage detection method [2]. The dynamic
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Fig. 6.2 Traditional double L-shaped stiffener versus the new stiffener concept developed by Stork–Fokker AESP and the NLR. (a) Traditional stiffener, (b) new stiffener concept
response is often measured in the time domain [30, 34]. Subsequently, the data is transformed to the frequency domain in which the modal parameters are determined. This method always requires an excitation, either as impact or (chirp) frequency sweep [30]. Alternatively, the time signal of vibrations occurring during operation can directly be used to determine the mean strain energy, by averaging over a fixed time interval. Choi and Stubbs [8] and Kim et al. [23] showed that this method can be used to calculate the strain energy damage index and detect and localize damage equally well compared to the modal strain energy damage index.
6.3 T-Beam with T-Joint Stiffener The structure investigated here is a composite skin-stiffener section. This type of stiffening is frequently used in aerospace components to increase the bending stiffness of the component without a severe weight penalty. Recently, a new type of skinstiffener connection, depicted in Fig. 6.2, was developed by Fokker Aerostructures, in collaboration with the Dutch National Aerospace Laboratories (NLR). A PEKK (polyehterkatonekatone) injection molded filler is used as a connection. PEKK is a semi-crystalline thermoplastic plastic. The fabrication process for this type of skinstiffener joint is presented in [29] and is referred to as a T-joint. The base structure analyzed in this paper is shown in Fig. 6.3. This T-beam with the dimensions as indicated in the figure, is used for the experimental verification of the model. The composite is built from 16 individual plies of uni-directional coconsolidated carbon AS4D reinforced PEKK. A [0/90/0/90/0/90/0/90]s lay-up was used. A typical damage occurring to composite structures is delamination. The location with the highest risk of failure of the structure is the injection molded thermoplastic T-joint profile which connects the base to the stiffener. Hence, the delamination is assumed to be located at the interface between the base and the stiffener.
6.4 Theory of the Modal Strain Energy Damage Index Algorithm The strain energy of a vibration mode is referred to as the modal strain energy of that mode. Consequently, the total modal strain energy is the sum of the modal strain
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Fig. 6.3 [0/90/0/90/0/90/0/90]s laminate lay-up and dimensions. The global coordinate system xyz is indicated as well as the material orientations 123 in the skin and the stiffeners
energy contributions of all modes considered. The modal strain energy is calculated by linking the deformation of a structure to the strain. A distinction must be made between axial, flexural and torsional deformation–strain relations. Only the bending and the torsion strains are analyzed, since the T-joint is a slender structure. The mechanical relations read ∂ 2 ux/y My/x = , 2 EI y/x ∂z
∂θxy Tz = , ∂z GJ xy
(6.1)
with u the displacement, θ the rotation, M the bending moment, T the torque, EI and GJ the bending and torsion rigidity of the beam respectively. The cartesian global coordinate system xyz is defined in Fig. 6.3 (the local material coordinate systems are indicate by 123). The subscript ‘(x/y)’ refers to either x or y. The strain energy is found by integrating the squared strains over the length l of the structure [13] 2 ∂ ux/y 2 ∂θxy 2 1 l 1 l EI y/x dz, UT = GJ xy dz. (6.2) UB = 2 0 2 0 ∂z ∂z2 Consider the structure to be vibrating in the nth bending mode. The displacement amplitude for the mode shape is u(n) x/y (z). As a result, the modal strain energy of the nth mode is written as 2 (n) ∂ ux/y (z) 2 1 l (n) UB = EI y/x dz. (6.3) 2 0 ∂z2 Subsequently, the structure is discretized in N elements in axial (z) direction. The (n) strain energy UB,j , due to the nth mode and associated with the j th element is then given by 2 (n) N ∂ ux/y (z) 2 1 zj (n) (n) (n) UB,j = (EI y/x )j dz with U = UB,j . (6.4) B 2 zj −1 ∂z2 j =1
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Similar quantities can be defined for a damaged structure, using the mode shapes u˜ (n) of the damaged structure. The derivation for the torsion modal strain energy components follows the same route. The rotation angle θxy is defined as ∂uy (x, z) ∂ux (y, z) + . (6.5) ∂x ∂y The displacements in the x-direction ux are small compared to those in the ydirection, as is the first derivative. The second term is therefore omitted from the definition of the strain energy. The strain energy of the nth torsional mode associated with the j th element hence reads 2 (n) ∂ uy (x, z) 2 1 zj (n) UT ,j = (GJ xy )j dz. (6.6) 2 zj −1 ∂x∂z θxy (z) =
Note that information of the change of the displacement in x- and z-direction is required to calculate the torsional strain energy, whereas the change of the displacement in z-direction suffices to calculate the bending strain energy. The local fractional strain energies, as defined by Cornwell et al. [11], are (n) FB/T ,j
=
(n) UB/T ,j (n) UB/T
(n) F˜B/T ,j =
,
(n) U˜ B/T ,j (n) U˜ B/T
(6.7)
,
for the intact and damaged structure respectively. The fractional strain energy remains relatively constant in the undamaged element, under the assumptions that the (n) ˜ (n) damage is primarily located at a single element (FB/T ,j = FB/T ,j ) and that the rigidity does not change over an element [2, 11]. It can then be derived for element j that (EI y/x )j
zj ∂ 2 u(n) x/y (z) 2 ∂z2
zj −1
dz
(n)
(GJ xy )j
UB zj ∂ 2 u(n) y (x,z) 2 zj −1
∂x∂z
dz
(n)
UT
=
=
y/x )j (EI
xy )j (GJ
zj ∂ 2 u˜ (n) x/y (z) 2 zj −1
∂z2
dz
(n) U˜ B zj ∂ 2 u˜ (n) y (x,z) 2 zj −1
∂x∂z
(n) U˜ T
(6.8)
, dz .
(6.9)
These equations are rearranged to obtain the quotient of the flexural and torsional stiffnesses, assuming the change of the total stiffness is negligible [2, 11], (n) f˜B,j (EI y/x )j ≡ (n) , y/x )j (EI fB,j
f˜T(n) (GJ xy )j ,j ≡ (n) ,
xy )j (GJ fT ,j
(6.10)
with the fractions fB,j and fT ,j (with and without ˜ ) equal to the integrals zj 2 (n) 2 l 2 (n) 2 ∂ uy (z) ∂ uy (z) dz dz, (6.11) fB,j = 2 ∂z ∂z2 zj −1 0 l 2 (n) zj 2 (n) ∂ uy (x, z) 2 ∂ uy (x, z) 2 dz dz. (6.12) fT ,j = ∂x∂z ∂x∂z zj −1 0
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The local damage index β for the j th element can be obtained by using the definition (n) proposed by Stubbs et al. [36], which is a summation of the fractions fB/T ,j over the number of modes considered N N (n) (n) ˜ βB/T ,j = fB/T ,j fB/T ,j . (6.13) n=1
n=1
The modal strain energy damage index algorithm also offers the possibility to estimate the severity of the damage. The contribution of each mode to the damage index value, as defined in (6.13) can be determined by (6.10). The higher the relative participation of the lower modes to the damage index β, the higher the damage severity, since the effect of the damage is affecting a larger section of the structure. The damage severity is defined by the parameter α [2] α=
1 − 1. β
(6.14)
The subject of damage severity is not further explored in this chapter.
6.5 Finite Element Model The numerical model is based on a finite element (FE) model. The objective was to develop a comparatively simple model with low computational requirements, allowing a large number of cases to be evaluated. Hence, a balance must be established between accuracy of the solution and complexity of the model. Experimental results, presented in Sect. 6.6 and more elaborately in [30], were used for a basic validation and to assess the validity of the approximations made in the model. The first approximation concerns the geometry. The PEKK injection molded filler for the T-joint is not incorporated in the model. The geometry of the filler is relatively complex to model. Additionally, the exact dimensions change during the autoclave cycle, in which the skin and stiffener are co-consolidated and obtain their final shape [29]. Effectively, the thermoplastic profile will add a minor amount of stiffness and omitting the T-joint will only result in slightly lower frequencies for the torsional natural frequencies. The bending natural frequencies will be hardly affected, due to the comparatively large contribution of the bending stiffness of the stiffener to the overall bending stiffness. Both the skin and the stiffener consist of 16 layers of fabric but have a small thickness compared to their length and width. The structure analyzed here, is built from uni-directional layers allowing the use of the classical laminate theory (CLT) [40] to calculate the homogenized material properties of the composite laminate straightforwardly. The composite laminates are modelled using a single layer of shells, with three integration points over the thickness and an orthotropic material definition. The material data, provided by Fokker Aerostructures, is obtained from static tests performed with samples of the batch used for the manufacturing of the T-beam that was measured [30]. The resulting homogenized material data used in the model is listed in Table 6.1.
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Table 6.1 Homogenized material properties of the uni-directional composite, based on the measured material data E1 = E 2
E3
ν12 = ν13
ν23
G12 = G13 = G23
ρ
72 GPa
10.5 GPa
0.30
0.45
5.56 GPa
1590 kg m−3
Fig. 6.4 Cross-section of the T-beam, that is extruded to obtain the T-beam model
Modelling a delamination generally involves non-linearities, due to the opening and closing of the delamination during cyclic deformation of the structure and friction between the two faces of the delamination.1 This requires explicit solvers and time-domain simulations. Small time-steps must be used, given the frequency range of interest (up to 1000 Hz), as is also indicated by Ullah and Sinha [39]. Alternatively, a ‘free mode’ model [12] can be used, which sets an upper bound for the effect of a delamination on the dynamic response of the structure: The nodes of the skin and the stiffener at the delamination share the same location, but there is no interaction defined between the stiffener and the skin. Hence, the elements of the skin and stiffener can penetrate each other at the location of the delamination, effectively causing a lower stiffness and absence of damping induced by contact between the skin and stiffener during cyclic closing of the delamination. However, it allows linear theory to be applied and simulations can be performed in the frequency domain, employing a linear perturbation based analysis. This saves a considerable amount of computation time. Evidently, the validity of this assumption must be checked by a comparison with experimental results or alternatively a non-linear numerical simulation in the time domain. The dimensions of the first T-beam modelled correspond to the T-beam used in the experiments [30] (see Fig. 6.3). The model is created from an extrusion of a T-shaped cross-section, depicted in Figs. 6.4 and 6.5. Nodes are positioned at three lines along the length of the T-beam (point O—(x, y) = (0, 0)—and points s1 and s2 —(x, y) = (±0.041, 0) m—in Fig. 6.4), corresponding to the positions where the velocities are measured [30]. The dimensions in Figs. 6.4 and 6.5 are collected in 1 Damage
propagation is not accounted for, as the time scale of damage propagation must be significantly smaller than the time scale of the vibration. If not, the damage detection will coincide with actual failure of the structure.
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Fig. 6.5 Bottom view of the T-beam in the xz-plane Table 6.2 Dimensions and number of elements in the different sections of the T-beam. Note that the width to the right and left are equal for a symmetric T-beam Dimension
[mm]
Section
# Elements
Edge length [mm]
W1 W2 H
50 41 40
W2 to W1 0 to W2 0 to H1
2 8 10
4.5 4.3–5.6 3.75–5.0
Ls Ld LT
300:700 10:100 1000
0 to Ls Ls to Ls + Ld Ls + Ld to LT
2:20
10–13 5 10–13
Fig. 6.6 Cross-sectional view in the yz-plane at x = 0 of the T-beam showing the delamination. The open circles represent the nodes connected to both the skin (thick line at the bottom) and the stiffener, the filled circles represent two nodes, one connected to the skin, the other to the stiffener. Note that only nodes at the skin-stiffener interface are indicated by markers
Table 6.2. The clamping area is not included in the model (see Fig. 6.3). The T-beam is fully constrained at the clamped end of the beam (at z = 0). As mentioned, the delamination is modelled by defining nodes that share the same location, but are either attached to the skin or to the stiffener. The region of the delamination is schematically depicted in Fig. 6.6, which presents a crosssectional view in the yz-plane at x = 0. The open circles represent single nodes connected to both the skin and the stiffener, whereas the filled circles represent double nodes, which are either connected to the skin (visualized by the thick line at the bottom) or to the stiffener and hence form the delamination. The distance from the left end (z = 0) to the start of the delamination Ls is varied, as is the length of the delamination Ld . Only nodes at the interface between the skin and the stiffener are marked with a circle.
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A structured mesh of 4-node Shell elements with reduced integration (A BAQUS© element type S4R) and three integration points over the thickness (Simpson’s rule for shell section integration) is used. The width of the base is divided in 20 elements, the height of the stiffener in 10 elements, resulting in element edge lengths from 3.75 mm to 5.6 mm. The element edge length in z-direction is approximately 10– 13 mm (compared to a total length (LT ) of 1000 mm, see also Table 6.2), resulting in element aspect ratios ranging from ∼1.8 to ∼3.5. Dynamic effects of for example the delamination will be more accurate if the element edge length decreases. Hence, the detection of delaminations can depend on the element edge size. The element edge length in z-direction is kept at 5 mm for all delamination lengths to avoid this effect. As a consequence, the number of elements in the delamination increases proportional with the increase in length.
6.6 Experimental Analysis of the T-Beam Vibration measurements were performed on an intact and a damaged T-beam. The damaged T-beam contains a 100 mm long artificial delamination right under the Tjoint. The delamination starts at 500 mm from the clamping and ends at 600 mm from the clamping (see Fig. 6.5). The delamination was created by inserting a 0.1 mm thick Polyimide film before consolidating the beam in the autoclave. An elaborate discussion on the experimental set-up and the results obtained is found in [30]. The experimental set-up, its most relevant characteristics and results are discussed in this section. The complete dynamic set-up and data acquisition system is presented in Fig. 6.7. The T-beam is clamped at one side, employing the hydraulic clamps of an Instron 8516 fatigue system. The robustness of frequency response function (FRF) measurements was analyzed by changing the clamping conditions [30]. The T-beam was excited by a spring suspended shaker, which was connect to the beam at a fixed point with a stinger. A chirp excitation [34] was applied to the structure. A laser-vibro meter was used to measure the vibrations of the structure. The laservibro meter was mounted on a traverse system with a range of 590 mm. The FRFs between a fixed point of excitation and 30 measuring points along 3 parallel lines over the length of the T-beam were measured (points L1/M1/R1 to L30/M30/R30, a total of 90 points), as indicated in Fig. 6.8. Note that both the measured data points and the evaluation points used in the numerical model are located at x = 0 mm and x = ±41 mm (s1 and s2 in Figs. 6.4 and 6.5 and W2 in Table 6.2). The velocities at all grid points were recorded by a S IGLAB© data acquisition system. A frequency range of 25–1025 Hz, with a resolution of 0.313 Hz was used. A measurement at each grid point consists of 20 averages, without overlap. The multi point FRFs measured are employed to estimate the modal parameters: the natural frequencies, the damping values and the mode shapes [34]. The experimental modal parameters for the structure are obtained by fitting orthogonal polynomials on a set of FRFs (experimental modal analysis (EMA) with global curve fitting method [33]). ME’ SCOPE© modal analysis software is employed for
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Fig. 6.7 Dynamic set-up and data acquisition for the experimental investigation
this task. A typical FRF (single point from R1, see Fig. 6.8) of the intact and delaminated T-beam is shown in Fig. 6.9. The FRF of the delaminated beam (gray dashed line) clearly differs from the FRF of the intact beam for frequencies above approximately 350 Hz. The second natural bending frequency (FN ≈ 215 Hz) is hardly affected, whereas the fourth natural bending frequency exhibits a shift of approximately 20 Hz (FN ≈ 735 Hz versus F˜N ≈ 715 Hz). The seventh torsion frequency (FN ≈ 350 Hz) is hardly affected. This applies to nearly all torsion modes in contrast to the bending modes. The natural frequencies of the intact and delaminated beam are collected in Table 6.3. A distinction is made between the natural bending and torsion frequencies. The viscous damping coefficients ζ were also determined. Not all natural frequen-
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Fig. 6.8 Location of the excitation point and the 3 × 30 measurement grid points at the T-beam (bottom view in the xz-plane). The 60 mm wide light gray area at the right-hand side of the T-beam is the clamped area
Fig. 6.9 Comparison of the frequency response functions (FRF) of the intact and delaminated T-beam. A distinct shift in the higher natural bending frequencies can be observed, whereas the natural frequencies of the torsion modes remain relatively unaffected. (Single point FRF from point R1)
cies could be measured. First of all, the first natural torsion frequency is lower than the lower limit of the excitation frequency (25 Hz). Other natural frequencies, for example the 6th intact natural bending frequency and the 6th natural torsion frequency (both intact and delaminated), could not be determined due to the low response. The point of excitation almost coincides with a point of zero amplitude in that case. Coupling between bending and torsion modes was observed in some cases, in particular for the 2nd bending and 5th torsion modes of the delaminated T-beam. The viscous damping coefficients ζ are also included in the analysis, as it is expected that the damping is also affected by the damage. This increase of damping is to be attributed to loss of energy mainly due to contact between the surfaces of the delamination. The lower natural bending frequencies indeed show a higher damping, but this trend is not confirmed by the damping values of the higher natural
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Table 6.3 Experimentally determined natural bending and torsion frequencies FN [Hz] and viscous damping coefficients ζ [%] for the intact and delaminated T-beam nr
Bending modes FNintact
1
37.2
ζ 0.272
Torsion modes FNdelaminated 36.7
ζ
FNintact
0.414
–
ζ
FNdelaminated
–
–
ζ –
2
217
0.194
211a
0.387
65.5
1.014
63.1
0.917
3
506b
0.018
493b
0.079
117.8
0.602
116.1
0.532 1111
4
736
0.239
714b
0.197
163.7
1.124
162.1
5
840
0.254
816
0.200
212
0.494
211a
0.387
6
–
–
886
0.048
–
–
–
–
7
934
0.131
920
0.198
351
0.210
348
0.095
8
994
0.172
996
0.209
0.942
–
–
444
9
616
0.726
601
0.619
10
699
0.124
699
0.018
11
826
0.256
827b
0.001
12
971
0.273
973
0.538
a Clear
coupling between this bending and torsion mode
b Bending modes showing some coupling effects, effects caused by point of excitation or other kind
of distortions
bending frequencies and the natural torsion frequencies. This is ascribed to inaccuracies in calculating the damping, also because a relatively large amount of scatter was observed in the damping values measured during the various experiments. Finally, the FRF data was used to determine the mode shapes. The ith mode shape is constructed by evaluating the FRF data of all points measured at the ith natural frequency FN(i) [34]. The mode shape of the 5th bending frequency for the intact and delaminated T-beam is shown in Fig. 6.10. Note that the mode shapes are not complete, as there are no data points between L30/M30/R30 and the clamped end of the beam. The stiffener is also not visible in the mode shape plots, as there are no data points on the stiffener. The mode shapes clearly differ at the location of the delamination. However, there are also smaller differences in the amplitude of the mode shape further away from the location of the delamination: the global behavior of the T-beam is affected by the local delamination in this case. These experimental results show that the distinction can be made between an intact and delaminated T-beam. However, the location of the delamination and the damage severity can not be determined based on the information in the FRFs and the differences between the FRFs. The information contained in the change of the mode shape is included to accomplish this task. Here, the modal strain energy damage index algorithm, as presented in Sect. 6.4, is implemented. The results are discussed in the next section, where the numerical and experimental results are compared.
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Fig. 6.10 Normalized mode shapes (amplitude A) of the 5th bending mode. The delamination between 500 and 600 mm from the clamping can be easily observed. Iso-view on top, yz-view at the bottom. (a) Intact, FN = 840 Hz, (b) delaminated, FN = 816 Hz
6.7 Results and Discussion A number of geometrical parameters in the model was changed in order to assess the capabilities of the MSE-DI algorithm to detect and localize a delamination. Firstly, the size of the delamination is varied between 10 mm and 100 mm (1% to 10% of the total length of the T-beam). Secondly, the location of the delamination is varied between 300 mm and 700 mm from the clamping. Thirdly, the perpendicular distance between the delamination and the evaluation points used to determine the damage index is assessed. Subsequently, the number of evaluation points is varied. Finally, the shape of the T-beam is varied by increasing the width asymmetrically. This increases the influence of a delamination on the torsion mode shapes. The nodal displacements u in all three cartesian directions at (x, y) = 0 and x = ±W2 (see Fig. 6.4 and Table 6.2) are collected for each natural mode. The general procedure to calculate the damage index from the nodal displacements of the natural modes is presented in Fig. 6.11. The number of nodes in z-direction equals 101. The nodal displacements are mapped on an equidistant grid of 65 points in z-direction and the spline is evaluated at 129 points (the total number of elements N in (6.4) equals 128). A cubic spline is fit, employing the standard spline function available in M ATLAB© . The first 6 bending modes and the first 12 torsion modes of the intact and delaminated T-beam were used to calculate the damage indices (βB and βT in (6.13) respectively). Erroneous peaks are observed if higher bending modes are included. The cause of these disturbances is not investigated since the first 6 modes contain sufficient information and including higher modes also requires a larger number of data points to be captured, which is against the objective to reduce the number of evaluation points.
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Fig. 6.11 General procedure to calculate the modal strain energy damage index, based on the nodal displacements of each natural mode shape of a structure
Inherently to the definition, the damage index β equals unity if no damage is present, whereas the value is larger than unity in case of a loss of stiffness indicating damage. The height of the peak is a measure for the severity of the damage. However, the damage index β does not provide absolute values. The value of the damage index β depends on the various parameters, amongst which are the distance from the damaged location and the number of evaluation points used to interpolate the spline functions. In addition to this, the signal generally includes noise. Consequently, a threshold must be set to distinguish the difference between noise related peaks and damage related peaks. The threshold should be set such that the probability of detection is maximized (number of false negatives is minimized) while the probability of false alarms (number of false positives) is restricted to an acceptable level [2]. No noise is added to the data here, as the objective is to investigate the capabilities of numerical parametric studies. However, the effect of noise is taken into account by demanding that the peak value of the damage index must be significantly higher than the mean value to detect the data.
6.7.1 Validation of Numerical Model A basic validation of the numerical model is performed by comparing the measured natural frequencies and mode shapes with the ones predicted by the numerical model. The natural bending frequencies in the plane of the base (xz-plane) were not measured and are therefore omitted from the comparison. The natural bending frequencies and the natural torsion frequencies below 1000 Hz, measured and calculated are presented in Fig. 6.12. The experimental values are obtained from Ooijevaar et al. [30]. The averaged standard deviation for the measured natural frequencies was found to be 0.3 and 0.75 for bending and torsion modes respectively [30]. The numerical results show only small differences compared to the measured natural frequencies.
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Fig. 6.12 Comparison between calculated (n) and measured (e) natural bending and torsion frequencies. The light gray bars refer to the intact T-beam, whereas the dark gray bars refer to the delaminated T-beam. (a) Natural bending frequencies, (b) natural torsion frequencies
The natural frequencies and the relative error are presented in Table 6.4. The relative error is defined as experimental
ε=
FNnumerical − FN
experimental
FN
· 100%.
(6.15)
The measured natural frequency is taken as the reference, since the aim here is to compare the numerical results with the experimental results and adapt the numerical model if necessary. It is recommended to change the definition of the error with the numerically calculated frequencies FNnumerical of the intact T-beam (‘base configuration’) as the reference if a numerical analysis is performed. This applies in particular if a parametric numerical study to the changes in the natural frequencies is performed. The maximum absolute error εmax = max |ε| ,
(6.16)
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Table 6.4 Natural frequencies [Hz] calculated by the numerical model, including the relative error [%] with respect to the experimentally determined natural frequency [30] (the natural frequency is not measured in case no error value is given). The absolute maximum error is indicated by the bold-face numbers, the mean value is based on the absolute error values nr
Bending modes FNintact
Torsion modes FNdelaminated
ε
FNintact
ε
FNdelaminated
ε
ε
1
39.05
4.97
39.00
6.27
20.87
–
20.87
2
219.04
0.94
218.27
3.45
63.91
−2.43
63.9
3
504.76
−0.25
496.55
−0.72
110.91
−5.85
110.88
−4.41
4
723.33
−1.72
682.38
−4.43
163.85
0.09
163.81
−0.12
5
831.76
−0.98
778.67
−4.57
224.51
5.90
224.28
6
889.66
–
874.49
–
294.29
7
934.87
0.09
901.48
−2.01
374.30
373.70
7.39
8
981.57
−1.25
977.26
−1.88
465.25
–
465.08
4.75
– 6.64
294.21
– 1.75
5.79 –
9
567.98
−7.80
566.65
−6.49
10
682.68
−2.33
682.52
−2.36
11
809.89
−1.95
808.61
−2.22
12
949.96
−2.17
949.47
−2.42
ε¯
1.46
3.08
3.91
3.87
is indicated by bold-face numbers in Table 6.4. The mean value is based on the absolute error values ε¯ =
N 1 |εi |, N
(6.17)
i=1
with N the number of natural frequencies for which the error is calculated. The error ε is defined as a signed number rather than an absolute value, to identify whether the numerically calculated frequency is higher or lower than the measured frequency. The maximum relative error is between 5% and 7.5% for all cases, but the mean relative error is larger for the torsional frequencies compared to the bending frequencies. This is explained by imperfections in the clamping [30] in the experimental set-up and the omittance of the thermoplastic profile of the T-joint in the numerical model. The imperfections in the clamping result in lower and the omittance of the profile in higher natural frequencies for the measurements compared to the numerical results. Hence the relative error can positive or negative. The mean relative error for the bending modes of the delaminated T-beam is slightly larger compared to that of the intact T-beam (3.08% versus 1.46%), which is attributed to the simplifications in the modelling of the delamination. This difference is not observed for the torsion modes, but it is also noted that the torsional natural frequencies are hardly affected by the delamination (light gray bars versus dark gray bars in Fig. 6.12). The model is not optimized to improve the correspondence between experimental and numerical results.
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Fig. 6.13 Damage indices for delamination lengths varying between 10 and 100 mm. A peak in the damage index indicates damage. Delamination starting at (a) 300 mm from the clamping, (b) 500 mm from the clamping, (c) 700 mm from the clamping
6.7.2 Length and Starting Point of Delamination The damage index β, Eq. (6.13), for delamination lengths Ld varying from 10 mm to 100 mm is shown in Fig. 6.13. The displacements uy at the center grid line (x = 0) are used. The three graphs show the results for three different starting points of the delamination Ls : 300 mm, 500 mm and 700 mm. The delamination can be clearly detected and localized if the length of the delamination is at least 20 mm (2% of the T-beam’s length). The peak in the damage index graph for a delamination of 10 mm at a starting point of 700 mm from the clamping (Fig. 6.13(c)) is too small to guarantee a successful detection if noise would be added to the model. The highest damage index value is found for delaminations of 30–50 mm (3rd to 5th rows in Fig. 6.13). The higher value of the damage index can be caused by several factors. Firstly, a relatively high damage index is likely be found if the location of the delamination coincides with or is close to a maximum amplitude of one or more of the modes included. Secondly, the largest discontinuities in the displacements are found at the ends of the delamination. A peak value is found if the edge element j for which the damage index is determined (βB/T ,j , (6.13)) coincides with one of the ends of the delamination. However, the damage index is not an absolute value, as discussed in the introduction of this section. No conclusions
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Fig. 6.14 Damage indices for delamination lengths varying between 10 and 100 mm, measured at 0.041 m from the center line of the T-beam. The tip of the delamination is located at 500 mm from the clamping
can be drawn from the height of the damage index other than that the probability of detection is higher for higher values of the index. There are no significant differences found in the damage indices of delaminations at varying distance from the clamping, as shown in Fig. 6.13. A difference could be expected, since the strains close to the clamping are higher (higher curvature) compared to the strains further away from the clamping. However, this effect is diminishing for increasing frequency, whereas the contribution to the damage index increases with increasing mode number; the damage is in that case always relatively close to the maximum amplitude of at least one of the modes shapes included.
6.7.3 Position of Evaluation Points One of the main advantages of the MSE-DI algorithm is that damage can be detected and localized without measuring at the damaged spot itself. The results presented in Sect. 6.7.2 were obtained by analyzing the nodal displacements at x = 0, i.e. coinciding with the location of the delamination. Here, the damage index is determined using the nodal displacements at x = 0.041 m (corresponding to the x-coordinate at which the velocities were measured by Ooijevaar et al. [30]). The resulting damage indices for delaminations varying between 10 mm and 100 mm and all starting at 500 mm from the clamped side of the beam are shown in Fig. 6.14. The peak values of the damage index drop significantly if it is based on the displacements measured at x = 0.041m, as can be concluded from a comparison between Figs. 6.14 and 6.13(b). Only delaminations of 70 mm and longer are likely to be detected. The probability that a delamination of 60 mm will be detected if noise is included, is low. The damage index shows a significantly more smooth peak compared to the damage index at x = 0. The maximum values are found halfway the delamination rather than at the both ends of the delamination. The distortion in the displacement field rapidly damps and smooths with increasing distance from the delamination. The ability to detect a delamination as a function of the perpendicular distance from the location of the delamination is analyzed. The height of the peaks are not a proper measure, as they do not provide an absolute value. This problem is overcome by summing all damage index values exceeding a preset threshold, hence capturing the damage index values at the location of the delamination only. The threshold should be higher than the signal to noise ratio. A threshold of 2 was used. The results are shown in Fig. 6.15.
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Fig. 6.15 The sum of the damage index values exceeding a threshold value of 2 for delamination lengths of 10–100 mm plot as a function of the perpendicular distance x to the delamination. The gray area covers the cases in which the probability of detection is low
It is expected that the sum of damage index values over the delamination is maximal at x = 0. Surprisingly, a drop in the value is observed for delamination lengths higher than 50 mm. This is possibly caused by differences in the deformation of the cross-section (xy-plane) of the intact versus delaminated T-beam, but further investigation is required to state more firm conclusions on this phenomenon. The lines in the graph show that the ability to detect the delamination drops nonlinearly with increasing distance from the delamination. It is difficult to set a threshold for the summed damage index βthrs=2 . However, an approximate threshold value of 20 is estimated for this case by comparing the results with Figs. 6.13 and 6.14. The gray area is the area below the threshold. The probability of detection is low for the cases which summed β index lies in this area.
6.7.4 Number of Evaluation Points The number of evaluations points is an important parameter in order to assess the practical feasibility of the method. The more evaluation points are used, the higher the accuracy of the mode shapes described. This results in an improved detection and localization of the damage. However, the aim is to minimise the number of evaluation points which improves the feasibility of the method. A compromise is needed to satisfy both contradicting demands as good as possible. The number of equidistant grid points (see Fig. 6.11) used for the spline interpolation is reduced to assess the effect of decreasing the number of sensors. Initially, the full set of 65 equidistant grid points in longitudinal direction of the beam are used. Subsequently, the number of grid points used is reduced to 33 and 17 equidistant points respectively. Note that this corresponds to 64, 32 and 16 elements as used in for example (6.4). The results for varying delamination lengths are shown in graphs of Fig. 6.16 (see Fig. 6.13 for the full set results). The center grid line (x = 0) is used here. The difference between the results based on 65 and 33 evaluation points is minimal. The peak heights are the nearly the same. The main difference is that
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Fig. 6.16 Damage indices for delamination lengths varying between 10 and 100 mm, using equidistant data points. A peak in the damage index indicates damage. (a) Damage index based on 33 equidistant data points. (b) Damage index based on 17 equidistant data points
the maximum value of the damage index is found in the middle of the delamination, rather than at one of the two ends of the delamination. This implies that the steepest gradients, found at the ends of the delamination are not captured by the spline interpolation. The disturbances on either side of the peaks are slightly higher for the case of 33 evaluation points, compared to the case of 65 evaluation points, but remain small compared to the peak height. The results based on 17 evaluation points shows significant differences. The peak height is lower (∼50%) and only comparatively large delaminations can be detected (>70 mm). Delaminations smaller than the distance between the evaluation points (∼66 mm in the case of 17 points) can not be detected. This implies that at least one data point must be present in the delaminated zone, hence disregarding the concept that a local damage affects the global behavior. This effect is attributed to the smoothing of the spline interpolation; the discontinuities in the strain energy, resulting in the high peaks in the damage index, do not appear in the spline interpolations. A clustered set of data points can be used instead of an equidistant set of points. Clustering of the data points can increase the performance significantly. Clusters of 3 evaluation points are distributed evenly over the length of the T-beam. Clusters of 2 points are used at each end of the beam. Two cases were studied: 7 internal clusters (25 points) and 3 internal clusters (13 points). The results are shown in Fig. 6.17. Again, the nodal displacements of the center line x = 0 are used for the analysis. The peak heights are similar if not higher, but the ability to detect is less trustable. The results can be significantly poorer if the delamination is between two clusters of sensors. The likelihood of false positives hence increases drastically. The localization is more inaccurate compared to using an equidistant set of data points, due to deviations in the spline interpolations and due to the widening of the peaks caused by the smoother spline functions. Smoother spline interpolations result in diminishing of the distinction between smaller and larger delaminations.
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Fig. 6.17 Damage indices for delamination lengths varying between 10 and 100 mm, using clustered sets of data points. A peak in the damage index indicates damage. (a) Damage index based on 25 clustered data points (3 per cluster). (b) Damage index based on 13 clustered data points (3 per cluster)
6.7.5 Incorporation of Torsion Modes One of the objectives of the current research is to maximize the amount of data extracted from a minimum number of data acquisition points. Torsion modes are a potential source of information in addition to the bending modes of the structure. The rotation of the beam, defined in (6.5), is required to determine the strain (n) energy UT ,i (6.6) associated with the nth mode. This angle is obtained by fitting a least square function through the three points (x = ±0.041, x = 0) at each zcoordinate on the grid. The first derivative in x-direction of the square function at x = 0 equals the rotation angle θxy (z), see (6.5). Subsequently, the strain energy and damage index are calculated employing Eqs. (6.6) and (6.9)–(6.13). The resulting damage indices are shown in Fig. 6.18(a). Firstly, it is observed that the peak height is significantly lower in the case of torsional modes, reducing the probability of detection to a critical level. Delaminations smaller than 60 mm (6% of the length of the T-beam) can not be detected. Secondly, the length of the delamination is not predicted accurately. This is not a critical issue, as the detection is more important than an accurate prediction of the size of the delamination. An analysis of the deformation of the cross-section of the T-beam at various values of the z-coordinate revealed that the presence of a delamination mainly affects the angle between the base and the stiffener. However, the rotation angle of the base is measured, which apparently is not a good measure in this case. An alternative formulation for the torsion angle θxy (z) is complicated and has not been found yet. The only conclusion that can be drawn at this point is that the torsion modes do contain information on the damage, but that the angle as calculated here is not able to capture this information in a sensible way. An alternative use of the information enclosed in the torsion modes is found by reconsidering the definition of the modal strain energy. The squared root of the
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Fig. 6.18 Damage indices for delamination lengths varying between 10 and 100 mm, using the first 12 torsion modes. Note the different scales for the axis of β. (a) Damage index using the torsion strain energy formulation (6.9). (b) Damage index using the bending strain energy formulation (6.8)
second derivative of the displacements (the curvature) is employed, effectively canceling the sign of the displacement. The displacements uy (z) of the nodes at x = 0 of the torsion modes are therefore treated in the same way as the displacements of the bending modes (i.e.: using (6.8) rather than (6.9)). The result is shown in Fig. 6.18(b). The damage index for delamination of 40 mm and longer are likely to be detected. The location, including the length, is also predicted accurately. Some noise is observed in case of small delaminations. However, using both bending and torsion modes will enhance the probability of detection. The peak height will increase at the location of the delamination as both the bending and the torsion modes contribute to these peaks, whereas the noise for smaller delaminations will be suppressed as only torsion nodes contribute to this noise.
6.8 Prospects for the Future The research discussed in this chapter is a step towards the application of a complete online vibration based structural health monitoring system, including identification algorithm, sensing system and diagnostics for composite aerospace structures. Evidently, there are still a number of steps to be taken, before this goal will be achieved. This leads to research activities in a number of research fields. The case studied here, is based on a slender structure with a single stiffener, with a geometrically well-defined delamination damage, located at the interface between the skin and the stiffener. The first research activities concern the extension to a plate structure with multiple stiffeners and an impact induced delamination of arbitrary shape, size and location. Note that the delamination can also occur between two plies in the skin or stiffeners. Hence, the method of VB-SHM will become more general. Moreover, the role of the numerical model will be more prognostic: here it was merely used in a more diagnostic sense as a virtual experimental space. The numerical models will develop towards models explaining the measured (change in)
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response in terms of potential damage locations, including the size and severity of the damage, while accounting for changes in the environment such as temperature and structural loading. This requires a baseline FE model of the structure being monitored. A second branch of research is the measurement of the vibrations. The experiments described are performed using a laser-vibro meter. This is not the most suitable method to be implemented in a real structure. An overview of the more recent developments in the area of sensor technology for structural health monitoring applications is given by Takeda [37]. One solution, gaining more and more attention is the application of optical fibre sensors (with fibre bragg grating—FBG), either embedded or on top of the structure [19]. Relevant research topics concern the development of the optical fibre itself (for example the bandwidth, sensitivity, multiplexing capabilities), the embedding of the fibre in the structure and the number of measuring points needed, including ways to achieve this. The resulting flow of data is substantial, in particular if larger structures are observed. The acquisition of this data and the analysis require dedicated hard- and software applications. For the hardware, miniaturization is one of the most critical issues, whereas for the software the challenges are found in the data interpretation and decision-making programs. The interpretation of the data measured results in values such as the damage index β (6.13) and the damage severity α (6.14) upon which a decision is made if a warning for a potential damage must be issued. Noise and external conditions, such as environmental conditions, can affect the measured response of the structure and consequently the damage index. The performance of the decision-making algorithm is based on the number of false negative and false positive damage identification calls. This topic is already addressed by various authors [2, 10, 25, 26], although the current result is still limited to noise only. Summarizing, the following steps in the future research can be recognized: • Experimental and numerical work on structures more closely resembling existing aircraft structures; • Development of smart sensing systems such as optical fibres (FBG); • Development of data acquisition and decision-making hard- and software leading to robust and trustworthy damage identification algorithms. Currently, a number of projects are either started, or about to start, in which the topics above will be investigated. It is expected that over the next year the first demonstrators will be built, with an online VB-SHM system integrated. From there on, the developments will be directed towards issues such as costs, robustness, reliability, repair-ability to achieve the technology readiness level (TRL) [28] required for aerospace technologies.
6.9 Summary It was shown in this chapter how the combination of the modal strain energy damage index (MSE-DI) algorithm and a finite element model can be used to evaluate
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the detection and localization of damage in a thin-walled, slender composite structure. The parametric study provides information on the ability of the algorithm to detect and localize damage for varying delamination lengths and locations, as well as for a variable number of data acquisition points. Using numerical models avoids the need for costly and time consuming experimental analysis, although it should be emphasized that an experimental validation of the model is important. Hence, the experimental set-up and results obtained with it were addressed as well in this chapter. The results of the measurements were found to be sufficiently accurate to identify the location and size of the delamination. The number of evaluation points is shown to be a critical parameter. A balance must be found between the desire to use higher frequency modes, as they are able to capture the effect of relatively small damages, and the desire to reduce the number of sensor elements, as they increase the cost and complicate the data acquisition systems required. This balance also depends on the size of the damage one wishes to detect. It was also shown that the damage index based on the bending modes (i.e. using (6.8) to calculate βB as defined in (6.13)) contained more information compared to the damage index based on the torsion modes (i.e. using (6.9) to calculate βT (6.13)). This is partly attributed to the location of the delamination with respect to the shear center of the structure. However, it was also observed that treating the curvatures in the torsion modes in the same way as the curvatures in the bending modes, effectively calculating βB using torsion modes, provides better results than calculating βT . Numerical models, as presented here, are shown to support the development of the MSE-DI algorithm. On one hand more complex models will be employed to analyze the response to be expected, for example models predicting the effect of non-linearities such as contact in the delamination. On the other hand, models will aid in the optimization of sensor placement and configuration, type of excitation and also will provide a virtual modelling space to assess effects that are difficult, time-consuming and/or costly to test, amongst which environmental conditions.
6.10 Selected Bibliography A number of topics have been addressed in this chapter. The focus was on the modal strain energy damage index (MSE-DI) algorithm. A list of references is included to support the theoretical derivations and the interpretation of both the experimental and numerical work. However, reading this chapter may require knowledge on neighboring research fields. A number of book and articles is listed here for those who are not fully acquainted with this area or are simply interested in deepening their knowledge on these subjects. The algorithm discussed here, the MSE-DI algorithm, is just one of the many vibration based damage identification algorithms available. The publications of Fan and Qiao [14], Montalvão et al. [27], Worden and Dulieu-Barton [42] and Carden and Fanning [7] provide extensive overviews of other methods available to assess
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the damage in a structure. A thorough overview of structural health monitoring, including the dynamic aspects such as vibrations of solids, structures and plates is written by Giurgiutiu [17]. Despite the different field of application it provides the reader with an thorough explanation of the dynamics of vibrating structures. The book of Adams [1] is another valuable contribution, in particular since it includes a number of case studies and exercise material for instructors and students studying the field of health monitoring. Staszewski et al. [35] composed a book dedicated to structural health monitoring of aerospace structures, with contributions ranging from application of SHM to sensor technologies applied and damage detection using ultrasonics. Those interested in sensing technologies for structural health monitoring are referred to the book of Balageas et al. [6] for a broad overview of available sensing technologies and to Gliši´c [18] for the fibre optic sensing technology in particular. A basic, but clarifying introduction into experimental modal analysis is provided by a series of online documents by Avitabile [5] and his publications in the sound and vibration magazine [3, 4]. The nature of these publications is instructive and can either be used by instructors or those who are not familiar with the terminology and methods of experimental vibration analysis.
References 1. Adams, D.: Health Monitoring of Structural Materials and Components. Wiley, New York (2007) 2. Alvandi, A., Cremona, C.: Assessment of vibration-based damage identification techniques. J. Sound Vib. 292(1–2), 179–202 (2006) 3. Avitabile, P.: Experimental modal analysis. Sound Vib. Mag. (2001). http://www.sandv.com/ downloads/0101avit.pdf 4. Avitabile, P.: Teaching experimental structural dynamics applications. Sound Vib. Mag. (2007). http://www.sandv.com/downloads/0711avit.pdf 5. Avitabile, P.: Modal space in our own little world. Tech. rep., University of Massachusetts Lowell (1998–2008). http://macl.caeds.eng.uml.edu/umlspace/mspace.html 6. Balageas, D., Fritzen, C.P., Güemes, A. (eds.): Structural Health Monitoring. ISTE (2006) 7. Carden, E., Fanning, P.: Vibration based condition monitoring: A review. Struct. Health Monitor. 3, 355–377 (2004) 8. Choi, S., Stubbs, N.: Damage identification in structures using the time-domain response. J. Sound Vib. 275, 557–590 (2004) 9. Choi, S., Park, S., Stubbs, N.: Nondestructive damage detection in structures using changes in compliance. Int. J. Solids Struct. 42, 4494 (2005) 10. Choi, S., Park, S., Park, N.H., Stubbs, N.: Improved fault quantification for a plate structure. J. Sound Vib. 297, 865–879 (2006) 11. Cornwell, P., Doebling, S., Farrar, C.: Application of the strain energy damage detection method to plate-like structures. J. Sound Vib. 224(2), 359–374 (1999) 12. Della, C., Shu, D.: Vibration of delaminated composite laminates: A review. Appl. Mech. Rev. 60(1–6), 1–20 (2007) 13. Duffey, T., Doebling, S., Farrar, C., Baker, W., Rhee, W.: Vibration-based damage identification in structures exhibiting axial and torsional response. J. Vib. Acoust. 123(1), 84–91 (2001) 14. Fan, W., Qiao, P.: Vibration-based damage identification methods: A review and comparative study. Struct. Health Monitor. (2010). doi:10.1177/1475921710365419
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15. Farrar, C., Jauregui, D.: Comparative study of damage identification algorithms applied to a bridge: I. Experiment. Smart Mater. Struct. 7(5), 704–719 (1998) 16. Farrar, C., Jauregui, D.: Comparative study of damage identification algorithms applied to a bridge: II. Numerical study. Smart Mater. Struct. 7(5), 720–731 (1998) 17. Giurgiutiu, V.: Structural Health Monitoring: With Piezoelectric Wafer Active Sensors. Elsevier, Amsterdam (2007) 18. Gliši´c, B., Inaudi, D.: Fibre Optic Methods for Structural Health Monitoring. Wiley, New York (2007) 19. Grouve, W., Warnet, L., de Boer, A., Akkerman, R., Vlekken, J.: Delamination detection with fibre bragg gratings based on dynamic behaviour. Compos. Sci. Technol. 68(12), 2418–2424 (2008) 20. Kim, B., Stubbs, N., Park, T.: Flexural damage index equations of a plate. J. Sound Vib. 283, 341–368 (2005) 21. Kim, J.T., Stubbs, N.: Model uncertainty impact and damage-detection accuracy in plate grider. J. Struct. Eng. 121(10), 1409–1417 (1995) 22. Kim, J.T., Stubbs, N.: Improved damage identification method based on modal information. J. Sound Vib. 252(2), 223–238 (2002) 23. Kim, B., Stubbs, N., Park, T.: A new method to extract modal parameters using output-only responses. J. Sound Vib. 282, 215–230 (2005) 24. Kumar, M., Shenoi, R., Cox, S.: Experimental validation of modal strain energies based damage identification method for a composite sandwich beam. Compos. Sci. Technol. 69, 1635– 1643 (2009) 25. Li, H., Yang, H., Hu, S.L.: Modal strain energy decomposition method for damage localization in 3D. J. Eng. Mech. 132(9), 941–951 (2006) 26. Li, H., Fang, H., Hu, S.L.: Damage localization and severity estimate for three-dimensional frame structures. J. Sound Vib. 301, 481–494 (2007) 27. Montalvão, D., Maia, N., Ribeiro, A.: A review of vibration-based structural health monitoring with special emphasis on composite materials. Shock Vib. Dig. 38(4), 295–326 (2006) 28. NASA: Definition of technology readiness levels. http://esto.nasa.gov/files/TRL_definitions. pdf 29. Offringa, A., List, J., Teunissen, J., Wiersma, H.: Fiber reinforced thermoplastic butt joint development. In: Proceedings of International SAMPE Symposium and Exhibition (2008) 16p 30. Ooijevaar, T., Loendersloot, R., Warnet, L., de Boer, A., Akkerman, R.: Vibration based structural health monitoring of a composite t-beam. Compos. Struct. 92(9), 2007–2015 (2009) 31. Pandey, A., Biswas, M.: Damage detection in structures using changes in flexibility. J. Sound Vib. 169(1), 3–17 (1994) 32. Pandey, A., Biswas, M., Samman, M.: Damage detection from changes in curvature mode shapes. J. Sound Vib. 145(2), 321–332 (1991) 33. Richardson, M., Schwarz, B.: Modal parameter estimation from operating data. Sound Vib. Mag., 8 (2003). http://www.sandv.com/downloads/0301rich.pdf 34. Schwarz, J., Richardson, M.: Experimental Modal Analysis. Vibrant Technology Inc. (1999) 35. Staszewski, W., Boller, C., Tomlinson, G. (eds.): Health Monitoring of Aerospace Structures. Wiley, New York (2004) 36. Stubbs, N., Kim, J., Farrar, C.: Field verification of a nondestructive damage localization and severity estimation algorithm. In: Proceedings of the 13th International Modal Analysis Conference, pp. 210–218 (1995) 37. Takeda, N.: Recent developments of structural health monitoring technologies for aircraft composite structures. In: Proceedings of the 26th International Congress of the Aeronautical Sciences, p. 12 (2008) 38. Toksoy, T., Aktan, A.: Bridge-condition assessment by modal flexibility. Exp. Mech. 34(3), 271–278 (1994) 39. Ullah, I., Sinha, J.: Dynamic study of a composite plate with delamination. In: Proceedings of the Third International Conference on Integrity, Reliability and Failure (2009). S1145_P0506, 15p
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40. de Vries, R., Lamers, E., Wijskamp, S., Rodriguez, B.V., Akkerman, R.: The university of Twente micromechanics modeller. Tech. rep., University of Twente (2004). http://www.pt.ctw.utwente.nl/organisation/tools/ 41. Whitherell, C.: Mechanical Failure Avoidance: Strategies and Techniques. McGraw-Hill, New York (1994) 42. Worden, K., Dulieu-Barton, J.: An overview of intelligent fault detection in systems and structures. Struct. Health Monitor. 3, 85–98 (2004) 43. Zimmerman, D., Kaouk, M.: Structural damage detection using a minimum rank update theory. J. Vib. Acoust. 116(2), 222–231 (1994)
Chapter 7
An Efficient Sound Source Localization Technique via Boundary Element Method A. Seçgin and A.S. Sarıgül
Abstract The boundary element method (BEM) is a widely used technique in vibro-acoustics. This method is effective not only in the determination of exterior and interior sound fields but also for the sound source localization of complex systems. In this chapter, the Helmholtz integral equation formulation and its boundary element discretization are presented. Half-space algorithm and half-space-contact version of this algorithm feasible for most machine locations are introduced. Some theoretical examples, for a dilating sphere in half-space, are presented. The chapter continues with a case study: Sound source identification and characterization of a refrigerator, via a cost-effective and easy-to-use technique, based on surface velocity measurements and BEM computation of surface and field acoustic pressures.
7.1 Introduction Acoustic field theory has an extensive application area in science and engineering including underwater acoustics and aviation. Especially from the point of view of mechanical engineering, it is important to predict the sound radiation characteristics of mechanical systems for preventing the possible noise problems in the design stage. Although analytical solutions are available for sound sources having uniform geometry [26, 27, 29, 43], closed form solutions are not generally possible with regard to arbitrary shape and vibration behavior of the source. Therefore, in most cases, it becomes inevitable to obtain the solutions by using numerical techniques. Helmholtz integral equations constitute the foundation of many studies accomplished by numerical methods [4, 6, 9, 11, 22, 28, 32–34, 37, 42]. In the course of application of Helmholtz integral to acoustic radiation problems, it is necessary to know the acoustic pressure and normal velocity distributions on the harmonically vibrating sound source. Generally, in practice, normal velocity is the pre-known variable that may be obtained from structural analyses or direct measurements. In this A. Seçgin () · A.S. Sarıgül Department of Mechanical Engineering, Dokuz Eylül University, 35100 Bornova, Izmir, Turkey e-mail:
[email protected] A.S. Sarıgül e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_7, © Springer Science+Business Media B.V. 2011
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case, surface Helmholtz integral equation is applied first for the solution of acoustic pressure distribution on the surface of the sound source by using boundary conditions. Predicted surface pressures are then used for the computation of sound pressure field around the source via exterior Helmholtz integral equation. In general, methods that are based on the numerical solution of a surface integral, such as Helmholtz integral, are known as Boundary Integral Equation (BIE) method or Boundary Element Method (BEM). These methods resemble the finite element method due to its properties; discretization of the body surface, use of shape functions and obtaining a set of algebraic equations at the end of these numerical approaches. The primary advantage of BEM is the decrease of the problem in one dimension, due to inclusion of only the points on the body surface. In the isoparametric element formulation technique used in the vibroacoustic applications in this chapter, surface geometry and acoustic variables such as pressure and velocity are represented by high-degree shape functions (an explanation about isoparametric element formulation and shape functions is given in Appendix A). It has been shown that this technique gives sufficiently sensitive results even with a low number of elements [23, 31, 41, 42]. In most practical cases, sound sources don’t take place in a three-dimensional full space. They are located at least in a half-space, as is the case of machines sitting on the floor. Therefore, using the half-space algorithms for Helmholtz integral developed by Seybert and Soenarko [39] and Seybert and Wu [40] substantially facilitates the numerical solution. As a result of technological development, machinery has become much more complicated in order to perform certain tasks effectively and to satisfy customer expectations. Decreasing energy consumption and reducing the human dependency in operation are mostly required improvements on machinery. However, these treatments may directly cause many complex dynamic problems. For instance, in order to decrease the fuel consumption rate for a vehicle, the most straightforward way in the design stage is to lighten the body weight, however, reducing the weight leads to higher vibration amplitudes in the body and dependently increases noise levels. Furthermore, vibration of any structural part of machinery may interact with radiated sound waves in enclosures and, for such a complex system, distinguishing the noise sources together with their dynamic characteristics becomes much more difficult. In vibro-acoustic analysis, being able to localize sound sources in a complex system is one of the most informative approaches. This information can be used in active, passive vibration and acoustic treatments during and after design process. Performing appropriate measurements processed by some numerical techniques is generally a convincing strategy for this purpose. The BEM, with its traditional or modified versions, is effectively used for sound source localization purposes. Its traditional use may be a practical tool for engineers in industrial vibro-acoustic applications. This chapter is devoted to analytical, numerical and experimental background of this procedure via a case study for a refrigerator. Here, numerical solution of Helmholtz integral equation is carried out by BEM, feeding surface velocity data of the sound sources. Rigid wall reflection effects are taken into account by
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the half-space-contact formulation. An in-house code has been developed for the main analysis and for the post-processing in order to display self-explanatory figures. As the first problem a dilating sphere in half-space is considered; and its near sound field is presented for different cases. Secondly, as a case-study, a brand-new refrigerator is considered; sound sources of the refrigerator are detected and examined by using surface vibration measurements, surface and near field sound pressure patterns. Local positions of sound sources and their radiation characteristics are determined.
7.2 Overview of the State of the Art Noise source identification and characterization have become a powerful tool for the design and production of modern and noiseless machines and systems. Various techniques have been developed for this purpose. In general, measurements of surface velocity and exterior sound pressure are more practical ways to perform a simple source localization analysis. The capability of these measurement approaches is very restricted due to the fact that it directly depends on the accuracy of the test set-up and expertise of the measurement technician. In the identification of acoustical characteristics of sound sources, some methods are available for researchers. These are, inverse and reciprocity methods [14, 44, 45] presenting characteristics of sources and transmission paths; pseudo-forces method [20, 21] where the internal excitation in a source component is reproduced by fictitious forces on the outer surface; and Inverse Boundary Element Method (IBEM) [36, 53] that is a three-dimensional holography method joining boundary element modeling used in direct (forward) solution with inverse methods. The more industrialized techniques are, Spatial Transformation of Sound Fields (STSF) [15, 16] that bases on measurements on a flat image of the source scanned on a reference plane; Near-field Acoustic Holography (NAH) [25, 46], an approach to reconstruct the acoustic field on the surface of a planar source based on Helmholtz integral equation and the two-dimensional spatial Fourier transform; and beamforming [7, 10], a technique that measures acoustic pressure by means of microphones arrays and locates sound sources by post-processing the measured signals. Different commercial software based on these techniques compatible to measurement hardware can be directly supplied by test equipment sellers. In addition to these briefly explained techniques some other approaches such as, Statistically Optimal NAH (SONAH) [18, 19]; Non-Stationary STSF [17]; Helmholtz integral equation, BEM with Singular Value Decomposition (SVD) [2, 47]; Helmholtz Equation with Least Squares method (HELS) [49, 51]; Fast multipole BEM (FMBEM) [30, 52], are also available. All these aforementioned methods have their own advantages and disadvantages regarding on the efficiencies of considered methodologies; accuracy, operation time and expenses, required equipment and software costs and consistency of measurements and computations. Table 7.1 presents the benefits and limitations of some of these methods. This table is mainly compiled from a comprehensive work provided by [3].
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Table 7.1 Benefits and limitations of some sound source determination procedures Method
Benefits
Limitations
• Time consuming. • Low equipment cost. Sound Pressure • Scalar based technique, no Level Measurements • Easy measurement technique. determination of sound direction • The frequency limitation is based or acoustic directional on the microphones or the analyzer. fluctuations. • Measurements cannot be performed in the near-field. • Sound pressure measured is a summation of the actual sound source plus many other acoustic contributions. • Measurements are only relevant on stationary sources. Sound Intensity Techniques
• Because sound intensity is a vector quantity, the acoustic field can be represented as amplitude and direction. • True measure of acoustic radiation by determination of energy flow. • Possible determination of sound power from sound intensity measurements. • Portable technique (can be used with a two channel sound level meter).
Spatial Transformation of Sound Fields (STSF)
• • • •
• Frequency limitations due to the pressure approximation gradient. • Relatively time-consuming. • Accurate equipment required. • Sources need to be stationary.
Ease of use. • Limited frequency range Fast and very reliable. (depending on the spacing of the Excellent resolution. measurement points in the grid), Provides a complete 3D description typically valid up to 6.4 kHz. of both the near-field and far-field. • Can only be used with stationary • Can be used in tougher sound fields. environmental conditions. • Possible suppression of uncorrelated background noise thanks to a precise cross-spectral measurement principle. • Provides calibrated maps of sound intensity, pressure and particle velocity close to the source. (continued on next page)
7.3 Helmholtz Integral Equation and Boundary Element Method In this section, Helmholtz integral equation is derived; and related BEM algorithm is presented for different space conditions. Some of the space cases for an acoustic radiation analysis such as full-space, half-space (and half-space-contact) and quarter-
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Table 7.1 (continued) Method
Benefits
Limitations
Non-Stationary STSF
• Very fast technique. • Requires a large number of • Easy to use. microphones and measurement • Excellent spatial and temporal channels. resolution. • Requires free-field conditions. • All time domain averaging • Very intensive algorithms can techniques are possible, so it can lead to long calculation times. show both where and when noise is radiated. • Projection of sound fields both in near- and far-fields. • Sound source ranking allows for filtering before the averaging process for the complete sound field parameters.
Beamforming
• Fast technique as all measurement channels are recorded simultaneously. • Measurements on large objects (provides a 60° opening angle). • High frequency range (>20 kHz). • Good resolution. • Excellent for wind-tunnel applications. • Possibility of pass-by or transient applications.
• Sound pressure maps are not calibrated (no calibration absolute levels near the source are obtained, as opposed to NAH/STSF techniques); only shows the relative contributions of the sound field at the array position. • Requires a multi-channel data acquisition system.
Inverse Boundary Element Method (IBEM)
• Possible estimation of the vibration velocity and the sound intensity right on the surface of an irregularly-shaped source. • Very precise noise source location and ranking.
• Much more difficult to solve than the forward problem because of the intrinsic ill-posed nature of the problem. • Requires additional algorithms for ill-posed matrix
space are demonstrated in Fig. 7.1. A machine sitting on the floor is an example of half-space (specifically half-space-contact) case; whereas a machine sitting on the floor with a wall behind, is an example of quarter-space case. The term full-space denotes three-dimensional free space, i.e., no bounding plane around the source; therefore there is no back contribution to the source itself. However, for half- or quarter-space conditions, sound radiation characteristics of the source have to be modified in order to take into account one or two bounding planes, respectively. The bounding plane is theoretically infinite and may be a soft, impedance or rigid surface. A soft surface is one that has no acoustic pressure, such as a fluid surface. An impedance or a rigid surface partially or completely, respectively, reflects the acoustic waves impinging on it.
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Fig. 7.1 Representation of (a) full-space, (b) half-space, (c) quarter-space and (d) half-space-contact geometries
In this regard, the implementation of BEM to Helmholtz integral equations depends on space conditions and if they exist, and on the reflection characteristics of the bounding planes. Here, only the formulations of full- and half-space (and halfspace-contact) cases are presented since furthermore would be straightforward.
7.3.1 Full-Space Case The derivation of the Helmholtz integral equation starts with the three-dimensional wave equation, ∇ 2p −
1 ∂ 2p = 0, c2 ∂t 2
(7.1)
where p is the acoustic pressure and c is the speed of sound. Assuming a harmonic time dependence, ei2πf t , acoustic pressure may be written as p = pe ¯ i2πf t ,
(7.2)
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where f is the wave frequency. Substituting Eq. (7.2) in Eq. (7.1), the wave equation takes the form 2 (7.3) ∇ + k 2 p = 0, where k = 2πf/c is the wavenumber. Equation (7.3) is known as Helmholtz equation. If the Helmholtz operator is defined as L¯ = ∇ 2 + k 2 ,
(7.4)
the non-homogeneous Helmholtz equation is given as ¯ = −4πδ(Q − P ), LG
(7.5)
where δ is the Dirac-delta function, Q and P represent two points in the medium. The fundamental solution of Eq. (7.5) in three-dimensional space is the free-space Green’s function, G(R, k) =
e−ikR . R
(7.6) √ Here, R is the distance between points Q and P (R = |Q − P |), i = −1. Green’s theorem relates the surface integral over S to the volume integral over V bounded by S for any two smooth and non-singular functions such as p and G(R, k): 2 ∂G(R, k) ∂p p dS, (7.7) − G(R, k) p∇ G(R, k) − G(R, k)∇ 2 p dV = ∂n ∂n V S where n is the outward normal of the surface S . Forming the difference ¯ ¯ and by using Eqs. (7.3) and (7.4), yields p LG(R, k) − G(R, k)Lp ¯ ¯ = p LG(R, ¯ p LG(R, k) − G(R, k)Lp k) = p∇ 2 G(R, k) − G(R, k)∇ 2 p.
(7.8)
If two sides of Eq. (7.8) are integrated over the volume V and Green’s theorem in Eq. (7.7) is used, yields ∂G(R, k) ∂p ¯ p dS. (7.9) p LG(R, k) dV = − G(R, k) ∂n ∂n V S Substituting Eq. (7.5) in Eq. (7.9) and using the following property of the Diracdelta function, p(Q)δ(Q − P ) dV = p(P ), (7.10) V
the following relation is obtained: ∂G(R, k) ∂p(Q) p(Q) dS. − G(R, k) −4πp(P ) = ∂n ∂n S
(7.11)
In order to derive exterior Helmholtz integral equation, Q is taken as any point on the surface S; P is taken as any point in the domain V between the surface S and
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the surface at the infinity, as shown in Fig. 7.1(a). The surface at the infinity may be represented by a sphere with infinite radius (R = R → ∞). The integral over the surface S in Eq. (7.11) may be written by the summation of two integrals over the surfaces S and since S = S ∪ . By using the relation ∂/∂n = ∂/∂R , the contribution of surface may be written as ∂G(R , k) ∂p −p dS. (7.12) + G(R , k) lim R →∞ ∂R ∂R By using spherical coordinates, dS = R 2 sin θ dθ dφ = R 2 d
(7.13)
may be written. Here is space angle. By making the differentiations and using the value of free-space Green’s function, Eq. (7.12) takes the form R (∂p/∂R ) + ikp + (p/R ) e−ikR d. (7.14) lim R →∞
As seen in Eq. (7.14), if the condition lim R (∂p/∂R) + ikp = 0 R→∞
(7.15)
is satisfied, the contribution of the points at infinite distance will be zero. This is known as Sommerfeld radiation condition. This condition states that there is no acoustic source at the infinity, no reflection from the infinity and the function p will vanish when R → ∞. By using the Sommerfeld radiation condition, the surface integral in Eq. (7.11) may be written only over the body surface S. Since the outward normal of the surface S is n = −n , Eq. (7.11) may be written as ∂G(R, k) ∂p(Q) 4πp(P ) = p(Q) − G(R, k) dS. (7.16) ∂n ∂n S This equation is called as exterior Helmholtz integral. If S is a smooth surface, by approaching the point P to the surface, Eq. (7.16) may be written as [12] ∂G(R, k) ∂p(Q) p(Q) − G(R, k) dS. (7.17) 2πp(P ) = ∂n ∂n S This equation is known as surface Helmholtz integral. If the point P belongs to internal domain D , the interior Helmholtz integral is obtained as ∂p(Q) ∂G(R, k) − G(R, k) dS. (7.18) 0= p(Q) ∂n ∂n S If Eqs. (7.16)–(7.18) are thought together, in general the Helmholtz integral equation may be written as ∂p(Q) ∂G(R, k) C(P )p(P ) = − G(R, k) dS. (7.19) p(Q) ∂n ∂n S
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The coefficient C(P ) takes different values in accordance with the position of the field point P : If P is in the body, C(P ) is equal to 0; if P is out of the body it is equal to 4π ; if P is on the surface of the body and the surface is smooth it is equal to 2π . However, if the surface S is not smooth, C(P ) is given as [42] ∂(1/Rb ) dS. (7.20) C(P ) = 4π + ∂n S Using the particle velocity u and density ρ0 of the fluid in the medium, the momentum equation is written as ρ0
∂u = −∇p. ∂t
(7.21)
Assuming a harmonic change for the acoustic particle velocity, u = U ei2πf t , we have ∂u = i2πf u. (7.22) ∂t Substituting Eq. (7.22) in Eq. (7.21), we obtain the equality ∂p (7.23) = −izo kun , ∂n where un is the normal surface velocity and zo = ρo c is the characteristic impedance of the medium. When Eq. (7.23) is substituted in Eq. (7.19), Helmholtz integral is expressed in terms of the acoustic pressure and velocity as ∂G(R, k) p(Q) (7.24) + izo kun (Q)G(R, k) dS. C(P )p(P ) = ∂n S In the boundary element method, the surface of the body is discretized by using several number of elements. If the surface integrals in Eqs. (7.20) and (7.24) are written as the summation of a L number of boundary element integrals, the surface Helmholtz integral equation becomes L L
∂G(Rb , k) ∂(1/Rb ) 4π + p(Q) dSb dSb p(P ) − ∂n ∂n Sb Sb b=1
=
L
b=1
izo kG(Rb , k)un (Q) dSb .
(7.25)
b=1 Sb
Here the subscript b represents boundary elements. After transforming global coordinates to local ones and using Gaussian quadrature technique, in order to numerically perform ordinary integrals on surfaces Sb , Eq. (7.25) can be written in matrix form as Ep − Kp = Hun ,
(7.26)
where p and un are N × 1 vectors including the acoustic pressure and normal velocity of the N nodes used in the boundary element mesh, respectively. The coefficient
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matrix of the pressure vector, E, is a N × N diagonal matrix including geometrical data of the coefficient C(P ), K and H are N × N square matrices formed by the mathematical operations on the second term of the left-hand side and the term on the right-hand side of Eq. (7.25), respectively. The matrix K contains the coefficients of surface pressures whereas H includes the coefficients of surface velocities. The surface pressures are predicted by using prescribed surface velocities obtained from measurements or structural analysis programs. The field (exterior) pressures are determined by substituting the calculated surface pressures p into the exterior Helmholtz integral equation in the boundary element matrix form pe = kTe p + hTe un ,
(7.27) (·)T
denotes the transpose where pe is the exterior field pressure and the symbol operator. The elements of the vectors kTe and hTe are composed by the appropriate rows of matrices K and H, respectively, divided by 4π ; the constant value of the coefficient C(P ) for the exterior Helmholtz integral.
7.3.2 Half-Space Case The half-space case is examined for two situations: half-space and half-space contact conditions. In this sub-section, some modifications in the assumed solution of Helmholtz integral equation (Green’s function) and in the boundary element implementation for these conditions are presented.
7.3.2.1 Half-Space Condition In the case where there exists an infinite reflecting plane that makes the acoustic domain V a half-space, the surface S is given by the summation S = So ∪ Sp . Here, So is the surface of vibrating body and Sp is the surface of the infinite plane that forms the half-space, as shown in Fig. 7.1(b). Here, P is the image point of P ; Q is any point on the surface of the body S or on the rigid plane Sp . For a reflecting plane Sp with acoustical impedance Z, the impedance boundary condition can be written as ikp +
Z ∂p = 0. zo ∂n
(7.28)
The solution of Helmholtz integral Eq. (7.25) requires the evaluation of surface integrals over the entire boundary S. The integral over the infinite plane Sp , puts forward some numerical modeling and computational difficulties. The BEM provides considerable flexibility in the selection of kernel functions to the Helmholtz integral. Any regular solution may be added to the usual freespace Green’s function and the resulting function may be used instead of G(R, k) in Eq. (7.25). It is required to select an appropriate Green’s function, so that the
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integration over the infinite plane Sp is removed. Therefore, boundary element discretization can only be implemented to the vibrating body itself. The proper Green’s function GM (R, R , k) leads to the following integral over Sp , ∂GM (R, R , k) p(Q) (7.29) + izo kun (Q)GM (R, R , k) dSp = 0, ∂n Sp and also GM (R, R , k) satisfies Eq. (7.28), yielding ikGM (R, R , k) +
Z ∂GM (R, R , k) = 0. zo ∂n
(7.30)
By using Eq. (7.29), the Helmholtz integral in Eq. (7.25) can be written for the half-space case considering only the integration over So , yielding ∂GM (R, R , k) + izo kun (Q)GM (R, R , k) dSo . (7.31) C(P )p(P ) = p(Q) ∂n So Equation (7.30) may be satisfied for two important special cases: 1) Sp is a rigid plane (∂p/∂n = 0, un = 0, Z = ∞ on Sp ). 2) Sp is a free surface (p = 0, Z = 0 on Sp ). For these two conditions, the modified Green’s function may be chosen as [40] GM (R, R , k) =
e−ikR (P ,Q) e−ikR(P ,Q) + Rp , R(P , Q) R (P , Q)
(7.32)
which is formed by the addition of a second term to the usual Green’s function. This term arises due to the image P of the point P with respect to the infinite plane. R is the distance between points Q and P (R = |Q − P |). Rp is the reflection coefficient of the plane. Equation (7.30) is satisfied for point Q on Sp for the following two cases: 1) A rigid plane when Rp = 1, since ∂GM /∂n = 0; 2) A free surface when Rp = −1, since GM = 0. The function in Eq. (7.32) is usually known as the “half-space Green’s function” and used in theoretical acoustics, for example in the determination of the field produced by a point source near a reflecting plane. Seybert and Soenarko [39] have successfully applied the half-space Green’s function to finite body problems for the first time. The boundary element discretization and matrix formation in half-space condition are the same as in full-space case.
7.3.2.2 Half-Space Contact Condition Half-space contact condition is a special case of the half-space position. If the vibrating body is sitting on the infinite plane as shown in Fig. 7.1(d), some modifications are required in the half-space formulation. In this case, the boundary S of the body is divided into two parts: a first part, Sc , in contact with Sp , and a second part,
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So , exposed to the acoustic medium. Helmholtz integral Eq. (7.31) is valid again; however, So represents not all but a part of the surface. In this case, the term C(P ) in Eq. (7.20) is written as follows [40]: If point P is on So but not in contact with Sp , we have ∂(1/R) (7.33) d(So + Sc ); C(P ) = 4π + ∂n So ∪Sc If point P is not only on So but also in contact with Sp , yields
∂(1/R) C(P ) = (1 + Rp ) 2π + d(So + Sc ) ; ∂n So ∪Sc More specifically, for a rigid plane, Eq. (7.34) leads to
∂(1/R) C(P ) = 4π + 2 d(So + Sc ) . ∂n So ∪Sc
(7.34)
(7.35)
The elements on Sc are called “dummy elements”. These elements are used only for integrations in Eqs. (7.33) and (7.34) in order to obtain geometrical data. However, there isn’t any vibrational and acoustical variable associated with these elements. By using Eqs. (7.31) and (7.34), the surface Helmholtz integral equation may be written, for a point P in contact with surfaces So and Sp , as
∂GM (R, R , k) ∂(1/R) dSh p(P ) − p(Q) dSo (1 + Rp ) 2π + ∂n ∂n Sh So = izo kGM (R, R , k)un (Q) dSo , (7.36) So
where the domain Sh = S0 ∪ Sc . After boundary element discretization, Eq. (7.36) may be represented in the form L
∂(1/Rb ) d(Sh )b p(P ) (1 + Rp ) 2π + ∂n b=1 (Sh )b L
∂GM (Rb , Rb , k) − p(Q) d(So )b ∂n b=1 (So )b L
= izo kGM (Rb , Rb , k)un (Q) d(So )b . (7.37) b=1 (So )b
After making all the necessary operations in Eq. (7.37), an algebraic set in the form of Eq. (7.26) is obtained. However, for half-space contact algorithm, the content of the coefficient matrices changes due to the presence of dummy elements; and the size of the matrices reduces. Here, the node number N represents the number of total nodes excluding those at the dummy elements. The presented algorithms are applied and matrix equations are solved by using an “in-house” BEM code. The validity and precision of the code have been proved under different circumstances [1, 31].
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Fig. 7.2 (a) A sphere in half space; (b) two spheres in full space
Fig. 7.3 Boundary element discretization of the sphere
7.4 Theoretical Examples: Sound Field Determination As a main theoretical source, dilating (uniformly vibrating) sphere in half-space is examined. The first problem is the computation of the sound field around a dilating sphere of radius a, located at a distance 3a from a rigid plane. Although this is a half-space problem, its full-space analog is two spheres of radius a with centers 6a apart (Fig. 7.2). It may be anticipated that the half-space algorithm provides a numerically feasible solution. The spherical source is modeled by using 24 boundary elements and 82 nodes as shown in Fig. 7.3. Second-order isoparametric quadrilateral elements with 8 nodes are considered here. The ordinary surface integrals are evaluated by using 16-point Gaussian quadrature. Since the sphere is near a rigid surface the solution is performed via BEM with half-space algorithm. Equal pressure contours computed at the z = 0 plane of the sphere for Helmholtz number (non-dimensional wavenumber) ka = 1 is shown in Fig. 7.4 where only one side of the symmetrical distribution is presented. The numbers on the contours indicate the corresponding values of the nondimensional pressure amplitude p/z0 U0 , where U0 is the uniform velocity amplitude of the sphere. If the sphere were in a free field, equal pressure contours would
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Fig. 7.4 Near field of a dilating sphere near a rigid surface (z = 0, b = 3a, ka = 1)
Fig. 7.5 Near field of a dilating sphere near a rigid surface (z = 0, b = 2a, ka = 1)
be in the form of uniform circles. Due to its reflection effect, the rigid surface deforms the field. For example, the maximum non-dimensional sound pressure 0.7 appears only around the left surface of the sphere. This situation indicates the back scattering characteristics of the pressure waves in this region. The contours representing high pressures vanish as going toward the right surface of the sphere. The second problem is the analysis of the sound field around the same sphere but located at a distance 2a from a rigid plane. Figures 7.5, 7.6 and 7.7 show equal pressure contours in different horizontal planes, for ka = 1. The contours at z = 0 plane are shown in Fig. 7.5; whereas at z = a/3 and z = 2a/3 planes in Figs. 7.6 and 7.7, respectively. It can be seen that acoustic pressures decrease as one moves up from the z = 0 plane. Figures 7.8 and 7.9 show equal pressure contours at z = 0 plane, for ka = 0.1 and ka = 2, respectively. When Figs. 7.5, 7.8 and 7.9 are compared, the effect of frequency on the pressure distribution is clearly realized. As the frequency increases the wavelength decreases; reflection effects increases; and sound field gets more complicated.
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Fig. 7.6 Near field of a dilating sphere near a rigid surface (z = a/3, b = 2a, ka = 1)
Fig. 7.7 Near field of a dilating sphere near a rigid surface (z = 2a/3, b = 2a, ka = 1)
7.5 Case Study: Sound Source Localization Beside of quality and price, low-noise is an important characteristic of a machine or system. Since noise caused by household appliances can have a huge impact on our daily lives, researches continue to evolve towards fewer noisy products in order to satisfy consumer expectations. The case study presented here is part of a product processing and development research. The product is a refrigerator and a vibro-acoustic analysis of the refrigerator is performed for the purpose of sound source identification and characterization. The analysis procedure is a combination of experimental and numerical operations, as explained in this section. The refrigerator shown in Fig. 7.10 is of class A; has two lids and dimensions of 71 cm × 186 cm × 69 cm.
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Fig. 7.8 Near field of a dilating sphere near a rigid surface (z = 0, b = 2a, ka = 0.1)
Fig. 7.9 Near field of a dilating sphere near a rigid surface (z = 0, b = 2a, ka = 2)
7.5.1 Surface Velocity Measurements A simple vibration test system that is shown in Fig. 7.11, is set up for the surface velocity measurements. The system is composed of a vibration meter (Brüel & Kjær, 2511), a tunable band-pass filter (Brüel & Kjær, 1621), an accelerometer (Brüel & Kjær, 4332) and a data-logger (Hioki, 8421-51). As shown in Fig. 7.11, the accelerometer is placed on the rectangular meshes drawn on surface panels of the refrigerator. Measurements are performed while the compressor and fan are running.
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Fig. 7.10 The tested refrigerator
Fig. 7.11 Measurement system
The following conditions are satisfied in the measurements with respect to standard ISO 8187:1991—Dictating characteristics and test methods of refrigerators, namely: • All shelves are fixed; • The internal temperature of the refrigerator should be 5 ± 2 °C (this temperature measurement is performed by using two T type thermocouples each possessing 10 g weight); these data are collected by the data-logger;
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Fig. 7.12 Variation of internal temperature of the refrigerator with time
• The environmental conditions consist of a temperature of 22 ± 3 °C, relative humidity of 50 ± 20% and atmospheric pressure of 96 ± 10 kPa; • The measurements are performed at the optimal operation condition (i.e., set to medium cooler-medium freezer temperatures); • The variation of internal temperature with time is gathered from the data-logger and presented in Fig. 7.12. This characteristic displays the operation condition of the compressor. In the filled region of the figure the compressor is “in operating condition” and the internal temperature satisfies the given standards. Therefore, all surface velocity measurements are accomplished when the refrigerator is working in the filled 3–7 °C temperature region. The frequency analysis is accomplished by setting the filter to 1/3-octave band center frequencies. The 50 Hz and 100 Hz are determined as the dominant frequencies in this analysis. Therefore, the surface velocity distribution on the five panels of the refrigerator is measured at 50 Hz and 100 Hz frequencies; this data is fed to BEM code for the acoustic surface pressures to be computed.
7.5.2 Boundary Element Operations The boundary element model of the refrigerator is formed by using 192 elements with 578 nodes. Again, second-order isoparametric, quadrilateral, 8-noded elements and 16-point Gaussian quadrature are used as in the case of the dilating sphere examples in Sect. 7.4. Figure 7.13 shows the layout of the model. Due to the half-space contact algorithm, 16 elements and 33 nodes at the bottom surface are regarded
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Fig. 7.13 Layout of the boundary element model
as “dummy” for which no vibrational and acoustical variables are specified. As a necessity of the BEM solution, the data concerning nodal coordinate (Table 7.B1) and incidence matrices (Table 7.B2) exemplified in Appendix B should be fed to the computer. Nodal coordinate matrix includes global rectangular coordinates of the nodes; whereas incidence matrix is composed of node arrays in each element. These two matrices are automatically generated by using computer codes developed for this purpose. The conformity of the generated matrices defining the nodal surface geometry of the refrigerator model is also presented in Appendix B by an “in-house BEM code” interface (Fig. 7.B1). Since the refrigerator is sitting on the floor the solution is performed via BEM with the half-space-contact algorithm given in Sect. 7.3.
7.5.3 Sound Source Identification and Characterization In order to perform sound source localization and to make a more complete determination, vibration velocity distribution, surface and exterior pressure predictions should be considered all together. Measured surface velocity distribution is displayed on the model layout in Fig. 7.14. Surface sound pressure distribution predicted by the BEM is displayed on the model layout in Fig. 7.15. Near-field sound radiation patterns at the horizontal cross-sections through the compressor and fan
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Fig. 7.14 Measured surface velocity distributions at (a) 50 Hz and (b) 100 Hz
Fig. 7.15 Predicted surface pressure distributions at (a) 50 Hz and (b) 100 Hz
are given in Figs. 7.16 and 7.17, respectively. By examining these distributions, the following observations and outcomes may be drawn: • Vibrations at 50 Hz are more critical than 100 Hz vibrations (Fig. 7.14); • Higher velocities on top-front and top-rear regions imply vibrations of fan and air-circulation due to fan blades (Fig. 7.14); • Higher velocities on bottom-rear region connotes compressor surface vibrations (Fig. 7.14); • Reasons for panel vibrations are (Fig. 7.14): – Transmission of compressor vibrations by joints fixing the compressor to the refrigerator chassis at both 50 Hz and 100 Hz;
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Fig. 7.16 Near field sound radiation patterns at the horizontal cross-section through the compressor at (a) 50 Hz and (b) 100 Hz; zoomed view at (c) 50 Hz and (d) 100 Hz. (a = 0.71 m, b = 0.69 m)
•
•
• •
– Vibration of cooler grid at 100 Hz; – The effect of fan at both 50 Hz and 100 Hz; The compressor dominates the sound radiation for both of the two frequencies; the near-field patterns imply also that the compressor is the most important sound source (Figs. 7.15, 7.16, 7.17); Compressor vibrations not only cause higher surface sound pressures on the rear side, but also influence the other sides (Figs. 7.15 and 7.16); this situation may be explained by the coupling phenomena leading to structure-structure and structureacoustic interaction; Although coupling is observed at both of the two frequencies, it is more obvious at 100 Hz, despite low surface pressure levels. (Fig. 7.15(b)); Cooler grid is affected by the vibrations of the compressor and fan. Therefore, it behaves as a planar sound source at 100 Hz;
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Fig. 7.17 Near field sound radiation patterns at the horizontal cross-section through the fan at (a) 50 Hz and (b) 100 Hz; zoomed view at (c) 50 Hz and (d) 100 Hz (a = 0.71 m, b = 0.69 m)
• Sound radiation pattern at the horizontal cross-section through the fan is almost at the same level for all sides at 50 Hz (Fig. 7.17(c)); this condition states that circulated air causes uniform sound radiation to all sides; • The radiation pattern at 100 Hz is mostly originated from the cooler grid vibrations (Fig. 7.17(d)). Consequently, three main sound sources can be determined for this refrigerator, as evidenced from the analysis of Figs. 7.14, 7.15, 7.16, 7.17: 1. Compressor (point source at 50 and 100 Hz); 2. Fan (point source at 50 and 100 Hz); 3. Cooler grid (planar source at 100 Hz). The displayed results expose that the compressor is the dominant sound source for the refrigerator. Therefore, in order to recover the acoustical performance, it should be considered at first. In practice, using a flexible connection between the joints of compressor and refrigerator chassis will prevent direct transmission of compressor vibrations to the panels. Refrigerator itself behaves as if it is a point
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source due to circulated air by fan blades in the far field. The cooler grid vibrates and radiates sound due to the structural interaction.
7.6 Prospects for the Future Although BEM offers extreme modeling facilities, this method requires formation and solution of large non-symmetric matrices. Therefore, the applicability of BEM is restricted to low frequencies. On the other hand, at some critical frequencies corresponding to the eigen-frequencies of the sound source, numerical solution of the surface Helmholtz integral equation possesses ill-conditioning [13, 35]. This problem, known as “non-uniqueness” and occurring towards the higher frequencies, is another low-frequency limitation of BEM. Although there are some researches performed in order to tackle this problem, BEM does not promise yet for the solution of medium and high frequencies. There are some high frequency techniques in the literature, such as the Statistical Energy Analysis (SEA) [24], Energy Flow Analysis (EFA) [48], with its finite element and boundary element implementations [5], Ray Tracing Method (RTM) [8] and Discrete Singular Convolution (DSC) [38, 50]. These methods are currently under development and most of them have been applied to vibration analysis. Vibroacoustic implementations are in limited number and need to be generalized and disseminated. In view of this, the future prospect will be the need for an effective implementation of the present methods to vibro-acoustic applications and/or the development of high frequency methods viable to be used in coupled vibro-acoustic design and analysis of state of the art structures, such as aircrafts.
7.7 Summary In this chapter, an effective procedure on sound source localization is introduced. The theory at the background and the related bibliography are presented. The chapter continues with theoretical examples, that demonstrate the capability of the given procedure for academic problems; and culminates by a real application. The reallife problems require surface velocity measurement of the vibrating body that may be performed by a simple instrumentation. The velocity data is fed to a BEM code for the surface and exterior pressures to be computed. The fairly simple procedure is a well composition of experimental and numerical techniques; and may be confidentially advised to engineers and young researchers.
7.8 Selected Bibliography • P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Mc-Graw Hill, 1953. This classical book presents the mathematics at the background of many physical phenomena, specifically in vibration and acoustics. The reader who intends further searching the theory at the basis of this chapter may examine the topics:
174
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Fig. 7.A1 A boundary element
Green’s functions, integral equations and the wave motion. The book has been written for physicists and engineers. • R.D. Ciskowski, C.A. Brebbia (Eds.), Boundary Element Methods in Acoustics, Computational Mechanics Publications, 1991, ISBN 1-85312-104-5. This book focuses exclusively on major active areas of research and applications involving the use of boundary element method to solve problems in acoustics. The book consists of three parts: a historical survey, basic formulations and fundamental concepts, and applications. The intended audience for this book is engineers and scientists in research and practice, and also postgraduate students. • E.G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, London: Academic Press, 1999. This book presents a very good introduction to sound radiation, particularly in the context of NAH. The issue is treated in a manner of progressively increasing complexity, first in rectangular, then in cylindrical and finally in spherical coordinates. It introduces clear algorithms for intended code developers as far as an NAH software concern.
Appendix A An 8-noded quadrilateral boundary element with local coordinates ξ and η is shown in Fig. 7.A1. In the isoparametric element formulation technique, the global Cartesian coordinates, x, y and z, acoustic pressure, p, and normal velocity, un , of any point on an element are expressed in terms of the nodal values by using the same shape functions Nα (α = 1, 2, . . . , 8 shows the node numbers in each element). Accordingly, we have xi (ξ, η) = yi (ξ, η) = zi (ξ, η) =
8
α=1 8
α=1 8
α=1
Nα (ξ, η)xiα , Nα (ξ, η)yiα , Nα (ξ, η)ziα ,
(7.A1)
7 An Efficient Sound Source Localization Technique
pi (ξ, η) =
8
175
Nα (ξ, η)piα ,
(7.A2)
Nα (ξ, η)(un )iα ,
(7.A3)
α=1
(un )i (ξ, η) =
8
α=1
where xiα , yiα , ziα are the global coordinates of the nodes of the ith element, and piα and (un )iα are the acoustic pressure and normal velocity of the nodes of the ith element, respectively. xi (ξ, η), yi (ξ, η), zi (ξ, η) are the global coordinates of any point on the ith element with local coordinates (ξ, η). pi (ξ, η) and (un )i (ξ, η) are the acoustic pressure and normal velocity of this point, respectively. The nodal values of this 8-noded quadrilateral element are given in terms of the local coordinates by the quadratic polynomial x = a0 + a1 ξ + a2 η + a3 ξ η + a4 ξ 2 + a5 η2 + a6 ξ 2 η + a7 ξ η2 .
(7.A4)
Therefore, the quadratic shape functions of this element, that as called by Zienkiewicz [54] belongs to the serendipity class, are N1 (ξ, η) = (ξ + 1)(η + 1)(ξ + η − 1)/4, N2 (ξ, η) = (η + 1) 1 − ξ 2 /2, N3 (ξ, η) = (ξ − 1)(η + 1)(ξ − η + 1)/4, N4 (ξ, η) = (ξ − 1) η2 − 1 /2, N5 (ξ, η) = (1 − ξ )(η − 1)(ξ + η + 1)/4, N6 (ξ, η) = (1 − η) 1 − ξ 2 /2,
(7.A5)
N7 (ξ, η) = (ξ + 1)(η − 1)(η − ξ + 1)/4, N8 (ξ, η) = (ξ + 1) 1 − η2 /2. It is known that a complete quadratic polynomial of two functions (say ξ and η) has nine terms and the corresponding boundary element includes nine nodes. However, the quadratic polynomial of serendipity class element, Eq. (7.A4), includes eight terms and the element has eight nodes. The term involving ξ 2 η2 multiplication is omitted. It has been shown that this simple and widely used element gives sufficiently sensitive results [23, 31, 41, 42].
176
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Appendix B Table 7.B1 Computed coordinates of front surface of the refrigerator Node
X
Y
Z
Node
X
Y
Z
1
292.500
−355.000
−937.500
76
292.500
88.750
0.000
2
292.500
−266.250
−937.500
77
292.500
355.000
−93.750
3
292.500
−177.500
−937.500
78
292.500
355.000
0.000
4
292.500
−177.500
−843.750
79
292.500
266.250
0.000
5
292.500
−177.500
−750.000
80
292.500
−177.500
93.750
6
292.500
−266.250
−750.000
81
292.500
−177.500
187.500
7
292.500
−355.000
−750.000
82
292.500
−266.250
187.500
8
292.500
−355.000
−843.750
83
292.500
−355.000
187.500
9
292.500
−88.750
−937.500
84
292.500
−355.000
93.750
10
292.500
0.000
−937.500
85
292.500
0.000
93.750
11
292.500
0.000
−843.750
86
292.500
0.000
187.500
12
292.500
0.000
−750.000
87
292.500
−88.750
187.500
13
292.500
−88.750
−750.000
88
292.500
177.500
93.750
14
292.500
88.750
−937.500
89
292.500
177.500
187.500
15
292.500
177.500
−937.500
90
292.500
88.750
187.500
16
292.500
177.500
−843.750
91
292.500
355.000
93.750
17
292.500
177.500
−750.000
92
292.500
355.000
187.500
18
292.500
88.750
−750.000
93
292.500
266.250
187.500
19
292.500
266.250
−937.500
94
292.500
−177.500
281.250
20
292.500
355.000
−937.500
95
292.500
−177.500
375.000
21
292.500
355.000
−843.750
96
292.500
−266.250
375.000
22
292.500
355.000
−750.000
97
292.500
−355.000
375.000
23
292.500
266.250
−750.000
98
292.500
−355.000
281.250
24
292.500
−177.500
−656.250
99
292.500
0.000
281.250
25
292.500
−177.500
−562.500
100
292.500
0.000
375.000
26
292.500
−266.250
−562.500
101
292.500
−88.750
375.000
27
292.500
−355.000
−562.500
102
292.500
177.500
281.250
28
292.500
−355.000
−656.250
103
292.500
177.500
375.000
29
292.500
0.000
−656.250
104
292.500
88.750
375.000
30
292.500
0.000
−562.500
105
292.500
355.000
281.250
31
292.500
−88.750
−562.500
106
292.500
355.000
375.000
32
292.500
177.500
−656.250
107
292.500
266.250
375.000
33
292.500
177.500
−562.500
108
292.500
−177.500
468.750
34
292.500
88.750
−562.500
109
292.500
−177.500
562.500
35
292.500
355.000
−656.250
110
292.500
−266.250
562.500
36
292.500
355.000
−562.500
111
292.500
−355.000
562.500
(continued on next page)
7 An Efficient Sound Source Localization Technique
177
Table 7.B1 (continued) Node
X
37
292.500
38
292.500
39
292.500
40
292.500
41
Y
Z
Node
X
Y
Z
266.250
−562.500
112
292.500
−355.000
468.750
−177.500
−468.750
113
292.500
0.000
468.750
−177.500
−375.000
114
292.500
0.000
562.500
−266.250
−375.000
115
292.500
−88.750
562.500
292.500
−355.000
−375.000
116
292.500
177.500
468.750
42
292.500
−355.000
−468.750
117
292.500
177.500
562.500
43
292.500
0.000
−468.750
118
292.500
88.750
562.500
44
292.500
0.000
−375.000
119
292.500
355.000
468.750
45
292.500
−88.750
−375.000
120
292.500
355.000
562.500
46
292.500
177.500
−468.750
121
292.500
266.250
562.500
47
292.500
177.500
−375.000
122
292.500
−177.500
656.250
48
292.500
88.750
−375.000
123
292.500
−177.500
750.000
49
292.500
355.000
−468.750
124
292.500
−266.250
750.000
50
292.500
355.000
−375.000
125
292.500
−355.000
750.000
51
292.500
266.250
−375.000
126
292.500
−355.000
656.250
52
292.500
−177.500
−281.250
127
292.500
0.000
656.250
53
292.500
−177.500
−187.500
128
292.500
0.000
750.000
54
292.500
−266.250
−187.500
129
292.500
−88.750
750.000
55
292.500
−355.000
−187.500
130
292.500
177.500
656.250
56
292.500
−355.000
−281.250
131
292.500
177.500
750.000
57
292.500
0.000
−281.250
132
292.500
88.750
750.000
58
292.500
0.000
−187.500
133
292.500
355.000
656.250
59
292.500
−88.750
−187.500
134
292.500
355.000
750.000
60
292.500
177.500
−281.250
135
292.500
266.250
750.000
61
292.500
177.500
−187.500
136
292.500
−177.500
843.750
62
292.500
88.750
−187.500
137
292.500
−177.500
937.500
63
292.500
355.000
−281.250
138
292.500
−266.250
937.500
64
292.500
355.000
−187.500
139
292.500
−355.000
937.500
65
292.500
266.250
−187.500
140
292.500
−355.000
843.750
66
292.500
−177.500
−93.750
141
292.500
0.000
843.750
67
292.500
−177.500
0.000
142
292.500
0.000
937.500
68
292.500
−266.250
0.000
143
292.500
−88.750
937.500
69
292.500
−355.000
0.000
144
292.500
177.500
843.750
70
292.500
−355.000
−93.750
145
292.500
177.500
937.500
71
292.500
0.000
−93.750
146
292.500
88.750
937.500
72
292.500
0.000
0.000
147
292.500
355.000
843.750
73
292.500
−88.750
0.000
148
292.500
355.000
937.500
74
292.500
177.500
−93.750
149
292.500
266.250
937.500
75
292.500
177.500
0.000
178
A. Seçgin and A.S. Sarıgül
Table 7.B2 Incidence matrix for the front surface of refrigerator Element No
Nodes forming the elements
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1 3 10 15 7 5 12 17 27 25 30 33 41 39 44 47 55 53 58 61 69 67 72 75 83 81 86 89 97 95 100 103 111 109 114 117 125 123 128 131
2 9 14 19 6 13 18 23 26 31 34 37 40 45 48 51 54 59 62 65 68 73 76 79 82 87 90 93 96 101 104 107 110 115 118 121 124 129 132 135
3 10 15 20 5 12 17 22 25 30 33 36 39 44 47 50 53 58 61 64 67 72 75 78 81 86 89 92 95 100 103 106 109 114 117 120 123 128 131 134
4 11 16 21 24 29 32 35 38 43 46 49 52 57 60 63 66 71 74 77 80 85 88 91 94 99 102 105 108 113 116 119 122 127 130 133 136 141 144 147
5 12 17 22 25 30 33 36 39 44 47 50 53 58 61 64 67 72 75 78 81 86 89 92 95 100 103 106 109 114 117 120 123 128 131 134 137 142 145 148
6 13 18 23 26 31 34 37 40 45 48 51 54 59 62 65 68 73 76 79 82 87 90 93 96 101 104 107 110 115 118 121 124 129 132 135 138 143 146 149
7 5 12 17 27 25 30 33 41 39 44 47 55 53 58 61 69 67 72 75 83 81 86 89 97 95 100 103 111 109 114 117 125 123 128 131 139 137 142 145
8 4 11 16 28 24 29 32 42 38 43 46 56 52 57 60 70 66 71 74 84 80 85 88 98 94 99 102 112 108 113 116 126 122 127 130 140 136 141 144
7 An Efficient Sound Source Localization Technique
179
Fig. 7.B1 An interface of ‘in-house BEM code’ conforming the co-ordinates and incidence matrix defining the geometry of the refrigerator
References 1. Avsar, E.: Determination of the acoustic radiation characteristics of vibrating bodies near reflecting surfaces. M.Sc. thesis, The Graduate School of Natural and Applied Sciences, Dokuz Eylül University, ˙Izmir, Turkey (2010) 2. Bai, M.R.: Application of BEM (boundary element method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries. J. Acoust. Soc. Am. 92(1), 533–549 (1992) 3. Batel, M., Marroquin, M., Hald, J., Christensen, J.J., Schuhmacher, A.P., Nielsen, T.G.: Noise source location techniques—simple to advanced applications. Sound Vib. 37(3), 24–38 (2003) 4. Bell, W.A., Meyer, W.L., Zinn, B.T.: Predicting acoustics of arbitrarily shaped bodies using an integral approach. AIAA J. 15(6), 813–820 (1977) 5. Bitsie, F.: The structural-acoustic energy finite element method and energy boundary element method. Ph.D. thesis, Purdue University Graduate School, USA (1996) 6. Burton, A.J., Miller, G.F.: Application of integral equation methods to numerical solution of some exterior boundary-value problems. Proc. R. Soc. Lond. A 323(1553), 201–210 (1971) 7. Castellini, P., Martarelli, M.: Acoustic beamforming: analysis of uncertainty and metrological performances. Mech. Syst. Signal Process. 22(3), 672–692 (2008) 8. Chae, K.S., Ih, J.G.: Prediction of vibrational energy distribution in the thin plate at highfrequency bands by using the ray tracing method. J. Sound Vib. 240(2), 263–292 (2001) 9. Chertock, G.: Sound radiation from vibrating surfaces. J. Acoust. Soc. Am. 36(7), 1305–1313 (1964) 10. Christensen, J.J., Hald, J.: Beamforming. Bruel Kjaer Tech. Rev. 1, 1–48 (2004) 11. Copley, L.G.: Integral equation method for radiation from vibrating bodies. J. Acoust. Soc. Am. 41(4), 807–816 (1967) 12. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York (1962) 13. Dokumaci, E., Sarigul, A.S.: Analysis of the near-field acoustic radiation characteristics of two radially vibrating spheres by the Helmholtz integral-equation formulation and a critical-
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22. 23. 24. 25. 26. 27. 28. 29. 30.
31.
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A. Seçgin and A.S. Sarıgül study of the efficacy of the chief overdetermination method in two-body problems. J. Sound Vib. 187(5), 781–798 (1995) Fahy, F.J.: Some applications of the reciprocity principle in experimental vibroacoustics. Acoust. Phys. 48(2), 217–229 (2003) Ginn, K.B., Hald, J.: STSF—practical instrumentation and applications. Bruel Kjaer Tech. Rev. 2, 1–27 (1989) Hald, J.: STSF—a unique technique for scan-based near-field acoustic holography without restrictions on coherence. Bruel Kjaer Tech. Rev. 1, 1–50 (1989) Hald, J.: Non-stationary STSF. Bruel Kjaer Tech. Rev. 1, 1–36 (2000) Hald, J.: Patch near-field acoustical holography using a new statistically optimal method. Bruel Kjaer Tech. Rev. 1, 40–50 (2005) Hald, J.: Combined NAH and beamforming using the same array. Bruel Kjaer Tech. Rev. 1, 11–39 (2005) Janssens, M.H.A., Verheij, J.W.: A pseudo-forces methodology to be used in characterization of structure-borne sound sources. Appl. Acoust. 61(3), 285–308 (2000) Janssens, M.H.A., Verheij, J.W., Loyau, T.: Experimental example of the pseudo-forces method used in characterisation of a structure-borne sound source. Appl. Acoust. 63(1), 9– 34 (2002) Koopmann, G.H., Benner, H.: Method for computing the sound power of machines based on the Helmholtz integral. J. Acoust. Soc. Am. 71(1), 78–89 (1982) Latcha, M.A., Akay, A.: Application of the Helmholtz integral in acoustics. J. Vib. Acoust. Stress Reliab. Des. 108(4), 447–453 (1986) Lyon, R.H., DeJong, R.G.: Theory and Application of Statistical Energy Analysis, 2nd edn. Butterworth, Boston (1995) Maynard, J.D., Williams, E.G., Lee, Y.: Nearfield acoustic holography: 1. Theory of generalized holography and the development of NAH. J. Acoust. Soc. Am. 78(4), 1395–1413 (1985) Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953) Morse, P.M., Ingard, K.U.: Theoretical Acoustics. McGraw-Hill, New York (1968) Piaszczyk, C.M., Klosner, J.M.: Acoustic radiation from vibrating surfaces at characteristic frequencies. J. Acoust. Soc. Am. 75(2), 363–375 (1984) Pierce, A.D.: Acoustics: An Introduction to Its Physical Principles and Applications. McGrawHill, New York (1981) Sakuma, T., Yasuda, Y.: Fast multipole boundary element method for large-scale steady-state sound field analysis. Part I: setup and validation. Acta Acust. United Acust. 88(4), 513–525 (2002) Sarigül, A.S.: The computation of multi-body acoustic fields by Helmholtz integral equation. Ph.D. thesis, The Graduate School of Natural and Applied Sciences, Dokuz Eylül University, ˙Izmir, Turkey (1990) (in Turkish) Sarigül, A.S.: Sound attenuation characteristics of right-angle pipe bends. J. Sound Vib. 228(4), 837–844 (1999) Sarigül, A.S., Kiral, Z.: Interior acoustics of a truck cabin with hard and impedance surfaces. Eng. Anal. Bound. Elem. 23(9), 769–775 (1999) Sarigül, A.S., Seçgin, A.: A study on the applications of the acoustic design sensitivity analysis of vibrating bodies. Appl. Acoust. 65(11), 1037–1056 (2004) Schenck, H.A.: Improved integral formulation for acoustic radiation problems. J. Acoust. Soc. Am. 44(1), 41–58 (1968) Schuhmacher, A., Rasmussen, K.B., Hansen, P.C.: Sound source reconstruction using inverse boundary element calculations. J. Acoust. Soc. Am. 113(1), 114–127 (2003). Seçgin, A., Sarigül, A.S.: Acoustic Design Sensitivity Analysis for Vibrating Bodies. Lambert Academic Publishing, Saarbrücken (2010) Seçgin, A., Sarigül, A.S.: Vibration Analysis of Plates by Discrete Singular Convolution. Lambert Academic Publishing, Saarbrücken (2010) Seybert, A.F., Soenarko, B.: Radiation and scattering of acoustic-waves from bodies of arbitrary shape in a three-dimensional half-space. J. Vib. Acoust. Stress Reliab. Des. 110(1), 112–117 (1988)
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40. Seybert, A.F., Wu, T.W.: Modified Helmholtz integral-equation for bodies sitting on an infinite-plane. J. Acoust. Soc. Am. 85(1), 19–23 (1989) 41. Seybert, A.F., Soenarko, B., Rizzo, F.J., Shippy, D.J.: Application of the BIE method to sound radiation problems using an isoparametric element. J. Vib. Acoust. Stress Reliab. Des. 106(3), 414–420 (1984) 42. Seybert, A.F., Soenarko, B., Rizzo, F.J., Shippy, D.J.: An advanced computational method for radiation and scattering of acoustic-waves in three dimensions. J. Acoust. Soc. Am. 77(2), 362–368 (1985) 43. Skudrzyk, E.: Foundations of Acoustics: Basic Mathematics and Basic Acoustics. Springer, Vienna (1971) 44. Verheij, J.W.: Inverse and reciprocity methods for machinery noise source characterization and sound path quantification. Part 1: sources. Int. J. Acoust. Vib. 2, 11–20 (1997) 45. Verheij, J.W.: Inverse and reciprocity methods for machinery noise source characterization and sound path quantification. Part 2: transmission paths. Int. J. Acoust. Vib. 3, 103–112 (1997) 46. Veronesi, W.A., Maynard, J.D.: Nearfield acoustic holography (NAH): 2. Holographic reconstruction algorithms and computer implementation. J. Acoust. Soc. Am. 81(5), 1307–1322 (1987). 47. Veronesi, W.A., Maynard, J.D.: Digital holographic reconstruction of sources with arbitrarily shaped surfaces. J. Acoust. Soc. Am. 85(2), 588–598 (1989) 48. Wang, S.: High frequency energy flow analysis methods: numerical implementation, applications and verification. Ph.D. thesis, Purdue University Graduate School, USA (2000) 49. Wang, Z.X., Wu, S.F.: Helmholtz equation least-squares method for reconstructing the acoustic pressure field. J. Acoust. Soc. Am. 102(4), 2020–2032 (1997) 50. Wei, G.W., Zhao, Y.B., Xiang, Y.: A novel approach for the analysis of high-frequency vibrations. J. Sound Vib. 257(2), 207–246 (2002) 51. Wu, S.F.: On reconstruction of acoustic pressure fields using the Helmholtz equation least squares method. J. Acoust. Soc. Am. 107(5), 2511–2522 (2000) 52. Yasuda, Y., Sakuma, T.: A technique for plane-symmetric sound field analysis in the fast multipole boundary element method. J. Comput. Acoust. 13(1), 71–85 (2005) 53. Zhang, Z.D., Vlahopoulos, N., Raveendra, S.T., Allen, T., Zhang, K.Y.: A computational acoustic field reconstruction process based on an indirect boundary element formulation. J. Acoust. Soc. Am. 108(5), 2167–2178 (2000) 54. Zienkiewicz, O.C.: The Finite Element Method, 3rd edn. McGraw-Hill, London (1977)
Chapter 8
Dispersion Analysis of Acoustic Circumferential Waves Using Time-Frequency Representations R. Latif, M. Laaboubi, E.H. Aassif, and G. Maze
Abstract The dispersion estimation is an important objective in acoustic processing. This work presents a study of the group velocity dispersion of some circumferential waves propagating around an elastic tube. The dispersive character of the circumferential waves is theoretically known but the experimental measurement of the group velocity in a dispersive medium is still a complex operation. We have determined the characteristics of the circumferential waves dispersion for aluminium tube using a time-frequency representations. Among these time-frequency techniques the Wigner-Ville distribution, the spectrogram and the reassigned spectrogram are used here for its interesting properties in terms of acoustic applications. These techniques are applied on a measured acoustic signal scattered by an air-filled tube immersed in water. From these time-frequency images the symmetric (S0), antisymmetric (A1) and the symmetric (S1) circumferential waves are identified and the dispersion group velocities are also determined for these circumferential waves. The obtained image of the reassigned spectrogram technique allows a better visibility of the circumferential waves which are propagated around the tube through the good resolution of the time-frequency image. From the obtained image, one can go up easily with certain characteristics of the circumferential acoustic waves such as the group velocity and the cut-off frequencies of each circumferential waves. The results obtained from the time-frequency image of the reassigned spectrogram are
R. Latif () ESSI, National School of Applied Science, Ibn Zohr University, Agadir, Morocco e-mail:
[email protected] M. Laaboubi · E.H. Aassif LMTI, Faculty of Science, Ibn Zohr University, Agadir, Morocco M. Laaboubi e-mail:
[email protected] E.H. Aassif e-mail:
[email protected] G. Maze LOMC, Le Havre University, Le Havre, France e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_8, © Springer Science+Business Media B.V. 2011
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in good agreement with those calculated values using the theoretical methods given in the scientific literature.
8.1 Introduction The analysis of acoustic signals scattered by elastic shells, such as spheres, infinite cylinders and tubes, is a topic that has received large attention for the last ten years [1, 27, 29, 30]. In previous studies, the characterization of the scattering problem is mainly performed in the frequency domain. But this spectral analysis is not sufficient to describe the fundamental characteristics of circumferential waves such as velocity dispersion. Some studies have shown the interest of using timefrequency representations where the acoustic energy of the scattered pressure field can be analyzed simultaneously in the time and frequency domains, and the velocity dispersion characteristics of the circumferential waves are clearly displayed. The experimental measurement of the group velocity of these waves propagating around an elastic tube is a still complex operation [15–17]. In this study, we show that the dispersion velocity can be determined from the time-frequency representations. These representations of a signal jointly in time and frequency present a very important interest for the non-stationary signal processing such as the acoustic signal. But the presence of the interference terms through to the bilinear nature of the distribution (Wigner-Ville representation [20–24]) or the broadening spectrum of the signal components (spectrogram representation) makes difficult the choice of a technique of time-frequency representation among those of the Cohen class [7, 11, 12, 18, 26]. The interference terms or broadening spectrum allows the reduction of the readability of time-frequency image hence the interest of trying to choose the time-frequency technique to improve readability of the image. Many works were then devoted to the improvement of the readability of the Cohen class distributions such as the technique of reassignment temporal and frequency components. The technique of reassignment was the result of the work carried by Kodera, Gendrin and of Villedary in 1970. Its principal aim consists of the rearranging of the distribution values for improve the localization of the components in the time-frequency plan [8, 28, 32]. In particular, the reassignment spectrogram representation consists on replacing a whole of points in time and frequency by their gravity center of maximum energy in the time-frequency plan [8, 28, 32], which improves the legibility of the image and increase the precision of the required parameters. Time-frequency techniques, such as the Smoothed Pseudo Wigner-Ville distribution, the spectrogram distribution and the reassignment spectrogram, are used in this work to analyze the experimental acoustic signals scattered by aluminium and copper tubes with radius ratio b/a = 0.9 (b: internal radius and a: external radius), immersed in water. The aim of this study is to visualize the frequency evolution versus the time of the symmetric and the anti-symmetric circumferential waves (S0, A1, S1, . . .) and to identify them. The principle of the reassignment technique, used in this work, consists in replacing all points in time and frequency by their gravity center of maximum energy in the time-frequency plan. In the case of our study, the time-frequency
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technique of the reassigned spectrogram permits to obtain the localization of the trajectories of the circumferential waves (S0, A1 and S1). This localization allows to give a good readability of the time-frequency image and to thus increase the precision of the required parameters. The obtained representations are the synthetic images of the wave trajectories propagating around the tube with a good readability. Also, this technique permits to visualize the circumferential wave trajectories of S0, A1 and S1, to determine their cut-off frequencies and to estimate their group velocities dispersion. The first section of this chapter will describe the time-frequency representations such as the Wigner-Ville, the spectrogram and the reassignment spectrogram. The acoustic experimental signal scattered by an elastic tube is performed in the second section. In the latter sections, some circumferential waves (S0, A1 and S1) are observed on the time-frequency images obtained. The group velocities dispersion of these waves are determined from these images.
8.2 Overview of the State of the Art Our present interest has been focused on the circumferential waves propagating around the circumference of the tube, and in particular the lowest two Lamb-type modes (referred to as S0, A1 and S1 modes, respectively), because of their special properties and potential applications in other areas. This work is in fact a continuation and extension of the previous research published by our laboratory. Properties of the acoustic modes in a plane plate have been extensively studied since they were first discussed by Rayleigh and Lamb a century ago [2–4, 14, 19]. Study of the circumferential waves in an elastic cylindrical shell, as a counterpart of the plate case, has also been carried out for decades and is still found active in recent years [31, 34, 36]. Most of the research reported in the literature is in the context of underwater acoustics, where incident plane impinge are scattered from the cylindrical shells. While extensive theoretical analysis and numerical computations based on various thin-shell models have been presented, and full dispersion curves in different cases were given, experimental dispersion curves are rarely found in the literature and usually only a few discrete points are available. Many studies have focused on problems where a cylindrical shell is immersed in water with inside filled with air. The circumferential wave phenomenon is commonly analyzed by the resonance scattering theory, with experimental data obtained from evaluating the measured parameters of the scattered field in the media surrounding the shell. Among the numerous theoretical dispersion curves published, whether the modes presented could actually exist in a liquid-loaded tube or not is not clear. On the other hand, a major concern, for example, has been the identification of the modes corresponding to the observed signals in particular experimental configurations. In our study we have used the timefrequency representations to analyze the velocity dispersion characteristics of the circumferential waves in particular S0, A1, S1 modes which are propagated around the circumference of the elastic tube. The time-frequency techniques, such as the Wigner-Ville distribution, the spectrogram distribution and the reassigned spectrogram, are used in this work to analyze the experimental acoustic signals scattered
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by an aluminium and copper tubes with radius ratio b/a = 0.9 immersed in water and also to study the dispersion curves of the circumferential waves (S0, A1, S1). The time-frequency techniques used in this work permit to obtain the localization of the circumferential waves (S0, A1 and S1) trajectories. This localization allows obtaining a better legibility of the time-frequency in the case of the reassigned spectrogram image and also to thus increase the precision of the required parameters. The synthetic time-frequency images obtained permit to visualize and to determine the dispersion curves of the circumferential wave S0, A1 and S1 and to determine their cut-off frequencies.
8.3 Time-Frequency Representations In the scientific literature there are two types of time-frequency representations; parametric and nonparametric techniques. In this chapter we have used only the nonparametric time-frequency representation. The majority of the nonparametric time-frequency methods are represented by the Cohen class [7, 11, 12, 17, 18, 26]. Among the large number of the time-frequency representations of this class, we have retained the Wigner-Ville, the spectrogram and his modified version (reassigned spectrogram) for their interesting properties. To avoid the covering of frequential components in the time-frequency representation, in this study we use in the place of the real signal x(t) the analytical signal xa (t), defined by the expression (8.1) xa (t) = x(t) + j H x(t) , where j2 = −1, x(t) is the signal with real values and H{x(t)} its Hilbert transform. The spectrum Fa (k) of the analytical signal xa (t) is given by ⎧ ⎨ 2X(k) if 0 < k < N/2, Fa (k) = X(0) if k = 0, N/2, (8.2) ⎩ 0 if N/2 < k < N, where X(k) is the Fourier transform of the original signal x(t) and N is the number of points.
8.3.1 Wigner-Ville Distribution The time-frequency representation of Wigner-Ville distribution (WVD) associated to a signal xa (t), of finite energy, is the function WVD{xa (t)}(ω) depending on the temporal (t) and frequency (ω) parameters. This distribution is given by the expression [7, 11, 12, 17, 18, 26] +∞ τ ∗ τ −jωτ WVD xa (t) (ω) = xa t + dτ, (8.3) xa t − e 2 2 −∞ where xa∗ (t) is the complex conjugate of the analytical signal xa (t).
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The Smoothed Pseudo Wigner-Ville (SPWV) is implemented to attenuate the interference terms presented between the inner components figuring in Wigner-Ville representation. The SPWV uses two smoothing windows h and g. These smoothing windows are introduced into the definition of the Wigner-Ville distribution in order to allow a separate control of the interference, either in time (g) or in frequency (h). The expression of this representation is defined by [7, 11, 12, 17, 18, 26] SPWV xa (t) (ω) +∞ 2 +∞
τ
τ ∗ τ −jωu
h u + u − g(t − u)x du dτ, (8.4) = x e a
2
2 a 2 −∞ −∞ where h(t) is a smoothing frequency window and g(t) is a smoothing temporal window.
8.3.2 Spectrogram Distribution The short time Fourier transform (STFT) can be interpreted as a Fourier analysis of successive sections of the signal weighted by a temporal window such as Gabor, Hamming and Blackman. The STFT is given by [7, 11, 12] +∞ h xa (τ )h∗ (τ − t)e−jωτ dτ, (8.5) STFT xa (t) (ω) = −∞
where h is the analytical window. The energy density function or spectrogram (SP) of the signal corresponding to the window h is given by
h h 2 SP xa (t) (ω) = STFT xa (t) (ω) . (8.6) The spectrogram representation can be expressed as a smoothed version of the Wigner-Ville distribution of the signal xa (t), yielding [7, 11, 12] h dτ dξ SP xa (t) (ω) = WV xa (t) (τ, ξ )WVh (τ − t, ξ − ν) , (8.7) 2π where WVh is the Wigner-Ville distribution of the window h.
8.3.3 Reassignment Spectrogram 8.3.3.1 Principle of the Heisenberg Uncertainty The principle of Heisenberg says that it is impossible to obtain an analyzing function h(t) localized in time and frequency simultaneously, it is a function with which the scalar product is calculated.
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Fig. 8.1 The time-frequency plan decomposed into small rectangular windows, called the Heisenberg boxes
The figure above indicates the Heisenberg boxes in the time-frequency plan. Its length represent broadening in time σt of the analyzing function, around an average time (tm ) and its width represents broadening in spectrum frequency σω of the analyzing function, around an average frequency, called center frequency (ωm ). The temporal average (tm ) is then defined by the expression [6, 33] +∞
2 t h(t) dt, (8.8) tm = −∞
and the expression of the center frequency (ωm ) is given by +∞
2
1 ω H (ω) dω, ωm = 2π −∞
(8.9)
where H is the Fourier transform of the analytical window h. Knowing the average time and the average frequency, we obtain the variances, in time and frequency, around of tm and of ωm , by using the two following formulas [6, 33], +∞
2
(t − tm )2 h(t) dt, (8.10) σt2 = −∞
σω2
=
+∞ −∞
2
(ω − ωm )2 H (ω) dω.
(8.11)
The minimal value of the product σt σω is reached when h(t) is Gaussian. These Heisenberg boxes give an image of the decomposition of the time-frequency plan. In the case of the spectrogram, the analysis provides constant resolution for all frequencies since it uses the same window for the analysis of the inspected signal x(t) (Fig. 8.1) and the localization of the spectra of energy of this image is related to the choice of the analysis window and its width. However, if one wishes a good frequential localization the H (f ) window must be short. But as h(t) short led to broad H (f ) and vice versa, there will be always a compromise between the temporal and the frequential resolutions. To overcome this limitation we have used the reassignment method.
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8.3.3.2 Reassignment Time-Frequency Image The energy of Wigner-Ville distribution is concentrated around each one of the variation laws of the frequency according to time. Despite it’s advantage the bilinear nature of this distribution causes the appearance of the cross-terms which materialize by the oscillating structures between the positive and negative values which comes to complicate the reading of the time-frequency image. In the spectrogram distribution the cross-terms are well attenuated by the smoothing of the analysis window [7, 11, 12]. On the other hand, this smoothing causes the broadening spectrum of the energy distribution, which permits for the spectrogram with a loss of resolution and contrast. The reassignment method allows to solve the problem of the broadening spectral and the cross-terms to have a readable and localized distribution in the timefrequency plan. The principle is to rearrange the spectrogram on the distribution of the time-frequency energies given by the Wigner-Ville distribution [5, 13, 32]. That consists to move of the spectrogram values of their point of calculation towards a new position (tx a (t, ω), ωx a (t, ω)) given by a barycenter evaluated on the WignerVille distribution of the signal of the time-frequency plan defined by the smoothing core WVh (t, ω) in the following way [5, 9, 10, 13, 32]: 1 dτ dξ , (8.12) tx a (t, ω) = h τ WVxa (τ, ξ )WVh (τ − t, ξ − ω) 2π SPxa (t, ω) 1 dτ dξ , (8.13) ωx a (t, ω) = h ξ WVxa (τ, ξ )WVh (τ − t, ξ − ω) 2π SPxa (t, ω) where SPhxa is the spectrogram of the signal xa using the analyzed function h. The reassignment spectrogram RSP, obtained from the spectrogram values in the time-frequency plan by making the sum if two quantities arrive at the same place, is given by the relation [5, 9, 10, 13, 32] dτ dξ SPhxa (τ, ξ )δ t − t (τ, ξ ) δ ω − ω (τ, ξ ) , (8.14) RSPxa (t, ω) = 2π where δ is the two-dimensional Dirac function. The reassignment operators in time (reassigned times) and in frequency (reassigned frequencies) may be computed by a simple manner using the STFT [32]. , for the spectral component from the short-time analysis The reassigned time tt,ω window centered at time t can be expressed as [5, 8, 13, 26, 32] =t − tt,ν
∗ STFTth xa (t, ν) × STFTxa (t, ν)
SPxa (t, ν)
,
(8.15)
where STFTth xa (t, ν) is the short-time Fourier transform of xa (t) using the timeweighted window function th(t), STFT∗xa (t, ν) is the complex conjugate of the short-time Fourier transform of xa (t) and {·} is the real part of the bracketed ratio.
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, corresponding to the same component is The corrected frequency, νt,ν [5, 9, 10, 13, 32]
∗ STFTdh xa (t, ν) × STFTxa (t, ν) νt,ν = ν + , (8.16) SPxa (t, ν)
where STFTdh xa (t, ν) is the short-time Fourier transform of xa (t) using the frequencyweighted window function dh(t) = dtd h(t) and {·} is the imaginary part of the bracketed ratio. for the spectral component from the short-time The discrete reassigned time tn,k analysis window centered at time n [5, 9, 10, 13, 32] is tn,k =n−
∗h STFTth xa (n, k) × STFTxa (n, k)
SPhxa (n, k)
.
(8.17)
, corresponding to the same component is The discrete corrected frequency, ωn,k given by [5, 6, 13, 26, 33] ωn,k
∗h STFTdh xa (n, k) × STFTxa (n, k) . =k+ SPhxa (n, k)
(8.18)
8.3.3.3 Reassignment Vectors We defined the reassigned operators as the co-ordinates of the gravity center calculated on the Wigner-Ville distribution of the signal taken in a vicinity of the timefrequency plan. The concentrated energy around this point is evaluated by the vector of displacement (reassignment vector) [5]. For calculation of the reassignment vectors one can assimilate the time-frequency plan in the complex plan to have a field of the displacement vectors, then one reducing this vector by his image in the complex plan, which is given by the relation [5, 8, 13] r=
t − t ω − ω +j , th ωh
(8.19)
where th and ωh are the temporal and the frequential bands of the analysis window h. One can determine the expression of the reassignment vector using the STFT of the signal by employing the analysis windows of the th(t) and the dtd h(t) with the expression [5, 8, 13] r=
d
1 1 STFTth STFT dt h(t) + j . th ωh STFTh STFTh
(8.20)
Figure 8.2 shows the field form of the reassignment vector orientations of the energy distribution in the time-frequency plan.
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Fig. 8.2 Field form of the reassignment vector orientations of the energy distribution in the time-frequency plan
Fig. 8.3 Experimental setup
8.4 Acoustic Measured Signal Backscattered by an Elastic Tube 8.4.1 Experimental Setup The experimental study is led in a parallelepipedic tank. To obtain exploitable signals, it is necessary to avoid the reflexing of the tank sides. The experimental system, presented in Fig. 8.3, is constituted of a pulse generator Sofranel 5052PR, a digital oscilloscope LeCroy 9310M-300MHz, a personal computer, a transducer and an air-filled cylinder with closed extremities immersed in water. The water density is ρwater = 1000 kg/m3 and the acoustic velocity wave is cwater = 1470 m/s measured at the ambient temperature (20 °C) [17]. The broadband transducer placed opposite of the tube is used successively as emitter and receiver and is excited by a short electrical pulse generated by the pulse generator via the transducer. The sample used is an aluminium and copper tubes, with a radius ratio b/a = 0.9 and the transducer used, with a wide-band, is a Panametrics type and its central frequency is 10 MHz and it’s diameter is 10 mm. The electric excitation signal of the transducer can be adjusted by selection of the repetition frequency by using a pulse generator and excited the target with a normal incidence. The pulse generator is synchronized by the square wave generator. After the propagation in water the incident acoustic wave
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Fig. 8.4 Experimental signal backscattered by an air-filled aluminium tube immersed in water with radius ratio 0.9; (b) is a zoom of the selection part of the signal in (a)
is partly reflected from the tube and is partly propagated around the circumference of the tube. The backscattered acoustic signal, composed of a series of echoes, is converted into an analogical electrical signal. The received electrical signal is transmitted through the transmission line and arrives at the Transmitter/Receiver from which the transmission signal was sent. This signal is amplified to obtain a measurable level of electric tension, which is then visualized on the screen of the digital oscilloscope. This instrument is equipped with signal processing software. This software makes it possible to carry out the Fast Fourier Transform of the received signal. The digital oscilloscope is connected with a personal computer by an IEEE 488.2 interface with signal processing tools not included in the digital oscilloscope. A software implemented in the laboratory ESSI allows to store the experimental signals in the memory of the personal computer, in order to analyze them thereafter using other softwares.
8.4.2 Measured Acoustic Response Figure 8.4 shows an example of the experimental signal backscattered by an aluminium tube of radius ratio b/a = 0.9. This signal is composed of a sequence of echoes related to the circumferential waves propagating around the circumference of the tube. This figure presents, a first echo, called specular reflection echo (large amplitude and short duration). This echo is taken as a starting point in time. On the same figure, we can observe several waveform packets with low amplitude associated with the different circumferential waves (S0, A1, S1, . . .) [17, 18]. Also, the observation of this signal shows a succession of more or less distinct components that
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Fig. 8.5 Mechanism of echoes showing the specular reflection (1), circumferential waves (2) and Scholte wave (3)
Fig. 8.6 Resonance spectrum for the signal given in Fig. 8.4
one seeks to identify. The different echoes end up overlapping and, in these conditions, the identification and the measurement of the time of onset of echoes (this time depends on the radii of tubes a and b) becomes difficult perhaps even impossible. This constitutes a major disadvantage of the temporal approach. Figure 8.5 shows the formation mechanism of the different echoes; the specular reflection echo (1), the circumferential wave echoes (2) (S0, A1, S1, . . .) and the Scholte wave echo (3) (A wave) [17, 18]. The observation of the signal in Fig. 8.4(b) shows the presence of these echoes. These echoes are related to the propagation of the wave around the circumference of the cylindrical shell and correspond to the Scholte wave A and the circumferential waves S0, A1 and S1, which are recognizable in this signal.
8.4.3 Resonance Spectrum If we remove the first echo from the measured signal the spectrum obtained, by the application of the fast Fourier transform of the remaining signal, is called resonance spectrum. This spectrum is presented in function of frequency on Fig. 8.6. The circumferential waves are also recognizable on the resonance spectrum. On this resonance spectrum the peaks maxima which correspond to the waves A, S0, A1 and S1 are in relation to the proper modes of the tube (resonance frequency) [1, 27, 29, 30]. In this frequency domain, the analysis of an acoustic signal backscattered
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by a tube allows us to isolate and identify the frequency resonances but cannot give any information about the time of these frequencies.The use of the spectral method is very interesting in the field of the scattering acoustic. In return, a temporal analysis of the acoustic signal allows to separate some events (as successive arrivals of different waves) but cannot provide any information on their frequency structure. Thus, the time-frequency methods such as the Wigner-Ville, spectrogram and reassigned spectrogram are used in this study allowing us to overcome the limitations of the mono-dimensional methods. The time-frequency analysis takes into account both the time and the frequency parameters leading to synthetic images that allow us to follow the evolution of the frequency content of a wave as a function of time.
8.5 Time-Frequency Images of Experimental Acoustic Signal 8.5.1 Spectrogram and Wigner-Ville Images The spectrogram time-frequency images, obtained of the temporal signal of Fig. 8.4 are represented on the following figures. The first images are obtained using the Gaussian window (Gabor transform). In this case, the value of the Gaussian window parameter α = 0.01 and the second images are obtained using the Blackman window. These images are presented in function of time and the reduced frequency ka (without units) is given by the following expression [26, 27, 30]. ka =
2πf a , cwater
(8.21)
where f is the resonance frequency and cwater is the acoustic velocity wave. The temporal signal of Fig. 8.4 is analyzed using a function family called the Gabor atoms. This function is based on the complex exponentials of various frequencies windowed by changing the frequency of the underlying complex exponential. These atoms can represent different frequency features in the signal. In addition, the Gaussian window function gives the atoms localization in time. The result coefficients of the Gabor transform therefore provide information not only about the frequency content of the signal, as in the case of the Fourier transform, but also information about how the frequency content of the signal changes with time. From the time-frequency representations presented in the below figures, the circumferential waves S0, A1 and S1 are isolated and localized. It returns to some extent to a decomposition of the acoustic temporal signal in its elementary components in the time-frequency plane, which are the various types of circumferential waves. This study allows then, to recognize the contribution of each circumferential wave to the total temporal signal (Fig. 8.4). Figures 8.7 and 8.8 display time-frequency images corresponding to the SP and SPWV given in Eqs. (8.4) and (8.6) of the impulse response. On the each of these images (Figs. 8.7 and 8.8), we clearly observe the evolution of the symmetric wave S0 which is in the range of the reduced frequencies comprised between 20 and 60, and which shows many trajectories. These
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Fig. 8.7 Time-frequency spectrogram images (Gabor transform) of an experimental signal backscattered by an aluminium tube b/a = 0.9 (the window used is Gaussian with α = 0.01 and the window size is 65 points for (a) and (c) and is 200 points for (b) and (d) Fig. 8.8 Time-frequency Wigner-Ville image of the experimental signal backscattered by an aluminium tube with radius ratio 0.9 (the h window size is 256 points and the g window size is 3 points)
trajectories are due to the wave propagating around the circumference of the tube of radius ratio b/a. The antisymmetric wave A1 comprise between 60 and 120, and the symmetric wave S1 which is in the range of the reduced frequencies higher than 120
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is also illustrated on these time-frequency images. However, the spectrogram transform also suffers from significant problems. The important problem is the choice of the windows duration (i.e.: Gaussian, Blackman). If the window is chosen to be too short, the low frequency is not adequately represented and hence the low frequency information in the signal will be poorly resolved (see Figs. 8.7(a) and 8.7(c)). In contrast, if the window function is too long, the high frequency atoms will have poor localization in time and will not represent the short duration transients in the signal very well (see Figs. 8.7(b) and 8.7(d)). The SPVW time-frequency image is used in this study to compare and to valid the results. Figure 8.8 shows the SPWV image of the same signal. This time-frequency image shows the frequency evolution of the circumferential waves S0, A1 and S1 over time. This synthetic image presents the interference terms between the trajectories of the circumferential wave which is materialized by oscillating structures between the positive and negative values. These terms reduce the interpretation of the time-frequency image. In this study we use the reassigned spectrogram to exceed the limitation of the Wigner-Ville representation (interference terms) and to localize the energy spectra.
8.5.2 Reassigned Spectrogram Image The three Blackman analysis windows employed in reassigned Short-Time Fourier Transform analysis are shown in Fig. 8.9. If the derivative of the window function is unknown, then dtd h(t) can also be computed numerically. The derivative theorem for Fourier transforms, denoted here by F {·}, states that if the relation H (ω) = F h(t) , (8.22) then
d h(t) . jωH (ω) = F dt
(8.23)
We can therefore construct the time derivative window used in the evaluation of the frequency reassignment operator by computing the Fourier transform of h(t), multiplying by jω, and inverting the Fourier transform [5, 13]. That is d h(t) = F −1 jωH (ω) = − F −1 ωH (ω) , dt and so, in discrete time
−1 2πk H (k) . dh(n) = − F N
(8.24)
(8.25)
Figure 8.10 displays the time-frequency image corresponding to the reassigned spectrogram given by Eq. (8.14) of the same impulse response (Fig. 8.4). On these images (Figs. 8.10(a) and 8.10(b)), we clearly observe the evolution of the symmetric wave S0 which is in the range of the reduced frequencies comprised between 20
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Fig. 8.9 The Blackman analysis windows employed in the three Short Time Fourier Transforms used to compute reassigned times and frequencies. Waveform (a) is the original window function, waveform (b) is time weighted window function, and waveform (c) is the frequency-weighted window function
and 60, and which shows many trajectories. These trajectories are due to the wave propagating around the circumference of the tube of radius ratio b/a. Figure 8.10(a) shows the vector field of displacement. Figure 8.10(b) presents the reassigned spectrogram which shows a high resolution in time-frequency plane. This figure is more concentrated, have less interference and the localization of the trajectory waves is more improved. On these images, only the trajectories related to the symmetric wave S0, the anti-symmetric wave A1 and the symmetric wave S1 are presented. The trajectories related to the wave S0 are slightly descending, it means that the group velocity of this wave decrease when the reduced frequency increases. Starting the time-frequency images in Figs. 8.7, 8.8 and 8.10, the cut-off frequency of the wave A1 is ka = 66.5, starting from this
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Fig. 8.10 Time-frequency spectrogram images of the signal backscattered by an aluminium tube; (a) is the image with vector field of displacement and (b) is the reassigned spectrogram image
Fig. 8.11 Time-frequency images of the signal backscattered by a copper tube with radius ratio 0.9; (a) is the spectrogram and (b) is the reassigned spectrogram images
frequency, the anti-symmetric wave A1 is appeared. The low frequency part of this wave arrives more tardily than the part high frequency (dispersive wave). The cut-off frequency of the wave S1 is ka = 130. Starting from this frequency, the symmetric wave S1 is appeared. The part low frequency of this wave arrives more tardily than the part high frequency (dispersive wave). In order to show the benefits of this technique, we have changed the aluminium tube by a copper tube. Figures 8.11(a), 8.11(b) and 8.12 show the time-frequency image corresponding to the spectrogram, the reassigned spectrogram and the Wigner-Ville images respectively of the signal backscattered by a copper tube with b/a = 0.9. These time-frequency images are given by using Eqs. (8.4), (8.6) and (8.14). On these images (Figs. 8.11(a), 8.11(b) and 8.12), we clearly observe the evolution the Scholte wave A which is in the range of the reduced frequencies comprise between 0 and 20, the evolution of the symmetric wave S0 which is in the range of the reduced frequencies comprise between 25 and 45 and which shows many trajectories. These images show also the evolution of the anti-symmetric wave A1 which is in the range of the reduced frequencies comprise between 50 and 98 and
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Fig. 8.12 Time-frequency Wigner-Ville images of the signal backscattered by a copper tube with radius ratio 0.9 (the h window size is 256 points and the window g size is 20 points)
which shows many trajectories. The evolution of the symmetric wave S1 which is in the range of the reduced frequencies higher than 99 is also illustrated on these time-frequency images. The trajectories illustrated in these time-frequency are due to the waves propagating around the circumference of the tube of radius ratio b/a. Starting the time-frequency images in Figs. 8.11(a), 8.11(b) and 8.12, the cut-off frequency of the wave A1 is ka = 50. The low frequency part of this wave arrives more tardily than the high frequency part (dispersive wave). The cut-off frequency of the wave S1 is ka = 99.5.
8.6 Dispersion of the Circumferential Waves 8.6.1 Determination of Dispersion Curves of Circumferential Waves by the Theoretical Method One of the most important points is to find out some parameters that carry most of the information available from the response of the shell. Such parameters may be found the velocity dispersion of the circumferential waves, since it is directly related to the geometry and to the physical properties of the target. The phase velocity cph is calculated from the resonance frequencies which correspond to circumferential waves propagating around the surface of the shell. Thus, for each resonance frequency ω, an integer number of wavelengths fits the circumference of the cylindrical shell, and the following relation holds [25], cph (ω) =
ωa , n
(8.26)
where n is the circumferential wave mode. In addition, group velocity dispersion cg of the circumferential wave is deduced from the phase velocity cph using the relation [25] cg = cph (ω) + ω
∂cph (ω) . ∂ω
(8.27)
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Fig. 8.13 The group velocity dispersion curve of the symmetric circumferential wave S0
Fig. 8.14 The group velocity dispersion curve of the anti-symmetric circumferential wave A1
Figures 8.13, 8.14 and 8.15 show the examples of the group velocity dispersion curves of the symmetric and anti-symmetric circumferential waves S0, A1 and S1 propagating around the aluminium tube.
8.6.2 Determination of Dispersion Curves of Circumferential Waves by the Reassigned Spectrogram Image From the time-frequency image, the group delay can be determined, for a given frequency, by the first order moment along the time axis [7, 11, 17]. For an acoustic application, this property is of great interest since the group velocity of circumferential waves can be directly determined using the group delay τg2 and τg1 of successive wave packets [35], cg (ω) =
2πa . − τg1
τg2
(8.28)
The paths obtained by the reassigned spectrogram time-frequency analysis represent the dispersion of the group velocity of the circumferential waves. In this work, we have estimated the values of the group delay form the obtained image. The group
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Fig. 8.15 The group velocity dispersion curve of the symmetric circumferential wave S1
velocity in function of the frequency is determined using Eq. (8.26). The values of these velocities are compared with those computed theoretically using Eq. (8.27). Figures 8.13, 8.14 and 8.15 show these dispersion curves for the aluminium tube with radius ratio 0.9. Figures 8.13, 8.14 and 8.15 are relative to the symmetric S0, the anti-symmetric A1 and the symmetric S1 waves, respectively. A good agreement between the two estimation methods is obtained. These comparisons permit to identify the different circumferential waves (S0, A1 and S1) displayed on the time-frequency images thanks to the estimation of group velocity dispersions. We can, however, notice that resonances with very close frequencies are linked to very different circumferential waves. These are scattered around the circumference of the tube, following very different velocities. The same remark goes for time: two circumferential waves with close propagation velocities can correspond to welldistinct frequencies. These two remarks suggest that we make a simultaneous analysis according to the two variables time and frequency in order to separate waves on the time-frequency plan, since we can not separate them neither in time nor in frequency. For the A1 wave, Figs. 8.7, 8.8 and 8.10 previously presented show two packets that can be interpreted as successive arrivals of the acoustic wave around the shell circumference. From a qualitative viewpoint, we can observe that the wave loses its energy every lap. We notice that this loss is more important in high frequency than in low frequency. From time-frequency images obtained, we can also notice the inclination of temporal motive. This inclination is closely related to the dispersion curves of group velocity of circumferential waves.
8.7 Prospects for the Future The application of other time-frequency techniques, given in the scientific literature, such as wavelet transform, Rihaczech, Choi-Williams and Margenau-Hill distributions are envisaged in the prospects for the future of this work. This future work is in fact a continuation, a comparison and extension of the previous research. These time-frequency techniques are used, in the future work of our research, to analyze
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and to characterize the measured acoustic signals backscattered by an elastic tube constituted by other materials and with other radius ratio b/a. The continuation of this work is the identification and characterization of the different circumferential waves such as A2, S2, A3 and S3 using time-frequency techniques such as the Rihaczech, Choi-Williams, Margenau-Hill, Wigner-Ville distributions, the spectrogram, the reassigned spectrogram and the wavelet transform. The determination of the dispersion phase velocity of these circumferential waves is examined. The application of the parametric time-frequency representations based on parametric models, such as AR, MA and ARMA filters are also envisaged in the prospects for the future of this work. These parametric representations are more desirable for presenting highly precise time-frequency domain information due to its high-resolution property. The comparisons between the parametric and nonparametric time-frequency representations are also examined. Physiologic signals, especially the electrocardiogram ECG, contain important information about the existence of cardiac complications under the form of anomalies for a patient. In the clinical level, the ECG signals interpretation is usually done in the time domain. The different time-frequency techniques developed in previous work of our research are adopted and applied on the normal and noised ECG signals in the future work. The analysis and the identification of different components of the ECG signals are envisaged. The perspectives and the main goal of the last study is to provide the time-frequency techniques robustness against noise to detect different components originals of the ECG signal.
8.8 Summary The time-frequency analysis of the acoustic signals scattered by a thin elastic cylindrical shell immersed in water allows understanding the particular structure of the waves which are propagated around the circumference of the target. It results a better comprehension of the mechanism of the formation of the backscattered signal. Indeed, it becomes possible to identify and characterize certain circumferential waves by determining the propagation range of each wave. In this work, we have been able to show, in a concrete case, the interest of the time-frequency representations of the Wigner-Ville distribution, the spectrogram and the reassigned spectrogram. These techniques are used to analyze the measured acoustic signals backscattered by an aluminium and copper tubes with radius ratio 0.9. From the obtained images, we had access to many parameters. The cut-off frequencies of the circumferential waves are accessible from the time-frequency images. However, the reassigned spectrogram presents the advantage of attenuating the cross-terms interferences and permits to decrease the broad spectral which ensures a better localization of the energy spectra of each circumferential waves (S0, A1 and S1). This localization allows to have a better legibility of the image and to thus increase the precision of the required parameters. Finally, the time-frequency representation of the reassigned spectrogram gives access the several types of qualitative and quantitative information. Among
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qualitative information, we showed that the reassigned image permits to follow the evolution of the frequential contents of the circumferential waves (S0, A1, S1) versus time. The dispersion group velocity of these waves and the reduced cut-off frequency are quantitative information which can be given starting from a reassigned image.
8.9 Selected Bibliography The work of Flandrin [11] is a good starting point for those how want to initiate them in the time-frequency methods. Afterwards the reading of Flandrin’s book it is advised [13] where one can find the description and the mathematical development of the reassignment time-frequency methods. Besides the mathematical description of the algorithms of the reassignment timefrequency methods, the Niethammer work [28] contains the theoretical and experimental results of many benchmark examples. In parallel, it is recommended the reading of Latif work [18] and Magand work [25], in order to initiate the interested reader in the analysis of the circumferential acoustic waves. There are many works containing the theoretical and experimental results of the circumferential acoustic waves backscattered by a tube, such as the ones of Maze [14] and Veksler [36]. Concerning the analysis of the dispersion of circumferential acoustic waves, papers in references [23] and [31] are suggested. In these articles the dispersion curves of circumferential acoustic waves propagating around the tube are presented in order to understand the dispersion phenomena.
References 1. Ahmad, F.: Acoustic scattering from elastic cylinders. Acoust. Lett. 23(12), 247–252 (2000) 2. Auger, F., Flandrin, P.: Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Trans. Signal Process. 43, 1068–1089 (1995) 3. Auld, B.A.: Acoustic Fields and Waves in Solids, vol. 2, 2nd edn. Krieger, Melbourne (1990) 4. Bao, H.L., Franklin, R.K., Raju, P.K., Uberall, H.: The splitting of dispersion curves for plates fluid-loaded on both sides. J. Acoust. Soc. Am. 102, 1246–1248 (1997) 5. Chassande-Mottin, E., Auger, F., Flandrin, P.: On the statistics of spectrogram reassignment. Multidimens. Syst. Signal Process. 9, 355–362 (1998) 6. Cheeke, J.D.N., Li, X., Wang, Z.: Observation of flexural lamb waves (A0 mode) on waterfilled cylindrical shell. J. Acoust. Soc. Am. 104, 3678–3680 (1998) 7. Cohen, L.: Time-Frequency Analysis. Prentice Hall, Englewood Cliffs (1995) 8. Dabirikhah, H., Turner, C.W.: The coupling of the A0 and interface Scholte modes in fluidloaded plates. J. Acoust. Soc. Am. 100, 3442–3445 (1996) 9. Fitz, K., Haken, L.: On the use of time-frequency reassignment in additive sound modeling. J. Audio Eng. Soc. 50, 879–893 (2002) 10. Fitz, K., Haken, L., Christensen, P.: Transient preservation under transformation in an additive sound model. In: Proceedings International Computer Music Conference, Berlin, pp. 392–395 (2000)
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11. Flandrin, P.: Time-Frequency. Hermes, Paris (1998) 12. Flandrin, P.: Time-Frequency/Time-Scale Analysis. Wavelet Analysis and its Applications, vol. 10. Academic Press, San Diego (1999) 13. Flandrin, P., Auger, F., Chassande-Mottin, E.: Time-frequency reassignment: from principles to algorithms. In: Suppappola, A.P. (ed.) Applications in Time-Frequency Signal Processing, pp. 179–203 (2003) 14. Flandrin, P., Gonçalvès, P.: Geometry of affine time-frequency distribution. Appl. Comput. Harmon. Anal. 3, 10–39 (1996) 15. Gao, W., Glorieux, C., Thoen, J.: Study of circumferential waves and their interaction with defects on cylindrical shells using line-source laser ultrasonics. J. Appl. Phys. 91, 303–310 (2002) 16. Kley, M., Valle, C., Jacobs, L.J., Qu, J., Jarzynski, J.: Development of dispersion curves for two-layered cylinders using laser ultrasonics. Acoust. Soc. Am. 106, 582–588 (1999) 17. Latif, R., Aassif, E., Moudden, A., Décultot, D., Faiz, B., Maze, G.: Determination of the group and phase velocities from time-frequency representation of Wigner-Ville. NDT&E Int. 32, 415–422 (1999) 18. Latif, R., Aassif, E., Maze, G., Décultot, D., Moudden, A., Faiz, B.: Analysis of the circumferential acoustic waves backscattered by a tube using the time-frequency representation of Wigner-Ville, Meas. Sci. Technol. 11, 83–88 (2000) 19. Latif, R., Aassif, E., Moudden, A., Faiz, B.: High resolution time-frequency analysis of an acoustic signal backscattered by a cylindrical shell using a modified Wigner-Ville representation. Meas. Sci. Technol. 14, 1063–1067 (2003) 20. Latif, R., Aassif, E., Moudden, A., Faiz, B., Maze, G.: The experimental signal of a multilayer structure analysis by the time-frequency and spectral methods. NDT&E Int. 39, 349–355 (2006) 21. Latif, R., Aassif, E., Moudden, A., Maze, G.: Analyse des caractéristiques acoustiques d’une plaque élastique immergée dans l’eau à partir de l’image temps-fréquence. Acta Acust. United Acust. 92, 549–555 (2006) 22. Latif, R., Aassif, E., Laaboubi, M., Maze, G.: Identification and characterization of the Scholte A and circumferential S0 waves from the time-frequency analysis of an acoustic experimental signal backscattered by a tube. In: 155th Meeting, Paris, France, Acoustical Society of America, 29 June–4 July 2008 23. Latif, R., Aassif, E., Laaboubi, M., Maze, G.: Analyse de la dispersion des ondes circonférentielles à partir des représentations temps-fréquence d’un signal acoustique rétrodiffusé par un tube. In: 10ème Congrés Français d’Acoustique SFA, Lyon, France, 13–16 April 2010 24. Latif, R., Aassif, E., Laaboubi, M., Maze, G.: Mesure de la vitesse de phase de propagation à l’aide de la technique temps-fréquence. In: Third International Metrology Conference, CAFMET2010, Cairo, Egypt, 19–23 April 2010 25. Magand, F., Chevret, P.: Time-frequency analysis of energy distribution for circumferential waves on cylindrical elastic shells. Acta Acust. 8, 707–716 (1996) 26. Matula, T.J. and Marston, P.L.: Energy branching of a subsonic flexural wave on a plate at an air-water interface. J. Acoust. Soc. Am. 97, 1389–1398 (1995) 27. Maze, G.: Acoustic scattering from submerged cylinders MIIR: experimental and theoretical study. J. Acoust. Soc. Am. 89, 2559–2566 (1991) 28. Maze, G., Izibicki, J.L., Ripoche, J.: Resonance of plate and cylinders: guided waves. J. Acoust. Soc. Am. 77, 1352–1357 (1985) 29. Maze, G., Lecroq, F., Decultot, D., Ripoche, J., Numrich, S.K., Uberall, H.: Acoustic scattering from finite cylindrical elastic objects. J. Acoust. Soc. Am. 90, 3271–3278 (1991) 30. Maze, G., Decultot, D., Lecroq, F., Ripoche, L., Bao, X.L., Uberall, H.: Resonance identification of a solid axisymmetric finite length target. J. Acoust. Soc. Am. 96, 944–950 (1994) 31. Metsaveer, J., Klauson, A.: Influence of the curvature on the dispersion curves of a submerged cylindrical shell. J. Acoust. Soc. Am. 100, 1551–1560 (1996) 32. Niethammer, M., Jacobs, L.J., Qu, J., Jarzynski, J.: Time-frequency representation of Lamb waves using the reassigned spectrogram. J. Acoust. Soc. Am. 107, 19–24 (2000)
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33. Qian, S., Chen, D.: Joint Time-Frequency Analysis. Prentice Hall, New York (1998) 34. Quinquis, A., Ioana, C.: Suppression of Wigner Ville interferences-terms using extended libraries of bases. In: XXVIII Session de Communications Scientifiques de l’Académie Technique Militaire de Bucarest, Section 8, Bucarest, Romania, 21–22 October 1999, pp. 5–12 (1999) 35. Sessarego, J.P., Saheloli, J., Degoul, P., Flandrin, P., Zakharia M.: Analyse temps-fréquence des signaux en milieux dispersifs: application à l’étude des ondes de Lamb. J. Acoust. 3, 273– 280 (1990) 36. Veksler, N.D., Izbicki, J.L. and Conoir, J.M.: Bending A waves in the scattering by circular cylindrical shell: its relation with the bending free modes. J. Acoust. Soc. Am. 96, 287–293 (1994)
Chapter 9
Viscoelastic Damping Technologies: Finite Element Modeling and Application to Circular Saw Blades C.M.A. Vasques and L.C. Cardoso
Abstract A great deal of information on viscoelastic damping technologies, comprising surface mounted or embedded viscoelastic damping treatments, is nowadays available for the practical noise reduction of machinery, in general, and circular saw blades, in particular, for woodworking operations. Among the most efficient and appellative noise reduction techniques for low-noise woodworking circular saw blades demonstrated during the last decades, the use of viscoelastic damping technologies is an interesting possibility which did not receive sufficient attention and dissemination so far. These technologies are analyzed in this chapter in order to gain a preliminary insight into the interest of this noise control solution to further continuing developing more refined and efficient viscoelastic-based noise reduction designs towards the widespread use of low-noise circular saw tooling and industrial practices by woodworking companies. For that purpose, a more comprehensive and tutorial approach to the field is presented in this book chapter. Emphasis is put not also on the specific application to circular saws, which is used to illustrate the interest, applicability and design procedures of such technologies, but also on practical engineering aspects related with the use of computational tools and finite element (FE) modeling software for the mathematical modeling, design and assessment of the efficiency of damping treatments. In particular, different configurations of damping treatments, spatial FE modeling and meshing approaches, mathematical descriptions of viscoelastic frequency-dependent material damping and their implementation into FE frameworks and the use of different solution methods and commercial FE software are discussed.
C.M.A. Vasques () · L.C. Cardoso INEGI, Universidade do Porto, Campus da FEUP, R. Dr. Roberto Frias 400, 4200-465 Porto, Portugal e-mail:
[email protected] L.C. Cardoso e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_9, © Springer Science+Business Media B.V. 2011
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9.1 Introduction The environmental awareness and the ever increasing demand for technology improvement is nowadays raising the concerns of people to promoting more appropriate machines and components which do not cause health damage to its operators. One such example occurs in woodworking companies, where excessive employee exposure to noise is a very important public health issue. In these companies, band and circular sawing is one of the most common woodworking operations and, in general, although large band saws can still produce significant levels of noise, circular saws are responsible for the great majority of the employee noise overexposures in woodworking companies. It is common sense that, whenever possible, a beneficial relationship between environmental protection, public health and economically technological sustainable growth should be achieved. Proven damping technology has been used for that purpose in order to attenuate the unpleasant noise effects of operating circular saws. Techniques employing constrained viscoelastic damping layers are among the most efficient and appellative noise reduction techniques for low-noise woodworking circular saw blades demonstrated during the last decades. However, they did not receive sufficient attention and dissemination so far. Among the several requirements for adequately designing viscoelastic damping treatments, the viscoelastic material properties need to be accurately measured and characterized and their constitutive and phenomenological damping behaviors must be properly known and taken into account in the underlying mathematical models and simulation tools. In general, the elastic and dissipative properties of viscoelastic materials depend on the frequency, operating temperature, amplitude and type of excitation, to name just a few. These dependencies of the viscoelastic material properties make a mathematical description of the viscoelastic constitutive behavior and material damping more difficult and complicated to obtain, and might turn the underlying FE implementations and solution methods more troublesome and difficult endeavors. Thus, for simplicity, the amplitude and type of excitation effects are often overlooked, isothermal conditions are usually assumed and only the frequency dependency of the viscoelastic constitutive behavior is usually taken into account. In addition, despite all those simplifications, quite often the modeling and design task easily become troublesome for people with no experience in the filed since in the design procedure several “unconventional” approaches and care must be taken to avoid some pitfalls and misinterpretation of unrepresentative results. In view of the foregoing, viscoelastic damping technologies are addressed in this chapter in terms of finite element (FE) modeling, analysis and design aspects. The orientation followed is quite general and indeed can be useful for any type of structural designer and is not limited to the specific application to a circular saw blade. The key idea is to give not only an insight into the interest and applicability of viscoelastic damping technologies for circular saw blades but also to instruct potential end users and designers of the required knowledge to model and assess different designs using conventional CAD/CAE commercial computational frameworks. All the different applications of these damping technologies share the same modeling
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and simulation needs; the circular saw blade is presented as a potential area and application and is taken as a design example. For that purpose, a more comprehensive and tutorial approach to the field is presented in this book chapter. It starts with an overview of the state of the art and a discussion on typical configurations of viscoelastic damping treatments. Practical engineering aspects, of interest to commercial FE software end-users and structural designers, related with the use of computational tools and FE modeling software for the mathematical modeling, design and assessment of the efficiency of damping treatments are presented. In particular, FE models of viscoelastic structural systems are addressed regarding the use of different deformation theories and spatial FE modeling and meshing approaches, and mathematical descriptions of the viscoelastic constitutive behavior, their implementation into FE frameworks, the use of different solution methods and the main features available in commercial FE software are discussed. Lastly, emphasis is put also on the specific application to circular saw blades damping, which is used to illustrate the interest, applicability and design procedures of such technologies.
9.2 Overview of the State of the Art Viscoelastic materials are rubber-like polymers possessing stiffness and damping characteristics which, among other factors, vary strongly with temperature and frequency. These polymers are materials composed of long intertwined and crosslinked molecular chains, each containing thousands or even millions of atoms. The internal molecular interactions that occur during deformation in general, and vibration in particular, give rise to macroscopic properties such as stiffness and energy dissipation during cyclic deformation, which in turn characterizes the mechanism of viscoelastic damping [61, 145]. This particular, and in some way, unique property of viscoelastic materials has been used to develop surface mounted or embedded damping treatments, which take advantage of the damping capabilities of the viscoelastic materials. Seminal developments on viscoelastic damping technologies took place in the 1950s. Nowadays, they are well established and constitute a widespread means of controlling the dynamics of structures, reducing and controlling structural vibrations and/or noise radiation in several technological areas (e.g. automotive, aeronautical and aerospace industries), as attested in reference textbooks [74, 84, 113, 123, 172]. Regarding noise control applications, it is well known that viscoelastic materials can be used as an effective means of controlling the dynamics of structures, reducing and controlling not only structural vibrations but also the underlying noise radiation. Noise radiation control is a somewhat distinct problem, which involves also different types of applications, but that shares a common ground with vibration control since vibratory response needs also to be controlled (and not necessarily eliminated) in order to achieve “quiet” structural designs. One such example of application are circular saw blades used in woodworking companies.
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Circular saws are generally thin structures which in operation are usually subjected to an elevated rotating speed which strongly dictates the dynamic characteristics of the system. In view of this, at the modeling stage, important aspects to consider, strongly related with the rotating speed, involve gyroscopic and centrifugal stiffening/softening effects, as discussed for example by Gupta and Meek [71, Chap. 5] in the context of general FE analysis and in [44, 58] in the context of disks and circular plates (saws). These effects, though, must be carefully taken into account at the modeling and design stages of saws with viscoelastic damping solutions, at the cost of increasing the complexity of the analysis. For simplicity, these effects are usually neglected which, for a preliminary simple (mainly qualitative) analysis, is a reasonable and admissible modeling simplification which may suffice for the development of an appropriate “quiet” saw design. Noise produced in circular saw blades is generally accepted to involve two main sources: (i) aerodynamic sources, involving the tooth and gullet area of the blade during idle and (ii) structural vibration noise, that is produced by blade and workpiece related sources [124]. The understanding of the noise generation mechanisms and strategies to reducing the noise produced by circular saw blades has for a long time motivated a great deal of research efforts. On the one hand, research has dealt with the aerodynamic aspects, mainly consisting of studies on airflow disturbances created by rotating disks with various types of openings (gullets) cut into the periphery. On the other hand, research efforts have been directed towards the vibration behavior of the saws and the resultant noise radiated by the blade. There are many different geometries of circular saws in the market nowadays, with all of them being somewhat prone to induce noise problems. As previously discussed, the generated noise can be a consequence of many different aspects and physical phenomenon, like the vibration of work-piece, the cut impact, the vibration of the saw or aerodynamic sources during idle. The focus of this chapter is put on the noise generated by the vibration of the saw. Among the many sources causing vibration of the saw, the most important are the aerodynamic fluctuations [41] and the self excited vibration [176] caused by the cut. These sources excite the structure, and the resulting vibrating structure radiates unwanted sound. Since in woodworking the workpieces are wood, which in general have a significant amount of damping that makes them inefficient noise radiators and vibration conveyors to the supporting and fixing apparatus, workpiece related sources are considered not too much relevant. Regarding the aerodynamic ones, they are not of concern in this chapter and the attention is driven to structure-borne noise generated by the saw blade. The structure-borne noise produced by a circular saw blade may result from a resonant blade vibration response, whether resulting from aerodynamic excitation sources during idle or from tooth impact forces during cutting, or from a forced vibration response, which is caused by the periodical tooth impact forces occurring only during cutting. Therefore, in idle, the radiated noise of a “whistling” blade typically consists in an intense pure tone noise resulting from blade resonant vibration which results from excitation forces providing energy in a frequency range where at least one easily excitable mode exists; in this case, damping plays a central role,
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which for lightly damped saws may result in extremely high level of pure tone noise usually of an efficient structural acoustics mode. In a different way, during cutting, the tooth impact excitation produces broadband vibrational energy as well as energy at the rotating circular frequency (and harmonics) of the saw. Therefore, resonant response during cutting may involve more than one resonant frequency. Lastly, considering forced-blade vibration, the response at non-resonant frequencies is not as sensitive to damping as the resonant case, turning resonant reduction techniques not appropriate, and the noise is characterized by strong frequency peaks at the tooth passage frequency and harmonics [124]. Neglecting the effects of the complex geometry of the teeth and cuts typically encountered in circular saws, due to the resemblance of circular saw blades with annular plates the comparison and investigation of related open literature is brought into play here. Pioneering works in the study of the vibration behavior of rotating disks can be found in [93, 167] while some of the earlier works about the structural acoustic behavior can be found in [54, 90, 125]. Subsequently, the noise radiation of annular plates (circular saws) has been widely studied by many researchers and in different ways. Lee and Singh [95] studied analytically the modal radiation efficiency of an annular rotating disk in a rotating frame of reference, which is a restrictive simplification since the human observer, for most applications (e.g. circular saws, compact and computer disks, etc.), is obviously not in that frame; the rotation effect in the modal radiation efficiency was considered through a model of centrifugal forces. Regarding the centrifugal forces effects, as reported by Sinha [159], the resonant frequencies of a structure in the stressed state, as is the case of a rotating saw under centrifugal forces, are different from that one in the unstressed state. In fact, the shift in the resonant frequencies depends upon the nature and magnitude of the stresses induced by the rotation, which, for thin structures, usually manifests itself due to the non-linear coupling between the membrane and flexure stresses. As far as the type of reference frame is concerned, Côté et al. [44] investigated more recently the forced sound radiation of a rotating disk with the observer in rotating and non-rotating frames of reference, where the effects of rotation are considered through the gyroscopic and centrifugal effects. “Low-noise” saw blades designs available in the market are most effective in reducing resonant blade vibration noise during idle and have mainly been based in trying to somewhat introduce damping; the use of laser cut slots and plugs causing localized “disruption” of wave propagation of certain modes and damping through the scrubbing that occurs in the laser slots and plugged holes has been proposed. Along this line of thought, most studies on vibration control strategies for circular saw blades have been directed towards the alteration of the saw dynamics through the development of improved passive designs, as demonstrated by numerous patents in the field such as [94, 163, 173]. In view of this, methods regarding tensioning techniques [151, 153], the use of high damping alloys [75, 76], vibration damping due to air film [182], the use of slots [126, 158], variable distance between successive teeth and/or depth of gullets [28, 162], improved aerodynamic performance of teeth design [29], stiffening by thermal membrane stress [120], use of guided [96, 152] and asymmetric [89, 203] saw blades designs, to name a few, have been
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reported and assessed through numerous analytical, numerical and experimental approaches. Review literature in the field for further information also pointing out pertinent issues and results can also be found in [124, 127, 168, 169, 198]. On the other hand, non-contacting damping methods [6] and/or active control methods for the reduction of noise and vibration of circular saws have also been proposed. A pioneering work in the area was carried out by Ellis and Mote in the 1970s [52] and more recently electromagnetic actuators have also been used by Wang et al. [196]. The use of semi-active shunt damping electrical networks to dampen saw blades with surface mounted piezoelectric patches has also been proposed in [140]. Alternatively, laminated blade designs making use of the principle of constrained layer damping have been proposed since the 1970s as effective means to reduce both resonant vibration during idle and blade vibration noise during cutting. Although not so effective, free layer damping treatments have also been considered [45]. However, regarding the forced-blade structure-borne noise, aside from the highly effective laminated blade construction discussed above, typical damping treatments were reported to provide only minor noise reduction under those forcing conditions [124]. Although the use of viscoelastic damping technologies applied to the noise control of saws is a damping solution of considerable interest and applicability, a very limited number of works have been published on this topic. Contrarily, several patents regarding the use of viscoelastic damping technologies in rotating disk-like members, such as saw blades, gear wheels and rotary grinding or polishing heads, have appeared in the last decades [20]. The use of continuous and/or segmented constrained layer damping treatments has been patented in the 1970s by Tsunoda [185] and a more sophisticated design providing an abrasion-resistant edge coating around the edge of the constrained layer damping device by Caldwell [31]. Along a similar line of thought, laminated designs of saw blades comprising at least two preferably metallic discs interconnected by any binding means (e.g. a viscoelastic binding layer) are discussed, mainly from a production perspective, by Wikner and Josefsson [197]. Increased damping through the combined use of radial slots in the blade, to perform blade segments with interrupted circumferential vibration, and damping collars axially aligned with the blade to provide a sink for vibrational energy dissipation from the segmented blade are patented by Stewart in 1980 [171]. More recently, the application of viscoelastic damping treatments and the study of the vibratory response of viscoelastically damped rotating circular plates are performed in [40, 155, 193, 204]. Furthermore, constrained layer damped circular plates with moving loads were investigated by Yu and Huang [204]. An alternate design consisting of constrained layer damping to reduce the transverse vibrations of spinning disks and expand the region of stability applied to computer disk drives is presented by Seubert et al. [155]. Along the same line of thought, Marui et al. [109] investigated a thin plate structure as a model of a circular saw and the efficiency of various damping foils inserted between the collars and the saw blade model. Damping achieved by the saw blade clamping system can significantly reduce the noise level at idling. The cost of these clamping systems is fairly low, increasing interest in such research. To enhance the
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vibration-proof capacity of a thin-bladed too such as circular saw blades for woodcutting application, Marui et al. [109] experimentally investigated the effects of the clamping condition on damping of the saw. Using a simple thin plate structure of cantilever type as model, bonded, adhesive, wire, projection, non-bonded and solid type clamping mounting conditions of the plate and shims were considered and their performance in terms of damping and stiffness were assessed and compared. The optimal improvement resulted from the insertion of strips of adhesive tape between the shims and the plate or from the projection (through the use of rugged shims) plate arrangement. The use of these latter strategies, in general resembling the use of viscoelastically damped collars, to enhance damping capability of thin disk structures was partially experimentally demonstrated and its success envisioned for the use with rotating disks. As shown in [21], saw blades without radial slots emit a high-level noise during idling caused by self-excited vibrations. For such saw blades the noise level cannot be changed by using rubber damping rings. Whistling noise also occurs, but rarely, with saw blades having radial slots. By inserting rubber damping rings between the collars and the saw blade with radial slots, the whistling noise can be completely suppressed. In the rotational frequency range in which the saw blades were tested, the saw blades with copper corks at the end of radial slots produced no whistling noise at all. This difference can be explained by the aerodynamic sound that originates from the slots and holes at the end of the slots (because the slots are made by punching). In conclusion it was found that the insertion of rubber damping rings between the saw blade and collars in combination with a slotted circular saw may be an effective and inexpensive way to eliminate the whistling noise of idling circular saws. Although such “low-noise” saw blades have been available on the market for several years, as reported by Maue and Hertwig [110] they have not been used very frequently to date. The probable reason reported is related with the relative insufficient knowledge about them. Tests on various low-noise saw blades designs are presented in [110] where a conventional saw blade with fine radial expansion slots on the outer edge is compared with one with additional fine laser cuts in the middle region and another one considering a sandwich construction with a viscoelastic docked cover plate. A noise reduction of approximately 3 to 10 dB(A) have been reported without any notable financial expense (i.e. the price of low-noise saw blades is comparable to that of conventional blades). To sum up, the current the state of the art previously discussed allows to conclude that despite the unequivocal potential to attenuate vibration and structure-borne noise from circular saw blades, there is a lack of studies on the detailed modeling and design optimization in the field, the main conclusions being usually driven from experimental evidence of the vibroacoustic damping phenomenon in damped saws.
9.3 Configurations of Viscoelastic Damping Treatments Over the last decades, the particular, and in some way unique, damping property of viscoelastic materials [61, 145] has been used to develop surface mounted or
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Fig. 9.1 Viscoelastic damping treatments configurations: (a) unconstrained layer damping (ULD); (b) passive constrained layer damping (PCLD); (c) PCLD with the strain magnifying effect of a spacer-layer; (d) active constrained layer damping (ACLD)
embedded damping treatments, which take advantage of the damping capabilities of the viscoelastic materials. The different and most significant damping configurations that have been proposed, along with the different alternatives to model them, both in terms of their phenomenological behavior and transverse spatial distribution are shortly reviewed in what follows. Passive viscoelastic damping treatments have been extensively used in engineering to reduce vibration and/or noise radiation [74, 84, 113, 114, 123, 172]. From a phenomenological point of view, when a damping layer is applied to a structural system, it dissipates energy by virtue of the heat generated by the strain field that it undergoes. The strain field varies over the surface and depth of the structure and is proportional to the fluctuating extensional strains, due to the membrane and bending loads, and shear strains, due to the shear loads. These damping treatments are usually designed and incorporated so that the energy is dissipated primarily by one type of strain or the other, yielding different configurations and material properties for each type in order to achieve the greatest damping efficiency. The simplest form of employing passive viscoelastic damping treatments is the one where a layer of viscoelastic material is attached to the outer surfaces of the structure. When the layer is attached in this way, the outer surface of the damping layer is left free and unconstrained by any shear stress. This configuration is therefore known as unconstrained layer damping (ULD) and allegedly it was originally proposed by Liénard [100] and Oberst [130] in the early 1950s (Fig. 9.1(a)). When the structure vibrates, energy is dissipated by the viscoelastic layer, mainly due to the heat generated by the extensional strains which, as the distance of the damping layer to the neutral surface increases, become greater. Increasing the thickness and/or the distance from the neutral surface and length of the viscoelastic treatment in an appropriate manner would increase the energy dissipation and consequently the damping. However, in applications where the weight and size of the structure
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and damping treatments are of critical importance, these constraints demand a more efficient treatment, and other alternatives to increase damping must be found. Alternatively, a damping treatment designed to dissipate energy mainly by virtue of shear strain can be achieved by the inclusion of an elastic constraining layer sandwiching the viscoelastic layer between itself and the host structure. Since the maximum shear strain usually occurs close to the neutral surface and is zero at the outer surfaces, it is logical finding configurations where the viscoelastic layer is incorporated close to the neutral surface. The resultant high shear stress and strain values developed in the damping layer justify the use of the damping layer in a sandwich configuration (Fig. 9.1(b)). However, it is not essential for the damping layer to be close to the neutral surface to achieve a good damping performance. Provided that it is sandwiched between an host structure and a stiff constraining layer, reasonable levels of shear stress and strain are developed which allow a significant damping to be achieved. This configuration of damping treatment is known as passive constrained layer damping (PCLD). Some examples of their use can be found in [25, 47, 107]. While the ULD treatment performance mainly depends upon the distance from the damping layer to the neutral surface and the value of the so-called loss extensional modulus, PCLD treatments are more complex to design since their best damping performance is achieved by a trade-off between different parameters. Therefore, optimal values for the so-called storage shear modulus of the viscoelastic material and elastic properties and thicknesses of the remaining layers in the configuration must be carefully chosen in order to have optimal damping performance [113]. When comparing the ULD and PCLD damping layers thicknesses, the latter usually requires smaller thicknesses and therefore the influence of the damping treatment in the dynamic properties of the structural systems where it is applied is mainly devoted to the damping increase and smaller mass and stiffness modifications are introduced. However, while passive damping treatments, constrained or not, can greatly improve the damping of the system, there are some limitations. The frequency- and temperature-dependent mechanical properties of the viscoelastic materials can make the damping change, bringing limitations to the effective temperature and frequency range of the treatment over which it is efficient. In order to provide adequate damping over a broad frequency band and different temperature ranges, different viscoelastic materials must be simultaneously applied in the same damping treatment, which often complicates the analysis and design of the system. Therefore, while viscoelastic treatments are easy to apply, the damping is often of limited frequency bandwidth and temperature range. From the early 1990s, so-called active constrained layer damping (ACLD) treatments have been analyzed and applied to structures [3, 15, 156]. These are hybrid treatments with active constraining layers made of piezoelectric materials (Fig. 9.1(d)) which have the unique feature of having the ability to be used both as distributed sensors and/or actuators [30]. If utilized as actuators, and according to an appropriate control law, an active constraining layer can actively increase, in a convenient way, the shear deformation of the viscoelastic layer and therefore overcome
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some of the PCLD limitations by adapting the treatment to the temperature and spectral content of the undergoing operating conditions. Thus, ACLD treatments combine the high capacity of passive viscoelastic materials to dissipate vibrational energy at medium to high frequencies with the active capacity of piezoelectric materials at low frequencies. Therefore, in the same damping treatment, a broader control over the temperature and frequency ranges of interest is achieved benefiting from the advantages of both passive (simplicity, stability, fail-safe and low-cost) and active (adaptability, observability, controllability and high-performance) systems. Some examples can be found in the works of Baz and his colleagues [18, 133, 143]. Various configurations of active and passive layers surface mounted or embedded in an host structure have been proposed in an attempt to improve damping performance even further. In general, hybrid active-passive damping (YAPD), or arbitrary ACLD, treatments, involving arbitrary arrangements of constraining and passive layers, integrating piezoelectric sensors and actuators, may be utilized [10, 16, 135, 146, 174, 178, 180, 181]. Surveys on the advances in hybrid active-passive vibrations and/or noise control via piezoelectric and viscoelastic constrained layer treatments can be found in [22, 134, 170, 178]. Damping treatments with a stand-off layer (SOL) also constitute nowadays an attractive alternative to conventional constrained layer damping (CLD) treatments. The SOL is a slotted spacer-layer, in general assumed in idealistic physical conditions having infinite stiffness in shear and zero stiffness in bending, which is sandwiched between the viscoelastic layer and the base structure (Fig. 9.1(c)). In essence, it acts as a strain magnifier that considerably amplifies the shear strain and hence the energy dissipation characteristics of the viscoelastic layer. Accordingly, more effective vibration suppression can be achieved by using SOL as compared to employing CLD [38, 200, 201], however with some mass and stiffness modifications in exchange.
9.4 Viscoelastic Constitutive Behavior The viscoelastic constitutive behavior is identified using experimental procedures which can resort on the use of time- and frequency-domain approaches. The former is established by means of differential or integral equations forms where the typical creep and relaxation behavior of viscoelastic materials are directly observed and identified from experimental measurements. However, since one of the simplest ways of experimentally determining the viscoelastic properties is to subject the material to periodic dynamic oscillations, the latter dynamic representation of the material properties by means of a complex modulus is a current practice and is perhaps the most commonly used approach when modeling the material in the frequency domain. These two approaches are briefly described in what follows. The reader is referred to Vasques et al. [190, 191] for a more complete and detailed discussion and survey on the identification and mathematical representation of the linear viscoelastic constitutive behavior.
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Viscoelastic materials are sometimes called materials with infinite “memory”, in the sense that their actual mechanical response is modulated by the past history. Therefore, the viscoelastic constitutive behavior relies on the assumption that the current value of the stress tensor depends upon the complete past history of the strain tensor components. This latter assumption therefore implies that the behavior of any viscoelastic material may be represented by a hereditary approach. Thus, considering an isotropic viscoelastic material under isothermal conditions and under small (infinitesimal) deformation conditions, the theory of linear viscoelasticity [42, 63] states that the constitutive relationship for a generic one-dimensional isotropic stress-strain system (e.g. in shear or extension) can be given by a Riemann convolution integral, t ∂ε(τ ) Grel (t − τ ) dτ, (9.1) σ (t) = ∂τ −∞ where σ (t) and ε(t) are the time-dependent stress and arbitrary strain component histories and Grel (t) is called the constitutive time-varying (shear or extension) characteristic relaxation function of the material (the stress response to a unit-step strain input) which is also utilized in the literature under many different names, such as damping kernel, retardation, hereditary or after-effect function. Equation (9.1) expresses an essential feature of linear behavior of viscoelastic materials known as Boltzmann’s superposition principle which, according to Tschoegl [183], is documented in [27]. Since the lower limit of integration is taken as −∞, it is to mean that the integration is to be taken before the very beginning of the motion. Thus, if the motion starts at time t = 0, meaning that the stress and strain are equal to zero up until time zero where the loading begins, i.e., σ (t) = ε(t) = 0 for t < 0, and discontinuous strain histories with a step discontinuity at t = 0 are to be considered, Eq. (9.1) reduces to t ∂ε(τ ) Grel (t − τ ) dτ, (9.2) σ (t) = Grel (t)ε(0) + ∂τ 0 where ε(0) is the limiting value of ε(t) when t → 0 from the positive side. The first term in Eq. (9.2) gives the effect of the initial disturbance and it arises from allowing a jump of ε(t) at t = 0. Furthermore, Eq. (9.1) is written tacitly assuming that ε(t) is continuous and differentiable (see [63] for further details). The hereditary integral time-domain representation in Eq. (9.2) is in fact the departure point for modern time-domain modeling and solution techniques based on the previous knowledge (measurement) of the time varying relaxation stress history to a step strain stimulus which is obtained from a relaxation test applied to a viscoelastic material (see [183] for further details). However, as previously stated, there are practical situations in which structures with viscoelastic materials may be subjected to steady-state oscillatory forcing conditions. Thus, considering nil initial conditions, i.e., ε(0) = 0, the Laplace transform of Eq. (9.2) yields σ (s) = G(s)ε(s) ≡ sGrel (s)ε(s),
(9.3)
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where G(s) ≡ sGrel (s) is a characteristic material function, which should be experimentally identified somehow. Under these conditions, the characteristic (shear) material function previously defined in the stress-strain relationship in Eq. (9.3) is defined by assuming s as a pure imaginary variable (or similarly through its Fourier transformation), so that G(jω) = G(s)
for s = jω,
(9.4)
where ω is the circular frequency, yielding the so-called complex (shear or extensional) modulus in the form G(jω) = G (ω) + jG (ω),
(9.5)
where G (ω) is the storage modulus, which accounts for the recoverable energy, and G (ω) is the loss modulus, which represents the energy dissipation effects [42, 84, 123]. The loss factor of the viscoelastic materials is defined as η(ω) =
G (ω) , G (ω)
which alternatively allows writing Eq. (9.5) as G(jω) = G (ω) 1 + jη(ω) .
(9.6)
(9.7)
The latter complex modulus representation of the linear, homogeneous and isotropic viscoelastic constitutive behavior in Eq. (9.7) is certainly the most commonly used approach when identifying the material in the frequency domain and is also the departure point for performing a frequency analysis, this approach for the representation of the constitutive behavior being also known as a complex modulus approach (CMA) as discussed in the following sections. Lastly, it should be mentioned that the time- and frequency-domain constitutive representations in Eqs. (9.2) and (9.7) may be used both for the definition of a shear or extensional modulus. One can be obtained from the other through the use of a simplified real time- or frequency-independent Poisson’s ratio [39, 92, 141, 184] through the well known elastic relationship between the two.
9.5 Finite Element Modeling of Viscoelastic Structural Systems As reported in the previous sections, according with the operating thermal conditions and spectral content of the excitation of the host structure, the optimal performance of viscoelastic damping treatments demands alternative configurations to be tested and designed. In the limit case, these configurations might comprise arbitrary arrangements and stacking sequences of a single or multiple kinds of viscoelastic materials with different thicknesses and different locations. For shear-type damping mechanisms, an accurate representation of the shear strain and stress fields inside the viscoelastic constrained layer(s) is a key issue in the accurate physical representation which must be properly taken into account
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in the mathematical modeling and dynamic simulation of structural systems with viscoelastic damping treatments. In the development of the mathematical model the transverse spatial distribution of the damping treatments is a special requirement which demands specific modeling strategies to be developed and utilized. These modeling strategies are special in the sense that the extensional and/or shear strains and stresses developed in the viscoelastic layers must be accurately modeled in order to obtain good damping estimates and in the sense that the treatments are very often discontinuous which makes analytical models difficult to use. It is well known that a general approach able to circumvent these complications corresponds to the use of the FE method, which can be used as a reliable and useful tool to characterize the structure response, analyze the effects of any damping treatment configuration and assess the influence of the design parameters upon the structural dynamic performance. However, despite the obvious and well known capabilities of the FE method, there are a few main issues that still need to be carefully considered in its application: 1. The specific deformation patterns developed in viscoelastic laminated structures demand a special and dedicated deformation theory able to capture the complex shear deformation pattern developed in the viscoelastic layers during the structure vibration; 2. The meshing and transverse spatial modeling of the damped structural system demand specific and dedicated discretization procedures, employing 2D or 3D meshes, according to the spatial model approach utilized involving whether “composite” assemblies of standard FEs available in commercial FE software or the development of specific and dedicated new discrete-layer FEs; 3. Different descriptions of the viscoelastic material constitutive behavior may be considered and implemented in a FE modeling framework; the viscoelastic constitutive model and solution method are often associated but a clear distinction between them exists and should be made; 4. In the analysis of viscoelastically damped structures the choice of the most adequate solution method is strongly dictated not only by the applied time- and frequency-dependent viscoelastic constitutive model but also by the desired type and accuracy of the system responses. The former issues are discussed in what follows. The reader is referred also to Vasques et al. [190, 191] for a more complete and detailed discussion and survey on the FE implementation of viscoelastic damping which somewhat complements the current chapter.
9.5.1 Some Comments on Deformation Theories The deformation theory typically applied to structural laminates (see for example Reddy’s textbook [144]), i.e., the classical laminate theory (CLT), cannot be used for this purpose because it is not able to represent any shear strain distribution within the layers. Similarly, equivalent single-layer (ESL) theories and even
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high-order theories (HOT), do not take into account the interlaminar and intralaminar effects and therefore suffer from lack of accuracy and generalization. These issues were in fact analyzed by Kosmatka and Liguore [91] that report a situation where plate/solid/plate “composite” (assembled) approaches accurately predict both the resonant frequencies and the loss factors of a viscoelastic sandwich plate, while ESL approaches based in layered (or composite) single solid or plate FEs generally overpredict and underpredict the resonant frequencies, respectively. Despite the assumed continuity of the displacement field between the viscoelastic and the stiff constraining layers, the continuity of the out-of-plane stress components is not usually considered and the localized effects in the highly sensitive shear strained viscoelastic layers are not accurately captured. However, refined deformation theories taking into account the so-called Cz0 -requirements described by Carrera [32], can be used to accurately model layered structures with arbitrary configurations of damping layers. A comprehensive overview and assessment of some refined deformation theories of plates and shells, e.g. first- or higher-order discretelayer (or layerwise) or zig-zag theories (ZZT), in some cases using also mixed variational principles, in static and dynamic situations, can be found for example in Carrera’s works [33, 34, 36].
9.5.2 Spatial Modeling and Meshing Regarding the solution method utilized, as previously stated, the FE method is the most interesting approach which, as far as the transverse (out-of-plane) spatial modeling is concerned, is often utilized considering two main strategies: (i) by connecting multiple FEs (i.e., beam, plate, shell and/or solid FEs) into a single “composite” FE; (ii) or developing new dedicated higher-order discrete-layer (or zig-zag) models. Therefore, in what follows, the underlying spatial modeling and meshing approaches, related with the aforementioned deformation theories, are discussed from a FE method implementation standpoint.
9.5.2.1 “Composite” Elemental Models In the past, researchers developed various FE approaches for structures with constrained layer damping treatments. The easiest way of doing that was utilizing the modeling capabilities of existing FE codes, such as NASTRAN, ANSYS, ABAQUS, etc., to analyze constrained layer damped beams, plates and shells. Since these structural viscoelastic damped models involve different physical layers with different material properties, the aforementioned first approach, in an attempt to represent the real physical distribution of the layers, was therefore often obtained by combining the existing FE modeling capabilities available at that time into a “composite” FE model involving various FE types. Let us consider the sandwiched or viscoelastically constrained generic plate depicted in Fig. 9.2. As previously discussed, the CLT is not adequate to accurately
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Fig. 9.2 Generic viscoelastically damped sandwich or constrained plate and elemental volume
Fig. 9.3 “Composite” FE plate (or shell) models of viscoelastically damped structures using existing commercial or in-house FE codes/elements
describe the shear deformation of the viscoelastic layer and therefore it is necessary to use a different model. In order to overcome the aforementioned issues, special layered schemes of beam, plate and brick FEs, used also in conjunction with rigidlinks, were utilized in conjunction with commercial FE software to model the host structure, viscoelastic and constraining layer (cf. [84, 123, 172]). Several distinct approaches available in the open literature are depicted in Fig. 9.3. For simplicity, only three-layered sandwiched plate structures have been considered to illustrate the “composite” modeling approach. However, the underlying modeling idea and concept could also be employed and extended to multilayered plate and shell structures using also higher-order solid FEs. The “composite” and mesh models main
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characteristics, including the most relevant works where these approaches are first developed and utilized, are summarized in Table 9.1. The first model, model 1, due to its simplicity and easiness of implementation with commercial FE codes (e.g. NASTRAN, ANSYS, ABAQUS), benefiting from the usual standard discretization procedures available in these codes, was probably the first approach to be utilized. It uses standard linear (or higher-order) solid brick elements to model all the layers and, for a three-layered configuration, yields four planes (layers) of nodal points. In this model, since a 3D formulation is used, a 3D stress-strain state is considered and all the components, i.e., the in-plane and out-ofplane extensional and shear components, are considered and a full 3D stress-strain state is captured. However, from a computational cost point of view, this model yields a “brute-force” formulation with too many degrees of freedom (DoFs), making it uninteresting for large-scale problems. Since this is a very common approach, it is difficult to point out the first work using it. The reader is referred to some of the more relevant works in the last decade in [72, 117, 205], fully representative of the results obtained with model 1. As mentioned in [117], when using modeling approaches considering solid brick FEs to model the viscoelastic layer, special care must be taken in order to avoid numerical pathologies such as shear (or in the case of shells also membrane) locking [108]. A comparative and assessment study, using conventional FEs from NASTRAN, between models 1, 2 and 4, looking at their limitations, accuracy and applicability, was performed by Moreira and Rodrigues [117]. All the three models used in that study share a common FE representation of the viscoelastic layer which is performed using solid brick FEs (HEXA8). The base plate and the constraining layer of the surface treatments, or the skin plates of the sandwiched layered configuration, are both modeled by either plate (QUAD4) or brick (HEXA8) FEs. It is shown that the relatively low storage modulus of the viscoelastic layers, when compared with the stiff materials used in the host structure and constraining layer, attenuates the locking effects. However, the locking effects can also be a problem if the host structure and constraining layer are also modeled with solid brick elements, as is the case in model 1. According to some authors [82, 91, 101], numerical pathologies in viscoelastically damped plates occur mainly due to the shear locking effects within the viscoelastic core for aspect ratios of the layer (length to thickness ratio of the layer) higher than 5000. However, for the constraining layers the limit aspect ratio where shear locking effects become more pronounced is 100, when using selective reduced integration or other shear correction mechanisms (e.g. MITC, EAS, etc.), and its value is limited to 5 when normal integration schemes are used. Obviously, the previous conclusions hold only for planar structures. Nevertheless, they illustrate well the difficulties encountered when using FE solution methods for planar structures and it is expected to a have a somewhat different behavior when considering more complex curved structures such as shells or curved beams, due to the more complex stress state developed where membrane locking may also occur [108]. As shown in [117], in general, when using FEs from NASTRAN, the convergence rate of model 1, considering selective reduced integration only for the viscoelastic layer, is better than the one obtained with model 4. Furthermore, elements of the external layers should also use adequate shear locking reduction strategies when they
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Table 9.1 Main features of the “composite” and discrete-layer elemental models Models
Strengths
Weaknesses “Composite” elemental models
Model 1 (s/s/s) • Simplicity and easiness of implementation • Standard and readily available FEs • Standard discretization procedures • 3D stress-strain state may be captured
• “Brute-force” modeling • High computational cost (many DoFs) • Uninteresting for large-scale problems • Numerical pathologies may occur for thin layers
Model 2 (p/r/s/r/p)
• Moderate computational cost • Less sensible to numerical pathologies for thin external layers • Ideal for sandwich structures • Possibility to simulate bonding failure
• Rigid links impose more complexity • Non-standard discretization and assemblage procedures • Less-standard FEs
Model 3 (p/b/p)
• It exempts the use of brick FEs • Better accuracy with less DoFs • Insensitive to numerical pathologies due to issues related with the core thickness
• Requires special procedures to define the beam element properties • Non-standard discretization and assemblage procedures
Model 4 (p+o/s/p+o)
• Less computational cost (allows the reduction of DoFs) • Less sensible to numerical pathologies for thin external layers
• Requires off-setting of plate DoFs • Requires external constraint equations • Non-standard discretization and assemblage procedures
Model 5 (p/n+i/p)
• Lower computational cost with only plate DoFs • Less sensible to numerical pathologies for thin external layers • More standard discretization and assemblage procedures
• Requires an a priori coupled-kinematics formulation • Non-standard composite FE • Not readily available
Discrete-layer elemental models Model 6 (single-sided)
• High degree of automation in spatial • Not readily available modeling and mesh generation • Segmented double-sided damping • Best suited for optimization configurations can not be considered procedures • Non-standard coupled-kinematics • Requires a standard 2D mesh formulation
Model 7 (double-sided)
• The same as model 6 • Segmented symmetric and asymmetric double-sided configurations can be considered • Highest spatial modeling generalization capabilities
• Not readily available • Non-standard coupled-kinematics formulation • More cumbersome formulation
(continued on next page)
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Table 9.1 (continued) Models
Strengths
Weaknesses
Model 8 (regenerated)
• Same as 6 and 7 • More straightforward to formulate and implement • Accurate with excellent trade-off with complexity • Higher degree of versatility
• Same as 7 • High degree of technical expertise required to the formulation
Notes: ‘s’, solid; ‘p’, plate; ‘r’, rigid link; ‘b’, beam; ‘o’, offset; ‘n’, no slip; ‘i’, interpolation
become too thin in order to make the response less dependent of the mesh refinement. Another more recent alternative to model 1 is the use of models 2 and 4; these latter two models have in common the use of solid brick elements to model the viscoelastic core. However, regarding the outer stiffer layers modeling, there are two distinct approaches. In the first one, model 2, the translational degrees of freedom of the plate are connected to the brick ones by means of rigid links [14, 105]. Using this model, the most complex one, it is possible to simulate bonding failure between the viscoelastic layer and the adjacent plates by simply removing those links in specific nodes of the FE method mesh. However, as in model 1, there are still four planes of nodal points but the locking effects of the stiff thin external layers might be more easily circumvented. Other structures, such as a continuously constrained damped beam and ring (curved beam) structures, are also modeled with similar and analogous procedures in [104]. The second approach, model 4, proposed by Johnson and Kienholz [82] and further employed in [91, 138, 139, 155], is not as straightforward to use because when attempting to attach plate elements to a solid element surface there are two problems that must be overcome: (i) the first problem involves off-setting the nodes from the plate mid-plane to the plate surface adjacent to the solid element; (ii) the second problem involves developing and incorporating a set of constraint equations used to correct continuity discrepancies between the plate and solid elements. However, since there are coincident nodes and translational DoFs for the plate and the adjacent face of the solid element, it allows the reduction of the total number of nodal points requiring only two nodal planes. A similar approach to model 4 is also performed in [166] where a dedicated code called MAGNA is used to analyze constrained layer damping treatments using eight-node thin shell elements or penalty function shell elements to model the constraining layers and the base structure, respectively. Although providing accurate results two decades ago, FE solutions using solid brick elements to model the thin viscoelastic layer used to require a great deal of computational time. Therefore, when trying to study more complex structural problems, that approach was limited by the current computing capabilities at that time. These limitations motivated Killian and Lu [88] to develop and propose the ap-
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proach depicted in model 3. They had in mind that it was needed to introduce an approximation which is simple and yet provides accurate predictions. Their objective was to present a FE approximation with simple elements to replace solid elements which are not really necessary for linear vibratory systems. The modeling approach is the same as that given in model 2 for the host structure and the constraining layer. However, the brick elements to model the thin viscoelastic damping core are not used and, instead, the thin viscoelastic layer is approximated by viscoelastic beams connecting the base plate and the constraining plate. The structural properties and damping for these beams are then specified in order to simulate the shear and extensional damping provided by the thin viscoelastic layer. With this approach, the number of nodal points for the composite system is reduced from four rows to two, and the rigid link elements used in model 2 to assure the geometric compatibility are also eliminated. When compared with model 2, model 3 has been shown to provide better results than those obtained by using 3D solid elements, not only predicting the gross features of the system behavior but also the resonances and anti-resonances [88]. Although a very interesting and creative approach at the time of its creation, in the forthcoming years, and with the fast development of modern computing capabilities, since its use in commercial FE codes is not straightforward, requiring special procedures to define the beam element properties and assembling process, the approach in model 3 has become less attractive. The last model, model 5, the so-called degenerate constrained layer element, was proposed by Jeung and Shen [80] for plate and shell structures. In their work they adopt the degenerated shell element proposed by Ahmad [4] for the constraining layer and the base shell structure. The displacement field of the viscoelastic layer is linearly interpolated through no-slip boundary conditions at the layer interfaces, therefore creating pseudo-nodes in the viscoelastic layer. In fact, this model is very similar in principle to model 2; the main difference lies in the fact that the rigid link effect is imposed a priori at the formulation level by no-slip conditions, yielding a composite FE that does not need to be assembled at the elemental level and which requires a 2D global assemblage. This type of composite approach is slightly different from the others in the sense that it requires special efforts to the user to formulate a new element and implementing it in commercial or in-house FE codes. For thin plate structures, numerical results have shown that the present approach can predict resonant frequencies, loss factors, and mechanical impedances that are as accurate as the ones obtained with NASTRAN, however requiring substantially fewer elements. Also, the application of model 5 to thin shell structures is possible if appropriate numerical remedies and procedures are implemented to control the spurious modes effects which occur due to the underlying numerical pathologies. Other models also based in the use of existing FEs of commercial codes, applied to damped beams, plates and shells, based on ESL approaches [91] or on the previous models with slight modifications [104, 106, 122], can still be found in the open literature.
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Fig. 9.4 Discrete-layer (layerwise) 2D FE plate (or shell) models of viscoelastically damped structures using new dedicated FE codes/elements
9.5.2.2 Discrete-Layer (Layerwise) Elemental Models Some general drawbacks of the composite spatial models previously described are the higher intervention of the designer during the spatial modeling task, which has to decide which combination of FEs is more appropriate and to develop special elemental assembling procedures for the elements to be combined, and the need to completely redefine the 3D FE mesh when changing the geometric parameters of the damping treatments such as the thickness of the layers or the location(s) of the damping treatment(s). Therefore the latter approach is not well suited for design optimization procedures of single and multi-layer viscoelastic damping treatments. Furthermore, when using this discretization approach to obtain the spatial model of structures, a cumbersome and time consuming 3D FE mesh generation process is required. The aforementioned reasons motivate the use of alternative methods circumventing these limitations. A more efficient approach using new dedicated FEs, which are not readily available in commercial FE codes, which inherently yields an higher degree of automation in the spatial modeling and mesh generation processes, can alternatively be used to model viscoelastically damped structures. These new FEs can vary substantially in terms of nodal topology because for their formulation different assumptions regarding deformation theories, kinematics, constitutive behavior, number of layers, interpolation degree, shape, etc., may be considered. Following that line of thought, special FEs based on first- or higher-order discrete-layer (or layerwise) or zig-zag theories (Fig. 9.4), have been proposed and successfully applied to beam, plate and shell-type structures with active and/or passive damping treatments [25, 35, 47, 78, 107, 119, 150, 174, 187, 189]. As far as the deformation theory and kinematics are concerned, with a discrete-layer approach each physical layer (or sub-laminate) is treated individually according with the desired accuracy. Different deformation theories of different orders may be postulated for each layer, and by the use of “simple”
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or “mixed” variational principles and a priori assemblages at the elemental level or layered formulations with coupled kinematics [186], the elemental equations of motion and the correspondent layerwise FE formulation are established in a 2D sense. Due to the specificity of these formulations, the introduction of these new and dedicated FEs into general commercial FE codes for the composite, sandwich materials and damping treatments end-users, is progressing well but far from being fully accomplished. These layerwise FEs allow a significant simplification on the spatial model generation because the out-of-plane description, which depends on the morphology of the locally applied damping treatment (coverage, numbers of layers, layering scheme and material of the layers), is defined during the FE formulation, due to the coupled-layer kinematics, or during the a priori assemblage at the elemental level, going from a single- to a multi-layer FE. Thus, this modeling technique enables the external definition of the damping treatment parameters through a user defined data structure containing the properties which can be dynamically modified or updated in a numerical optimization process analysis, which is an impractical feature when using the previously presented layered composite models. As mentioned before, the out-of-plane description of the damping treatment configuration is performed either by coupled-layer kinematics or a priori assemblage at the elemental FE level so that the 3D continuum assemblage, at the global FE level, is reduced to a 2D discretization. In general, the damping treatments might be sandwiched or surface mounted in a single or both sides of the host structure. For those cases, and since the continuity of the host structure must be assured, dedicated single- or double-sided layerwise FEs using coupled-layer kinematics must be employed to model the equivalent 2D continuum. These modeling approaches are represented by models 6 and 7, respectively. An alternative and more general approach proposed by Vasques [186] in the context of piezo-visco-elastic shell FEs, instead of using formulations based on coupled-layer kinematics for an arbitrary number of layers, as presented by Vasques et al. [189], utilizes an a priori assemblage of single-layer FEs before performing the global assemblage of the equivalent 2D continuum. That approach, first proposed by Vasques and Rodrigues [187], is depicted in model 8 and represents a “regenerated” layerwise 2D approach where, in opposition to the well known Ahmad’s “degenerated” approach [4], some transformations of the elemental DoFs are performed allowing the displacement and stress DoFs of different layers to be assembled imposing not only displacement continuity but also shear stress continuity across the interfaces of multi-layer FEs, making it suitable for assemblage of elemental matrices from single- to multi-layer level.
9.5.3 Damping Modeling and Solution Approaches As previously referred, in general, the elastic and dissipative properties of viscoelastic materials may depend upon the frequency, operating temperature, amplitude and
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type of excitation [61, 84, 123, 145]. These dependencies of the viscoelastic material properties make a mathematical description of the viscoelastic constitutive behavior and material damping more difficult and complicated to obtain, and might turn the underlying FE implementations and solution methods more troublesome and difficult endeavors. Thus, for simplicity, the amplitude and type of excitation effects are often overlooked, isothermal conditions are usually assumed and only the frequency dependency of the viscoelastic constitutive behavior is usually taken into account. The direct frequency response (DFR) method, which is based upon the complex modulus approach (CMA) that makes use of a constitutive relationship as the one presented in Eq. (9.7), early presented in the 1950s by Myklestad [121] and subsequently further discussed and utilized by Snowdon [164] and Bert [23], is a frequency domain method that utilizes a time-domain based model which is limited to steady-state (time-harmonic) vibrations. This corresponds to a simple way of modeling viscoelastic damping effects where complex material properties are continuously updated for each discrete frequency value [117]. However, the structural model can alternatively be formulated in the frequency domain, yielding the so-called wave models, which are based on the definition of a dynamic stiffness matrix. These wave methods can also be extended to discontinuous structures and damping treatments, for example by the use of the so-called spectral finite element method proposed by Doyle [51] which utilizes dynamic interpolation functions. These wave-based methods, which also allow viscoelastic damping effects to be accurately considered by defining the material constitutive behavior for the specific frequency of the current waves, have been used, for example, in [17, 194, 195]. This solution method is usually denoted as a wave propagation method and it appears that Douglas [49, 50] was the first to explore wave solutions in order to implicitly account for the frequency-dependent complex modulus of viscoelastic components in the solution method. The CMA is also the basis of the so-called modal strain energy (MSE) method, first derived and utilized by Mead [112] and later popularized by Johnson et al. [83], where the loss factor of each individual mode is determined from the ratio between the dissipated modal strain energy of the viscoelastic counterpart and the storage modal strain energy of the whole structural system. However, MSE based methods are known to lead to poor viscoelastic damping estimation for highly damped structural systems. Following also a CMA but using an iterative framework, two other alternative iterative approaches have been recently systematized, assessed and compared with other damping modeling strategies in [190, 191]. The first is an iterative MSE approach, denoted as iterative modal strain energy (IMSE) method, successfully used also by [179] for moderate damping values, where an iterative procedure is used to estimate the modal parameters with the modal loss factor determined through a MSE approach. The second approach is denoted as iterative complex eigensolution (ICE) method, where, similarly to the IMSE method, the modal parameters are determined through an iterative solution, usually requiring only a few iterations (typically less than 10). In this case, a complex eigenvalue problem is considered instead of a real one, which allows a direct estimation of the loss factor and damped natural frequency from the complex eigenvalue.
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In more recent years, and in opposition to frequency-domain-based, i.e. CMAbased, approaches, the time domain models, examples being the Golla-HughesMcTavish (GHM), after Hughes and his colleagues [69, 111], anelastic displacement fields (ADF), after Lesieutre and co-workers [98, 99], and others presented by Yiu [202] and more recently by Silva [157], utilizing additional internal (or dissipation) variables [81], have been successfully utilized and shown to yield excellent damping estimates, irrespective to the amount of damping of the structural system. Comparing the methods described so far, the ICE approach was shown to be robust and to present some advantages over the internal variables, GHM and ADF, models: it does not need to increase the size of the problem and the eigensolution can consider only a small number of modes necessary to build the required truncated modal model. However, it has the drawback of requiring several complex eigenproblem (of a smaller size than the GHM and ADF approaches) calculations during the iterative process until convergence is achieved. When compared with the ICE approach, the IMSE method requires iterative calculations of a real eigenvalue problem at the cost of, at least for highly damped systems, yielding bad estimates of the modal damping ratios. Alternatively, the use of fractional calculus models, initially proposed by Bagley and Torvik [11, 12] and recently revisited, among others, in [53, 64–66, 84, 154], provides a simpler and more economic descriptor of the complex constitutive behavior of viscoelastic materials, being able to represent the storage and loss modulus of frequency-dependent viscoelastic materials using a low order parameter set formed usually only by one series of parameter and based upon the use of fractional derivatives. It has the drawback, though, of generating a “non-standard” FE time domain formulation, with a more burdensome characteristic solution procedure making use of some special solution artifices, but which still allows to obtain good damping estimates of the system. Other relevant contributions for damping, in general, and viscoelastic damping modeling, in particular, were given, among others, by Adhikari and Woodhouse [2] and the references therein on non-viscous damping in discrete linear systems, Balmès and his co-workers [13, 137, 138], who have used different variations of modal subspace models specially defined and assumed to be representative of the damped system response, and in [86, 102], that have used perturbation method approaches. The interested readers are referred to the aforementioned references for further details on the methods used. The ultimate aim of all these viscoelastic damping models and solution approaches is to be able to simulate the time and frequency response of viscoelastically damped structural systems. While the frequency response is straightforward to obtain, whether directly obtained (as is the case for the DFR approach) or obtained through modal models derived from the spatial model either by iterative frequencydependent eigensolutions (IMSE and ICE) or from the augmented spatial model eigensolution (GHM and ADF), the time domain response can be obtained from the spatial model either by direct integration methods or by the modal models using the superposition principle (see Fig. 9.5). With the advantages and disadvantages men-
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Fig. 9.5 FE-based viscoelastic time and frequency domain solution alternatives
tioned thus far, all the approaches can be used to build a truncated modal model of the damped structural system, whether from the spatial model or from a FRF model generated with the DFR method, which can be used to estimate both frequency and time domain responses. However, it is important to emphasize that when considering structures with a high modal density, possessing modes not well separated and strongly damped, the modal identification methods may not be accurate and efficient, rendering the identification procedure more troublesome, if not impossible. To sum sup, the solution and analysis methods can be divided into frequency and time domain based solution strategies, which clearly can admit distinct specific constitutive models. Frequency domain techniques comprises the use of CMA-based approaches, namely the DFR, IMSE and ICE, where the FE spatial model is used by re-calculating the complex viscoelastic stiffness matrix for each discrete frequency value, in the case of the DFR, or during the iterative process, in the case of the IMSE and ICE. These approaches are more straightforward to use and implement at the global FE level. That is the reason why the CMA-based approach, DFR, is the most common approach implemented in commercial FE codes incorporating viscoelastic damping modeling capabilities. However, when general transient responses are required, time domain models are more suitable and versatile and they might represent better alternatives than the CMA-based frequency domain methods, since they allow the reduction of the computational burden due to the re-calculation of the stiffness matrix for each discrete frequency value (DFR) and the use of iterative eigenproblem calculations (IMSE and ICE). For further details on the FE implementation of the mathematical damping modeling and solution methods previously discussed, the reader is referred to Vasques et al. [190, 191]. Further comparative analysis and sur-
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veys of modern methods for modeling frequency-dependent damping in FE models can also be found in [22, 161, 178, 179].
9.5.4 Frequency- and Time-Domain Implementations Considering the frequency-dependent constitutive behavior of viscoelastic materials previously discussed and adopting the complex modulus representation in Eq. (9.7), the time-dependent matrix FE equation of motion of a general viscoelastically damped structural system may be written as ¨ + Du(t) ˙ + KE + KV (jω) u(t) = f(t), Mu(t) (9.8) where M and D are global mass and viscous damping matrices, KE and KV (jω) are elastic and complex frequency-dependent viscoelastic stiffness matrices, and u(t) and f(t) are displacement DoFs and applied loads vectors. For generalization two types of damping models are considered in the previous equation: (i) an arbitrary frequency-dependent hysteretic (or viscoelastic) damping type, represented by Im[KV (jω)], whose terms are frequency-dependent, representing the viscoelastic dissipation (relaxation) behavior; and (ii) a viscous damping type, which is described by D, representing other general sources of damping (e.g. air-based damping, energy dissipation in the supports, etc.) which are assumed to be proportional to the velocity. In addition, a frequency-dependent stiffness term of the viscoelastic counterpart, represented by Re[KV (jω)], is also implicitly considered with this mathematical realization. Using Eq. (9.8) as it is, the frequency-dependent matrix definition implies that its use and analysis can only be performed in the frequency domain, based on the CMA, where the material properties of the stiffness matrix of the viscoelastic parts are defined for each discrete frequency value [117, 188, 189]. In view of this, considering simple harmonic excitation, with f(t) = Fejωt , where F is the amplitude vector of the applied mechanical forces, the steady-state (time-harmonic) response of the system can be written as u(t) = U(jω)ejωt , where U(jω) is the complex response vector (displacements phasor), yielding (9.9) K(jω) + jωD − ω2 M U(jω) = F, where K(jω) = KE + KV (jω), from which the complex solution vector U(jω) can be obtained. For the use of a steady-state (time-harmonic) model of a general viscoelastically damped structural systems as given by Eq. (9.9), the DFR is the most appropriate frequency domain method where the frequency response model [57] can be generated in a straightforward and direct manner from the results of many discrete frequency calculations of the equation of motion, for which the complex stiffness matrix of the viscoelastic parts is re-calculated at each frequency value of the desired discrete frequency range. Alternatively, resorting to the model in Eq. (9.8), as depicted in Fig. 9.5 the CMA-based IMSE and ICE methods may also
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be used to generate the frequency response model with a less straightforward and indirect approach which first requires the iterative definition of the modal model [190, 191]. However, for the general analysis of transient responses the use of time-domain models is more suitable and versatile than frequency-domain methods based on the use of a complex modulus. For this purpose the IMSE and ICE approaches may also be used but we should bear in mind that a specific bandwidth should be fixed for which all modes should be determined, the solution becoming less interesting for systems with high modal densities and damping. Thus, a time-domain implementation of the viscoelastic constitutive behavior into the standard FE equations of motion of the general viscoelastically damped structural system in Eq. (9.8) is obtained by factoring out the shear modulus of the viscoelastic stiffness matrix. Thus, assuming that the structural system possesses only one type of viscoelastic material, following the hereditary stress-strain law given in Eq. (9.2) yields ¯ V u(0) ¨ + Du(t) ˙ + KE u(t) + Grel (t)K Mu(t) t ¯ V ∂u(τ ) dτ = f(t), Grel (t − τ )K + ∂τ 0
(9.10)
¯ V is the remaining viscoelastic stiffness term after factoring out the shear where K modulus; both shear and extensional stiffness terms may be considered in the viscoelastic stiffness matrix by assuming a frequency-independent Poisson’s ratio and the appropriate extensional and shear modulus relationship for isotropic materials. The mathematical time-domain realization in Eq. (9.10) represents the departure point from which time-domain solution techniques may be used. A more direct approach may consider the numerical integration of the non-linear matrix equation of motion in Eq. (9.10). However, internal variables models, such as the GHM and ADF methods, may alternatively be used to linearize the damped system, at the cost of increasing the size of the problem but allowing to obtain a computationally tractable augmented system of linear ordinary differential equations that can be solved by standard numerical methods applicable to first- or second-order linear systems. Size reduction methods may also be used to make the problem more amenable if it increases too much. The interested reader can find further details in [177, 190].
9.5.5 Commercial FE Software During the last decades several commercial FE softwares have been used to study and design viscoelastic damping treatments. The use of commercial FE software to validate and compare FE and analytical models and experimental tests using, for example, NASTRAN can be found in [37, 72, 80, 82, 88, 104, 105, 107, 117], with ANSYS in [7, 38, 155, 199, 201, 205] and with ABAQUS in [7, 106, 160]. Attempts to use the GHM and ADF internal variables models with commercial FE software such as NASTRAN are presented in [26, 68]. A general discussion regarding the
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basic procedures and major capabilities of commercial FE analysis software packages, such as ABAQUS, ADINA, ALGOR, ANSYS, COSMOSWorks, NASTRAN, ABAQUS/Explicit, DYNA3D and LS-DYNA, for vibration analysis is presented in [165]. In addition, an overview on FE software capabilities under a more general perspective is also presented in [147]. Apart from the accuracy and representativeness of the structural model, the damping modeling capability is a key ingredient in the design of damping treatments. In fact, an adequate damping design is strongly influenced by the degree of accuracy of how the damping quantification factors are determined and how they are utilized to optimally define the damping treatment location, geometry and thicknesses of the layers. It is well known that the structural model has received substantial attention by the scientific community in order to refine the structural models and that these developments have reached a relative stage of maturity, with some of these refined models being adopted and implemented by the major part of the FE software (an exception in the field of structural mechanics is for example the use of HOT and layerwise elements to the study of composite laminated structures which is not a common tool in commercial FE software). However, damping modeling is a more uncertain and specific area of interest (mainly to dynamicists) which has not reached the maturity stage yet and where the transition of contemporary damping modeling approaches to commercial softwares is far from being done. As a result of that, when intending to design viscoelastic damping treatments using FE software, the available damping models of the commercial FE codes are an important aspect to take into account. With that in mind, a compilation of the damping modeling capabilities and general features available in some well known softwares is presented in Table 9.2. In addition to the aforementioned, it is worthy to mention the fact that when using more complex high-order or layerwise FEs with segmented viscoelastic damping treatments in composite laminated structures, the pre- and post-processing capabilities of these softwares may not furnish the designer with the adequate tools to discretize and generate the mesh of the structure, along with special analysis options (e.g. modal strain energy distribution) which may turn this task more cumbersome for the design analyst. Lastly, since most FE softwares cannot extract complex modes from models with frequency-varying material properties, the analysis procedure may be subdivided into frequency ranges, as considered by Hambric et al. [72], and an incremental approach considering interpolation of estimated modal parameters might be used to estimate the modal parameters at other frequencies. This procedure, somewhat resembling the ICE method discussed earlier, when compared with approaches that allow for frequency-dependent material properties to be used in a FE complex modal analysis, such as the GHM and ADF methods, might be considered inefficient, since it requires multiple calculations of a complex eigenvalue problem while the internal variables models require only one, however considering a higher-dimensional problem.
G ENERAL DAMPING
Small deformation Small strain (linear viscoelasticity) Large deformation Small strain (visco-hypoelasticity) Large strain (visco-hyperelasticity) Time-domain Prony’s series Frequency-domain Complex modulus Curve-fitting to Prony’s series Creep test data Relaxation test data Frequency-dependent test data Temperature effects/ Shift-functions (TRM) Williams-Landel-Ferry (WLF) Tool-Narayanaswamy (TN)
N
N
N
V ISCOELASTIC C ONSTITUTIVE M ODELS
Viscous Hysteretic Composite modal damping
ABAQUS v6.7
ANSYS v11
N
N
N
MD NASTRAN R2
N
N N N
N
N N
ALGOR v20
N
N N
N
N
LS-DYNA v971
N
N N
N
N N
ADINA
Table 9.2 Damping modeling capabilities and general features available in some commercial FE softwares
N
N N
N
N
N
COSMOSWorks 2008
N
N
N N N
N
N
N
N
N N N
N
N
N
N
ACTRAN/VA 2007
(continued on next page)
LUSAS v14
234 C.M.A. Vasques and L.C. Cardoso
N N
N N
× × × ×
× × × ×
×
ABAQUS v6.7
×
ANSYS v11
N
N
× × N N
×
MD NASTRAN R2
N
N N
N N N N
× × N N
× N
ALGOR v20
N
N/ /N
N N N N
× × N N
× N
LS-DYNA v971
N
N
N N N
× × × ×
×
ADINA
N N
N/ N N
N N N N
× × × ×
N N
COSMOSWorks 2008
N
N/ N N
N
× × × N
× N
LUSAS v14
N
N N/
N
N N
N
N
ACTRAN/VA 2007
Notes: ‘’, procedure fully available; ‘×’, approximated procedure not admissible/accurate for frequency-dependent materials (however, feasible using internal variables models); ‘N’, not available or not known
2D/3D Beam/Truss Plate/Shell Layered (composite) Constitutive behavior Isotropy Orthotropy Anisotropy
V ISCOELASTIC FE T YPES
Eigenanalysis Undamped (real) Damped (complex) Direct frequency response Direct integration method Transient response Mode superposition method Transient response Frequency response Response spectrum analysis Random response analysis Other features Reduced solution methods Frequency spanning tools Substructuring analysis Subspace solution method
S OLUTION M ETHODS
Table 9.2 (continued)
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9.6 Vibroacoustic Simulation and Analysis In a vibroacoustic analysis the governing system of equations is particularly different from the more typical simple structural vibration analysis problem because each individual problem, namely the acoustic fluid and the structural problems, need to be considered simultaneously, so that the interaction that occurs between the two mediums is accurately accounted for. However, the former approach, know also as a two-way (or more generally, multiway) or strong interaction/coupling, can be relaxed into a more simple one-way or weak coupling if there is no feedback between the subsystems (e.g. if we ignore the effect of the fluid pressure in the structural equation of a vibrating radiating structure immersed in air and subjected to an external mechanical loading) [60]. Regarding the latter approach, different fluid-structure coupling strategies have been proposed somewhat combining the equations governing the physics of the two mediums (see [56, 116, 131, 132] for further information on this topic). For strong fluid-structure coupled analysis the FE formulation and discretization are usually performed using displacement formulations for the structure and multiple choices for the acoustic fluid which may consider either fluid displacement or different scalar fields such as pressure, displacement and velocity potential and combinations thereof. These alternative formulations may yield different nonsymmetric and/or symmetric formulations, the latter being usually preferred to make system solving and response computations easier. Concurrently to symmetrization procedures, modal decomposition procedures utilizing the eigenvectors of each subdomain, i.e., the acoustic fluid medium and the structure, to diagonalize and reduce the size of the coupled vibroacoustic problem, have also been successfully used mainly for transient analysis [149]. From the alternative FE formulations discussed above, for steady-state (timeharmonic) analysis probably the most appellative formulation considers displacement DoFs for the structure and the velocity potential (or a related potential), rather than the acoustic pressure, as the fundamental DoFs in the fluid region [55]. The main advantages of this formulation come from the fact that the solution of the problem involves the inversion of a fluid-structure frequency-dependent dynamic “stiffness” matrix which is also symmetric, irrespective of considering any viscous (velocity proportional) structure or fluid damping effects. This is in fact the approach implemented in commercial vibroacoustics software such as the software ACTRAN/VA [62] where the resultant symmetric system of governing equations for a generic interior problem considering also viscous damping terms for the structural and acoustic domains may be written as KS (jω) + jωDS − ω2 MS U(jω) FS jωCFS , (9.11) = T 2 FF (jω) jωCFS KF + jωDF − ω MF where the subscripts S, F and FS denotes structure, fluid and fluid-structure coupling related terms, KS (jω) = KES + KV S (jω) denotes the net structural stiffness matrix accounting for both elastic, KES , and complex frequency-dependent viscoelastic,
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KV S (jω), terms, U(jω) and (jω) = −jωρF P(jω) are the complex vectors of displacement and fluid velocity potential amplitudes, P(jω) being the vector of complex fluid pressure amplitudes, and FS and FF are appropriate structural and fluid forcing vectors. Since the structural matrix, KS (jω), in Eq. (9.11) is frequency-dependent, linear modal solvers can not be directly employed to solve weak or fully coupled systems of equations; as the degree of coupling increases, the higher the computational cost is and the more robust and sophisticated modal solvers are required. Alternatively, besides allowing the fully coupling being considered and the more straightforward consideration of frequency-dependent structural-acoustic damping effects, a DFR analysis of the matrix system of equations in Eq. (9.11) may be more straightforward to use and implement, this time the coupled displacement and pressure (or velocity potential) frequency responses being obtained for arbitrary structure and/or fluid loadings. In addition, the DFR approach may benefit also from the high computational efficiency of algebraic solvers such as the Krylov solver available in AC TRAN /VA. The latter solver is an iterative multifrequential solver which must be used in conjunction with a direct solver. It is aimed to be used to quickly calculate the frequency response of a system where all materials and boundary conditions have a simple relation with the frequency. Further, it becomes a more efficient solving solution as more individual frequencies are computed, making the procedure nearly independent of the number of individual frequencies in the frequency band under consideration [62]. When analyzing sound radiation and scattering problems it is quite frequent the need of considering exterior (unbounded) acoustic domains. Due to that, some additional difficulties arise with the use of FEs for the modeling of the acoustic infinite free field medium. If using a “conventional” FE approach to model the infinite free field it would be necessary to model the entire field, resulting in a prohibitive number of elements and computational cost. Alternatively, infinite element technologies, somewhat embodying the far field characteristics, have been successfully reported as a means to overcome this difficulty. The key idea is to divide the structural domain into proximity and far field counterparts, the latter being considered through appropriate “finite” (infinite) elements which extend from the proximity/far field boundary surface to infinity and that are designed to ensure that all acoustic waves propagate freely towards infinity so that no reflections occur at the boundary. The particularity of infinite elements is the use of exponential functions multiplying the shape functions which is able to reproduce the decay of the pressure in the infinite (far) field [8, 24]. Many other methods have been proposed to terminate the computational domain of exterior FE models based, for example, on the use of artificial local and non-local boundary conditions, absorbing and perfectly matching layers [73, 79, 175]. However, infinite element schemes have proved to be the most robust of these methods for commercial exploitation, mainly due to the simplicity of implementation with conventional FE programs, and are implemented in some of the major commercial codes available today, as is the case of ACTRAN/VA. An interesting overview on the foundations of numerical methods in acoustics modeling addressing this and other topics can be found in [9, 79].
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Alternatively to domain-based methods, such as the FE method, a natural way of modeling the acoustic region exterior to a scattering/radiating object is to introduce a boundary element discretization of the surface of the object by using surfacebased methods. That is the case of the boundary element method (BEM) which solves the acoustical quantities on the boundary of the acoustical domain instead of in the acoustical domain itself. The solution within the acoustical domain is then determined based on the boundary solution. This is accomplished by expressing the acoustical variables within the acoustical domain as a surface (boundary) integral equation over the domain boundary instead of expressing it as a differential equation satisfied in the entire acoustic domain (see [77] for a brief introduction to the BEM in acoustics). It only requires discretization of the surface of the body and automatically satisfies the required radiation condition at infinity, known as the Sommerfeld radiation condition at infinity (see [79, 136] for further details), which states that
∂p(x, t) 1 ∂p(x, t) + = 0 or lim r r→∞ ∂r c ∂t
∂P (x, jω) lim r + jkP (x, jω) = 0, (9.12) r→∞ ∂r where p(x, t) and P (x, jω) are the transient and complex amplitude of acoustic pressure, r = |x| is the radial distance from the origin of a source of sound to any position in a 3D space defined by vector x, c the speed of sound and k = ω/c the wave number. This condition requires that as r → ∞, then only outgoing (there is no reflection) waves resembling planar waves can exist [136]. Detrimentally, the application of the classical BEM for acoustic problems often requires the solution of large, dense, complex linear systems due to the nonlocal support of the fundamental solution which leads to high computational expense and storage requirements. Compared with the BEM, it has been shown that the numerical advantage of the FEM is that it leads to sparse matrices (containing several lines and columns with zeros), which by avoiding calculations on the zeros significantly speeds up computations and reduces memory requirements. However, contradicting this latter idea, we have attested the recent developments in fast multipole BEMs [103] which have been reported to significantly accelerate the required calculation time, making less clear which method presents the best efficiency. Notwithstanding this improvement, the FEM as a domain-based method still has the interesting advantage of allowing a more robust and natural integration with other discrete models in coupled problems through a simpler domain coupling procedure. In many practical situations a vibroacoustic analysis in the half-space may suffice to get a qualitative picture of the sound power being radiated to the surrounding acoustic medium. This is the case for example here where, for simplification, an easier assessment of the sound being radiated by one side of an annular plate, with and without a damping treatment, may be performed in order to qualitatively trace down the radiated sound power energy along the frequency spectrum. To allow that, we idealistically consider that the circular saw may be considered as a planar structure set in a plane baffle, i.e. a surface in which a limited portion (plate) has prescribed
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normal velocity and where the reminder of the surface is idealized as rigid. Under those circumstances, we disregard one of the radiation sides of the plate. It is known, though, that the consideration of more realistic acoustic boundary conditions allowing the sound field from one side to interact with that in the other side of the plate, known as a plate in unbaffled conditions, leads to a more complicated problem to solve (both analytically and numerically). In fact, these more realistic conditions more closely resemble the real acoustic conditions of a rotating circular saw, but they still disregard some aspects since the true real conditions, such as the aerodynamic effects of rotation, are not considered. The outcome of this effort is usually not significant since it is know that unbaffled conditions usually decrease the radiation efficiency, mainly due to the partial cancellation of the sound field between the two sides of the plate [142], the difference being more significant at low frequencies, making the simpler baffled case more attractive for design purposes since it is easily implemented and represents the worst scenario in terms of noise radiation efficiency. In addition to the foregoing, baffled plate conditions also allow handling acoustic radiation in the half-space through the use of the so-called Rayleigh integral, which is based in a particular case of the Kirchhoff-Helmholtz integral theorem [136]. The latter integral is a general mathematical theorem for exterior acoustic problems considering an isolated vibrating body (or a fixed surface enclosing a source) in an otherwise unbounded acoustic fluid, and is obtained using the Helmholtz equation, Green’s second identity and the Sommerfeld radiation condition. Through its use a general expression for the acoustic pressure outside the source can be derived which, when considering the simplifying features of a vibrating plate set in baffle, due to simplifying considerations such as equal normal velocities, canceling integrals over the surface pressure and equal (duplicated) contributions of velocity integrals in the front and back surfaces of the plate, is simplified into the Rayleigh integral, yielding v˙n (xS , t − R/c) ρ dS or p(x, t) = 2π S R Vn (xS , jω)e−j kR jωρ dS, (9.13) P (x, jω) = 2π S R where ρ is the density of the acoustic medium, x is the position vector of the observation point, xS is the position vector of a point on the radiating surface S having transient and complex amplitude normal velocities vn (xS , t) and Vn (xS , jω), respectively, the dot over the variables denotes time differentiation and R = |x − xS |. The evaluation of the latter integral is probably the conceptually simplest approach to calculating the sound field radiated by an area of a vibrating surface and represents the starting point for any sound radiation calculation based on the boundary integral representation for plane baffled structures. Essentially it evaluates the sum of the fields generated by a distribution of elemental sources, each having a complex volume velocity Vn (xS , jω) dS in the case of a time-harmonic analysis. This latter approach is handled by ACTRAN/VA through the numerical solution (integration) of the Rayleigh integral, avoiding the definition and discretization of the acoustic medium and only requiring the definition of a radiating surface, considering the
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boundary side of the saw that we wish to consider coplanar with the baffle. In this case, the problem is formulated with reference to a vibrating structure whose radiation surface is located in the plane of a rigid baffle. Alternatively, the fluid media may also be discretized and infinite element technology may be used to model the far-field acoustic media, at the cost of requiring both a structural and an acoustic domain mesh along with the definition of a separating surface mesh dividing the proximity and far field acoustic domains. One advantage on the latter modeling approach would be the consideration of more closely true unbaffled acoustic boundary conditions yielding more accurate and representative results of the real problem. As previous stated, the latter approach would complicate the analysis since it would require an higher computational effort due to the higher total number of DoFs, increased by the inclusion of the DoFs of additional acoustic elements (unnecessary when using a Rayleigh surface approach). In view of this, the latter approach will not be pursued in what follows. For further details on the theoretical foundations of the FE technologies available in ACTRAN/VA the reader is referred to the accompanying manual of the used vibroacoustic FE software [62].
9.7 Circular Saw Blades Damping: Modeling, Analysis and Design In order to illustrate the interest and the applicability of viscoelastic damping technologies and the possible modeling, simulation and design approaches previously discussed, the vibroacoustic study of the application of viscoelastic damping technologies to circular saw blades is performed in this section. For simplification an annular plate geometry is considered representative of the saw blade. A vibroacoustic numerical analysis is performed to the annular plate (saw) in order to infer about the advantages of incorporating a viscoelastic passive constrained layer damping (PCLD) treatment into circular saws. Emphasis is put in the modeling and analysis aspects considering a PCLD treatment. The commercial vibroacoustics FE software ACTRAN/VA [62] is considered here for that purpose.
9.7.1 Geometric and Material Properties of the “Saw” Let us consider a vibroacoustic system composed by a structural counterpart (annular plate–saw) and an acoustic counterpart (surrounding air) with the geometric and material properties presented in Table 9.3. For simplicity, the plate is considered homogeneous and to be made of an isotropic elastic material. Further, since the geometry of circular saws is very similar to the geometry of annular plates, the simplification of neglecting the teeth is considered here. In fact, this modeling simplification has also been considered previously and regarded as an admissible simplification for the representation of a saw [44].
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Table 9.3 Material and geometric properties of the vibroacoustic system Structural system Young modulus / GPa Plate (saw)
210 + 0.42j
Poisson Density Inner radius Outer radius Thickness ratio / kg m−3 / mm / mm / mm 0.30
7800
15
150
2.200
0.32
2708
30
120
0.254
Viscoelastic layer Eq. (9.15), Table 9.4 0.49
1140
30
120
0.127
Constraining layer 70
Acoustic system Speed of sound / Air (20 °C)
m s−1
Density / kg m−3
343.26
1.2041
Regarding the viscoelastic damping technology, it was previously discussed that in practice viscoelastic materials dissipate more energy when the predominant deformations are shear. Thus, the use of viscoelastic materials can be improved by using a constraining layer, which shifts the neutral axis towards the center of the viscoelastic layer [84, 113, 123]. In view of that, a PCLD treatment is applied to the saw, for the present case study being applied only one side of the plate. The geometry and material properties of the PCLD treatment are also specified in Table 9.3. The thicknesses of both viscoelastic and constrained layers, also denoted as 5 and 10 mil, respectively, in typical US industry units, correspond to a well known commercially available damping treatment provided by 3M company (3M™ Damping Foil 2552). The air properties at room temperature of 20 °C are also presented. The constitutive mechanical properties of the viscoelastic material used here in this example are considered only as frequency-dependent. Many authors have proposed different mathematical models to represent such constitutive behavior as previously discussed and thoroughly reviewed in [190, 191]. Here, an anelastic displacement fields (ADF) constitutive representation is used to represent the properties of the viscoelastic material. Thus, taking an ADF Laplace domain representation, the values of the complex shear modulus in the Laplace domain, G(s), can be obtained from
n i s G(s) = G∞ 1 + , (9.14) s + i i=1
or, alternatively, considering a pure imaginary variable, s = jω, in the frequency domain the shear modulus, G(jω), is given by
n ω2 + jω i , (9.15) G(jω) = G∞ 1 + i ω2 + 2i i=1 where G∞ is the so-called relaxed (also known as static) modulus, i is the inverse of the characteristic relaxation time at constant strain and i the correspondent relaxation resistance.
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Table 9.4 Identified GHM and ADF parameters for 3M ISD112 at 27 °C using three series (n = 3) Model i
GHM G∞ / MPa
αi
ωˆ i / rad s−1
ζˆi
G∞ / MPa
ADF i
i / rad s−1
1
0.1633
4.8278
28045
22.013
0.1789
3.5286
504.20
2
14.548
41494
3.1275
8.7533
4286.5
3
40.043
41601
0.6165
60.324
39313
To take into consideration the relaxation behavior, the entire ADF model itself may be comprised of several individual fields, where n series of ADFs are used to describe the material behavior. Given a set of measured values of the shear modulus in the form of a frequency-dependent complex modulus, G(jω) is determined through curve fitting techniques. The number of series of ADF parameters determines the accuracy of the matching of the measured material data over the frequency range of interest. The identified value of these parameters for the viscoelastic material used in this work, with commercial designation 3M ISD112, at 27 °C are presented in Table 9.4; for generality the values are presented here both for the GHM and ADF models [191], the ADF model being the one use here. Additionally, a frequency-independent Poisson’s ratio equal to 0.49 is considered. Regarding the fitting procedure of the viscoelastic material properties, for completeness, some further information is given here about it. One and three series of parameters were used to curve fit the identified complex modulus of the 3M ISD112 material at 27 °C over the frequency range 10–3000 Hz. The fitted curves of both models are presented and compared with the measured data in [191]. The results for the ADF model are presented in Fig. 9.6. As can be verified, the quality of the fit depends on the number of terms (series) retained in the ADF model. Using more series improves the accuracy of the model material description, however, at the cost of increasing the size of the model. Further, we can conclude that in the frequency range defined the ADF model fits the measured data with a satisfactory accuracy with 3 series of parameters (one series would not be enough accurate). It is worthy to mention that the values of the parameters determined are guaranteed to define the material properties with accuracy only over the frequency range specified in the fitting process, i.e. 10–3000 Hz.
9.7.2 FE Modeling and Vibroacoustic Media Discretization The FE modeling approach followed in this work is discussed in what follows. For simplicity of the modeling task, the rotating (centrifugal) effects, which may be considered through an equivalent initial stress field (pre-stress effects) in ACTRAN/VA, are not taken into account. Furthermore, for simplification also, the teeth of the saw are not considered, gyroscopic effects are neglected, the plate is considered to be in an infinite baffle, only one side of the saw is coupled with the acoustic domain,
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Fig. 9.6 Curve fitted ADF curves with 1 and 3 series of parameters at 27 °C
isothermal conditions are assumed and the inner radius cylindrical surface of the annular plate is considered as clamped. Regarding discretization issues, the structural mesh is externally generated and afterwards imported to, and manipulated, with the pre/post-processing software AC TRAN /VI. The prepared input file defining the problem to be analyzed is then used as the input information to be processed by ACTRAN/VA. From the processing task, several output files are produced which may then be pos-processed and analyzed with ACTRAN/VI. The annular plate (saw) and the constrained layer damping treatment are modeled with 20-noded hexahedron (brick) quadratic solid-shell FEs (HEX20); a discrete layer approach is used considering one layer of elements per each one of the three physical layers, i.e., the saw, the viscoelastic and the constraining layers (Fig. 9.7). Thus, the discretization of the viscoelastic and constraining layers is generated with the same type of elements used for the plate, namely solid-shell HEX20 FEs. These elements have three displacement and one pressure-related (velocity potential) DoFs per node, the latter being active only when/if considering interfaces with the acoustic domain. For simplicity, it is preferable that the three different physical layers (saw, viscoelastic and constraining layers) meshes be compatible (coincident), which means the nodes at the interfaces plate/viscoelastic layers and viscoelastic/constraining layers, must be the coincident to enforce displacement continuity. In more complicated cases, though, if required, ACTRAN/VA has also the capability of working with incompatible meshes. Worthy to mention is also the fact that due to the thinness of the plate and damping layers, numerical problems related with shear locking may compromise and deteriorate the accuracy of the re-
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Fig. 9.7 FE mesh and zoom of the discretized annular plate (light gray), viscoelastic damping layer (green) and constraining layer (dark gray), and clamped inner cylindrical surface (red)
sults. However, the element HEX20 has some built-in stratagems to avoid numerical pathologies such as locking. The reader is referred to the user’s manual of the software for further details and theoretical explanations [62]. Due to the spherical wave propagation nature of the sound radiated by the plate, the structural mesh is generated by revolving a 2D mesh around the axis of rotation of the annular plate. Thus, the generated mesh is regular, in the sense that the medium is discretized with equally spaced discretization steps of the radial and angular polar coordinates, containing eighteen elements in the radial direction and forty divisions in the circumferential (angular) coordinate. It is worthy to mention that the structural mesh should have a more refined discretization than the fluid mesh, whether if utilized directly through the consideration of a 3D acoustic fluid domain or by the use of a surface integral, due the propagating wave speed magnitude of the solid structural medium which in general is higher than the one in the air. Indeed, it is in fact the regular structural mesh discretization, in terms of the radial and angular steps, and the interpolation order of the FEs used, that dictate the useful bandwidth of the analysis (the finest the mesh is the higher the maximum frequency of the allowable bandwidth would be). However, in order to obtain an admissible computational effort, the mesh size was limited to a comfortable level of discretization for an analysis up to 1 kHz.
9.7.3 Results 9.7.3.1 Modal Analysis A preliminary modal analysis of the annular plate is carried out in order to determine the regions of larger deformations. In principle, these regions are the most attractive places to locate any damping treatment, in general, and the PCLD treatment, in particular, since the main idea of these treatments is to force them to absorb as much straining energy as possible in order to dissipate vibration energy through heat
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Fig. 9.8 Undamped (flexural) mode shapes and natural frequencies of the bare annular plate (saw); ∗ , duplicated modes; cold color (dark blue) meaning zero transverse displacement, as is the case for example in the clamped inner circle, and hot color (red) meaning relevant transverse flexural displacement
dissipation in the viscoelastic layer. However, we should not forget that a circular saw has regions which need to be free, such as the cut region and the clamping region, which introduces some limitations in the choice of the treatment location and covering extension of the treatment. A modal analysis is performed considering an uncoupled undamped annular plate with the aim of determining and evaluating the structure undamped natural frequencies and mode shapes. This information is important, in the one hand, to critically judge which modes might be prone to be more efficient noise radiators and the frequencies involved and, in the other hand, to inspect the modal strain energy distribution and to correlate it with the desired locations and covering extension of the damping treatment. The first nine modes of the annular plate are represented in Fig. 9.8. It can be observed that, in general, the modes with radial nodal lines (nodal diameters, m)
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have the lower frequencies, while modes with circumferential nodal lines (nodal circles, n) have higher frequencies. The mode shapes are denoted here by (m, n). Worthy to mention is the fact that the lowest mode, contrarily to what happens in circular plates, due to the annular nature of the structural system is the (1, 0) mode, immediately followed by the (0, 0) mode. This latter mode, from an acoustic radiation viewpoint, is generally identified as the most efficient mode contributing to the radiated sound; it occurs due to the lack of the noise canceling effect which occurs because of the air recirculation among neighboring cells with opposite normal velocity phase, as is the case of modes with an higher number of nodal diameters and circles. Regarding the choice of the location of the damping treatment, it would be convenient looking at the modal strain distribution, which is not presented here (the used version of ACTRAN/VA does not offer that capability). However, it can still be inferred that for the mode (0, 0) the highest curvature occurs close to the clamped inner cylindrical surface and that the appropriate damping distribution is independent on the angular coordinate. Higher order modes, however, may require different appropriate damping treatment configurations for optimal damping performance, which for thin structures may be considered as strongly correlated with the secondorder derivative of the transverse displacement. This important issues will not be further addressed here but the reader is suggested to recall its importance when designing damping treatments for structures such as this.
9.7.3.2 Forced Response As previously discussed, the use of direct solution methods when considering a structural model with viscoelastic materials may present some advantages over other solution methods [190, 191], since it demands a problem resolution in the frequency domain. In addition, even without the necessity of considering the frequencydependent constitutive behavior of the viscoelastic layer, modal analysis of coupled vibroacoustic systems is sometimes not possible to perform, as is the case of the version of ACTRAN/VA being used which does not offer that possibility. Thus, in order to assess the effects of the PCLD treatment for a non-rotating undamped and damped saw blade, it is taken advantage of the efficient frequency sweep solution method supported and promoted by ACTRAN/VA, i.e. the Krylov solver, and the vibroacoustic responses are thus presented next in terms of frequency responses of the displacement, mean-square velocity and radiated sound power. A forced frequency analysis, representative of the real response observed due to the in-operation excitation, demands a model for the input force (or excitation). Some authors have considered a rotating force and a non-rotating plate (circular saw). Conversely, other analysis are made which consider a non-rotating force, but a rotating plate. As previously discussed, the excitation can originate from many different sources, but the principal sources usually considered are the self-excited vibrations due to tooth impact occurring only during cutting and the aerodynamic fluctuations occurring during idle. The self-excited vibrations are caused by the phenomenon called stip-slick. The stip-slick phenomenon is caused by a variation of the
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coefficient of friction, between the saw and the cut piece. This variation generates a variation of the cut force applied. It can be noted that this phenomenon is selfgenerated, which is a characteristic of instable systems. These instabilities excite the resonant frequencies of the saw, producing sound radiation. The other source of force is due to the aerodynamic fluctuation, which due to the variation of the pressure on the tooth cause an excitation force which excites the resonant frequencies of the saw. Here both the rotating force and saw are considered to be non-rotating; the aerodynamic fluctuation and any tooth related effects are also disregarded. In view of this, the excitation typically produced by a circular saw in operation is simplified to a harmonic force represented by a unitary transverse point load applied on a node in one side of the outer diameter of the annular plate. This model of force, although maybe too much simplifying, is however expected to capture in an elementary way the physics of the underlying excitation and saw dynamics during cutting. Three types of curves of frequency responses per unit force applied at the outer radius are presented in Fig. 9.9 for the undamped and viscoelastically damped annular plate (saw) configurations: forced frequency response in terms of (i) transverse displacement measured at the same point of the application of the transverse force (driving-point receptance), (ii) mean-square velocity of the radiating surface (surface of one side of the plate) and (iii) radiated sound power. In this way, both the structural acoustics and vibratory behavior with and without the PCLD treatment are compared and the damping performance assessed. Regarding the receptance frequency response function plotted in the top graphic of Fig. 9.9, it represents in fact a qualitative correlated measure of the amount of “kerf” losses, which are due to the cuts or incisions made by a saw in a piece of wood during operation, strongly related also with the blade thickness and displacement vibration amplitudes (more important as we approach the teeth zone). For the first nine modes of vibration it is shown that the displacement amplitude at the outer diameter is significantly reduced with the PCLD treatment, the amplitude being bigger for the lower order modes, both for the undamped and damped cases. The values of the resonant frequencies, modal loss factors and response reductions are presented in Table 9.5. It is clear from the analysis of the data in the table that not only the resonant frequencies are slightly increased from one configuration to the other, the physical justification being that the PCLD treatment more significantly increases the net stiffness of the system than the mass (in fact the treatment is very thin), but also that a significant damping effect is encountered for the plate with the PCLD treatment, roughly speaking with an average modal loss factor of the order of 1–2%, as compared with the bare plate with roughly 0.2% of damping introduced in the model through the imaginary counterpart of the Young modulus of the saw defined in Table 9.3. These effects are pronounced in all modes but it is not difficult to conclude that modes with nodal circles are generally more damped (with values above 2%) than the remainder ones. Regarding the receptance reduction, low order modes are the easier excited ones with higher amplitudes and an average of 20 dB reduction being obtained for all modes, meaning that roughly speaking the amplitude reduction is of the order of 10 to 1.
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Fig. 9.9 Driving-point receptance, measured at the outer radius, mean-square (MS) velocity and radiated sound power per unit force applied at the outer radius of the undamped (dotted line) and viscoelastically damped (solid line) annular plate (saw) configurations
Regarding the radiated sound power, Wrad , by definition for harmonic fluctuations the time-averaged power may be expressed as 1 Wrad = 2
S
Re P (xS , jω)Vn ∗ (xS , jω) dS,
(9.16)
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Table 9.5 Resonant frequencies, modal loss factors and response reductions for the plate with and without the PCLD treatment Mode (m, n): m, nodal diameters; n, nodal circles (1, 0)
(0, 0)
(2, 0)
(3, 0)
(4, 0)
(0, 1)
(1, 1)
(5, 0)
(2, 1)
Resonant frequency / Hz Bare plate
85.67
103.8
137.4
303.7
532.5
621.1
679.4
816.9
902.7
Plate with PCLD
85.68
106.3
140.7
309.3
538.9
624.2
684.1
823.3
909.1
Modal loss factor / % Bare plate
0.26
0.34
0.23
0.20
0.20
0.32
0.21
0.20
0.20
Plate with PCLD
1.36
2.14
1.79
1.98
1.68
2.57
2.52
1.34
2.62
21.9
Reduction / dB Receptance
16.0
17.1
18.7
20.0
18.7
18.1
21.5
16.5
MS velocity
32.0
33.9
37.0
39.7
37.1
37.0
43.6
33.0
44.8
Sound power
15.9
16.8
7.79
21.3
14.7
19.1
23.7
17.9
–
where Re{·} denotes that only the real part is considered, and P (xS , jω) and V ∗ (xS , jω) are the complex amplitudes of acoustic pressure and conjugated velocity fields over the radiating structural surface S. However, classically, for an arbitrary structure with some time- (angle brackets) and space-averaged (over bar) square normal vibrational velocity, vn2 (xS , t), the radiated sound power has also been defined as [192] (9.17) Wrad = σρcS vn2 , where σ is the global (in opposition to the modal one, σmn ) radiation ratio (also sometimes referred to as radiation efficiency or radiation index) and vn2 is known as the average mean-square velocity [59]. In fact, the radiation ratio provides a powerful and useful relationship between the structural vibrations and the corresponding radiated sound power (see also [129, Chap. 3] for further details). Further, Eq. (9.17) is perfectly generic and can be used to describe any radiating structural system, provided the radiation ratios of different systems can be established and known. Then, estimating the subsequent radiated noise is a relatively easy process which is performed directly in terms of the radiating surface vibration levels, determined either theoretically (analytically or numerically) or experimentally. However, as is the case in the present study, the radiated sound power can be directly calculated using Eq. (9.16) and applied to more structural systems with complex-shapes to which we do not know a priori the correspondent radiation ratio. Although we can directly determine the radiated sound power with Eq. (9.16), the radiation efficiency has been widely used and discussed in the structural acoustics literature as a vibroacoustic modeling/analysis approach and a useful design parameter [46, 59, 85]. Furthermore, it embodies also a particular physical inference and reasoning approach about sound radiation phenomena which should not be overlooked here. The radiation ratio, σ , of an arbitrary structure is defined as the sound
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power radiated by the structure into the half-space (i.e. one side of the structure) divided by the sound power radiated by a large piston with the same surface area and vibrating with the same RMS velocity as the structure [192]. Thus, it describes the efficiency with which the structure radiates sound as compared with a piston of the same surface area with a unity radiation ratio, and can be either greater or less than unity. An important aspect to be mentioned is that the typical vibroacoustic behavior of radiating structures, examined for example through the analysis of the modal radiation ratios, where the radiation ratios associated with the different structural modes are analyzed separately [142, 192], evidences that there are critical frequency values for each mode (dissociated with the modal resonant frequencies) below which the radiated modes are inefficient. Additionally, the net sound radiation efficiency accounts for the individual and cross-coupled influence of all modes, with the coupling effect being more pronounced below the critical frequency value, with a somewhat destructive or constructive “interference” of the modes, a phenomenon know as radiation cancellation effect [59, p. 137]. This latter phenomena when analyzed from a modal radiation efficiency point of view allows a more straightforward reasoning of the modal interactions which occur and their relationship with the noise being radiated. Results of the variation of the mean-square velocity and radiated sound power with frequency are presented in the remainder graphics in Fig. 9.9. The results demonstrate that the PCLD treatment is able to reduce both the kinetic energy (related with the mean-square velocity), which is a spatial averaged quantity, and the radiated sound over all modes considered, as expected. However, the mean-square velocity does not give any information about the radiation efficiency of the modes. Conversely, it can be seen in the radiated sound power plot that the mode that most contributes to the sound radiation is the (0, 0) mode, being nearly 17 dB (50 times) higher than the (1, 0) mode, both in the treated and untreated configurations. It can also be noticed that the second most important mode in terms of sound radiation contribution is the (0, 1) mode. In fat, is not a strange result since the radiation cancellation effect is expected to be more significant for modes with higher interference, i.e. with several nodal diameters, as for example the (2, 0), (3, 0) and (4, 0) modes. A significant sound radiation attenuation is achieved for all modes, the attenuation increasing as the frequency increases, as expected, somewhat due to the higher damping capability of the viscoelastic materials at high frequencies and the fact that more complex states of modal deformations are induced for the higher order modes. Off-resonance, where more than one mode is significant and where the radiated power cannot be simply calculated using individual radiation ratios, the attenuation is not relevant, making the damping treatment only efficient on the resonances. This also may turn the treatment inefficient when considering a tonal disturbance due to the rotating speed (and harmonics) of the rotating saw; those effects are not considered is this study.
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9.8 Prospects for the Future Viscoelastic damping technologies for passive noise and vibration control have by the end of the last century achieved a relative stage of maturity. By then, their main features and engineering use were clear and satisfactorily well described and documented in the literature. Viscoelastic materials were well understood and the constrained configurations established as an effective means for adding damping to the systems. There was an indubitable proof-of-concept from a physical and phenomenological viewpoints attested by the numerous patents and use in different application areas. In terms of research trends, by the early 1990s the already stationary capabilities and features of passive viscoelastic-based damping technologies motivated researchers to find innovative ways of increasing their efficiency and adaptability. In view of this, extended solutions somewhat combining the interesting features of smart materials and technologies (e.g. piezoelectric and electromagnetic materials, shape memory fibers, shunted electrical networks, smart constraining layers, active damping composites) have been proposed by then [19]. Despite their increased performance for a broader frequency range, increased adaptability and fail-safe capabilities, these technologies are more complex and expensive, the trade-off apparently still not being good enough nowadays to attract a generalized application interest. Therefore, it is envisioned in the future that the inherent complexity (also in design) and cost of these technologies will be more competitive and will make them more appellative and feasible to use. There is still a lot to be done, though, before getting to that point. The previous adaptive, smart, multi-functional perspective is without any doubt appellative and worth to be followed. However, it appears to stem from there that the trend for innovation is not being followed up by experienced practitioners and engineers in passive methods but by newcomers which tend to disregard the previous knowledge on the design of passive viscoelastic damping technologies. However, anyone involved in the design of viscoelastic damping technologies does not take long to realize what the main limitations of the state of the art and use are and that there is still plenty or room and need for improvements. Certainly, these are a good starting point to perform a critical analysis in the past and present and envision what the future developments in the field will be. Accurate predictions of damping in general single- or multi-layered damping treatments are the essential requirement for the development of successful damping designs and solutions. For that, both accurate mathematical models and accurate material properties are required, the accuracy and modeling limitations of both having advanced immensely in the last half-century. Apparently, prior to the 1950s, attention was first given to the mechanical testing for observation of viscoelastic properties of rubber-like materials [128]. In the early 1950s, Oberst [130] and Liénard [100] have allegedly worked in the qualification and documentation of various free-layer damping coatings for automotive applications and in the developed of test methods which constitute the foundation of some methods used nowadays, no attempt to design better coatings having been done.
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Despite the good knowledge on methods of specifying the properties of viscoelastic materials [5], at that time there was reduced knowledge on models for single- and multi-layered damped structures and therefore no opportunity to include the properties for the viscoelastic coatings and perform better designs. Thus, following this first approach, investigation on the development and use of models was performed in the following decades up to the 1970s. These models were used to better understand the physics of the damping treatments but many authors have published results from their models without considering realistic measured viscoelastic material properties; instead, vibration testing was usually carried out to validate the models based on measurements on the “final” damped/coated structural system. Regarding the modeling approaches followed during that period, these are basically twofold, according to whether the damping material is mainly subjected to extensional or shear deformation. On the one hand, we have the analytical approach followed by Oberst for beams with free-, or unconstrained-, layer damping treatments. The modeling is simple, since it considers only two layers, and was derived in order to predict the modal loss factor for a given loss factor of the damping layer and to predict the direct flexural rigidity as a function of the flexural rigidity of the base beam, in order to predict the resonant frequencies [130]. However, these classical equations for free-layer treatment behavior are approximate, the main limitations being the requirements of uniform thickness, use of isotropic materials, the treatment must be uniformly applied to the full surface of the beam and the fact that shear strains are disregarded. On the other hand, analytical models of more complex treatment configurations considering significant shear deformations have been proposed, the first and still well known approach being that of Ross, Kerwin and Ungar (RKU), for beams and plates [148]. Similarly to Oberst approach, these models provide expressions for the loss factors and the effective flexural stiffness of a number of three-layered and multi-layered configurations. In fact, this equations can be seen as a more general ones from which the same equations obtained by Oberst can be seen as a particular case, some of its limitations being also applicable to the RKU equations. Additionally, they cannot be applied where the thickness of the constraining layer approaches the thickness of the host structure (plate) to be damped, they are more accurate for low frequencies and for small values of damping, the kinematics becomes also less representative if thin layers/structures are not considered. In spite of these limitations, the RKU theory is still sufficiently accurate to cover the design and scaling of a considerable range of practical damping configurations, at least as an initial guess for the design of more complex cases. Indeed, the RKU equations, design guidelines and conclusions are still applicable today to unbounded plates or beams or to finite plates/beams on simple supports, but they do not apply precisely to those with any other support conditions. The original work of RKU has triggered numerous further investigations with capabilities and accuracy defined by its inherent limitations. Due to lack of generality, since the complex sixth- higher-order differential equation governing the damped beam or plate motion was not derived, further developments were proceeded towards the establishment of an equation for a sandwich beam with any type of boundary condition [48, 50, 115]. These developments led to the conclusion
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that knowledge of damped normal modes can be used in an uncoupled modal analysis to compute the forced harmonic response of any hysteretically damped structure [114]. With the 1980s came the modal strain energy method (MSE) [82] which also allowed the damping analysis and design came within the reach of the general FE community. At that stage and benefiting from the accuracy and generality of the FE method, a large need for high-quality data on viscoelastic material properties suddenly appeared. However, the accuracy of the materials depended upon both the accuracy of the modal test and underlying models, Mead and Markus 6th order model being the most accurate at that time. It was soon recognized that the MSE method could give quite accurate results over a limited range of damping layer shear stiffnesses and for low damping but beyond this it can be quite erroneous. In view of that, by the late 1980s alternative ways of considering viscoelastic material properties into FE models through the inclusion of internal variables, such as the GHM and ADF models, took place and are becoming today more and more appellative due to the increased computational capability at our disposal in modern computers [190, 191]. By the early 1990’s, along with the recent developments in the FE modeling of laminates and the inclusion of viscoelastic materials properties into FE models through internal variables and fractional calculus models, the analysis in the time domain started to be more computationally attractive. In addition, the complex shear modulus had been measured accurately and cataloged for many viscoelastic materials available commercially and the viscoelastic identification methods were being perfected and extended to obtain more complete viscoelastic constitutive data for applications likely to require more material characterization than just the shear modulus. From that perspective, in the 2000s material properties were already possible to be conveniently considered into a FE framework, which by obvious reasons started to be the standard and preferred modeling and design approach of viscoelastic damping technologies. Notwithstanding the excellent representativeness of the available state-of-the-art FE technology in conjunction with the ever increasing computational power available in this era, quite frequently the analysis often require solving problems with many DoFs which for step-by-step frequency or time domain analysis can easily become unsupportable by the current hardware, this effect becoming more important as we direct the attention for vibroacoustics analysis usually requiring analysis in higher frequency ranges. In other words, we may conclude that the physics of these systems is well captured by a modeling strategy based in FE framework but that quite often some problems remain unsolved due an excessive computational effort. In view of this, some research efforts have been directed towards the model reduction [177] and better complex eigensolvers and solution techniques [1, 43]. These still remain as hot research topics which in conjunction with the development of modeling techniques for the mid-frequency range, where more of our knowledge of waves and wave behavior is somewhat incorporated into element-based modeling approaches (idealistically strongly reassembling and compatible with “conventional” FE frameworks), constitute important modeling trends
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which, if successful, will significantly increase the size and complexity of problems being designed nowadays. To sum up, research continues on passive and, more recently, active viscoelasticbased damping techniques. The current state-of-the-art analytical modeling approaches does not fit all applications but allows greatly simplify mathematical models so that candidate designs can be evaluated more quickly. However, over restrictive simplifying modeling assumptions can lead to erroneous results for some applications. On the other hand, by using state-of-the-art FE technology, more rigorous and accurate analysis may be performed allowing also to address some aspects remaining still somewhat unexplored. Some further examples are the apparent damping improvement by the artificial introduction of artificial cuts into the damping treatments [97], the assumption of linear and nonlinear full 3D viscoelastic material properties and its design implications, the extension to the robust design of mechanical structures where the uncertainties concerning both the variables related to the main structure and those pertaining the viscoelastic treatment are considered [70], to further study the effect of temperature and pre-stresses in damping treatments [87, 157], treatments having either a functionally graded viscoelastic polymer layer (for which the properties vary through the thickness of the layer) [67] or a multi-layer configuration [118] and their interest to increase the temperature range of effectiveness of the damping treatment.
9.9 Summary This chapter addresses FE-based mathematical strategies to model the damped constitutive behavior of viscoelastic materials used as surface mounted, constrained or embedded damping treatments in structures, in order to reduce vibrations and/or noise radiation. A particular emphasis is put in the modeling, analysis and design of viscoelastic damping technologies for circular saw blades. An overview of the state of the art on viscoelastic damping technologies, in general, and on the numerous possibilities that have been proposed to solve noise and vibration problems in circular saw blades, in particular, is performed. In order to better understand the implications of different configurations of viscoelastic damping treatments, their main features are described and discussed. These issues are also discussed from the point of view of the FE modeling approach to be followed in terms of deformation theories and spatial modeling and meshing possibilities. Both time and frequency domain based techniques are considered to model the constitutive behavior and the implementation of these approaches into FE solution procedures is discussed. The chapter also presents and discusses the mains features available nowadays in commercial FE codes which are relevant for the modeling of viscoelastic damping technologies into general structural systems. One important aspect to highlight is the different damping modeling and solution approaches available today. Time domain techniques regard the use of the GHM and ADF internal variables models which, unfortunately, are not yet readily available in
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the commercial software. One major disadvantage in using internal variables models, such as the GHM or ADF, is the creation of additional dissipation variables increasing the size of the coupled damped FE model. Frequency domain techniques comprise the use of CMA-based methods, namely the DFR, IMSE and ICE, where the FE spatial model is used by re-calculating the complex viscoelastic stiffness matrix for each discrete frequency value, in the case of the DFR, or during the iterative eigensolution process, in the case of the IMSE and ICE. These approaches are more straightforward to use and implement at the global FE level. That is the reason why the CMA-based method, DFR, is the most common approach implemented in commercial FE codes incorporating viscoelastic damping modeling capabilities. The latter DFR approach may however be speed up through an improved fast frequency response synthesis as the one provided by ACTRAN/VA using the Krylov solver. The study on the application of viscoelastic damping technologies to circular saw blades is considered here in the framework of the vibroacoustic commercial FE software ACTRAN/VA. It is important to highlight also that vibroacoustic systems are multiphysics problems, which means that different physical domains are involved and interrelated in the analysis. In this case, there are different mediums which may be needed to be discretized, namely the structural (circular saw blade) and acoustic fluid (air) media. The circular saw is considered to be in free field, so that the far field wave attenuation effect needs to be considered in the analysis. This latter issue may be tackled by modeling the far field acoustic fluid medium with infinite element technology while the reminder proximity fluid field and structural system are modeled through solid and solid-shell 3D elements available in ACTRAN/VA. Alternatively, some modeling simplifications, considering the structure set in a rigid baffle and the analysis of the sound radiation into the half-space (one side of the saw), are considered allowing also the simplification of the analysis and the use of a Rayleigh integral surface approach which avoids the need to discretize the acoustic domain and the inclusion of additional pressure related DoFs. It is shown that the use of a PCLD treatment in the saw attenuates the radiated sound power and allows reducing the “kerf” losses due to the blade thickness and vibration. Constrained layers are more effective than pure viscoelastic layers due the increase of the shear deformation in the viscoelastic material. Thus, the effect of a commercial constrained layer treatment applied to the circular saw is considered and assessed. The comparison of the vibroacoustic behavior of a standard circular saw blade with and without viscoelastic damping treatments in terms of the reduction of the radiated sound power and vibratory response is shown to yield in general good attenuation of all modes, being more significant in the higher ones and with more complex states of deformation patterns. It is also shown that the constructive and destructive “interference” of neighboring “cells” of each individual mode dictates de individual mode radiation efficiency. Therefore, at the design stage of the damping treatment, the modal interference of the vibration modes should be considered to attenuate the resultant radiation modes. With this preliminary simple analysis the previously discussed topics on modeling and simulation approaches of structures with viscoelastic damping technologies
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are recalled in the context of a specific problem, a solution and design being proposed. However, several simplifications are considered in the analysis. The damped and undamped performance are qualitatively compared to gain preliminary insight into the modeling, analysis and design of a solution to the problem. For an effective solution design, it might be required to proceed to a more detailed research of the use of viscoelastic damping technologies applied to general circular tooling. Effects due to the gyroscopic and centrifugal nature of the rotating system due to the rotation of the saw may originate significant modifications of the resonant frequencies and damped vibroacoustic behavior. In order to account for these effects, the centrifugal load may be represented by distributed forces in the volume, which generates stiffening in the structure and consequently causes the increase of the resonant frequencies. In addition, thermal effects need to be considered, more realistic geometries (including the teeth) and vibroacoustic boundary conditions need also to be somewhat incorporated. These issues constitute topics of relevant nature for a more detailed and accurate analysis which may be pursued by any designer in the field. However, the key steps, ideas, and design philosophy are already somewhat covered in this text, which are certainly useful for anyone initiating or already experienced and working in the field. Acknowledgements The joint funding scheme provided by the European Social Fund and from Portuguese funds from MCTES through POPH/QREN/Tipologia 4.2 are gratefully acknowledged.
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139. Plouin, A.S., Balmès, E.: Steel/viscoelastic/steel sandwich shells computational methods and experimental validations. In: 18th International Modal Analysis Conference (IMAC XVIII), vol. 1, pp. 384–390. Society for Experimental Mechanics, San Antonio (2000) 140. Pohl, M., Rose, M.: Vibration and noise reduction of a circular saw blade with applied piezoceramic patches and semi-active shunt networks. In: Adaptronic Congress, Berlin, June 19– 20, 2009 141. Pritz, T.: Measurement methods of complex Poisson’s ratio of viscoelastic materials. Appl. Acoust. 60(3), 279–292 (2000) 142. Putra, A., Thompson, D.J.: Sound radiation from rectangular baffled and unbaffled plates. Appl. Acoust. 71(12), 1113–1125 (2010) 143. Ray, M.C., Baz, A.: Optimization of energy dissipation of active constrained layer damping treatments of plates. J. Sound Vib. 208(3), 391–406 (1997) 144. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004) 145. Riande, E., Calleja, R.D., Prolongo, M.G., Masegosa, R.M., Salom, C.: Polymer Viscoelasticity: Stress and Strain in Practice. Marcel Dekker, New York (2000) 146. Ro, J., El-Ali, A., Baz, A.: Control of sound radiation from a fluid-loaded plate using active constrained layer damping. In: Ferguson, N.S., Wolfe, H.F., Mei, C. (eds.) Proceedings of the Sixth International Conference on Recent Advances in Structural Dynamics, Southampton, pp. 1252–1273 (1997) 147. Robert, G.: Commercial software. In: Ewins, D.J., Rao, S.S., Braun, S.G. (eds.) Encyclopedia of Vibration, pp. 243–256. Academic Press, Oxford (2001) 148. Ross, D., Ungar, E.E., Kerwin Jr., E.M.: Damping of plate flexural vibrations by means of viscoelastic laminae. In: Structural Damping, pp. 49–88. ASME Publication, New York (1959) 149. Sandberg, G., Wernberg, P., Davidsson, P.: Fundamentals of fluid-structure interaction. In: Sandberg, G., Ohayon, R. (eds.) Computational Aspects of Structural Acoustics and Vibration, pp. 23–101. Springer, Udine (2009) 150. Saravanos, D.A.: Integrated damping mechanics for thick composite laminates and plates. J. Appl. Mech. 61(2), 375–385 (1994) 151. Schajer, G.S.: Understanding saw tensioning. Holz Roh Werkst. 42(11), 425–430 (1984) 152. Schajer, G.S.: Why are guided circular saws more stable than unguided saws. Holz Roh Werkst. 44(12), 465–469 (1986) 153. Schajer, G.S., Mote, C.D.: Analysis of optimal roll tensioning for circular-saw stability. Wood Fiber Sci. 16(3), 323–338 (1984) 154. Schmidt, A., Gaul, L.: Finite element formulation of viscoelastic constitutive equations using fractional time derivatives. Nonlinear Dyn. 29(1–4), 37–55 (2002) 155. Seubert, S.L., Anderson, T.J., Smelser, R.E.: Passive damping of spinning disks. J. Vib. Control 6(5), 715–725 (2000) 156. Shen, I.Y.: Bending-vibration control of composite and isotropic plates through intelligent constrained-layer treatments. Smart Mater. Struct. 3(1), 59–70 (1994) 157. Silva, L.A.: Internal variable and temperature modeling behavior of viscoelastic structures— a control analysis. Ph.D. thesis, Virginia Tech, VA, USA (2003) 158. Singh, R.: Case-history: the effect of radial slots on the noise of idling circular saws. Noise Control Eng. J. 31(3), 167–172 (1988) 159. Sinha, S.K.: Determination of natural frequencies of a thick spinning annular disk using a numerical Rayleigh-Ritz’s trial function. J. Acoust. Soc. Am. 81(2), 357–369 (1987) 160. Slanik, M.L., Nemes, J.A., Potvin, M.J., Piedboeuf, J.C.: Time domain finite element simulations of damped multilayered beams using a Prony series representation. Mech. TimeDepend. Mater. 4(3), 211–230 (2000) 161. Slater, J.C., Belvin, W.K., Inman, D.J.: Survey of modern methods for modeling frequency dependent damping in finite element models. In: 11th International Modal Analysis Conference (IMAC XI), vol. 1923, pp. 1508–1512. Society for Experimental Mechanics, Kissimmee (1993)
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C.M.A. Vasques and L.C. Cardoso tural Dynamics, and Materials Conference, vol. 11, pp. 7529–7579. American Institute of Aeronautics and Astronautics, Newport (2006) Vasques, C.M.A., Mace, B.R., Gardonio, P., Dias Rodrigues, J.: Analytical formulation and finite element modelling of beams with arbitrary active constrained layer damping treatments. Technical Memorandum No. 934, Institute of Sound and Vibration Research, Southampton, UK (2004) Vasques, C.M.A., Mace, B.R., Gardonio, P., Rodrigues, J.D.: Arbitrary active constrained layer damping treatments on beams: finite element modelling and experimental validation. Comput. Struct. 84(22–23), 1384–1401 (2006) Vasques, C.M.A., Moreira, R.A.S., Dias Rodrigues, J.: Viscoelastic damping technologies— part I: modeling and finite element implementation. J. Adv. Res. Mech. Eng. 1(2), 76–95 (2010) Vasques, C.M.A., Moreira, R.A.S., Dias Rodrigues, J.: Viscoelastic damping technologies— part II: experimental identification procedure and validation. J. Adv. Res. Mech. Eng. 1(2), 96–110 (2010) Wallace, C.E.: Radiation-resistance of a rectangular panel. J. Acoust. Soc. Am. 51(3), 946– 952 (1972) Wang, H.J., Chen, L.W.: Axisymmetric vibration and damping analysis of rotating annular plates with constrained damping layer treatments. J. Sound Vib. 271(1–2), 25–45 (2004) Wang, G., Wereley, N.M.: Frequency response of beams with passively constrained damping layers and piezo-actuators. In: Davis, L.P. (ed.) Smart Structures and Materials 1998: Passive Damping and Isolation, Bellingham, WA, US. SPIE, vol. 3327, pp. 44–60 (1998) Wang, G., Wereley, N.M.: Spectral finite element analysis of sandwich beams with passive constrained layer damping. J. Vib. Acoust. 124(3), 376–386 (2002) Wang, X.G., Xi, F.J., Daming, L., Zhong, Q.: Estimation and control of vibrations of circular saws. In: Proceedings of the IEEE International Conference on Control Applications, vol. 1, pp. 514–520 (1999) Wikner, K.G., Josefsson, F.P.: Laminated saw blade (1976). US Patent 3990338, November 9, 1976 Yao, T., Duan, G.L., Cai, J.: Review of vibration characteristics and noise reduction technique of circular saws. Zhendong yu Chongji/J. Vib. Shock 27(6), 162–166 (2008) Yellin, J.: An analytical and experimental analysis for a one-dimensional passive stand-off layer dampind treatment. Ph.D. thesis, Mechanical Engineering Department, University of Washington, Washington, US (2004) Yellin, J.M., Shen, I.Y., Reinhall, P.G., Huang, P.Y.H.: An analytical and experimental analysis for a one-dimensional passive stand-off layer damping treatment. J. Vib. Acoust. Trans. ASME 122(4), 440–447 (2000) Yellin, J.M., Shen, I.Y., Reinhall, P.G.: Experimental and finite element analysis of standoff layer damping treatments for beams. In: Wang, K.W. (ed.) Proceedings of the SPIE, San Diego, CA, US, vol. 5760, pp. 89–99 (2005) Yiu, Y.C.: Finite element analysis of structures with classical viscoelastic materials. In: 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, La Jolla, CA, US, vol. 4, pp. 2110–2119 (1993) Yu, R.C., Mote, C.D.: Vibration of circular saws containing slots. Holz Roh Werkst. 45(4), 155–160 (1987) Yu, S.C., Huang, S.C.: Vibration suppression of a CLD treated plate subject to a harmonic traveling load. J. Chin. Inst. Chem. Eng. 25(6), 627–638 (2002) Zhang, S.H., Chen, H.L.: A study on the damping characteristics of laminated composites with integral viscoelastic layers. Compos. Struct. 74(1), 63–69 (2006)
Chapter 10
Vibroacoustic Energy Diffusion Optimization in Beams and Plates by Means of Distributed Shunted Piezoelectric Patches M. Collet, M. Ouisse, K.A. Cunefare, M. Ruzzene, B. Beck, L. Airoldi, and F. Casadei Abstract This chapter proposes a synthesis of different new methodologies for developing a distributed, integrated shunted piezo composite for beams and plates applications able to modify the structural vibro acoustical impedance of the passive supporting structure so as to absorb or reflect incidental power flow. This design implements tailored structural responses, through integrated passive and active features, and offers the potential for higher levels of vibration isolation as compared to current designs. Novel active and passive shunting configurations will be investigated to reduce vibrations such as distributed Resistance Inductance and Resistance with negative Capacitance circuits.
M. Collet () · M. Ouisse FEMTO-ST Institute, Applied Mechanics, University of Franche-Comté, 25000 Besançon, France e-mail:
[email protected] M. Ouisse e-mail:
[email protected] K.A. Cunefare · B. Beck G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA K.A. Cunefare e-mail:
[email protected] B. Beck e-mail:
[email protected] M. Ruzzene · L. Airoldi · F. Casadei School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, USA M. Ruzzene e-mail:
[email protected] L. Airoldi e-mail:
[email protected] F. Casadei e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_10, © Springer Science+Business Media B.V. 2011
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10.1 Introduction Research activities in smart materials and structures are very important today and represent a significant potential for technological innovation in mechanics and electronics. The growing interest of our society for sustainable development motivates a broad research effort for optimizing mechanical structures in order to obtain new functional properties such as noise reduction, comfort enhancement, durability, decreased ecological impact, etc. In order to realize such a multi-objective design, new methods are now available which allow active transducers and their driving electronics to be directly integrated into passive structures. The number of potential applications of these approaches is growing in many industrial fields such as civil engineering, aerospace, aeronautics, ground transportation, etc. The main research challenge today deals with the development of new multi-functional structures integrating electro-mechanical systems in order to optimize their intrinsic mechanical behavior to achieve desired goals. In the past few years, a technological revolution has occurred in the fields of integrated Micro Electro Mechanical Systems (MEMS) that offers new opportunities for smart structures design and optimization. We know today that the mechanical integration of active smart materials, electronics, chip sets and power supply systems is possible for the next generation of smart composite structures. By using such an integrated active or hybrid distributed set of electro mechanical transducers, one can attain new desired vibroacoustic functionalities. In this sense, one can speak of integrated distributed smart structures. This chapter proposes a synthesis of different new methodologies for developing a distributed, integrated shunted piezo composite for beams and plates applications able to modify the structural vibro acoustical impedance of the passive host structure so as to absorb or reflect incidental power flow. This design implements tailored structural responses, through integrated passive and active features, and offers the potential for higher levels of vibration isolation as compared to current designs. Novel active and passive shunting configurations, such as distributed ResistanceInductance (RL) and Resistance with negative Capacitance (RCneg) circuits, will be investigated to reduce vibrations. The main idea presented is to work with a periodic distribution of shunted piezo transducers integrated on a passive supporting structure. An optimization process is implemented for the design of the electronic shunt circuits so as to achieve a desired dynamical impedance of the passive/active composite beam or plate structure for wave absorption or reflection. The results indicate great potential of such a technique by increased robustness and efficiency of the strategy. The experimental tests show the real potential of such new approach for structural stabilization or isolation. Finally, energy balance assessments, absolute efficiency and robustness are considered. It constitutes a first step in designing new hybrid integrated composite for vibroacoustic optimization.
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10.2 Overview of the State of the Art Piezoelectric shunt damping (Fig. 10.5) is a promising passive technique for vibration control of flexible structures. Contrary to active control, the only external element to be used is a Passive Electrical Network (PEN) that is directly connected to the electrodes of the piezoelectric device. With this approach, the sensing element is not needed and the use of a passive network guarantees the stability of the coupled system. Furthermore, shunted piezoelectric systems offer a very attractive path for electro-mechanical integration and distribution. Numerous works have been published [1, 4, 5, 20, 22, 35, 38, 51, 57] that present analysis of the capability and efficiency of a single shunted piezoelectrical patch for structural stabilization and wave cancellation. With this approach, the sensing element is not needed and the use of a passive network guarantees the stability of the coupled system. Hagood and VonFlotow [35] provided the first analytical formulation for passive shunt networks. They demonstrated how a piezoelectric patch shunted through a single resistive-inductive (RL) circuit acts as a vibration absorber tuned at the resonance frequency of the circuit. Since then, more complex shunting circuits have been investigated to extend the effectiveness over broader frequency bands. For example, multi-mode techniques have been proposed by Wu [70, 71] who employed a series of blocking, inductive-capacitive (LC) circuits in parallel with an RL shunt circuit designed to attenuate a specific resonance frequency. Other methods of broadband suppression include state switching [18], synthetic impedance [26], and negative impedance circuits [6, 30, 56, 57]. The selection of negative impedance shunt parameters have been chosen by a few different tuning theories. Behrens selected the shunt parameters through investigation of the active control element of the negative capacitance and absorbing performances [6, 53]. Park and Palumbo decomposed the wave field on a beam to find a minimization of the reflected wave component [57]. By noticing the link between the wave based approach of Park and Palumbo and the power input to the system, Cunefare developed a parameter selection technique based on the reactive input power [21]. Negative capacitance shunts can be designed to work in conjunction with piezoelectric patches arrayed in a periodic fashion. Periodic structures have been shown to improve the broadband performance of control systems on structures [10, 62, 63]. On all these works, piezoelectric shunt damping also appears as promising passive technique for vibration control of flexible structures [4, 21]. Contrary to active control, the only external element to be used is a passive electrical network (PEN) that is directly connected to the electrodes of the piezoelectric device. Examples of these devices are well described in the patent of R.L. Forward [29]. Hybrid, active/passive piezoelectric networks are studied in [61]. These techniques allow to use a perfectly collocated system of transducers to control the mechanical structure. Thus, we can profit of the poles/zeros interlacing theorem to guarantee robust stability of the control system [58]. Furthermore, the analogical electrical circuits used for shunting the piezoelectric patches, do not introduce a large additional mass into the system nor a large time delay and also can be implement for a large frequency band. Depending onto the used shunt circuits, the necessary large reactive part of the electrical circuit can be obtained by
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Fig. 10.1 Schematic of a used piezo composite beam: (a) Shunted with a resistive (R) circuit; (b) shunt with R and negative Capacitance (RCneg) circuit
using suitable synthetic inductance or capacitance implying operational amplifiers [27, 28]. Based on these works, this first part of this chapter is focused on determining the design parameters for optimizing the fully electromechanical parameters behavior of shunted piezoelectric transducers for structural stabilization in a modal frequency domain. For designing the piezo-mechanical system one can use the modal effective electromechanical coupling coefficient, proposed in [13]. However, no closed form tuning solution has been found for these techniques and numerical optimization is the only way to simultaneously determine the values for the electrical components when several modes are taken into configuration. Thus, for the control of a large number of modes, numerical optimization may result in a complicated procedure, and the implementation of the circuit may become a difficult task due to the high order of the system. For designing the piezo-mechanical system one can use the modal effective electromechanical coupling coefficient, proposed in [13].
10.3 Classical Tools for Designing RL and RCneg Shunt Circuits 10.3.1 Piezoelectric Modeling and Shunt Circuit Design Let us consider a very simple system constituted of a single piezoelectric patch glued onto a supporting beam as described in Fig. 10.1. The piezoelectric material is assumed to be polarized along positive (Oz) direction. The bottom electrode (interface between piezoelectric domain and the support) is grounded.
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10.3.1.1 Modeling Aspects Different modeling approaches can be used for describing such systems. The survey paper of Benjeddou [7], and the papers by Noor [54] and Mackerle [45] analyze a wide variety of different works in this area. A numerous of piezoelectric finite elements methods for sandwich beams [16, 47], plates [3, 25, 33, 40–43, 60, 64, 65, 69], layered composite shells [3, 8, 40, 72] or volume elements, as those used in many finite element codes [44, 64, 67] can be found in literature. As noted in [7, 15], the most theoretically advanced finite elements has not been widely used for practical modeling of adaptive structural elements for “intelligent” or smart materials and structure applications. Because of the difficulty of building accurate and reduced models for multiphysics analysis, many authors use simplified modeling approaches in order to limit numerical complexity while focusing on physical design, computation and optimization [14, 19, 35, 52, 58, 59]. Modeling of standard piezoelectric problem is based on writing two sets of partial differential equations and their associated boundary conditions for introducing the elastodynamic equilibrium and in the other one the electrostatic one. A synthetic form of the 3D electromechanical dynamical equilibriums can be written as ¨ ρ w(x, t) − ∇σ (x, t) = f ∀x ∈ ∪ p , t ∈ R+ , (10.1) −∇D(x, t) = 0 ∀x ∈ p , t ∈ R+ , where is the structural material domain, p the piezoelectric material domain, x the coordinate vector and t the time. The linear piezoelectric constitutive equations are σ (x, t) = cE (x)ε(x, t) − eT (x)E(x, t),
(10.2)
D(x, t) = e(x)ε(x, t) + εS (x)E(x, t),
(10.3)
where σ (x, t) represents the Cauchy stress tensor, ε(x, t) = ∇sym w(x, t) the Green strain tensor, E(x, t) = −∇V (x, t) the electric field vector (V the voltage), cE (x) the elasticity tensor at constant electrical field, e(x) the piezoelectric coupling tensor and ε S (x) the dielectric permittivity at constant strain. Combining Eqs. (10.2), (10.3) and (10.1) leads to the standard general form of the set of partial differential equations describing the dynamic equilibrium of such coupled problem. To obtain a simple weak formulation limiting the numerical cost to treat the coupled problem, the classically used assumption, applied in most of the finite elements codes, is to consider through-thickness linear variation of the electrical potential [16, 34, 35, 58, 66]. This hypothesis leads to neglect the induced potential, while the electromechanical coupling will be only partially captured as described in [7, 13, 15]. This approach can yield up to 30% error in evaluating the equivalent piezoelectric capacity for small patches. In fact, it is known that the asymptotic electric potential for a short-circuited thin plate is quadratic in the thickness [8]. In this paper, we use two different modeling approaches: one using this standard assumption, which is a 1D composite beam modeling as described in [16, 34, 66], and a second one using a full 3D approach based on the methodology proposed in [15].
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10.3.1.2 Electric Shunt Design Parameter Whatever the modeling approach is, we always obtain a discretized system of the form ¨ + Cw(t) ˙ Mw(t) + Kw(t) + ewv V(t) = F(t),
(10.4)
−eTwv w(t) + CSp V(t) = Q(t),
(10.5)
where M, K and C stand respectively for mass, damping and open circuit stiffness matrices, ewv is the piezoelectric coupling matrix depending on piezoelectric material coefficients e but also on geometric and support characteristics, CSp is the diagonal matrix of each equivalent piezoelectric capacitances for zero strain. Furthermore, w represents the vector of mechanical degrees of freedom and V the vector of the applied upper electrode voltage, whereas Q is the dual measured current. The main difference between the full (or well condensed) 3D and the simplified beam or plate approaches is located in evaluation of CSp and ewv matrices. When optimizing shunt circuit for stabilizing that dynamical system, you have to design the suitable electrical relationship (i.e. the circuit) by generically finding optimal parameters of ordinary differential equations linking applied voltages V and the associated produced current Q, so that N i=0
d i V(t) d j Q(t) = bj . dt i dt j P
ai
(10.6)
j =0
Let us consider only three simple electronic circuits: • A Resistive circuit, for which dQ(t) , (10.7) dt that is represented in Fig. 10.1(a), where R represents the diagonal matrix of resistances individually connected on each used piezoelectric patch; • A serial Resistance/Capacitance circuit, for which dQ(t) −1 V(t) = − Cneg Q(t) + R , (10.8) dt V(t) = −R
represented in Fig. 10.1(b), where Cneg is the diagonal matrix of capacities individually connected in serial with the shunt resistance on each patch; • A serial Resistance/Inductance circuit, for which 2 dQ(t) d Q(t) +R V(t) = − L , (10.9) dt dt 2 where L represents the diagonal matrix of the inductance part of each shunt circuit. Resistive Shunt Circuit Let us consider a simple resistive circuit. Equations (10.4) and (10.5) and the feedback term can also be rewritten as
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Fig. 10.2 Root locus induced by a resistive shunt circuit
−1 −1 ¨ + Cw(t) ˙ Mw(t) + K + eTwv CSp ewv w(t) = −ewv CSp Q(t) + F(t), (10.10) S −1 −1 Cp Q(t) + CSp eTwv w(t) = V(t), (10.11) (10.12) −R−1 V(t) dt = Q(t). In that case, we obtain a typical system of ordinary differential equation corresponding to the implementation of an Integral Force Feedback strategy. This active damping control is well described in [52, 58]. Thus all design parameters and optimization criteria are well known since the nineties. When only one piezoelectric patch is considered, the matrix R−1 is reduced to only one value 1/R. If one can modify this shunt resistance to take all values in R+ , each complex eigenvalue of the obtained dynamical system follows a well known root locus depicted on Fig. 10.2. For lightly coupled system (see [52] for definition), the maximum induced damping ratio for a given mode i is given by ξimax =
ωoci − ωsci , 2ωsci
(10.13)
where ωoci corresponds to open circuit eigenfrequencies (i.e. the solutions of 2 M + (K + eT (CS )−1 e ))φ oc = 0) and ω (−ωoc wv sci to the zeros of system repwv p i i resented in Eqs. (10.11) and (10.12), that is to say the short circuit eigenfrequencies 2 M + K)φ sc = 0). (i.e. the solutions of (−ωsc i i In [13, 15], the effective modal piezoelectric coupling coefficient keffi is defined as 2 keff = i
2 − ω2 ωoc sci i 2 ωsc i
.
(10.14)
If one considers a low piezoelectric coupling effect meaning that the piezoelectric patch lightly modifies the corresponding structural eigenfrequency (ωi ), one 2 ≈ can demonstrate that (ωoci + ωsci )/2 ≈ ωi . So, under this assumption, keff i 2 2 2 (ωoci − ωsci )/ωi is a first order approximation of the modal effective electrome2 /4. Finally, a suitable criterion for chanical coupling coefficient. Thus, ξmax ≈ keff i
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Fig. 10.3 The effective shunted piezoelectric composite stiffness [35] as a function of the connected negative capacitance term
optimizing resistive shunt circuit for damping a specific mode i can be maximizing the positive function i (Lp , xp , e) defined as 2 2 i (Lp , xp , e) = ωoc − ωsc . i i
(10.15)
The form is directly linked to eigenvalue computation and avoid any convexity prob2 = (ω 2 − ω2 )/ω2 lems that can occur when one tries to directly optimize keff oci sci sci i or ξimax . Negative Capacitance Circuit If we now consider a serial RCneg shunt circuit, based on the system equilibrium Eqs. (10.4) and (10.5) and the electronic introduced feedback, the controlled equations are −1 ¨ + Cw(t) ˙ Mw(t) + K + eTwv CSp + Cneg ewv w(t) −1 = −ewv Cneg Cneg + CSp V(t) + F(t), (10.16) −1 −1 −eTwv Cneg Cneg + CSp w(t) + CSp Cneg Cneg + CSp V(t) = Q(t), (10.17) ˙ V(t) = −RQ(t). (10.18) By comparing with Eqs. (10.4), (10.11) and (10.12), we can show that this last system corresponds to a piezomechanical system in which we would have modified the initial short circuit stiffness K to K+eTwv (CSp +Cneg )−1 ewv . Thus if eTwv (CSp +Cneg ) is a negative matrix, we can easily demonstrate that we could decrease the corresponding short circuit eigenfrequencies and also increase the corresponding resistive feedback efficiency as previously demonstrated. To decrease the short circuit stiffness, we also need to use negative capacitance circuits connected on each piezoelectric patch built by using suitable synthetic circuit. This result corresponds to those
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Fig. 10.4 Different R induced mono-modal root loci with different negative capacitance part in the connected shunt Fig. 10.5 Plate with RL piezo-shunted device
obtained by Hagood Von Flotow with a very simple model in [35]. In this work, the authors introduce an effective stiffness for the shunted piezoelectric laminated composite that can be plotted as in Fig. 10.3. When only one patch is considered, the induced root loci by the resistive part of the shunt with various initial negative capacitance are plotted in Fig. 10.4. One can also materialize a real improvement for stabilizing the considered mode when negative capacitance tends toward −CpT (the piezoelectric capacitance at zero stress) from lower values. For Cneg between −CpT and −CpS the system is unstable as the effective stiffness becomes negative (see Fig. 10.3). For values superior to −CpS , the negative capacitance part does not introduce any improvement on the resistive induced modal damping. Resistance Inductance Circuit Hagood and VonFlotow [35] provided the first analytical formulation for passive shunt networks, focusing on resistive-inductive (RL) circuits. They observed analogies between the RL circuit and mechanical vibration absorbers and provided the equivalent tuning relations for the shunting PEN. If the vibration associated with the i-th mode is to be reduced, then the inductance of the RLC circuit is chosen as
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Fig. 10.6 (a) Piezo-shunted RLC effect on resonance; (b) example of multimodal electric circuit
L=
1 ωi2 Cp
,
(10.19)
where Cp is the inherent piezoelectric capacitance. The value of the shunting resistor R defines the damping ratio added to the coupled system and in general is chosen with a trial-and-error approach. Hagood and VonFlotow demonstrated, both analytically and experimentally, that RL networks are effective means of vibration absorption even if their effectiveness is limited to a narrow range of frequencies as shown in Fig. 10.6(a). The extension of the single-mode shunt damping technique
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Fig. 10.7 (a) Example of periodic structure; (b) associated band gaps diagram on which the blue domains correspond to evanescent waves propagation
to allow for multiple mode vibration suppression has been the subject of intense research in recent years. The goal is to obtain vibration reduction of multiple frequencies using a single piezoelectric transducer. A classical multi-mode technique is the one proposed by Wu [70] that employs a single RL network for each frequency to be damped, and adopts a series of current blocking LC circuits in order to decouple the resonances of each branch. The schematic of a three-mode shunt network is illustrated in Fig. 10.6(b). The main difficulty of this methods is that the order and complexity of the circuits increase rapidly with the number of modes to be damped and in general numerical optimization is the only way to simultaneously determine the values for the electrical components.
10.4 Controlling the Dispersion in Beams and Plates This research considers a different approach, where an array of independentlyshunted piezoelectric patches is distributed over the structure. This results in a periodic configuration (Fig. 10.7(a)), where equivalent sources of mechanical impedance mismatch are introduced through proper tuning of the shunting circuits. The resulting periodic assembly features frequency bands of wave attenuation, known as stop bands or band gaps, that can be tuned to suppress vibrations over a broad frequency spectrum. For example, Fig. 10.7(a) illustrates a 1D periodic rod where the alternation of mechanical impedance induces frequency band gaps. Figure 10.7(b) shows the dependence of the attenuation regions (i.e. the evanescent part of the corresponding as a function of the mechanical impedance √ √ wave numbers) ratio ζ defined as ζ = E1 ρ1 / E2 ρ2 and the normalized frequency = ωL1 /c1 where c1 is the corresponding wave phase velocity. In this work, the capabilities of controlling the impedance of a shunted piezo patch is used to tailor the dispersion properties of the assembly without altering other engineering functionalities. The propagation of waves in smart periodic rods and axisymmetric shells was first studied by Thorp et al. [62, 63] and in this work is extended to 2D periodic waveguides. Experimental results are also provided to demonstrate the effectiveness and
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robustness of the concept. According to Bloch’s theorem, the dispersion properties of this class of engineering structures are completely determined by the elementary unit cell that defines the periodic assembly [9]. This allows to implement an effective design strategy, discussed later, that involve only a small portion of the overall system, thus significantly reducing the computational effort.
10.4.1 Waves Dispersion Control by Using RL and Negative Capacitance Shunts on Periodically Distributed Piezoelectric Patches Resistive-inductive shunts are favorably used in this approach since they allow to simply select the frequency range of wave attenuation and to easily control the dispersion property of the system. Similarly to the single-shunt case, the value of shunted inductance defines the center of the electrically-induced attenuation regions, while the resistance controls the amplitude and width of the stop band. Periodic configuration of multimodal shunt circuits can also be adopted to achieve broadband attenuation over independent frequency bands as shown in Fig. 10.8(a). Significant performances can be achieved when a negative capacitance shunt network is adopted, as shown in Fig. 10.8(b). As a drawback, the practical implementation of this impedance requires active electronic components such as operational amplifiers. Casadei et al. validated with experimental tests the effectiveness of this distributed control strategy based on periodic shunted piezos [12]. As illustrated in Fig. 10.9, the experimental setup consists of an aluminum rectangular plate that hosts a periodic array of 4 × 4 piezo patches independently shunted with a resistiveinductive (RL) circuit. Figure 10.10 shows a significant attenuation of the frequency response when the plate is excited with a broadband noise by an electrodynamic shaker. An attenuation band of 500 Hz centered at the tuning frequency of the circuits is achieved along with 10 dB of maximum vibration reduction. Figure 10.11 shows how the spatial velocity distribution of the plate is altered when the piezo shunts are activated. The present control technique may also be improved by using on the same host structure an hybrid configuration of both passive RL and active negative impedance converter (NIC) shunts. Casadei et al. experimentally investigated the advantages of the hybrid approach [10]. Figure 10.12 highlights both broad vibration attenuation of the firsts structural modes due to the effect of the negative capacitance as well as attenuation in the tuned mode induced by the RL shunts.
10.4.2 Periodically Distributed Shunted Piezoelectric Patches for Controlling Structure Borne Noise Due to this appealing characteristics, periodic piezo shunts may be effectively used for the suppression of structure borne noise. Specifically, rotorcraft cabin noise is
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Fig. 10.8 Comparison of periodically distributed shunted piezoelectric patches with RL and RC (negative) circuits for controlling a plate. The upper part shows the band gaps estimation δ and the lower part the corresponding collocated FRFs: (a) Two bimodal RL circuits ruled according two sets of different blocking circuits; (b) RC (negative) circuits with two different resistors
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Fig. 10.9 Periodic piezoelectric plate and associated RL shunt circuits
Fig. 10.10 Spatial average of the plate velocity frequency response function with RL circuits tuned at ftun = 1720 Hz with two different resistors
characterized by random broadband noise and strong tonal component in the 500– 2000 Hz range. The application of periodic shunts as means for structure borne noise reduction was numerically and experimentally investigated by Casadei et al. [11]. A multi-physics FE code was developed to study the behavior of fully coupled piezo-structural-acoustic systems. Figure 10.13 clearly shows how strong attenuation of the plate vibrations directly leads to corresponding reductions in radiated noise into an acoustic enclosure. Casadei et al. [11] demonstrated that when air is considered the propagation characteristics of the in vacuo unit cell constitute a reasonable approximation of the dispersion properties of the fluid-loaded structure, so that a unit cell approach can still be applied to the problem. Contrary to the case of vibration control, the presence of a fluid medium introduces in the coupled system the presence of cavity-controlled natural modes that are almost insensitive to the presence of an external control on the mechanical domain. A preliminary set of experimental results on a plate-cavity system are also performed. In the experimental setup, shown in Fig. 10.14(a), an aluminum plate is clamped to a rigid frame fastened to a wooden supporting structure. A microphone rack is placed into the enclosure facing the plate to acoustically map an area corresponding to a 5×4 regular grid. The comparison of measured pressure levels at the grid points for the piezos in short circuit configuration and shunted
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Fig. 10.11 Visualization of the velocity distribution [m/(s V)] over the plate when vibrating at 1720 Hz in the short (a) and closed (b) RL circuit cases tuned at the same frequency
Fig. 10.12 Experimental FRF of the hybrid configuration with RL circuits tuned at 1150 Hz and R = 100
with tuning at 1600 Hz shows promising reductions in noise levels (Figs. 10.14(b) and 10.14(c)). Numerical and experimental analyses demonstrated that the tuning capabilities of RL networks can be effectively used within a periodic framework to obtain a broadband control effect. The behavior of the periodic piezo plate is characterized by energy dissipation due to wave interference resulting in a control mechanism different and more robust than classical vibration absorbers. Since both numerical and experimental results are very encouraging, periodic piezo shunts seems a promising control strategy both for vibration reduction and noise suppression applications.
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Fig. 10.13 Panel vibration and corresponding cavity pressure at 650 Hz with (a) short circuited patches and (b) RL shunted patches tuned at the same frequency
10.5 Optimizing Wave’s Diffusion in Beam A complementary approach for controlling waves dispersion onto complex structures is to control energy diffusion through a smart interface. The idea is to optimize the transmission or reflection at a given interface, for example in order to isolate the region of a structure from a disturbance. The basic concept is illustrated in Fig. 10.15, on which one can observe the structure A, which is typically equipped with distributed piezo shunts, connected to structure B, with incoming and reflected waves in both directions. The illustration is based on a beam-type structure considered as a wave guide but can be extended to more complicated systems as plates or shells. This part focuses on two main objectives: 1) to provide a suitable physical model of a coupled passive/patch-shunted beam structure (left/right part of Fig. 10.15), with the patches enabling suitable control of Medium Frequency (MF) dynamics; and 2) to obtain optimal shunt impedances for MF control strategies and to assess their effectiveness. A wave description of the phenomenon of interest underlies the approaches to both of these objectives. Indeed, from the knowledge of the wave content on a structure, one can construct a variety of optimization solutions. From the geometrical and mechanical properties of the structure, one can control those waves carrying most of the energy or that are generating excessive or unwanted sound. One can also control the wave diffusivity at a given interface. This can lead to the improvement of either transmission or reflection efficiencies allowing isolation of a region of a structure from a disturbance. This idea is employed for simple waves in acoustic treatments, for instance, with either passive or adaptive materials. Its extension to more complex wave cases today suffers from theoretical and physical difficulties, of which some are addressed in this section.
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Fig. 10.14 (a) Experimental set-up for structural acoustic control and comparison between (b) measured sound pressure levels without any connected circuit and with (c) RL shunt circuit tuned at 1600 Hz connected to the periodically distributed piezoelectric patches
In this study, a viscoelastic structure, treated as a wave guide, with bonded piezoceramic patches is considered. This smart structure comprises a periodic distribution of a three-layer unit of a core layer plus two symmetrically bonded piezoceramic layers. The finite Element Method (FEM) [31, 55] is the most commonly employed tool to deal with the dynamic behavior of such structures. The Wave Finite Element (WFE) technique appears especially well suited for the MF domain analysis as a system’s dynamical behavior can be accurately described using highly convergent reduced bases containing the essential wave motions [49]. Wave propagation in slender systems has been extensively studied previously. Analytical models give immediate insight into the physics of wave motions in simple systems and provide such information as the dispersion behavior (that is, the frequency evolution of the wavenumbers) and mode shapes. Examples of such an approach include that of Graf [32] and von Flotow [68] for elastic systems; Fuller [31] for pipes containing fluids and Banerjee [2] and Mahapatra [46] for multilayered structures. However, it should be emphasized that these analytical models are constrained by Low Frequency (LF) analytical assumptions including, among others, the plane wave description of the longitudinal, bending and torsional modes, which is not a-priori satisfied in the MF range [49]. Alternatively, numerical techniques, based on small size models discretization, have been developed for the MF wave
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Fig. 10.15 Concept overview and view of one cell of the periodic piezoelectric patch distribution
description of systems having simple and even complex cross-sections. Among others, an alternative numerical technique based on classic 3D finite element schemes was proposed in the work of Mead [48] and Zhong and Williams [73] allows the wave properties of periodic systems to be evaluated by solving either a quadratic or a linear eigenvalue problem. The numerical implementation of the technique, even when multi-layered structures and/or multi-physics problems are considered, appears rather simple as it is based on the classic finite element description [31, 55] of a typical cell extracted from the global system. In this sense, the method has been usually called the WFE formulation [24, 36, 49]. Finally, note that the formulation established by Zhong and Williams [73] is quite different from Mead’s work [48] as it is based on a dual state vector representation of the kinematic variables of systems (displacements and forces), which allows symplectic orthogonality properties to be
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considered: it has been shown that these properties appear appropriate for capturing the frequency evolution of the wave modes [36, 49]. In this section, LF and MF wave propagation in multi-layered slender elastic systems having periodically distributed segments is specifically formulated using the WFE approach. For this case, the underlying problem concerns the applicability of the formulation, as derived, for instance, by Zhong and Williams [73]: the issue is related to the size of the wave mode basis, given by the WFE formulation, which generally overestimates the cross-section dynamics. In this way, the formulation enforces linear dependencies between wave modes which lead to unexpected and non-physical wave motion couplings and cause severe problems for wave mode determination. The WFE formulation considered here follows that established by Mencik and Ichchou [49]. After recapping some classical results and the dedicated numerical procedures employed, the section presents the fundamental results obtained from the optimization of wave diffusion (transmission and reflection) using periodically distributed shunted piezoelectric patches.
10.5.1 Description and Modeling of a Periodic Beam System The system of interest is depicted in Fig. 10.15. It comprises an aluminum beam covered on each side with a periodically distributed set of piezoelectric patches. This section develops the description of the dynamical behavior of a slender active piezoelectric structure as illustrated in Fig. 10.15(a), considered to be a waveguide, and which is composed along a specific direction (e.g., the x-axis) of N identical substructures shown in Fig. 10.15(b). Note that this general description can be applied to any homogeneous system whose cross section is periodic along an axis normal to the cross section. Each substructure is presumed to be piezo-elastic or simply elastic, linear and lightly dissipative. For substructures with piezoelectric elements, we include provision for an external shunt circuit of arbitrary impedance across the piezoelectric electrodes. The dynamics of the global system is formulated from the description of the waves propagating along the x-axis. Consider a finite element model of a given substructure I (I ∈ 1..N ) in the waveguide. The left and right boundaries of the discretized substructure are assumed to contain n degrees of freedom (dof). The connecting kinematic variables are the structural displacement w and the adjoint forces F, defined on the boundaries, and denoted respectively by (wL , wR ) and (FL , FR ) on the left and right interfaces. The piezoelectric adjoint variables V and q are only used in the internal system and are not involved in the connection degrees of freedom. The resulting finite element model comprises 320 8-nodes volumetric Lagrange elements, for a total of 1785 dofs, including electric ones, and 75 dofs at the connecting interface. According to the standard variational form of a piezoelectric mechanical problem as given by Benjeddou [7], the periodic dynamical equilibrium of one piezoelectric cell is given by
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⎡
Mll
⎢ T ⎢ Mlr ⎢ T ⎢M ⎢ lc ⎢ ⎣ 0 0
⎡
⎤ w¨L ⎥ ⎥⎢ 0 ⎥ ⎢ w¨R ⎥ ⎥ ⎥⎢ ⎥ ⎢ 0⎥ ⎥ ⎢ w¨C ⎥ ⎥⎢ ¨ ⎥ 0 ⎦ ⎣ VC ⎦ 0 V¨I ⎤⎡
Mlr
Mlc
0 0
Mrr
Mrc
0
MTrc 0
Mcc 0
0 0
0
0
0
Kll
Klr
Klc
Elc
Krr
Krc
Erc
KTrc
Kcc
Ecc
−ETrc −ETri
−ETcc −ETci
Ccc
⎢ T ⎢ Klr ⎢ T +⎢ ⎢ Klc ⎢ ⎣ −ETlc −ETli
CTci
Eli
⎤⎡
wL
⎤
⎡
FL
⎤
⎥⎢ ⎥ ⎢ ⎥ Eri ⎥ ⎢ wR ⎥ ⎢ FR ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Eci ⎥ ⎥ ⎢ wC ⎥ = ⎢ 0 ⎥ , (10.20) ⎥⎢ ⎥ ⎢ ⎥ Cic ⎦ ⎣ VC ⎦ ⎣ 0 ⎦ QI VI Cii
where wL , wR and wC represent the generalized displacements of face SL , face SR and the interior of the domain . FR and FL are also the wL , wR adjoint applied forces on each connection interface SL and SR , respectively. VI is the scalar voltage applied to the driving electrodes and VC the unknown resulting electric field interior to each piezoelectric domain and QI represents the VI adjoint applied charges on the driving electrodes. According to the design shown in Fig. 10.15(b), the direct piezoelectric and mechanical coupling effects from interface SL to SR are zero. So, to simplify the notation, we can directly assume without loss of generality that Elc = Erc = Eli = Eri = 0 and Mlr = Klr = 0. By condensing internal mechanical and voltage degrees of freedom one can obtain in Fourier space the following set of equations (see [17] for details): wL FL BLI S¯ ll S¯ lr = + VI , (10.21) wR FR BRI S¯ T S¯ rr lr
and between voltage input and charge output as QI = BTLI wL + BTRI wR + DI VI ,
(10.22)
where T S¯ ll = Sll − Slc S−1 cc Slc , S¯ rr = Srr − Src S−1 ST ,
S¯ lr BLI BRI DI
cc rc T = −Slc S−1 S cc rc , = −Slc S−1 cc BI , = −Src S−1 cc BI , = CIo + BTI S−1 cc BI .
(10.23) (10.24) (10.25) (10.26) (10.27) (10.28)
The control impedances Z is then introduced as the shunt impedance linking the ˙ I so that VI = ZI . applied voltage VI to the output piezoelectric current I = −Q For a series resistor-capacitor (RC) circuit,
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Z=
1 + R, iωCc
(10.29)
and therefore the piezoelectric voltage is linked to the internal current as VI = −Z(iω)QI = −(1/Cc + iωRc )QI .
(10.30)
The impedance Z represents, in fact, a controlling complex number related to the ordinary differential time operator used to control the system behavior. This operator (or complex number in Fourier space) can be implemented through consideration of either series or parallel shunt circuits or other architectures. So, as long as 1 + Z(iω)DI = 0, the controlled system equation is Z(iω) T ¯ lr + Z(iω) BLI BT S B B S¯ ll + 1+Z(iω)D w LI L LI RI 1+Z(iω)D I I Z(iω) T ¯ rr + Z(iω) BRI BT wR S¯ Tlr + 1+Z(iω)D S B B RI LI RI 1+Z(iω)DI I Sall Salr wL FL = . (10.31) = a w F SaT S R R rr lr Equation (10.31) constitute the basis for our computation of characteristic wave solutions. For more details one should refer to [17]. The key point for dispersion curve computation may be found in Brillouin [9], corresponding to the assumption that any harmonic generalized kinematic field obeys the one-dimensional complex waveguide relationship w(x, y, z, iω) = W(y, z)e−ikx eiωt .
(10.32)
This equation states a harmonic dependency of the primary structural variables with respect to the main axis of propagation. The time dependency restricts analysis to steady-state movements. As the piezo-mechanically described substructure represents the unit cell of the periodic wave guide, one has WR (y, z) = λWL (y, z),
(10.33)
where λ = e−ikd in which d represents the x-axis characteristic length of the cell. For the sake of notational simplicity, the spatial dependency is omitted in the following by identifying WR (y, z), WL (y, z) and all state variables respectively with WR , WL or their vector label. By analyzing two adjacent identical cells designated, respectively, by the indices I and I + 1 as in Fig. 10.15, we can state that the dynamic compliance on the I -th and (I + 1)-th cells are equal so that S¯ I = S¯ I +1 . So, using the dynamic equilibrium of Eq. (10.31) and the connecting relations (I +1)
WL
(I )
= WR
and
(I +1)
−FL
(I )
= FR ,
(10.34)
it can be shown [36, 49] that (I )
(I )
FR = −λFL .
(10.35)
Substituting the respective expressions for WR and FR , given by Eqs. (10.33) and (10.35), into the cell’s dynamic equilibrium (10.31), and eliminating FL from
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this expression, leads to the quadratic eigenvalue problem (QEP) aT Slr + λ Sall + Sarr + λ2 Salr WL = 0.
(10.36)
This QEP permits numerical determination of eigenvalues λi as well as eigenvectors WLi . If λi is an eigenvalue, then 1/λi is an eigenvalue representing the reciprocity in the propagation mechanism. Expression (10.36) is close to the spectral problem proposed by Denke [23] and by Mead [48]. To avoid the ill-conditioning linked to the symplectic receptance matrix calculation, as proposed in alternative methods [36, 49], Eq. (10.36) can be directly used for computation of characteristic wave solutions. Obviously, from the WLi expression, one can establish the corresponding dual force quantities as (10.37) FLi = −Sall − λi Salr WLi . Relevant mathematical properties of this receptance matrix [36, 49] and the implications of the QEP include: • The state vectors of the dynamical equilibrium on each boundary of each cell are WL WR and UR = . (10.38) UL = FL FR The receptance matrix T(iω) is defined as the wave transfer matrix between the right and left boundaries, so that UR = TUL .
(10.39)
This matrix is obtained directly from relations given by Eq. (10.31) as explained in Mencik [49] and Houillon [36]. This matrix is usually very badly conditioned and cannot easily be used to compute wave eigen-solutions by directly solving the classical eigenvalue problem UR = λUL = TUL . • If WLi is a right eigenvector of the QEP (10.36), associated with λi , then WTLi is the left eigenvector of the QEP associated with 1/λi . The corresponding force is also given for this j -th eigen wave by FLj = (−Sarr − 1/λi SaT lr )WLi . • Introducing the state eigen solutions of Eqs. (10.36) and (10.37) as WLi i = , (10.40) FLi and the matrix
0 Jn = −In
In , 0
(10.41)
where In is the n × n identity matrix, then the symplectic orthogonality properties of matrix T of eigenvector i is given by [49],
Note that
Jn = TT Jn T.
Tj Jn i = 0
for λi = 1/λj ,
Tj Jn i = γi
for λi = 1/λj .
(10.42)
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• Since λj = e−ikj d (d being the length of the modeled cell), the corresponding wave number can be computed for each eigenvalue λj as kj = − ln(λj )/(i d). • The real part of kj , Re(kj ), represents the phase velocity of the corresponding waves, ω p . (10.43) vj = Re(kj ) If Re(kj ) > 0, the phase propagates in the positive direction. If Re(kj ) < 0, the phase propagates in the negative direction and if it is zero, kj corresponds to the wave number of a pure evanescent wave that only occurs when an undamped system is considered. • The imaginary part of the wave number, Im(kj ), represents the amplitude decay rate, or the wave dissipation. It is also linked to wave power flow defined as [48] 1 1 Re(−iωW∗Lj FLj ) = ω Im W∗Lj Sall + λj Salr WLj . (10.44) 2 2 The sign of the power flow determines the direction of energy propagation. It is g linked to the group velocity vj = ∂ω/∂ Re(kj ). For a damped system (Im(kj ) = 0), one can say that if Im(kj ) > 0 the power flow is positive and is transmitted toward increasing x; if Im(kj ) < 0 the power flow is negative and transmitted toward decreasing x. Stability occurs when Pj =
Im(kj )Pj ≥ 0.
(10.45)
Modes with different signs for phase and power transportation can occur. These modes correspond to complex evanescent waves. The set of 2n computed modes, n being the size of problem (10.36), can be ordered according to the magnitude of Im(kj ), and also distinguished between positive and negative power transport (toward increasing or decreasing x). • For a specific propagating power flow defined by the knowledge of its deformation W=
2n
αi WLi ,
(10.46)
αi FLi ,
(10.47)
i=1
and force F=
2n i=1
the associated mean power can be evaluated as 1 P {αj } j =1...2n = Re(−iωW∗ F) 2 2n 2n 1 ∗ αj WLj αl FLl . = Re −iω 2 j =1
l=1
(10.48)
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This expression can not be reduced on the modal basis and is not equal to 1 αAj 2 Pj , 2 2p
(10.49)
j =1
because the modal basis defined above is not a diagonal basis of the power operator. • Instability occurs when traveling waves with Im(kj )Pj < 0 exist, that is to say, power flow transport occurs with an exponentially increasing amplitude. The criterion of Eq. (10.45) is also assessed to verify stability when negative (resistive or capacitive) shunting impedance is permitted. • Regarding the sensitivity of the numerical results to discretization, slight differences between numerical and analytical dispersion curves may be observed and are attributed to numerical noise effects [37], as the WFE formulation suffers from a lack of convergence, of order O(k 2 d 2 ) for linear interpolation [39, 50], which grows as the frequency increases.
10.5.2 Optimization of Power Flow Diffusion by Negative Capacitance Shunt Circuits Consider now the optimization of the power flow diffusion between a purely passive beam (structure B of Fig. 10.15) and a shunted piezo composite beam (structure A of Fig. 10.15). Each semi-infinite structure is modeled as a periodic system. The piezo composite beam wave dispersion analysis was presented in the previous section. Analysis of the passive beam is treated in the same manner by only considering passive materials. The mesh of the unit cell presented in Fig. 10.15(b) includes 96 8-node volume Lagrange elements, for a total of 525 dofs (including 75 dofs at the connecting interface). To avoid the use of a generalized Lagrange multiplier for active/passive degrees of freedom connection [49], the two sets of interface dofs corresponding to parts A and B are chosen as compatible. The wave dispersion of the passive beam system is very similar to that of the shunted one and does not need to be exhibited. It is composed of propagating complex compressive, torsional and flexural waves as well as complex evanescent shear waves. A large set of highly damped propagating and complex evanescent waves are also presented. Concerning the computation of the diffusion operator and resulting power flow equilibrium, each set of state vectors describing the waves propagating in the two different semi-infinite structures A or B can be partitioned over the set of each eigen wave state solutions defined in Eq. (10.40), so that UR A=
2n i=1
Ai αAi =
n i=1
+ − − + Ai αAi + Ai αAi ,
(10.50)
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UL B =
2n i=1
Bi αBi =
289 n
+ − − + Bi αBi + Bi αBi ,
(10.51)
i=1
where (A,B)i represents each particular eigen wave vector on each structure A or B, and α(A,B)i are the complex corresponding wave amplitudes. To distinguish forward and backward power flow a standard decomposition between the symmetrical positive and negative eigen wave number ki and −ki is introduced in Eqs. (10.50) and (10.51), corresponding to each eigen solution + (A,B)i for forward power flow in structure A or B and − for backward power flow. (A,B)i L On the interface So between substructures A and B (see Fig. 10.15), UR A = UB . This relation simply represents displacement continuity and the force action/reaction conditions. A source term is added by imposing a non-zero incident wave, permitting computation of non-trivial solutions. This condition can be written as L n NA UR A + NB UB = Uo where NA , NB ∈ C are localization line vectors and Uo ∈ C is the imposed complex wave amplitude. This diffusion relationship between wave flow on the right and left part of the interface So can be imposed using a Lagrange multiplier ξ . The introduction of the incoming wave’s adjoint Lagrange multipliers allows one to solve a standard square algebraic problem. When using a reduced number of computed eigen wave solutions, the fact to remove these Lagrange terms leads to a standard rectangular minimization problem but when using the complete eigen wave basis leads to an ill-conditioned problem. Using a reduced number of 2p eigen solutions of Eq. (10.40) (2p 2n) and using orthogonality relationships of Eq. (10.42), the 2p space projection of the diffusion operator is obtained as ⎡ − +T + +T − +T ⎤ ⎡ + ⎤ 0 +T αA A Jn A − A Jn B − A Jn B NA ⎢ −T ⎥⎢ −⎥ + −T + −T − 0 − A Jn B − A Jn B 0 ⎥ ⎢ αA ⎥ ⎢ A Jn A ⎢ +T ⎥⎢ +⎥ +T − ⎢ Jn + +T Jn − ⎢ ⎥ 0 − B Jn B 0 ⎥ ⎢ B ⎥ ⎢ αB ⎥ A B A ⎢ −T + −T − −T + −T ⎥ ⎢ α − ⎥ ⎣ B Jn A B Jn A − B Jn B 0 NB ⎦ ⎣ B ⎦ +T −T ξ N 0 0 NB 0 ⎡A ⎤ 0 ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥ (10.52) =⎢ ⎢ 0 ⎥. ⎢ ⎥ ⎣ 0 ⎦ Uo − + T One might wonder why the complex amplitude vector [αA αB ] is directly used as unknown data. The reason is that the basis comprised of the first p forward and backward power flow modes does not diagonalize the power flow operator defined in Eq. (10.44), or any related energy operator. The proposed methodology is simple, deals with physical meaningful quantities such as power flow, and is related to the property of conservation of energy. Power conservation requires that the transmitted power in domain A, PAt , and the reflected power in domain B, PBr , are related by PA − PBr = Uo .
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Fig. 10.16 Ratio of the transmitted flexural power flux as a function of negative capacitance shunt at 30 Hz, 1500 Hz and 3000 Hz
To illustrate this methodology, Fig. 10.16 presents the ratio of the transmitted flexural power flow at 30 Hz, 1500 Hz and 3000 Hz as a function of the value of a negative capacitance shunted across each piezoelectric electrode of the piezocomposite structure A. This computation used 2p = 40 modes on each periodic structure and included a small hysteretic damping value of 0.1%. A specific zone of negative capacitance is observed which drives the shunted periodic structure to cancel flexural propagation. In this capacitance band the transmitted power ratio is very small leading to a reflected power ratio of close to one (see Fig. 10.16(b)). For any other shunt capacitance, the transmission ratio is close to unity, indicating a very high transmissibility. Nevertheless, for 30 Hz there is a point of maximum transmissibility as indicated in Fig. 10.16(a). For this example, we note some very interesting dynamical behavior that depends on the value of the shunt’s a negative capacitance. The equivalent dielectric capacitance of the
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pair of piezoelectric patches is 2525.5 pC V−1 . The indicated sensitive band of negative capacitance values are greater than 1942.7pC V−1 , which is very far from the electrical instability limit. This implies the possibility of physically realizing this configuration.
10.5.3 Optimization of Wave Reflection and Transmission 10.5.3.1 Wave Reflection Optimization Consider now the maximization of flexural wave reflection by means of actively shunted piezoelectric patches. A set of solutions representing unit power flow waves is used as a computational basis. By fixing Uo = 1 on the first propagative incoming flexural wave on substructure B, a unit power incoming propagating flexural wave on this substructure is imposed, interacting on interface So with the periodic piezocomposite beam. A space projection on the first 40 eigen waves on the shunted and passive structures (A and B) is performed, and hysteretic damping is fixed on both structures equal to 0.1% of the corresponding stiffness matrix. The performance criterion corresponds to the simple minimization of transmissibility such as (iω, Z) = P {αAi }i=1..2n , (10.53) where P ({αAi }i=1..2n ) is defined as the eigen wave power flow in Eq. (10.48), and αAi is the modal complex amplitude in substructure A, obtained from the diffusion Eq. (10.52). Consider the objective to decrease transmissibility by using two different shunt circuits, with or without resistance element. The shunt’s electrical impedance (10.30) is reduced to Z = 1/Cc or Z = 1/Cc + iωRc , where Cc is the shunt capacitance and Rc the resistor. The optimization procedure used here was a classical minimization algorithm based on the Nelder-Mead simplex method. Concerning the C-shunt, the maximal transmitted power flow is approximately 7.3% of the total incoming power with a maximum at 2300 Hz. Figure 10.17 presents the optimal capacitance values. We observe a small relative variation of the optimal negative capacitance for the considered frequency band. This underlines the robustness of the obtained solution with respect to variation of the optimal shunted capacitance. When a RC-shunt is considered, the minimal transmitted power flow is much smaller, less than 10−5 time the incoming power. The obtained reflection is also nearly complete and no flexural energy is coming into the shunted substructure A. Figure 10.18 presents the optimal resistance value of the piezoelectric shunt, while the optimal capacitance is almost always equal to −3.2 × 10−8 CV−1 , which is of the same order as those shown in Fig. 10.17, always larger than the piezoelectric equivalent capacitance DI . In both cases, the optimal resistance is negative over the whole frequency band. This indicates that perfect cut off of flexural transmission is reached only for zero equivalent wave damping. The applied negative resistance compensates for the assumed hysteretic damping. The optimal configurations obtained here are stable in the whole frequency band in the sense of Eq. (10.45).
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Fig. 10.17 Optimal shunt capacitance for reflection optimization (C-shunt)
Fig. 10.18 Optimal shunt resistance for reflection optimization (RC-shunt)
10.5.3.2 Wave Transmission Optimization After having verified the ability of the shunted periodically distributed piezoelectric beam to cancel flexural wave transmission, it can be shown that the system can also act on reflection to stabilize the passive connected semi-infinite beam. The main aim is to transmit all incoming energy from the passive structure B at the interface So into the shunted structure. As shown previously and depicted in Fig. 10.16, the semi-infinite periodic piezoelectric beam transmits passively almost all the incoming power in the whole frequency range, except in the band gap illustrated in
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Fig. 10.19 Criterion value vs. freq. for transmission optimization with RC shunt
Fig. 10.16(a). If one want to find the optimal electrical impedance to reach a full transmissibility, the optimization algorithm should use the criterion (iω, Z) = 1 − P {αAi }i=1..2n , (10.54) where P ({αAi }i=1..2n ) is defined as the eigen wave power flow in Eq. (10.48), and αAi the complex modal amplitude in substructure A, obtained from solution of the diffusion Eq. (10.52). A complementary interesting element to take into consideration is the damping properties of the transmitted flexural waves. Indeed, for physical applications only a finite set of periodically distributed piezoelectric patches are used, so the objective is to damp all the incoming energy with this finite set of transducers. To translate the constraint from the infinite approach, the n cells damped power function is defined as (10.55) Dn = P αAi λni i=1..2n − P {αAi }i=1..2n . This function simply describes the power balance on the first n cells of the piezoelectric beam following the connecting interface So . If the incoming flexural wave in structure B has a unit input power, this function represents the fractional power dissipated in the first n active cells. The mechanical control of waves in a finite set of cells requires maximization of this function. Introducing directly the criterion in Eq. (10.54) in an optimization procedure using the Nelder-Mead simplex method yields optimal resistance and capacitance values over the whole frequency band of interest, 30–3000 Hz. Figure 10.19 depicts the criterion values obtained. The very small values of the criterion, less than 1.2 × 10−5 , correspond to essentially total transmission of the incoming flexural wave into the shunted beam. As the previously presented results indicate, the ratio between the transmitted power and the incoming flexural power is constantly practically equal to 1 in the whole frequency band. Nevertheless, the ratio of the damped
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Fig. 10.20 10 cells damped power function for D10
power, Eq. (10.55) in the first 10 active cells and plotted in Fig. 10.20 indicates that a maximum of 24% of the incoming power flow is dissipated at 3000 Hz. The implication is that the system needs to be large enough (constituted of more than 10 cells) to totally damp the incoming energy. From a physical point of view the optimal architecture could be difficult to implement because of the large number of cells implied to reach this result. In this case, the very large values of the optimal shunting resistance indicates that the piezoelectric patch almost behaves as an open circuit (with 0 current). Indeed, these large resistance values do not induce a large damping effect as previously observed in Fig. 10.20, but alter the equivalent piezoelectric behavior from a zero voltage condition to a zero current one. This phenomenon is well known when an active damping setup is optimized as explained in Monnier [52] and Preumont [58]. The corresponding negative values for the shunt capacitance correspond to the same magnitude as those previously obtained for reflection optimization in Fig. 10.17 but exhibit an inverse dependency on frequency. Another way to optimize the periodic piezoelectric system for passive structural stabilization is to increase its damping power ratio in the first cells connected to the passive semi-infinite beam. This introduces large power pumping and localizes the effective active elements near the interface. The experimental embodiment of this approach should be facilitated because the number of required shunted cells should be reduced. The criterion used to maximize the total damped power ratio in the first 10 cells of the piezoelectric beam is (iω, Z) = |1 − D10 |,
(10.56)
where D10 is given by Eq. (10.55) for n = 10. As the incoming flexural wave in structure B has a unit power flow, the minimization of this criterion is intended to
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Fig. 10.21 Criterion value vs. freq. for transmission optimization with RC shunt
Fig. 10.22 10 cells damped power function: D10
impose total power damping of the incoming energy in the first 10 shunted cells. Figure 10.21 depicts the resulting criterion values versus frequency after optimization. We observe quite large values corresponding to 55% of input power damped at low frequency to rising 95% damped at 3000 Hz; complete damping is not obtained at any frequency. Nevertheless, note that the power ratio increase with the frequency and tends toward total absorption. The 10 cells dissipation ratio presented in Fig. 10.22 obviously corresponds to the previously mentioned results in Fig. 10.21. But, comparing the dissipated energy to the transmitted energy, it can be observed that when the frequency increases, the damped part tends to correspond to the total incoming power in the shunted sub-
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structure A. So, the piezoelectric setup tends to dissipate all the So incoming flexural power in only 10 cells. The proposed solution can be, also, physically speaking, considered more interesting than the previous one, even if it exhibits a lower absolute performance in term of absorption performance. The optimization in this case increases damping capability rather than changing its electrical boundary condition. The necessary resistance decreases with the frequency to reach 100 at 3000 Hz. The corresponding negative values for the shunt capacitance part, represents, for this case, a non-monotonic function. Its magnitude increases with frequency above 500 Hz. The capacitances are of the same order of magnitude as those previously obtained and are far from the electrical instability value(−DI = −2525.5 pC V−1 ).
10.6 Prospects for the Future Generally speaking, the research activities on Smart Materials and Adaptive Structures aims at optimizing with active or passive approaches some coupled multiphysical systems to obtain and guarantee more and more advanced physical functionalities. By the use of new materials or the addition of transducers and controlling systems, we intrinsically try to modify the physical behavior of structures or materials to reach the desired functional response. Up to now, our approach was to use a small number of transducers connected with a complex and energy consuming numerical system of control as for implementing ANC strategies. If it was not the case, as in structural applications, we restricted our systems by using collocated architecture to implement an improved simple property such as active damping. In both case, the practical applications of such systems were restricted because of a poor ratio between the cost and the efficiency/robustness properties of the proposed technological solution. The continuous improvement of our physical knowledge combined with the new technological opportunities offered by MEMS activities lead us to consider a direct management or control of the physical interaction inside materials and structures to obtain a desired functionality at the used scale. So, in spite of adding and optimizing an external layer of active/passive systems onto the controlled structure, we will consider a local and distributed modification of the physical material behavior. This physical behavior optimization can lead to a very wide range of macroscopic response and also, provide a very efficient way to drive responses of structural systems and so obtain the desired macroscopic functionality of interest to this research. This new materials can also be called metamaterial when the inner interactions are only passive but can be called active metamaterials when active interactions are implemented. This concept is also based on active/passive systems integrations and homogenization to the design of generalized active/passive composites. In this way, objectives of the future works aim to develop original tools and technology for robust design and optimization of new smart integrated materials for vibroacoustic control. The overall objective is also to provide original solutions concerning vibration and noise control using new optimized hybrid composite structures. There are
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a number of different approaches that can be adopted. Among these, one can use actively controlled smart materials to create opposing vibrations to counter or cancel those created by a noise and vibration source. As previously mentioned, many intelligent material systems are derived, characterized and applied on a laboratory scale without any impact yet on practical application of design rules for engineered structures. Such innovative material concepts suffer from a lack of: • Sufficient robust efficiency for industrial applications; • Sufficient level of integration onto the structure itself; • Comprehensive vision including conventional design and manufacturing processes in view of their optimal integration. Our proposed works are totally apart from this trend. In spite of distinguishing such classical vibroacoustical properties as transparency, insertion loss, isolation or stabilization, we would speak about explicit optimization of energy flow by introducing the concept of a generalized impedance operator. In spite of using a complex theoretical tool kit for designing optimal control of a distributed system by using a small set of transducers, we will seek to robustly optimize simple control laws but applied through a dense, distributed set of transducers integrated into a so-called smart composite structure. The particular studied technological systems will be constituted of a distributed set of piezoelectrical cells shunted by a semi-active electrical circuit (incorporating operational Amplifier Op-Amp). The low consuming shunt circuit may be connected on each piezo patches or between neighbor cells depending on the wanted controlling operator to implement. This proposed approach has the main advantage to offer low energy consumption and to be eventually implemented and integrated inside the composite by using micro electric ships. The specific scientific objectives also concern research in the following tasks: • To determine efficient theoretical active/passive optimization tools for controlling mechanical power flow in complex structures with respect to phenomenological criteria (transmission, absorption, damping) and the corresponding operator for technological implementation. The optimization will also adopt a design-oriented perspective in order to facilitate the choice and the integration of these solutions; • To understand and to analyze clearly multiphysical interactions between piezoelectric elements, supporting structures and shunt electrical circuits when dense distributed integration into composite structures is considered for physical implementation of optimal control operator; • To develop integrated electromechanical prototype for characterizing vibroacoustical properties of such new generation of hybrid smart materials; • To explore new concepts in passive, adaptive or active mechanical integrated composite interfaces with different kinds of electronic circuits and others electro mechanical transducers with a view toward MEMS integration in a near future; • To develop and to improve numerical models in view of the structural complexity of the components and their assemblies. Significant effort will be focused on the mid-frequency modeling of systems, including the development of accurate
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reduced-order models and the development of homogenization techniques in the presence of highly fluctuating kinematic fields; • Finally to produce design robust numerical tools for implementing such hybrid materials for industrial applications. This research program corresponds to the generic smart materials and structure framework. It is a multidisciplinary project. It covers various disciplines such as structural mechanics, mechatronic interaction, materials science and systems. This future works include some new and strategic issues, including new materials, reduced models, mid-frequency and multi-scale approaches and vibroacoustic design optimization tools. To achieve these goals, the generic used idea introduces the notion of mechanical hybrid interfaces often of a dissipative nature. This interface can be made of a simply layered material or multi-layered hybrid composite system including smart and passive materials, integrated electronic control devices, potentially fluids, etc. The considered interface is located between two solid media (as joints). It is made of shunted piezoelectric composite materials. This kind of interface acts on the whole structure as a generalized impedance linking power flow between each separated system. The optimization of its composite material behavior induces application of specific interface impedance able to confer specific properties for energy diffusion. This enables control of numerous desirable engineering properties such as insertion loss, absorption, reflection, damping usually considered as the design criteria for optimizing structural vibroacoustical behavior. This aspect demands to solve different theoretical problems regarding optimization and realization of complex differential and pseudo differential operators. The technological implementation of such concepts requires the integration, in the interface itself, of a hybrid distributed system including smart transducers (here piezoelectric elements) and electronic components (here semi-passive shunt devices). The global induced structural behavior should exhibit the desired vibroacoustical properties and should be able to guarantee efficiency in the face of structural alterations and modifications. The feasibility of developing and realizing such materials and these integrated distributed smart structures will be first studied through the physical implementations on simple examples. The paradigm application concerns the implementation of the proposed smart piezo composite interface into a beam and a plate for controlling its mechanical power flow diffusion. The considered piezoelectric control will be based on semi-active shunt circuit in a decentralized or non-decentralized architecture depending on the order of the corresponding optimal impedance operator. The investigated controlling operator consist of innovative shunt circuit connected to each individual piezoelectric cell (decentralized approach) or between two nearby cells (non-decentralized approach). By using the proposed new point of view based on implementation of distributed shunt circuit, we could also produce stable localized sub domain for sensitive system isolation, wave absorption for panel stabilization or mechanical energy concentration for energy harvesting device implementation. Low and mid frequency dynamical characterization of such smart piezoelectric composite structure will also be made in order to experimentally highlight the spe-
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cific vibro acoustical effect of such device for controlling the energy diffusion between the passive and active part of the system. Depending on the applied distributed impedance, absorption and/or reflection of flexural waves will be analyzed.
10.7 Summary The tuning capabilities of generalized shunt circuits applied to periodically distributed piezoelectric patches allow to obtain very interesting vibroacoustical properties for stabilization but also for optimizing energy diffusion. The resultant smart structure is found to be capable of significantly reduce vibration in low and medium frequency band without any complex control electronics. Preliminary FE simulations demonstrated the possibility to optimize the implemented electrical impedance even if the obtained function represent a non standard differential equation. The presented experimental implementations confirm numerical simulations and demonstrate the physical efficiency of the approach. This work enables research on more complicated systems using, for example, decentralized shunt circuits. It represents a step toward designing efficient solutions to development of controlled piezo composite structures with integrated electronic circuits. Finally, we note that the proposed strategy can be applied to numerous applications and can lead to new technological solutions for energy harvesting systems, isolation devices or structural health monitoring technologies. Acknowledgements This work is supported by a collaborative research agreement (ANR NT09 617542) between Georgia Tech, FEMTO-ST Institute and Ecole Centrale de Lyon. We gratefully acknowledge Georgia Tech and the French ANR and CNRS for supporting such international collaborations.
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Chapter 11
Identification of Reduced Models from Optimal Complex Eigenvectors in Structural Dynamics and Vibroacoustics M. Ouisse and E. Foltête
Abstract The objective of this chapter is to present some efficient techniques for identification of reduced models from experimental modal analysis in the fields of structural dynamics and vibroacoustics. The main objective is to build mass, stiffness and damping matrices of an equivalent system which exhibits the same behavior as the one which has been experimentally measured. This inverse procedure is very sensitive to experimental noise and instead of using purely mathematical regularization techniques, physical considerations can be used. Imposing the socalled properness condition of complex modes on identified vectors leads to matrices which have physical meanings and whose behavior is as close as possible to the measured one. Some illustrations are presented on structural dynamics. Then the methodology is extended to vibroacoustics and illustrated on measured data.
11.1 Introduction Being able to identify reduced physical models can help designers to understand the behavior of the system in a given frequency range, and orient design decisions in order to reach a given objective. Performing model reduction is quite usual in the field of numerical analysis [14, 29], in this case the objective is to find a model with a reduced number of degrees of freedom, which can be deduced from a large model, in order that the reduced model exhibits the same behavior as the full one in a frequency band of interest. An alternative to this model-based methodology could be based on experimental measurements. The basic idea is to identify from measurements the matrices describing the behavior of the system in order to help the designer to make proper decisions. The main difficulty in this kind of analysis is related to the very bad conditioning of the inverse procedure, since experimental conditions induce noise in the data, resulting in large changes in the final identified system matrices, in particular for the damping terms. M. Ouisse () · E. Foltête FEMTO-ST Institute, Applied Mechanics, University of Franche-Comté, 25000 Besançon, France e-mail:
[email protected] E. Foltête e-mail:
[email protected] C.M.A. Vasques, J. Dias Rodrigues (eds.), Vibration and Structural Acoustics Analysis, DOI 10.1007/978-94-007-1703-9_11, © Springer Science+Business Media B.V. 2011
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Several approaches have been proposed throughout the last decades to regularize the inverse problem on the field of structural dynamics. A brief overview of the state of the art is given in Sect. 11.2. Section 11.3 is dedicated to the so-called properness condition for structural dynamics and can be considered as a tutorial section. Original illustrations are presented to help the reader to understand the importance of the condition. An experimental test-case is given to illustrate application of the methodology on a real structure, on which a reduced model is directly derived from the experimental data. In Sect. 11.4, the properness condition is extended to vibroacoustics and new results about optimal correction of vibroacoustic complex modes are given. Several corrections techniques are described and illustrated on experimental data coming from vibroacoustic measurements on a guitar. Section 11.5 is dedicated to prospectives: some comments are given about the structural dynamics applications of the methodology, and some suggestions are given for improvement of the methodology concerning vibroacoustic applications. Section 11.6 gives some conclusions and a summary of the work presented in this chapter. The bibliography and a selection of additional references are finally given at the end of the chapter.
11.2 Overview of the State of the Art Identification of analytical models from measurements in structural dynamics is still an open question, in particular concerning the damping terms. Both stiffness and mass can be derived quite easily from models, or even from experiments with reasonable confidence. As far as the dissipative effects are concerned, there is still no consensus about the most reliable technique to obtain a physical description of damping which can be efficient for simulation. In this chapter we will mainly focus on techniques based on experimental data, that allow identification of second-order matrices corresponding to classical stiffness, mass and viscous damping terms of multi-degrees of freedom models. This topic has shown a growing interest over the last decades. The fundamental book from Lord Rayleigh [34] includes some considerations about sensitivity of eigenfrequencies and eigenvectors which are of first interest for system identification. Damping aspects have been at the center of several works, among which the famous papers from Caughey [11] including considerations about normal and complex modes, which are of first importance in the context of interest. Some review papers have been published [10, 15, 20], including many references to important works on damping related aspects. More recently, some papers have focused on the particular case of damping identification from measurement [33, 35, 36]. In these papers, the authors exhibit a large set of available methods, starting either from Frequency Response Functions (FRFs) or modal data to identify at least the damping matrix. These methods are applied and compared on given test-cases. An interesting point is that these papers do not lead to the same conclusions concerning the efficiency of the techniques for practical applications, which clearly means that there is still some work to do, even if among the available methods, some of them can provide quite confident results.
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One of the ways to obtain the system matrices is to start from identified complex modes. This chapter will be limited to this case, and will focus on a particular point, called properness condition, which is not addressed in the review papers referenced above. This condition, which has been mentioned in several publications [9, 22, 24, 41], is automatically verified by the exact complex modes of the system. When a full basis is extracted from experimental data to reconstruct a physical model, this condition should be enforced on the complex modes to obtain physical results. Balmès [7] has proposed a methodology to find optimal complex vectors which are as close as possible as initial identified vectors, while verifying the properness condition. Another way to obtain optimal complex vectors from measured ones has been proposed by Adhikari [1], but this method requires the knowledge of real modes, which is not necessarily the case in practical applications.
11.3 Properness Condition in Structural Dynamics The very classical matrix formulation used for structural dynamics is ¨ + Cq(t) ˙ + Kq(t) = f(t), Mq(t)
(11.1)
where q(t) is the vector of generalized displacements of the structure, M is the mass matrix of the structure, K is the stiffness matrix of the structure, C represents viscous losses and f(t) is the vector representing the generalized forces on the structure. One way to solve the system in Eq. (11.1) for steady-state harmonics is to use modal decomposition. This can be done using the space-state representation of the system, ˙ UQ(t) − AQ(t) = F(t), where
C M , U= M 0 f(t) . F(t) = 0
−K 0 , A= 0 M
(11.2)
Q(t) =
q(t) , ˙ q(t) (11.3)
The eigenvalues of this problem can be stored in the spectral matrix , so that (11.4) = λj . The j -th eigenvalue is associated to the eigenvector θ j such as (Uλj − A)θ j = 0, where θ j = {ψ Tj ψ Tj λj }T , ψ j being the complex eigenvector in the physical space (i.e. its components are related to q). Storing the eigenvectors (in the same order as the eigenvalues) in the modal matrix = [ T T ]T , the following relationship is verified: U = A.
(11.5)
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The orthogonality relationships can be written using 2n arbitrary values to build the diagonal matrix ξ = [ ξj ], T U = ξ
or T A = ξ .
(11.6)
The modal decomposition of the permanent harmonic response at frequency ω is finally −1 Q(t) = ξ (iωE2n − ) T F(ω)eiωt , (11.7) where E2n is a 2n × 2n identity matrix and F(ω) is the complex amplitude of the harmonic excitation. This relationship can also be written using the n degrees of freedom notation in the frequency domain as q(ω) = T f(ω), where =
(11.8)
1 . ξj (iω − λj )
(11.9)
In the following, without loss of generality, the eigenshapes are supposed to be normalized such as ξj = 1.
11.3.1 Properness of Complex Modes The properness condition is related to the inverse procedure: starting from the modal basis, the orthogonality relationships can be inverted to obtain the system matrices. Inverting relationships (11.6) leads to U−1 = T , or
C
M
M
0
−1 =
(11.10) M−1
0
−M−1 CM−1 T T = , T 2 T
M−1
(11.11)
and A−1 = T , or
−K 0 0 M
−1 =
=
(11.12)
−K−1
0
0
M−1
−1 T
T
T
T
.
(11.13)
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From these expressions, the mass, stiffness and damping matrices can be expressed as −1 (11.14) M = T , −1 K = − −1 T ,
(11.15)
C = − M2 T M .
(11.16)
These relationships are only valid if the complex modes verify the properness condition that directly comes from the zero terms in inverse matrices: T = 0.
(11.17)
It should be emphasized that this methodology leads to identification of matrices only if all modes of the system are identified. This is of course not realistic for continuous structures. Nevertheless, the reconstruction equations can lead to useful condensed model of the continuous structure if the number of identified modes is equal to the number of measured degrees of freedom, and if the locations of the sensors ensures physical meaning for the degrees of freedom of the reduced model. Some techniques are available to provide estimation of matrices when only a subset of the modes are identified, or even directly from measured FRFs. The readers are invited to refer to corresponding papers [12, 19, 21, 26, 28] or reviews [33, 35, 36] for more details. This chapter is limited to model reconstruction from a full set of complex modes.
11.3.2 Illustration of Properness Impact on Inverse Procedure When dealing with experimental data for matrices identification, it is clear that the input data (i.e. the complex eigenshapes) are polluted with random noise. In order to illustrate that point, a numerical example can be used: starting from exact solutions of a 4 degrees-of-freedom system, the eigenshapes are modified using a random noise of growing amplitude acting on amplitude and phase of vectors. This numerical noise does not necessarily represent exactly experimental noise, it is used here for a sake of simplicity, in order to illustrate impact of noise on properness condition. Experimental results will be shown later, on which the trends observed here will be confirmed. Figure 11.1 shows the impact of noise on the properness norm. The norm which is used here is the norm 2, i.e. the largest singular value of the matrix. It can clearly be observed that the properness norm grows up with the noise on inputs, which means that the inverse relations are no longer valid as soon as the properness condition is not verified. This is confirmed by Fig. 11.2, which shows the error on identified matrices, this error being defined from the ratio of norm 2 of the difference between identified and exact matrices to the norm 2 of the exact
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Fig. 11.1 Impact of noise on eigenvectors on properness norm
Fig. 11.2 Impact of noise on eigenvectors on error on identified matrices
matrices. It is clear from the figure that some very large errors can be obtained on matrices identification for small errors levels on inputs, in particular for the damping identification, while the identification of mass and stiffness matrices is quite robust, i.e. the level of error on outputs is of the same order as the level of error on inputs. Proper complex modes are then of first importance for correct damping estimation.
11.3.3 Properness Enforcement When the complex modes are available from experimental identification, Eqs. (11.14)–(11.16) can be used in order to find the reduced model which is supposed to have the same behavior as the measured system. In general, the complex modes do not verify the properness condition (11.17) and Ref. [7] proposes a
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Fig. 11.3 Eigenvectors of the first mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line)
methodology to enforce properness condition, in order to obtain optimal complex modes. The objective is to find the approximate complex vectors, which are as close as possible to the identified ones, and that verify the properness condition. It is shown that for structural dynamics, an explicit solution can be found, requiring only to solve a Riccati equation. This equation can be deduced from the problem ˜ minimizing ˜ − while ˜ ˜ T = 0. Find
(11.18)
Writing this problem using a constrained minimization approach leads to ˜ = [En − δδ]−1 [ − δ],
(11.19)
where δ is a Lagrange multiplier matrix, that can be found by solving the Riccati equation T
T
T − δ T − δ + δ δ = 0.
(11.20)
In the previous equations, is the conjugate of . After properness enforcement, the eigenvectors are typically mainly changed in phase, while the amplitude of vectors remains almost the same, as shown in Figs. 11.3, 11.4, 11.5 and 11.6, which present the eigenvectors of the four modes in complex plane. The figure exhibits three families of shapes: – the initial shapes, corresponding to those of the initial system; – the modified shapes, obtained after random changes of initial shapes; – the proper shapes, deduced from the modified ones after properness enforcement. It is quite clear from this picture that the proper modes are not those of the initial system, but the closest ones to modified ones that verify the properness condition. The case considered here for illustration is undoubtedly an extreme case, since it corresponds to the highest value of random noise used in Figs. 11.1 and 11.2, i.e. 30% in amplitude and phase. For practical applications, lower level of noise is expected, and the starting vectors should be closer to the “true” shapes of the system.
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Fig. 11.4 Eigenvectors of the second mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line)
Fig. 11.5 Eigenvectors of the third mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line)
In order to obtain the characteristic matrices of formulation (11.1), one has then to consider the following steps: – build FRFs from time domain measurements; – use complex curve fitting in order to find the complex modes in the frequency range of interest [17, 38]; – use the properness enforcement technique to obtain modified complex eigenvectors from identified ones; – use inverse relationships (11.14) to (11.16) to find the matrices of formulation (11.1).
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Fig. 11.6 Eigenvectors of the fourth mode in complex plane: initial shapes (dashed line), modified shapes (continuous line) and proper shapes (dashdot line)
Fig. 11.7 Experimental test-case: two bending beams coupled by common clamping device
11.3.4 Experimental Illustration In this section an experimental illustration of the methodology is presented. Figure 11.7 shows the experimental set-up which has been used. It is constituted with two bending beams which are coupled through their bases by a common “clamping” device. The frequency range of interest concerns the two firsts modes of the coupled system, which could be represented by a 2-degrees of freedom equivalent model, using points 1 and 2 indicated in Fig. 11.7 as reference points. These points are equipped with accelerometers and some contactless force transducers are used to excite the structure, with force sensors. An electrical intensity probe has also been used to check the value of the force sensors and to verify that the moving masses do not perturb the measured information.
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Fig. 11.8 Comparison of measured and synthesized FRF11
The complex eigenvectors which have been identified from the experimental FRFs are 0.0324 − 0.0377i 0.0283 − 0.0328i = . (11.21) 0.0249 − 0.0303i −0.0298 + 0.0350i The matrices which are deduced from original vectors are
0.5360 0.0348 M= , 0.0348 0.6320
(11.22)
−17.0 −1.29 C= , −1.29 −23.7
2.53 × 104 K= −1.00 × 103
−1.00 × 103 . 3.07 × 104
(11.23)
(11.24)
It is clear that the identified damping matrix is not physical. In order to improve its identification, the properness condition is enforced on the complex vectors
˜ = 0.0352 − 0.0349i 0.0304 − 0.0307i . (11.25) 0.0275 − 0.0277i −0.0325 + 0.0323i
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Fig. 11.9 Comparison of measured and synthesized FRF12
The amount of change in these vectors is clearly in the same order of magnitude as the one observed in the numerical illustration. These small changes in vectors clearly improve the matrices identification
˜ = 0.5330 0.0343 , M (11.26) 0.0343 0.6270
0.569 0.194 ˜ , C= 0.194 0.848 ˜ = K
2.52 × 104
−1.02 × 103
−1.02 × 103
3.05 × 104
(11.27) .
(11.28)
Changes associated to properness enforcement have a very limited impact on mass and stiffness identification, while they have a strong effect on the damping identification. The first observation that can be done is related to the numerical values in the damping matrix, which correspond to possible physical values. The second observation is that the identified values with properness enforcement are in accordance with the measured data, as indicated in Figs. 11.8, 11.9 and 11.10. These figures show the measured FRFs, the synthesized FRFs from complex modes
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Fig. 11.10 Comparison of measured and synthesized FRF22
identified using a curve fitting technique, the synthesized FRFs obtained from direct calculation using matrices coming from identified complex modes, and the corresponding ones after properness enforcement. The figures clearly show that: – the initial modal identification seems to be correct, since the associated synthesized FRFs are very close to the measured one; – if these identified modes are used for matrices identification, the bad conditioning of the problem leads to very large errors (as indicated above, mainly due to bad damping identification); – if these modes are slightly modified in accordance with the properness condition, the identified matrices are able to represent the behavior of the measured structure. For damping identification purposes, it is then clear that properness enforcement on complex vectors must be considered. This operation can be seen as a regularization technique based on physical considerations, instead of using purely mathematical methods. The procedure to enforce properness has been proposed some years ago [7], but unfortunately it is not widely used as it should be. The next section is dedicated to extension of properness for vibroacoustics.
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11.4 Extension of Properness to Vibroacoustics 11.4.1 Equations of Motion Discretizing an internal vibroacoustical problem using the natural fields for the description of the structure (those which can be directly measured), i.e. displacement for the structure and acoustic pressure for the cavity, leads to the matrix system [29] x¨ (t) x˙ (t) Cs 0 + ¨ ˙ 0 Ca p(t) p(t) LT Ma
0
Ms
¨ q(t)
M
+
Ks 0
C
˙ q(t)
Fs (t) −L x(t) = , ˙ a (t) p(t) Ka Q
K
q(t)
(11.29)
f(t)
where x(t) is the vector of generalized displacements of the structure, p(t) is the vector of acoustic pressures, Ms is the mass matrix of the structure, Ma is called “mass” matrix of acoustic fluid (its components are not homogeneous to masses, the name is chosen for analogy with structural denomination), Ks is the stiffness matrix of the structure, Ka is the “stiffness” matrix of fluid domain, L is the vibroacoustic coupling matrix, Cs and Ca respectively represent structural and acoustic losses. This formulation includes the hypothesis that there is no loss at the coupling between structural and acoustic parts, and that internal losses can be represented using equivalent viscous models. Fs (t) is the vector representing the generalized ˙ a (t) is associated to acoustic sources (volume accelforces on the structure, while Q eration) in the cavity. The non-self-adjoint character of the formulation induces difficulties for the resolution of this kind of problem using modal decomposition. Some research works have been done to find symmetric formulations dedicated to coupled vibroacoustic problems [16, 29], but up to now, these formulations are either not able to take into account dissipation in the fluid domain, or lead to full matrices which can not be efficiently used for large models. The technique which is widely used for model reduction in the field of numerical analysis is based on the use of two uncoupled bases (structural and fluid), and the solution of the coupled system is projected on these bases, even if some convergence problems can be found [37]. Being able to evaluate numerically the coupled modal basis in an efficient way is still a challenge, in particular for damped problems. On the other hand, starting from experimental data, it is possible to identify these modes [39], and one of the ways to build reduced models could be to follow the same methodology as the one used in structural dynamics, extended to vibroacoustics.
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11.4.2 Complex Modes for Vibroacoustics The system (11.29) can be solved for steady-state harmonics by modal decomposition. The non-symmetric character of the matrix system implies that right and left modes must be identified. This can be done using the space-state representation of the system ˙ UQ(t) − AQ(t) = F(t),
(11.30)
where U=
C
M
M
0 f(t) . 0
F(t) =
,
−K 0 A= , 0 M
Q(t) =
q(t) , ˙ q(t) (11.31)
The eigenvalues of this problem can be stored in the spectral matrix , =
λj .
(11.32)
The j -th eigenvalue is associated to: T ψ T λ }T . • a right eigenvector, θ Rj such that (Uλj −A)θ Rj = 0, where θ Rj = {ψRj Rj j Storing the eigenvectors (in the same order as the eigenvalues) in the modal matrix R = [ψRT ψRT ]T , the following relationship is verified,
UR = AR ;
(11.33)
T ψ T λ }T . • a left eigenvector θ Lj , such that θ TLj (Uλj − A) = 0, where θ Lj = {ψLj Lj j Storing the eigenvectors (in the same order as the eigenvalues) in the modal matrix L = [ψLT ψLT ]T , the following relationships are verified,
UT L = AT L
or TL U = TL A.
(11.34)
The orthogonality relationships can be written using 2n arbitrary values to build the diagonal matrix ξ = [ ξj ], TL UR = ξ
or TL AR = ξ .
(11.35)
The modal decomposition of the permanent harmonic response at frequency ω is −1 Q(t) = R ξ (iωE2n − ) TL F(ω)eiωt ,
(11.36)
where E2n is the 2n × 2n identity matrix and F(ω) is the complex amplitude of the harmonic excitation. This relationship can also be written using the n degrees of
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freedom notations in the frequency domain as Q(ω) = ψR ψLT f(ω),
(11.37)
1 . ξj (iω − λj )
(11.38)
where =
In the following, without loss of generality, the eigenshapes are supposed to be normalized such that ξj = 1. Each mode has its own response which is proportional to the right eigenvector, with a modal participation vector that includes the scalar product between the left eigenvector and the force exciting the system. In the case of a self-adjoint problem, right and left eigenvectors are equal. The non-self adjoint character of problem (11.1) is particular since extradiagonal coupling terms that appear in mass and stiffness matrices are linked. It can be shown [39] that the left eigenvectors are related to the right ones by the following relationship: Xj Xj If ψRj = then ψLj = , (11.39) Pj −Pj λ−2 j where X corresponds to the structural dofs of the eigenvectors, and P is related to the acoustic dofs. This point is fundamental for modal analysis of coupled system, since only extraction of right eigenvectors is required to derive the left ones. The previous relation can also be written as
X X . (11.40) then ψL = If ψR = P −P−2
11.4.3 Properness for Vibroacoustics The properness condition in the case of a non-self adjoint system can be derived from the orthogonality relationships (11.35): U−1 = R TL , or
C
M
M
0
−1
= =
0
(11.41)
M−1
M−1
−M−1 CM−1 ψR ψLT ψR ψLT
ψR ψLT
ψR 2 ψLT
,
(11.42)
and A−1 = R TL ,
(11.43)
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M. Ouisse and E. Foltête
−1
−K
0
0
M
= =
−K−1
0
0
M−1
ψR −1 ψLT
ψR ψLT
ψR ψLT
ψR ψLT
.
(11.44)
It is then clear that the properness condition for a non-symmetric second order system can be written as ψR ψLT = 0.
(11.45)
Once this relationship is verified, the matrices can be found using the inverse relations −1 M = ψR ψLT , (11.46) −1 K = − ψR −1 ψLT ,
(11.47)
C = − MψR 2 ψLT M .
(11.48)
For the particular vibroacoustic case, left eigenvectors are linked to right ones, and the properness condition can be written using only the right complex eigenvectors, XXT −X−2 PT = 0. (11.49) PXT −P−2 PT
11.4.4 Methodologies for Properness Enforcement 11.4.4.1 Structural Dynamics Based Strategy When the complex modes are available from experimental identification, one can use Eqs. (11.46) to (11.48) in order to find the reduced model which is supposed to have the same behavior as the measured one. The fact is that in general, the modes do not verify the properness condition (11.49). In the particular case of vibroacoustics, one can try to follow the same methodology as the one used in structural dynamics. The following constrained optimization problem should then be solved: ˜ and P˜ minimizing X ˜ − X and P˜ − P Find X while ˜X ˜ T = 0, X
(11.50) ˜ P˜ T = 0, X
˜ −2 P˜ T = 0, X
˜ −2 P˜ T = 0, P
where X and P are two given complex rectangular matrices and is a given diagonal complex matrix. This problem can be re-written using 4 Lagrange multipliers matrices δ j (j = 1 to 4), yielding
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⎧ ⎫ ˜⎬ δ2 ⎨ X
˜ X 1 δ 1 + δ T1 X − + P 2 δ T2 0 ⎩ P˜ ⎭ P˜ ⎧ ⎫ −2 ⎨ ˜ δ3 X ⎬ 1 0 = 0, − T T 2 δ 3 δ 4 + δ 4 ⎩ P ˜ −2 ⎭
(11.51)
˜X ˜ T = 0, X ˜ P˜ T = 0, X ˜ −2 P˜ T = 0, X ˜ −2 P˜ T = 0, P where the overbars correspond to complex conjugates. Solving this problem is clearly not easy because of the presence of the matrices that makes impossible to find explicitly the expression of multipliers versus the unknown vectors. An iterative procedure could be investigated but this is not the best way to obtain quick results that can be used in real-time during modal analysis. Some simplified methods have been proposed [30], among which one is called over-properness: considering the fact that the method developed for structural dynamics [7] is valid for all matrix Y subjected to a properness condition YYT = 0, one can use as Y matrix: ⎤ ⎡ X ⎥ ⎢ (11.52) Y = ⎣ P ⎦, −P−2
yielding
⎡ ⎢ YYT = ⎣
XXT
XPT
PXT
PPT
−2
−P
XT
−P−2 PT
−X−2 PT
⎤
⎥ −P−2 PT ⎦ .
(11.53)
P−4 PT
It can be observed that the four required terms of Eq. (11.49) are included in this matrix, while two of them are not theoretically required. Using this vector in the procedure detailed by Eqs. (11.18)–(11.20) leads to a so-called over-proper solution which includes more constraints than those required, but that includes the required ones.
11.4.4.2 Alternative Strategy Another thinkable way for obtaining matrices of system (11.29) is to use a leastsquare approach. Being given a set of measured frequency responses X corresponding to a set of measured excitations F, the matrices can be found by solving the minimization problem min ε(M, C, K) = −ω2 M + iωC + K X = F , (11.54) (M,C,K)∈A
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where A = M × C × K is the space of admissible matrices (whose topology correspond to a vibroacoustic problem). The function to minimize can be written using a linear system, ε(M, C, K) = Dα − G,
(11.55)
}T ,
where α = {M11 M12 . . . Knn while D includes terms coming from X and ω, and G includes terms coming from F. The matrices components can finally be found using pseudo-inverse for minimization of least-square error, −1 α = DT D DG. (11.56) This strategy can then be used to directly find the matrices without using the complex eigenvectors, which can be found in post processing stage by solving the eigenvalue problem. This approach implies undoubtedly a higher calculation cost than the previous strategies, in particular for systems with numerous degrees of freedom, while in the case of low order reduced models, this strategy could be appropriate.
11.4.5 Numerical Illustration The strategies which have been proposed here can be compared with a direct matrices reconstruction, i.e. without properness enforcement. The first test-case which is proposed here is a very simple 2-dofs numerical model, whose topology is the same as the one given in Eq. (11.29): 3.23 0 1.12 0 x¨ x˙ + −2 −3 p¨ p˙ −1.46 1.27 × 10 0 3.18 × 10
F (t) 1000 1.46 x . (11.57) + = ˙ a (t) 0 1.65 p Q Starting from this system, the complex eigenmodes are evaluated. Some noise is then added to the eigenfrequencies and eigenvectors (5% random noise on frequency, 10% on mode shapes), and the matrices of the system are evaluated using the three approaches: – direct reconstruction from complex modes (without properness enforcement); – reconstruction from complex modes (with over-properness enforcement); – reconstruction from least-square error on FRFs (the FRFs being generated with the noisy eigenvalues in order to keep the same noise level). Finally, the three results are compared by comparing the reconstructed matrices to the original ones (when available) or by plotting FRFs evaluated using each set of matrices. The direct approach, which is exact if no error exists in the identification procedure, is clearly very sensitive to noise, and final matrices can be very different from expected results. The corresponding matrices are:
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Identification of Reduced Models from Optimal Complex Eigenvectors
M=
3.29
−4.00 × 10−4
−1.55
1.31 × 10−2
K=
, 964
C= 1.37
−14.1 1.66
−1.53 −1.02 × 10−2 −1.28
6.10 × 10−3
321
, (11.58)
.
Estimation of mass and stiffness matrices is quite good, while the damping matrix is very badly reconstructed. Some clear improvements can be observed when the properness is enforced. The stiffness and mass matrices are almost unchanged, while the physical meaning of the damping matrix is improved when it is derived from the corrected eigenvectors: 2.62 −5.04 × 10−3 3.30 −4.12 × 10−4 , C= , M= −1.55 1.31 × 10−2 −0.653 −1.21 × 10−4 (11.59) 965 1.37 K= . −14.5 1.66 The negative damping term on the fluid part is balanced with its very small value compared to the (positive) value on the structural part. Finally, the least-square error (LSE) approach leads to a correct topology of matrices, with physical damping terms on both structural and acoustic parts: 3.31 0 0.740 0 M= , C= , −1.44 1.27 × 10−2 0 4.12 × 10−3 (11.60) 982 1.44 . K= 0 1.62 The three strategies can be compared using one of the corresponding FRFs in Fig. 11.11, on which the bad behavior of the direct method can be observed. One can also observe that, even if the topology of the over-proper solution is not exactly the right one, the global error on FRFs reconstruction is lower than in the case of LSE technique. Indeed, depending on the objective, one should evaluate matrices from both formulations and choose the ones which are the most appropriate.
11.4.6 Experimental Test-Case The second test-case which is proposed here corresponds to an experimental testcase based on measurements on a guitar given by F. Gautier from LAUM-Le Mans and J.-L. Le Carrou from LAM-Paris VI. In that case, only two degrees of freedom are considered, in order to represent the behavior of the guitar in the frequency range corresponding to the so-called A0 and T1 modes, which are of first interest in the design of the instrument [13, 18, 25]. The two degrees of freedom which have been
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Fig. 11.11 Methodologies for properness enforcement on numerical test-case
used in these measurements correspond to the structural transverse displacement of a point on the soundboard, and the acoustic pressure in the middle of the sound hole. A small impact hammer has been used for excitation on the structural degree of freedom. These two modes have been identified experimentally by a curve fitting technique, and the FRFs built from these two modes is considered as the reference in the following. The direct approach leads once again to bad estimation of damping terms: −2.23 2.19 × 10−6 3.10 × 10−2 2.10 × 10−9 , C= , M= 3.88 × 10−2 2.85 × 10−7 −3.68 −3.72 × 10−5 (11.61) 2.30 × 104 −3.59 × 10−3 K= . 705 1.28 × 10−5 The properness enforcement allows the damping terms to become more physical: 0.942 −1.52 × 10−6 3.09 × 10−2 1.88 × 10−9 , C= , M= 3.84 × 10−2 2.83 × 10−7 0.315 7.55 × 10−6 (11.62) 4 −3 2.27 × 10 −3.57 × 10 . K= 632 1.26 × 10−5
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Fig. 11.12 Methodologies for properness enforcement on guitar measurements
Finally, the least-square error (LSE) approach leads to a correct topology of matrices, with physical damping term on structural part, but not on the acoustic part: M=
2.91 × 10−2
0
3.44 × 10−2
2.57 × 10−7 K=
,
C=
1.37
0
0
−2.97 × 10−6
2.15 × 104
−3.45 × 10−3
0
1.15 × 10−5
, (11.63)
.
The comparison of FRFs, in Fig. 11.12, leads to the conclusions in accordance with both structural application and vibroacoustical numerical test-case. One can point out the fact that all methodologies lead to quite good estimation of mass and stiffness matrices, the critical point being the evaluation of damping matrix. The properness enforcement is not sufficient to obtain the correct topology, but the improvement is nevertheless clear, it can be seen as a regularization procedure for the inverse problem which is addressed here.
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11.5 Prospects for the Future As far as the structural dynamics applications are concerned, the properness enforcement technique leads to optimal complex modes that can be derived from identified ones. These modes can be efficiently used to reconstruct the system matrices when the full set of vectors is available. This should be considered in any application, since this operation acts as a regularization and helps to identify physical damping matrices. A challenge for the future is clearly to extend this notion for an incomplete set of identified complex vectors. Concerning the vibroacoustic extension, the properness condition has been derived. In this case no explicit solution can be found, obtaining optimal complex modes that verify the properness condition is still a challenge. Of course constrained minimization techniques could be applied, but they would certainly lead to a high calculation cost. The efficiency of the approach in the context of structural dynamics leads to similar expectations for the vibroacoustic case. Nevertheless some more research in this way are required to provide an efficient tool that works in any situation. The alternative way to achieve the expected goal is to extend advanced FRF-based methods referenced in the chapter to vibroacoustic applications. This could possibly lead to good results, since the least-square technique proposed here based on FRF data gives interesting results. This is undoubtedly a promising way to obtain efficient reconstruction of reduced vibroacoustic models.
11.6 Summary Damping matrices identification in the context of structural dynamics, starting from a full modal basis identified by measurements, is a topic which is quite clear today. The inverse procedure is very sensitive to noise on input data (i.e. on identified complex vectors), and some methods are available to provide regularization techniques based on physical considerations. Among them, the properness enforcement technique is undoubtedly very efficient, as shown on illustrative examples. The properness condition can be easily extended to vibroacoustics: this property must be verified by complex modes in order to be those of a physical system. Two techniques have been proposed to enforce the property on eigenshapes that do not verify it, leading to much better results than those corresponding to the use of initial identified vectors. The first technique is based on the structural dynamics procedure, leading to enforcement of more conditions than the theoretically required ones. The second one is based on a least square error minimization. None of the two methods exhibits perfect results, so it is clear that one of the next challenges in vibroacoustic reduced models identification based on experimental modal analysis will be the improvement of the properness enforcement methodology. Up to now, the two proposed methods can be applied for a given application and the user can choose between results depending of the efficiency of the identified reduced model.
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11.7 Selected Bibliography For a deeper insight into damping identification techniques, the following references are suggested. Adhikari and Woodhouse have written a very well documented paper in 4 parts [3–6], which constitutes a starting point for understanding the context and methodologies available for damping identification. The paper from Lin and Zhu [27] can be referenced as a good illustration of the relationship between viscous and hysteretic damping models, in particular to understand that viscous and hysteretic damping matrices are almost equivalent when the damping is distributed on the structure, while a correct choice of the damping model is of first importance for systems with distributed damping. In this context, Xu [40] has proposed an interesting formulation for computing explicit damping matrices for multiply connected, non-classically damped, coupled systems. It is nevertheless clear that most of the methodologies have been developed for viscous or proportional damping. In particular, the paper from Barbieri et al. [8] gives a comparison of three techniques for identification of proportional damping matrix of transmission line cables, and the paper from Pilkey et al. [32], dedicated to viscous damping, investigates some aspects of the damping identification procedure that are noise, spatial incompleteness and modal incompleteness. There are few papers to which the reader is invited to refer concerning identification of non-proportional damping, among which those by Adhikari [2] and Kasai and Link [23]. Almost all the methods that allows identification of damping matrix are related to measurements dofs: the size of the identified matrix is equal to the number of measurement points. The paper from Ozgen and Kim [31] compares two methods that can be used to expand the experimental damping matrix to the size of the analytical model. Acknowledgements The authors would like to thank Jean-Loïc Le Carrou from Laboratoire d’Acoustique Musicale (Paris VI) and François Gautier from the Laboratoire d’Acoustique de l’Université du Maine, for the fruitful discussions and for allowing us to use their measurements data, used in the last part of the chapter.
References 1. Adhikari, S.: Optimal complex modes and an index of damping non-proportionality. Mech. Syst. Signal Process. 18(1), 1–28 (2004) 2. Adhikari, S.: Damping modelling using generalized proportional damping. J. Sound Vib. 293(1–2), 156–170 (2006) 3. Adhikari, S., Woodhouse, J.: Identification of damping: Part 1, viscous damping. J. Sound Vib. 243(1), 43–61 (2001) 4. Adhikari, S., Woodhouse, J.: Identification of damping: Part 2, non-viscous damping. J. Sound Vib. 243(1), 63–88 (2001) 5. Adhikari, S., Woodhouse, J.: Identification of damping: Part 3, symmetry-preserving methods. J. Sound Vib. 251(3), 477–490 (2002)
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