Wave Propagation in Structures (Mechanical Engineering Series) [3rd ed. 2021] 3030596788, 9783030596781

Citation preview

Mechanical Engineering Series

James F. Doyle

Wave Propagation in Structures Third Edition

Mechanical Engineering Series Series Editor Francis A. Kulacki Department of Mechanical Engineering University of Minnesota Minneapolis, MN, USA

The Mechanical Engineering Series presents advanced level treatment of topics on the cutting edge of mechanical engineering. Designed for use by students, researchers and practicing engineers, the series presents modern developments in mechanical engineering and its innovative applications in applied mechanics, bioengineering, dynamic systems and control, energy, energy conversion and energy systems, fluid mechanics and fluid machinery, heat and mass transfer, manufacturing science and technology, mechanical design, mechanics of materials, micro- and nano-science technology, thermal physics, tribology, and vibration and acoustics. The series features graduate-level texts, professional books, and research monographs in key engineering science concentrations.

More information about this series at http://www.springer.com/series/1161

James F. Doyle

Wave Propagation in Structures

Third Edition

James F. Doyle School of Aeronautics & Astronautics Purdue University West Lafayette, IN, USA

ISSN 0941-5122 ISSN 2192-063X (electronic) Mechanical Engineering Series ISBN 978-3-030-59678-1 ISBN 978-3-030-59679-8 (eBook) https://doi.org/10.1007/978-3-030-59679-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 2nd edition: © Springer 1997 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to my father and mother, Patrick and Teresa Doyle. Thanks for the dreams. Had I the heavens’ embroidered cloths, Enwrought with golden and silver light, The blue and the dim and the dark cloths Of night and light and the half light, Would spread the cloths under your feet: But I, being poor, have only my dreams; I have spread my dreams under your feet; Tread softly because you tread upon my dreams. —W.B. Yeats [1]

Preface

The study of wave propagation seems very remote to many engineers, even to those who are involved in structural dynamics. One of the reasons for this is that the examples usually taught in school are either so simple as to be not applicable to real world problems or so mathematically abstruse as to be intractable. This book contains an approach, spectral analysis or frequency domain synthesis, that I have found to be very effective in analyzing waves. What has struck me most about this approach is how I can use the same analytical framework to examine the experimental results as well as to manipulate the experimental data itself. As an experimentalist, I had found it very frustrating having my analytical tools incompatible with my experiments. For example, it is experimentally impossible to generate a step function wave and yet that is the type of analytical solution often available. Spectral analysis is very encompassing—it touches on analysis, numerical methods, and experimental methods. I want this book to do justice to its versatility, so many subjects are introduced. As a result, some areas may seem a little thin, but I do hope, nonetheless, that the bigger picture, the unity, comes across. Furthermore, spectral analysis is not so much a solution technique as it is a different insight into the wave mechanics; consequently, in most of the examples an attempt is made to make the connection between the frequency domain and time domains. In writing the second edition, I strived to keep what was good about the first edition—that is, the combination of experimental and analytical results—but incorporated more recent developments and extensions at that time. The question not fully articulated in the first edition is: What should be different about a book on waves in structures ? This is the question that guided my reorganization of the material as well as the selection of new topics. It had become clearer to me that the essence of a structure is the coupling of systems and, this should be the central theme of a book on waves in structures. There are two readily recognized forms of coupling: mechanical coupling “at the ends” such as when two bars are joined at an angle, and differential coupling as when two bars are connected uniformly along their lengths by springs. Both couplings are intimately related to each other as seen from the example of a curved beam: it can be modeled as a collection of small vii

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straight segments connected end to end, or directly in terms of coupled differential equations. The two approaches, ultimately, give the same results but, at the same time, give quite different insights into the system behavior. The former leads to richer system response functions and its ultimate form is in the spectral element method. The latter leads to richer differential equations, which is manifested in very interesting spectrum relations. The variety of examples were chosen so as to illustrate and elaborate on this dual aspect of coupling. In writing this third edition, again I strived to keep intact what is good about the second edition but add recent developments. The two most important developments over the last two decades is the introduction of engineered materials in the form of metamaterials and nanostructures. They add a new level to the meaning of structure. The other development is the almost universal adoption of finite element (FE) methods as the “go to” tool for solving structural dynamics problems. I see spectral analysis as a tool to give insight into data be it obtained from physical experiment or computer experiment. In this edition, some tools are developed to extend the postprocessing of FE data using spectral analysis ideas. The organization of the chapters is mostly similar to that of the second edition, but each chapter is revised and updated with some new examples and references to the literature. One completely new chapter has been added. Chapter 9 deals with discrete and discretized structures. Spectral analysis methods are a very natural way to analyze the behavior of these systems. Examples are taken from nanotechnology and molecular dynamics. In recent years, desktop computers have become incredibly powerful and very affordable. Thus, problems that would not be tackled in the past can be accomplished very quickly, and solution schemes rejected in the past are now feasible. For example, in the first edition, the double summation was barely introduced, but now it plays a central role in the dynamics of plate and shell structures. Similarly, the eigenvibration analysis of large structures was limited to the lowest modes but now can be applied to the very high-frequency modes of relevance to wave propagation in discrete structures. Therefore, not only is the use of a computer implicit in all the examples, the solution strategies and techniques are also computer oriented. In a similar vein, I have tried to supplement each chapter with a collection of pertinent problems plus specific references that can form the basis for further study. A book like this is impossible to complete without the help of many people, but it is equally impossible to properly acknowledge all of them individually. However, I would like to single out helpers on the first and second editions: Brian Bilodeau, Albert Danial, Sudhir Kamle, Lance Kannal, Matt Ledington, Mike Martin, Steve Rizzi, Gopal Srinivasan, and Hong Zhang. Thank you guys. The errors and inaccuracies in all editions have been purely my own doing. West Lafayette, IN, USA August, 2020

James F. Doyle

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 6

1

Spectral Analysis of Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Continuous Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Fast Fourier Transform Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Space Distributions Using Fourier Series . . . . . . . . . . . . . . . . . . . . 1.2 Applications Using the FFT Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Explorations of the Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Experimental Aspects of Wave Signals. . . . . . . . . . . . . . . . . . . . . . . 1.3 Spectral Analysis of Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 General Functions of Space-Time and Spectrum Relations . 1.3.2 Some Wave Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Group Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Summary of Wave Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Propagating and Reconstructing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Integration of Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 12 18 20 22 23 26 31 31 34 37 39 40 40 44 46 47

2

Longitudinal Waves in Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Elementary Rod Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Equation of Motion and Spectral Analysis . . . . . . . . . . . . . . . . . . . 2.1.2 Basic Solution for Waves in Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dissipation in Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Distributed Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Viscoelastic Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Coupled Thermoelastic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 53 56 56 59 61 62 ix

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2.3.2 The Spectrum Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Blast Loading of a Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Reflection and Transmission of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Reflection from an Elastic Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Reflection from an Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Concentrated Mass Connecting Two Rods . . . . . . . . . . . . . . . . . . . 2.4.4 Interactions at a Distributed Elastic Joint . . . . . . . . . . . . . . . . . . . . 2.5 Distributed Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Periodically Extended Load Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Connected Waveguide Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 65 68 69 71 72 74 76 76 79 80 82

3

Flexural Waves in Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Bernoulli–Euler Beam Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Equations of Motion and Spectral Analysis . . . . . . . . . . . . . . . . . . 3.1.2 Basic Solution for Waves in Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Beam with Axial Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Bernoulli–Euler Beam with Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Beam on an Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Coupled Beam Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Reflection and Transmission of Flexural Waves . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Reflections from Simple Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Reflections and Transmissions at a General Joint . . . . . . . . . . . . 3.4 Curved Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Deformation of Curved Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Spectrum Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Impact of a Curved Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 89 93 94 95 98 104 105 107 112 112 115 119 121 122

4

Higher Order Waveguide Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Waves in Infinite and Semi-Infinite Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Navier’s Equations and Helmholtz Potentials . . . . . . . . . . . . . . . . 4.1.2 Constructing Potentials Appropriate for Boundaries . . . . . . . . 4.1.3 Forced Response of a Semi-Infinite Plane. . . . . . . . . . . . . . . . . . . . 4.1.4 Free-Edge Waves: Rayleigh Surface Waves . . . . . . . . . . . . . . . . . . 4.2 Waves in Doubly Bounded Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Forced Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Spectrum Relations for Free-Wave Responses . . . . . . . . . . . . . . . 4.2.3 Discussion of Lamb Waves in Structural Waveguides . . . . . . . 4.3 Variational Formulation of Dynamic Equilibrium . . . . . . . . . . . . . . . . . . . . 4.3.1 Work and Strain Energy in a General Body . . . . . . . . . . . . . . . . . . 4.3.2 Virtual Work and Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Illustrative Application of the Ritz Semi-direct Method. . . . .

123 123 124 127 132 135 137 137 141 145 146 146 148 152

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4.4

Refined Beam Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Timoshenko Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Adjustable Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Spectrum Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Impact of a Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Refined Rod Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Love One-Mode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Mindlin–Herrmann Two-Mode Model . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Three-Mode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 N-Mode Rod Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Kinematic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Spectrum and Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 158 159 161 163 164 167 169 173 174 177 180 182

The Spectral Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Structures as Connected Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Spectral Element for Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Dynamic Stiffness for Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Nodal Representation of Distributed Loadings . . . . . . . . . . . . . . 5.3 Spectral Element for Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Dynamic Stiffness for Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 General Frame Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Member Stiffness Matrix Referred to Global Axes . . . . . . . . . . 5.4.2 Structural Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Some Programming Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Periodic Structures and the Transfer Matrix . . . . . . . . . . . . . . . . . 5.4.6 Long Truss Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Spectral Super-Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Formulation of a Spectral Super-Element . . . . . . . . . . . . . . . . . . . . 5.5.2 Assemblage of Super-Element Stiffnesses . . . . . . . . . . . . . . . . . . . 5.5.3 Example of an Angle Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Recovery of Internal Displacements and Forces . . . . . . . . . . . . . 5.5.5 Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Impact Force Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Wave Propagation Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Frequency Domain Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Examples of Force Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 187 188 189 192 194 195 195 199 201 202 204 205 206 208 212 213 214 217 221 222 224 226 226 228 230 233 234

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Contents

Waves in Plates and Cylinders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Models of Plates in Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mindlin Plate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Flexural Behavior of Very Thin Plates . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Spectral Analysis of Thin Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Constructing Simpler Waveguide Models . . . . . . . . . . . . . . . . . . . . 6.2 Arbitrarily Crested Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Point Impact of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Point Impact Using Wavenumber Summations (pEL Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Reflection and Scattering of Flexural Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Waves Reflected from a Straight Edge . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Free-Edge Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Scattering of Flexural Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Waves in Cylinders and Curved Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Deformation of Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Wave Propagation Along a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Curved Plate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Spectrum Relation for Propagation in the Hoop Direction . . 6.4.5 Donnell Shell Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 238 241 244 250 254 255 260 263 263 268 270 275 275 280 283 286 288 291 292

7

Thin-Walled Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Spectral Elements for Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Membrane Spectral Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Flexure Spectral Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 A Simple Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Folded Plate Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Structural Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Computer Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Structural Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Spectral Elements for Curved Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Impact of an Infinite Curved Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 General Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Dynamic Stiffness Relation for a Curved Shell Element . . . . 7.3.4 Point Loading of a Complete Cylinder . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295 295 296 301 305 307 307 309 310 315 316 318 321 323 325 325

8

Structure–Fluid Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Plate–Fluid Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Linearized Acoustic Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Incident Plane Wave on a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327 327 328 332

Contents

9

xiii

8.2

Panel Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Line Loading of an Infinite Plate in a Fluid . . . . . . . . . . . . . . . . . . 8.2.2 Double Panel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Cylindrical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Comparison of Fluid Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Waveguide Modeling of Distributed Pressures. . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Free Wave Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Modified Spectrum Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Radiation from Finite Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Finite Plate Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Fluid Response from a Finite Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Far-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

336 336 338 342 345 347 347 351 353 353 355 357 359 360

Discrete and Discretized Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Wave Propagation in 1D Discrete Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Beaded String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Two-Mass String. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Chains with Internal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Wave Propagation in Anisotropic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Elastic Anisotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 2D Anisotropic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Atomic Lattice Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Lennard-Jones Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Atomistic Models of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Embedded Atom Models (EAM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Continuum Models for Discrete Systems . . . . . . . . . . . . . . . . . . . . 9.4 Spectral Analyses of FE Discretized Systems . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Spectrum Relations from Element Stiffness . . . . . . . . . . . . . . . . . 9.4.2 Real-Only Spectrum Relations and Spectral Shapes. . . . . . . . . Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 361 362 365 370 374 375 379 382 383 387 391 395 400 401 404 409 409

Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Bessel Equations and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Limiting Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413 413 413 415 417

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Notation

Roman letters a A ˆ B, ˆ C, ˆ D ˆ A, b B cg co cP , cS , cR CE D, D¯ E, E ∗ , Eˆ EI F gˆ i (x) ˆ G(t), G h Hn i I In Jn k, k1 , k2 kx , ky , kz K [ k ], [ K ] Kn

Radius Cross-sectional area Frequency dependent coefficients Thickness, depth Bulk modulus Group speed √ Longitudinal wave speed, EA/ρA Primary, Secondary, and Rayleigh wave speeds Specific heat Plate stiffness, D = Eh3 /12(1 − ν 2 ), D¯ = GH 3 /12 Young’s modulus, E ∗ = E/(1 − ν 2 ), viscoelastic modulus Beam flexural stiffness Member axial force Element shape functions Structural unit response, frequency response function Beam or rod height, plate thickness Hankel function, Hn = Jn ± iYn √ Complex −1 Second moment of area, I = bh3 /12 for rectangle Modified Bessel functions of the first kind Bessel functions of the first kind Waveguide spectrum relations 2-D wavenumbers Stiffness, thermal conduction Stiffness matrix Modified Bessel functions of the second kind

xv

xvi

L M, Mx n N p(t), pˆ P (t), Pˆ qu , qv , qw q r R t T u(t) u, v, w V W x, y, z Yn

Notation

Length of element, distance to boundary Moment Frequency counter Number of terms in transform Acoustic pressure Applied force history Distributed load Heat flux Radial coordinate Radius Time Time window, temperature Response; velocity, strain, etc. Displacements Member shear force Space transform window Rectilinear coordinates Bessel functions of the second kind, shape functions

Greek letters α β ij k δij η κ θ ν μ λ ρ σ,  ξ φ, φx , φy , Hz ψ ωn ωc

Coefficient of thermal expansion [ω2 ρA/EI ]1/4 , [ω2 ρh/D]1/4 Permutation symbol Kronecker delta, small quantity Determinant Viscosity, damping Plate curvature Angular coordinate Poisson’s ratio Shear modulus Lamê constant Mass density Stress, strain Space transform variable Rotation Helmholtz functions Lateral contraction, stress function Angular frequency Cut-off, coincidence, critical frequency

Notation

xvii

Special Symbols Rn ∇2

Random noise Differential operator,

V [ ] { }

Volume Square matrix Vector

∂2 ∂x 2

+

∂2 ∂y 2

Subscripts a m n 1, 2 P , S, R ,

Acoustic medium Space wavenumber counter Frequency counter Sensors, modes Primary, Secondary, and Rayleigh waves (Comma) derivative with respect to indicated variable

Superscripts ∗ ¯ ˙ ˆ ˜ 

Complex conjugate Bar, local coordinates Dot, time derivative Frequency domain (transformed) quantity Wavenumber domain (transformed) quantity Prime, derivative with respect to argument

Abbreviations BC, pBC DoF, SDoF CFT CST DKT EAM EoM EVP FB FE FRF FST MD MRT pEL TMM, NMM

Boundary condition, periodic BC Degree of freedom, single DoF Continuous Fourier transform Constant strain triangle element Discrete Kirchhoff triangular FE element Embedded atom model Equation of motion Eigenvalue problem Free body Finite element Frequency response function Fourier series transform Molecular dynamics Membrane with rotation triangular FE element Periodically extended load Three-mode model, N-mode model

xviii

Reference 1. Yeats, W.B.: Collected Poems of W.B. Yeats. Macmillan and Co., London (1960)

Notation

Introduction

We must gather and group appearances, until the scientific imagination discerns their hidden laws, and unity arises from variety; and then from unity we must re-deduce variety, and force the discovered law to utter its revelations of the future. W.R. Hamilton [4]

This book is an introduction to the spectral analysis method as a means of solving wave propagation problems in structures. The emphasis is on practical methods from both the computational and applications aspects, and reference to physical and computer FE experimental results is made whenever possible. While it is possible to solve structural dynamics problems by starting with the partial differential equations of motion and integrating, the task is horrendously large even for the biggest computers available. This would not be a good idea anyway because (and this is a point very often overlooked) a useful solution to a problem is one that also puts organization and coherence onto the results. It is not sufficient to be able to quote, say, the strain history at some location or even at thousands of locations; the results must be placed in some higher-order context, be seen as part of some larger unity. Notice that this aspect of the problem is present even when interpreting experimental results and is not just associated with analysis. One of the goals of this book is to provide such a unifying framework for the analysis of waves in structures. By consistently using the spectral analysis method for all problems a unity emerges. This unity is not only in the formulation but (when coupled with the fast Fourier transform FFT) is also among the formulation, the solution procedure, the solutions themselves, and the post-manipulations of the results.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8_1

1

2

Introduction

What Is Spectral Analysis? Over the years many analytical techniques have been developed for treating wave propagation problems. Central among these is the method of Fourier synthesis (or spectral analysis), where the behavior of a signal is viewed as a superposition of many infinitely long wave trains of different periods (or frequencies). The actual response is synthesized by a judicious combination of these wave trains. Thus the problem of characterizing a signal is transformed into one of determining the set of combination coefficients. These coefficients are called the Fourier transform of the signal. While the problem being tackled invariably simplifies when it is expressed in terms of the Fourier transform (Sneddon’s book [6], for example, shows a wide range of applications to both static and dynamic problems) the last step in the analysis involves performing an inverse transform (reconstructing the signal) and this, generally, is very difficult to do in an exact analytical manner. Consequently, many approximate and asymptotic schemes have usually been resorted to. These are quite adequate for determining the remote behavior (as is required in seismology, say) but can lose too much information when applied to structural impact problems. It should also be pointed out that analytical transforms are feasible only if the function to be transformed is mathematically simple—unfortunately this is not the case in any situation of practical interest and is certainly not true when dealing with experimental data. This inversion problem is the biggest impediment to a more widespread use of the transform methods. Spectral analysis forms the basis of this book, but the approach is different from the classical method in that, from the outset, the transforms are approximated by the discrete Fourier transform (DFT). In contrast to the continuous transform, this represents the signal by a finite number of wave trains and has the enormous advantage in that the fast Fourier transform (FFT) computer algorithm can be used for economically computing the transforms. Being able to do transforms and inversions quickly adds great heuristic value to the tool in which the waves can be actually “seen” and iterated on, and realistic signals (even ones that are experimentally based) can be treated. It should be pointed out that while the method uses a computer, it is not a numerical method in the usual sense, because the analytical description of the waves is still retained. As a consequence, the very important class of problems called inverse problems can be tackled. The approach presented in this book takes advantage of many of the techniques already developed for use in time series analysis and for the efficient numerical implementation of them. (Chatfield’s book [1], for example, is a readable introduction to the area of signal processing.) In fact, this aspect of spectral analysis is really part of the more general area of digital processing of the signals and herein lies one of the unifying advantages of the present approach—the programming structure is then already in place for the subsequent post-processing of the data. This is especially significant for the manipulation of experimental data.

Introduction

3

Structures and Waveguides A structure can be as simple as a cantilevered diving board or as complicated as an airplane fuselage constructed as a combination of thin plates and frame members. We view such structures as a collection of waveguides with appropriate connectivities. A waveguide directs the flow of wave energy and, in its elementary form, can be viewed as a hydraulic network analog, however, the entities transported are more complicated than water or oil. Perceiving a structural member as a waveguide is not always an easy matter. For example, it is reasonable to expect that a narrow bar struck along its length conducts longitudinal waves and intuition says that if it is struck transversely it generates flexural motion. However, on closer examination it turns out not to be that simple. When the bar is first impacted transversely, the waves generated propagate into a semi-infinite body and behave as if there is only one free surface. Only after some time has elapsed do the waves experience the other lateral surface, where they then reflect back into the body. On a time scale comparable to many transits of the wave, it is seen that a particle at some location further down the guide experiences a complex superposition of the initial wave plus all the new waves generated by reflections. This obviously is not flexural motion. The question then is: At what stage (both time and position) does the response resemble a flexural wave? The answer comes in two parts. First, it can be said that the transition depends on such factors as the duration of the pulse, the distance between bounding surfaces, and the transit time. That is, the longer the pulse and the smaller the depth, the sooner (both in time and position) the response resembles a flexural wave. However, it never does become a flexural wave. This leads to the other part of the answer. The point and success of waveguide analysis is to forego a detailed analysis of the waves and replace the three-dimensional model by a simpler one that has embedded in it the essential characteristics of the behavior as well as a reasonable approximation of the lateral boundary conditions. This model usually involves resultants on the cross section and is valid (within itself) for all time and positions not just at large times and distances. There are various schemes for establishing the waveguide model that ranges from the purely ad hoc “Strength of Materials” approaches, to reduced forms of the 3D equations, to using exact solutions; Redwood [5] gives a good survey of waveguide analysis for both solids and fluids. The approach taken in this book is to begin the analysis using an elementary model and then to add complexity to it—the formal procedure is via Hamilton’s variational principle combined with the Ritz method. This approach has the advantage of being quite intuitive because it is based on a statement of the deformation; in addition, it can show the way to approach as yet unformulated problems. Some exact solutions for waveguides are also developed. Generally, these are too cumbersome to be of direct use in structural dynamics, but they do aid considerably in gaining a deeper understanding of the nature of approximation used in the more familiar waveguide models.

4

Introduction

Wave Propagation and Vibrations A major aspect of this book is the persistent treatment of the effect of boundaries and discontinuities on the waves because real structures have many such terminations. This can be done efficiently because the quantities used in the analysis of the waveguide are also used to set up the connectivity conditions. As a result, wave solutions for structures more interesting than simple rods and beams can be pieced together successfully. Moreover, the way to solve problems of structures with many members and boundaries is then available. A connection not often investigated when studying structural dynamics is the relationship between wave propagation and vibrations. For many engineers, these are two separate areas with quite distinct methods of analysis. However, another advantage of the spectral approach to dynamics is that the close connection between waves and vibrations becomes apparent. Even the same language can be used, terms such as power spectral density, filtering, spectral estimation, convolution, and sampled waveforms have the same meaning. Consequently, many of the technologies developed over the last four decades for vibrations and modal analyses are directly applicable to the spectral analysis of waves. An exciting possibility (and one of the motivations for writing this book) is to facilitate the reverse process, that is, to transfer many of the wave ideas into vibration analysis. This should lead to a richer understanding of such topics as impulse testing, transient vibrations, and filtering in periodic structures. A number of examples throughout the book demonstrate the “evolution” of resonance as multiple reflections are included in the analysis.

Themes and Threads This book concentrates on wave propagation in the basic structural elements of rods, beams, and plates. These form a rich collection of problems and the intent is to show that they all can be analyzed within the same framework once the spectral analysis approach is adopted. Because of the structural applications (and because all structures are finite in extent), a primary theme is the interaction of the wave with discontinuities such as boundaries, junctions, and attachments. Supplemental themes involve the construction of the mechanical models, and the duality between the time and frequency domains. While the material of each chapter is reasonably self-contained, Graff’s book [3] can be used as an excellent supplemental reference on elastic wave propagation. The book by Elmore and Heald [2] gives a broader and simpler introduction to waves. Chapter 1 recapitulates the essence of the continuous Fourier transform and its approximation in the form of the discrete Fourier series. The factors affecting the quality of the approximation (or spectral estimate) are elucidated. It also discusses, in a general way, how spectral analysis can be used to solve differential equations

Introduction

5

and especially those associated with wave motion. Two concepts of significance emerge from this analysis. One is the idea of the spectrum relation (that unique relation between the frequency and the wavenumber) and is essentially the transform equivalent of the space-time differential equation. The other is that of multi-mode solutions. These are shown to play a fundamental role in the solution of actual boundary value problems even though all are not necessarily propagating modes. The following two chapters deal, respectively, with rods and beams, in nearly the same format. First, the governing differential equations are derived and then, by spectral analysis, the kernel solutions and spectrum relations are obtained. Having initiated a wave, how it interacts with structural discontinuities is then investigated. Each chapter includes an example of coupling both at the differential and mechanical levels. Chapter 4 addresses the question of the adequacy of the waveguide models and a foundation for the construction of higher-order waveguides is provided that readily allows extensions to be made to other specialized problems. Chapter 5 introduces the spectral element method as a matrix method approach to structural dynamics that combines aspects of the finite element method with the spectral analysis method. In a way, this chapter is the culmination of the connected waveguide approach and clearly makes the study of wave propagation in complicated frame structures practical. Chapter 6 expands the analysis to multiple dimensions to cover flexural wave propagation in plates and cylinders. Chapter 7 begins the development of a matrix methodology for plated structures including cylindrical shells. The final two chapters extend the spectral analysis to some specialist areas. Chapter 8 deals with the problem of plate/fluid interactions; Chap. 9 is new to this edition and focusses on discrete systems (atoms, for example), discretized systems (FE models, for example), and periodic structures (grilles and frame, for example). The examples are drawn from metamaterials, nanotechnology and focus on filtering actions. A number of threads run through all the chapters. The spectral methodology is an ideal companion for experimental analysis, and throughout the chapters summaries of some of its experimental applications are given. The emphasis is on how spectral analysis can extend the type of information extracted from the experimental data— for example, how a structural response can be used to infer the force history causing it. Another thread is the integration of finite element (FE) methods into the spectral analyses. They are used to enhance the examples through what are called computer experiments and vice versa how spectral analysis can be used to enhance and explain FE results. An important thread is the role played by elastic constraints. This is first introduced in connection with the simple rod encased in an elastic medium but reappears with increasing complexity in the subsequent chapters. Its ultimate expression is found in the coupled modes of the curved plate. Admittedly, a large range of problems have been left out even though most of them are treatable by the spectral methods. Consequently, an effort is made to supplement each chapter with a collection of pertinent problems plus specific references that indicate extensions of the modeling and the applications.

6

Introduction

A Question of Units The choice of any particular set of units is bound to find disfavor with some readers. Nondimensionalizing all of the plots is an unsatisfactory solution especially because experimental results are included. Furthermore, working with dimensional quantities help give a better “feel” for the results. The resolution of this dilemma adopted here is to do most of the examples using a nominal material with nominal properties. In the two major systems of units, these are given in the following table: Property Material Young’s Modulus, E Shear Modulus, G Mass Density, ρ Bar height, h Bar depth, b Bar length, L Plate thickness, h

Common (US) Aluminum 10 × 106 psi 4 × 106 psi 0.25 × 10−3 lb·s2 /in.4 1 in. 1 in. 100 in. 0.1 in.

SI Aluminum 70 GN/m2 28 GN/m2 2800 kg/m3 25 mm 25 mm 2500 mm 2.5 mm

The choice of shear modulus results in a Poisson’s ratio of ν = 0.25. One of the recurring material parameters is the ratio  co =

E = 200 × 103 in./s = 5000 m/s ρ

When a natural nondimensionalizing factor, such as co or the acceleration g, presents itself we use it. In some examples, we deviate from these nominal values and the dimensions are then made explicit. In other examples, where it is the relative waveform (shape) and not the absolute magnitude that is important, we do not state the units at all. Where ever possible, when FE results are quoted the complete specification of the problem (mesh density, BCs, and so on) is given.

References 1. Chatfield, C.: The Analysis of Time Series: An Introduction. Chapman and Hall, London (1984) 2. Elmore, W.C., Heald, M.A.: Physics of Waves. Dover, New York (1985) 3. Graff, K.F.: Wave Motion in Elastic Solids. Ohio State University Press, Columbus (1975) 4. Hamilton, W.R.: The Mathematical Papers of Sir W.R. Hamilton. Cambridge University Press, Cambridge (1940) 5. Redwood, M.: Mechanical Waveguides. Pergamon Press, New York (1960) 6. Sneddon, I.N.: Fourier Transforms. McGraw-Hill, New York (1951)

Chapter 1

Spectral Analysis of Wave Motion

It has long been known that an arbitrary time signal can be thought of as the superposition of many sinusoidal components, that is, it has a distribution or spectrum of components. Working in terms of the spectrum is called spectral analysis. In wave analysis, the time domain for a motion or response is from minus infinity to plus infinity. Functions in this domain are represented by a continuous distribution of components which is known as its continuous Fourier transform (CFT). However, the numerical evaluation and manipulation of the components require discretizing the distribution in some manner—the one chosen here is by way of the discrete Fourier transform (DFT). This has the significant advantage that it allows the use of the very efficient fast Fourier transform (FFT) computer algorithm. The goal of this chapter is to introduce the discrete Fourier transform for the efficient computation of the spectral content of a signal. Possible sources of errors in using it on finite samples and ways of reducing their influences are discussed. The Fourier transform is then applied to wave analysis. The crucial step is to set up the connection between the spectral responses at different space locations—we do this through the governing differential equations. In doing so, certain key ideas emerge which recur throughout this book; central among these ideas are that of a wave mode, its spectrum relation, and its phase and group speeds. In Fig. 1.1, movement of the amplitude corresponds to the group speed, movement of the zero crossings correspond to the phase speed.

1.1 Fourier Transforms The continuous transform is a convenient starting point for discussing spectral analysis because of its exact representation of functions. Only its definition and basic

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8_2

7

8

1 Spectral Analysis of Wave Motion

Fig. 1.1 Response showing the difference between the group and phase speeds. Movement of the envelope corresponds to group behavior while movement of the zero crossings corresponds to phase behavior

properties are given here; a more complete account can be found in Refs. [20, 21]. Background information on time series analysis can be found in Ref. [5]. The continuous Fourier transform is a powerful technique but has the drawback that the functions (signals) must be known analytically over the complete domain. This occurs in only rare cases making it unsuitable for practical situations, especially if the signals are experimental in origin. Discrete Fourier transforms are more suitable for our purposes.

1.1.1 Continuous Fourier Transforms The continuous Fourier transform pair of a function F (t), defined on the time domain from −∞ to +∞, is given as  2π F (t) =

∞ −∞

+iωt ˆ dω , C(ω)e

ˆ C(ω) =



∞

−∞

F (t)e−iωt dt

(1.1)

ˆ where C(ω) is the continuous Fourier transform (CFT), ω is the angular frequency, √ and i is the complex −1. The first form is the inverse transform while the second is the forward transform—this arbitrary convention arises because the signal to be transformed usually originates in the time domain. The factor of 2π is necessary so that a sequential use of the forward and inverse transforms recovers the original function. However, it should be pointed out that other forms for this factor can be found in the literature. The process of obtaining the Fourier transform of a signal separates the waveform ˆ into its constituent sinusoids (or spectrum) and thus a plot of C(ω) against frequency represents a diagram displaying the amplitude of each of the constituent sinusoids.

1.1 Fourier Transforms

9

ˆ The spectrum C(ω) usually has nonzero real and imaginary parts. We use the overhead “hat” to indicate the frequency domain spectrum of a function. By way of a simple example of the application of Fourier transforms, consider a rectangular pulse where the time function is given by F (t) = Fo

− a/2 ≤ t ≤ a/2

and zero otherwise. Substituting into the forward transform and integrating give    +a/2 sin (ωa/2) ˆ ≡ Cˆ o (ω) Fo e−iωt dt = Fo a C(ω) = ωa/2 −a/2 In this particular case the transform is real-only and symmetric about ω = 0 as shown in Fig. 1.2. The term inside the braces is called a sinc function, and has the characteristic behavior of starting at unity magnitude and oscillating with decreasing amplitude as its argument increases. It is noted that the value of the transform at ω = 0 is the area under the time function. This, in fact, is a general result as seen from  ∞  ∞ ˆ C(0) = F (t)e−i0t dt = F (t)dt −∞

−∞

ˆ A step function is therefore improper because C(0) is infinite. When the pulse is displaced along the time axis such that the function is given by F (t) = Fo

t o ≤ t ≤ to + a

and zero otherwise, the transform is then

real imag

0

to

to+a

to=-a/2

to=-a/5 to=+a/10

-100.

-50.

0.

50.

Fig. 1.2 Continuous transform of a rectangular pulse with a = 50 µs

100.

10

1 Spectral Analysis of Wave Motion

  sin(ωa/2) −iω(to +a/2) ˆ e = Cˆ o (ω)e−iω(to +a/2) C(ω) = Fo a ωa/2

(1.2)

which has both real and imaginary parts and is not symmetric with respect to ω = 0. On closer inspection, however, we see that the magnitudes of the two transforms are the same; it is just that the latter is given an extra phase change of amount ω(to + a/2). Figure 1.2 shows the transform for different amounts of shift. We therefore associate phase changes with shifts of the signal along the time axis. If a pulse is visualized to be at different positions relative to the time origin, then the amplitude of the spectra will be the same, but each will have a different phase change. That is, movement in the time domain causes phase changes in the frequency domain. Investigating these phase changes is the fundamental application of spectral analysis to wave propagation and is pursued later in this chapter. For completeness, we now summarize some of the major properties of Fourier transforms; more detailed accounts can be found in Ref. [21]. In all cases, the results can be confirmed by taking the example of the rectangular pulse and working through the transforms long hand. To aid in the summary, we refer to the transform pair ˆ F (t) ⇔ C(ω) where the symbol ⇔ means “can be transformed into.” The double arrowheads reinforce the idea that the transform can go in either direction and that its properties are symmetric. If the functions FA (t) and FB (t) have the transforms Cˆ A (ω) and Cˆ B (ω), respectively, then the combined function [FA (t) + FB (t)] has the transform [Cˆ A (ω) + Cˆ B (ω)]. That is, FA (t) + FB (t) ⇔ Cˆ A (ω) + Cˆ B (ω)

(1.3)

This is an essential property of the transform and it means that if a signal is composed of the simple sum of two contributions (say an incident wave and a reflected wave), then the transform is also composed of a simple sum of the separate transforms. This linearity property is at the heart of superposition. The function F (at) (where a is a nonzero constant) has the transform pair F (at) ⇔

1 ˆ C(ω/a) |a|

(1.4)

indicating a reciprocal scaling relationship for the arguments. That is, time domain compression corresponds to frequency domain expansion (and vice versa). For example, if the width of a pulse is made narrower in the time domain, then its extent in the frequency domain is made broader. It should also be noted that the amplitude decreases because the energy is distributed over a greater range of frequencies. If the function F (t) is time shifted by to , then it has the transform pair

1.1 Fourier Transforms

11 −iωto ˆ F (t − to ) ⇔ C(ω)e

(1.5)

This property was already seen in the last example where the rectangle was displaced from the origin. Of course, the property refers to any change at any position. There is a corresponding relation for frequency shifting. ˆ The transform pairs, Eqs. (1.1), are valid for both F (t) and C(ω) being complex, but the functions of usual interest in wave analysis are when F (t) is real. To see the effect of this, rewrite Eq. (1.1) in terms of its real and imaginary parts as  2π FR =  2π FI =

[Cˆ R cos ωt − Cˆ I sin ωt] dω , Cˆ R =

 [FR cos ωt + FI sin ωt]dt

 ˆ ˆ ˆ [CR sin ωt + CI cos ωt]dω , CI = − [FR sin ωt − FI cos ωt]dt

where the following decomposition was used cos θ = 12 [eiθ + e−iθ ] ,

sin θ = −i 12 [eiθ − e−iθ ]

(1.6)

From the above it is apparent that when F (t) is real-only, Cˆ R is even and Cˆ I is odd. Mathematically, this is expressed as Cˆ R (−ω) = Cˆ R (ω) , Cˆ I (−ω) = −Cˆ I (ω) or

ˆ C(−ω) = Cˆ ∗ (ω)

(1.7)

which says that the functions are symmetrical and antisymmetrical, respectively, about the zero frequency point. This can also be expressed as saying that the negative frequency side of the transform is the complex conjugate of the positive side. A very interesting property arises in connection with the products of functions. Consider the transform of two time functions  ˆ C(ω) = FA (t)FB (t)e−iωt dt Using the inverse transform relation for FA gives ˆ C(ω) =



¯ ¯ +i ωt d ω¯ FB (t)e−iωt dt Cˆ A (ω)e

This can be further rearranged as ˆ C(ω) =



¯ Cˆ A (ω)

 FB (t)e

−i(ω−ω)t ¯

 dt d ω¯ =

Cˆ A (ω) ¯ Cˆ B (ω − ω) ¯ d ω¯

and is expressed as the transform pair  FA (t)FB (t) ⇔

Cˆ A (ω) ¯ Cˆ B (ω − ω) ¯ d ω¯

(1.8)

12

1 Spectral Analysis of Wave Motion

This particular form is called a convolution. We use in explaining the effects of sampling and filtering on the computed transforms. For example, a signal truncated in the time domain can be thought of as the product of the original signal with the truncating function. The result in the frequency domain is then no longer a simple representation of either. There is a similar relation for products in the frequency domain, namely Cˆ A (ω)Cˆ B (ω) ⇔

 FA (τ )FB (t − τ ) dτ

(1.9)

This shows that a time domain convolution can be performed as frequency domain multiplication. Even though this involves both a forward and inverse transform it is computationally faster than performing the convolution directly. All the mechanical systems considered in the later chapters can be represented in the frequency domain as products of the input times the system response. In fact, the reason why the frequency domain is so useful for analyzing these systems is because the complicated convolution relations become simple algebraic relations.

1.1.2 Discrete Fourier Transform We discretize the CFT in two steps: first the frequency integrals are discretized giving the Fourier series transform (FST), and then the time integrals are discretized giving the Discrete Fourier transform (DFT). Let the time function F (t) be known only over a period T ; to apply the continuous Fourier transform to it, we must extend it somehow to the infinity limits. In the Fourier series representation, the function is assumed extended to plus and minus infinity as a periodic function with the period T . We can view this either as separate functions of duration T placed one after the other or as the superposition of separate functions of infinite duration but with nonzero behavior only over the period T . We adopt this latter view as shown in Fig. 1.3.

Fig. 1.3 Periodic extended signal as a superposition of infinite signals

1.1 Fourier Transforms

13

We saw from the rectangular pulse example that if Cˆ ∞ (ω) is the transform of a pulse, the transform of the same pulse shifted an amount T is Cˆ ∞ (ω)e−iωT . It is obvious therefore that the transform of the periodic signal can be represented as ˆ C(ω) = Cˆ ∞ (ω)[· · · + e+iω2T + e+iωT + 1 + e−iωT + e−iω2T + · · · ] This transform shows an infinite peak whenever the frequency is one of the discrete values ωn = 2π n/T . Under this circumstance, each of the exponential terms is unity and there is an infinity of them. (This result should not be surprising since the ˆ C(0) component is the area under the curve, and for the periodic rectangle of Fig. 1.3 we see that it is infinite.) At other frequencies, the exponentials are as likely to be positive as negative and hence their sum will be relatively small. Therefore our first conclusion about the transform of a periodic extended signal is that it shows very sharp spectral peaks. We can go further and say that the transform is zero everywhere except at the discrete frequency values ωn = 2π n/T where it has an infinite value. We represent this behavior by use of the delta function, δ(x); this special function is zero everywhere except at x = 0 where it is infinite. It has the very important additional property that its integral over the whole domain is unity. Thus ˆ C(ω) = Cˆ ∞ (ω)A[· · · + δ(ω + 2π/T ) + δ(ω) + δ(ω − 2π/T ) + · · · ]   δ(ω − 2π n/T ) = Cˆ ∞ (ω)A δ(ω − ωn ) = Cˆ ∞ (ω)A n

n

where A is a proportionality constant which we determine next. We reiterate that ˆ although the transform C(ω) is a continuous function of frequency it effectively evaluates to discrete nonzero values. The remainder of the transform pair is given by  2π F (t) =

∞ −∞

+iωt ˆ dω = C(ω)e



∞ −∞

 δ(ω − ωn )e+iωt dω Cˆ ∞ (ω)A n

Interchanging the summation and the integration, and using the properties of the delta function, gives  Cˆ ∞ (ωn )e+iωn t 2π F (t) = A n

Integrate both sides of this equation over a time period T , and realizing that all terms on the right-hand side are zero except the first, gives that 

T

2π 0

F (t) dt = ACˆ ∞ (0)T

Because Cˆ ∞ (0) is the area under the single pulse, we conclude that A = 2π/T .

14

1 Spectral Analysis of Wave Motion

We now have the representation of a function F (t), extended periodically, in terms of its transform over a single period. That is, n=+∞ 1  ˆ C∞ (ωn )e+iωn t , T n=−∞

F (t) =

Cˆ ∞ (ωn ) =



T

F (t)e−iωn t dt

(1.10)

0

Except for a normalizing constant, this is the complex Fourier series representation of a periodic signal. Consider the transform of the rectangular pulse of the last section. The coefficients are given by Cˆ n =



to +a

Fo e−iωn t dt = Fo

to



e−iωn t −iωn

to +a

 = Fo a

to

 sin(ωn a/2) −i(to +a/2)ωn e ωn a/2

This result is the product of three terms. The first, Fo a, is the size of the pulse as represented by the area. The third, the exponential, is a phase change due to the shifting of the pulse relative to the time origin. For example, if the pulse is symmetric about t = 0, then to = −a/2 and there is no phase change of the transform. The second term, sinc function, is the core of the transform and its amplitude is shown plotted in Fig. 1.4 for various periods. Notice that the spacing of the coefficients is at every 1/T hertz in the frequency domain. For the periods T = 500, 200, 100 µs this gives spacings of f = 2, 5, 10 kHz, respectively. Notice that in all the cases of Fig. 1.4, the Fourier series gives the exact values of the continuous transform, but it does so only at discrete frequencies. The discretization of the frequencies is given by ωn = 2π n/T Fourier series Continuous

-100.

-50.

0.

50.

100.

Fig. 1.4 Fourier series coefficients for a rectangular pulse of nonzero duration a = 50 µs extended with different periods T

1.1 Fourier Transforms

15

Fig. 1.5 Discretization scheme for an arbitrary time function

Therefore, for a given pulse, a larger period gives a more dense distribution of coefficients, approaching a continuous distribution in the limit of an infinite period. This, of course, is the continuous transform limit. The finite time integral causes the transform to be discretized in the frequency domain. The discrete coefficients in the Fourier series are obtained by performing continuous integrations over the time period. These integrations are now replaced by summations as a further step in the numerical implementation of the continuous transform. In reference to Fig. 1.5, let the function F (t) be divided into M, piece-wise constant, segments whose heights are Fm and base T = T /M. The coefficients are now obtained from Cˆ n ≈ Dˆ n =

M−1  m=0

= T

 Fm



tm + T /2

tm − T /2

e−iωn t dt

 sin ωn T /2  Fm e−iωn tm ωn T /2 m

We see that this is the sum of the transforms of a series of rectangles each shifted in time by t = tm + T /2. The contribution of each of these is now examined more closely. First look at the summation term. If n > M, that is, if n = M + n∗ , then the exponential term becomes e−iωn tm = e−inωo tm = e−iMωo tm e−in

∗ω t o m

= e−i2π m e−in

Hence the summation contribution simply becomes M−1  m=0

Fm e−in

∗ω t o m

∗ω t o m

= e−in

∗ω t o m

16

1 Spectral Analysis of Wave Motion

showing that it evaluates the same as when n = n∗ . More specifically, if M = 8 say, then n = −5, 9, 11, 17 evaluates the same as n = 3, 1, 3, 1, respectively. The discretization process has forced a periodicity into the frequency description. Now look at the other contribution; we see that the sinc function term does depend on the value of n and is given by sinc(x) ≡

sin(x) , x

x=

ωn T n T n =π =π 2 T M M

The sinc function is such that it decreases rapidly with increasing argument and is small beyond its first zero. The first zero occurs where x = π or n = M; if M is made very large, that is, the integration segments are made very small, then it is the higher order coefficients (i.e., large n) that are in the vicinity of the first zero. Let it be further assumed that the magnitude of these higher order coefficients are negligibly small. Then an approximation for the coefficients is Dˆ n ≈ T {1}

M 

Fm e−iωn tm

m

on the assumption that it is good for n < M and that Cˆ n ≈ 0 for n ≥ M. Because there is no point in evaluating the coefficients for n > M − 1, the approximation for the Fourier series coefficients is now taken as N −1 N −1 1  ˆ +iωn tm 1  ˆ +i2π nm/N Fm = F (tm ) ≈ = Dn e Dn e T T n=0

ˆ n ) ≈ T Dˆ n = D(ω

N −1  m=0

n=0

Fm e−iωn tm = T

N −1 

Fm e−i2π nm/N

(1.11)

m=0

where both m and n range from 0 to N − 1. These are the definition of what is called the discrete Fourier transform (DFT). It is interesting to note that the exponentials do not contain dimensional quantities; only the integers n, m, N appear. In this transform, both the time and frequency domains are discretized, and as a consequence, the transform behaves periodically in both domains. The dimensional scale factors T , 1/T have been retained so that the discrete transform gives the same numerical values as the continuous transform. There are other possibilities for these scales found in the literature. The discrete transform enjoys all the same properties as the continuous transform; the only significant difference is that both the time domain and frequency domain functions are now periodic. To put this point into perspective consider the following: A discrete Fourier transform seeks to represent a signal (known over a finite time T ) by a finite number of frequencies. Thus it is the continuous Fourier transform of a periodic signal. Alternatively, the continuous Fourier transform itself

1.1 Fourier Transforms

17

can be viewed as a discrete Fourier transform of a signal but with an infinite period. The lesson is that by choosing a large signal sample length, the effect due to the periodicity assumption can be minimized and the discrete Fourier transform approaches the continuous Fourier transform. To help better illustrate the properties of the discrete transform, we consider an eight-point sampled signal. This can also serve as the test case for any numerical implementation of the DFT. Let the real-only function be given by the following sampled values: F1 = F2 = 1,

F0 = F3 = F4 = F5 = F6 = F7 = 0

with T = 1, N = 8. This has the shape of a rectangular pulse if the points are connected by straight lines—remember, however, that the data is sampled and hence no information is actually known between the sampling points. Eight points are given, thus it is implicit that the function repeats itself beyond that. That is, the next few values are 0, 1, 1, 0, and so on. The transform becomes Dˆ n =

7 

Fm e−i2π nm/8 = F1 e−iπ n/4 + F2 e−iπ n/2

m=0

The first ten transform points, in explicit form, are Dˆ 0 = 2.0 Dˆ 1 = 0.707 − 1.707i Dˆ 2 = −1.0 − 1.0i Dˆ 3 = −0.707 + 0.293i Dˆ 4 = 0.0 Dˆ 5 = −0.707 − 0.293i Dˆ 6 = −1.0 + 1.0i Dˆ 7 = 0.707 + 1.707i Dˆ 8 = 2.0 Dˆ 9 = 0.707 − 1.707i The obvious features of the transform are that it is complex and that it begins to repeat itself beyond n = 7. Note also that Dˆ 4 [the ( 12 N + 1)th value] is the Nyquist value. The real part of the transform is symmetric about the Nyquist frequency, while the imaginary part is antisymmetric. It follows from this that the sum 12 [Dˆ n + Dˆ N −n ] gives only the real part, that is,

18

1 Spectral Analysis of Wave Motion

2.0,

−1.0,

0.707,

−0.707,

0.0,

−0.707,

···

while the difference 12 [Dˆ n − Dˆ N −n ] gives the imaginary part 0,

−1.707i,

−i,

+0.293i,

0,

−0.293i,

···

These two functions are the even and odd decompositions, respectively, of the transform. Also note that Dˆ 0 is the area under the function.

1.1.3 Fast Fourier Transform Algorithm The final step in the numerical implementation of the CFT is the development of an efficient algorithm for performing the summations of the discrete Fourier transform on a computer. The fast Fourier transform (FFT) is simply a very efficient numerical scheme for computing the discrete Fourier transform. This is not a different transform—the numbers obtained from the FFT are exactly the same in every respect as those obtained from the DFT. The intention of this section is to just survey the major features of the FFT algorithm and to point out how the great speed increase is achieved. More detailed accounts can be found in the Refs. [4, 7] and FORTRAN code is given in Ref. [19]. Consider the generic forward transform written as Sn =

N −1 m=0

Fm e−i2π nm/N ,

n = 0, 1, . . . , N − 1

We write this in the expanded form S0 = {F0 + F1 + F2 + · · · } S1 = {F0 + F1 e−i2π 1/N + F2 e−i2π 2/N + · · · } S2 = {F0 + F1 e−i2π 2/N + F2 e−i2π 4/N + · · · } .. . Sn = {F0 + F1 e−i2π n/N + F2 e−i2π n2/N + · · · } and so on. For each sum Sn , there are (N − 1) complex products and (N − 1) complex sums. Consequently, the total number of computations (in round terms) is on the order of 2N 2 . The purpose of the FFT is to take advantage of the special form of the exponential terms to reduce the number of computations to less than N 2 . The key to understanding the FFT algorithm lies in seeing the repeated forms of numbers. This is motivated by considering the special case of N being 8. First consider the matrix of the exponents −i2π( mn N ):

1.1 Fourier Transforms

19

⎡

0 ⎢0 ⎢ ⎢ 0 −i2π ⎢ ⎢ ⎢ N ⎢0 ⎢ .. ⎣.

0 1 2 3 .. .

0 2 4 6 .. .

··· ··· ··· ··· .. .

0 3 6 9 .. .

0 (N − 1) 2(N − 1) 3(N − 1) .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 (N − 1) 2(N − 1) 3(N − 1) · · · (N − 1)(N − 1) It is apparent that for an arbitrary value of N, 2π is not, in general, multiplied by an integer number. These exponents, however, can be made quite regular if N is highly composite. For example, if N is one of the following N = 2γ = 2, 4, 8, 16, 32, 64, 112, 256, 512, 1024, . . . then the effective number of different integers in the matrix is decreased. Thus if N = 8 we get ⎡

0 ⎢0 ⎢0 ⎢ −i2π ⎢ ⎢0 8 ⎢ ⎢0 ⎢0 ⎣ 0 0

0 1 2 3 4 5 6 7

0 2 4 6 8+0 8+2 8+4 8+6

0 3 6 8+1 8+4 8+7 16 + 2 16 + 5

0 4 0 8+4 16 + 0 16 + 4 24 + 0 24 + 4

0 5 8+2 8+7 16 + 4 24 + 1 24 + 6 32 + 3

0 1 2 3 4 5 6 7

0 5 2 7 4 1 6 3

0 6 8+4 16 + 2 24 + 0 24 + 6 32 + 4 40 + 2

⎤

0 7 ⎥ 8+6 ⎥ ⎥ 16 + 5 ⎥ ⎥ 24 + 0 ⎥ ⎥ 32 + 3 ⎥ ⎦ 40 + 2 48 + 1

which effectively evaluate to ⎡

0 ⎢0 ⎢0 ⎢ −i2π ⎢ ⎢0 8 ⎢ ⎢0 ⎢0 ⎣ 0 0

0 2 4 6 0 2 4 6

0 3 6 1 4 7 2 5

0 4 0 4 0 4 0 4

0 6 4 2 0 6 4 2

⎤

0 7⎥ 6⎥ ⎥ 5⎥ ⎥ 0⎥ ⎥ 3⎥ ⎦ 2 1

This comes about because the exponentials take on the simple forms e−i2π [0] = e−i2π [1] = e−i2π [2] = e−i2π [3] = · · · = 1 The regularity is enhanced even more if (N/2 = 4) is added to the latter part of the odd rows, that is, if it is written as

20

1 Spectral Analysis of Wave Motion

⎡

0 ⎢0 ⎢0 ⎢ −i2π ⎢ ⎢0 8 ⎢ ⎢0 ⎢0 ⎣ 0 0

0 1 2 3 4 5 6 7

0 2 4 6 0 2 4 6

0 3 6 1 4 7 2 5

(0 (0 (0 (0 (0 (0 (0 (0

0 1 2 3 4 5 6 7

0 2 4 6 0 2 4 6

⎤

0) + 0 3) + 4 ⎥ 6) + 0 ⎥ ⎥ 1) + 4 ⎥ ⎥ 4) + 0 ⎥ ⎥ 7) + 4 ⎥ ⎦ 2) + 0 5) + 4

We see that many of the computations used in forming one of the summations is also used in the others. For example, S0 , S2 , S4 , S6 all use the sum (F0 + F4 ). Realizing that e−i2π 4/8 = −1, then we also see that all the odd summations contain common terms such as (F0 −F4 ). This re-use of the same computations is the reason a great reduction of computational effort is afforded by the FFT. The algorithm sets up the bookkeeping so that this is done in a systematic way. The number of computations with and without the FFT algorithm is given by 3 2N

log2 N

versus

2N 2

When N = 8, this gives a speed factor of only 3.5:1, but when N = 1024, this jumps to over 100:1. It is this excellent performance at large N that makes the application of Fourier analysis feasible for practical problems. On a benchmark machine of 1 GFlops, say, a 1024-point transform takes less than one millisecond. The fast Fourier transform algorithm is so efficient that it has revolutionized the whole area of spectral analysis. It can be shown quite simply that it enjoys all the same properties of the continuous transform. Therefore, in the subsequent analyses, we assume that any time input or response can be represented in the spectral form F (t) =



Cˆ n e+iωn t

and the tasks of forward and inverse transforms are accomplished with a computer program.

1.1.4 Space Distributions Using Fourier Series In some of the chapters we have occasion to represent space distributions using Fourier transforms. Because, typically, only a limited number of points are of interest (think of strain gage or accelerometer locations) it is more efficient to use direct summations of Fourier series rather than the FFT algorithm. For this reason, we now summarize our use of the Fourier series transform. For concreteness, let the function be symmetric in the space window − 12 W < x < 12 W and represent it using a cosine series. That is,

1.1 Fourier Transforms

f (x) =

21

  M M 1 1 a0 + a0 + am cos(2π mx/W ) = am cos(ξm x) 1 1 W W

Multiply both sides by cos(ξn x) and integrate over the complete window. Because the cosine functions are orthogonal on the window, and the integrals of the squares are 12 W , the right-hand side is nonzero only if n = m and we get 

+W/2 −W/2

f (x) cos(ξm ) dx = {a0 , 12 am }

We thus represent it by the transform pair  M 1 a0 + f (x) = am cos(ξm x) , 1 W

 {a0 ,

1 2 am }

=

+W/2

−W/2

f (x) cos(ξm ) dx

If the function is antisymmetric on the window, then 1 M f (x) = bm sin(ξm x) , 1 W

 1 2 bm

=

+W/2 −W/2

f (x) sin(ξm ) dx

If the integrals are computed numerically (through quadratures), then for efficiency, both need only be computed on the half window and subsequently multiplied by two. A general function is represented as the superposition of these two representations. As an example, consider a rectangular function of width a symmetrically positioned at x = 0 as shown in Fig. 1.6. The coefficients evaluate to

-0.5

0.0

0.5

-10.

-5.

0.

5.

10.

Fig. 1.6 Fourier series representation of space distributions. (a) Rectangular distribution. Effect of window size and number of terms. (b) Representing a concentrated load on a large space window W = 800a

22

1 Spectral Analysis of Wave Motion



{a0 ,

1 2 am }

 2 = a, sin(ξm a/2) fo ξm

The figure shows the reconstructions. Doubling the number of terms improves the representation; the same effect is achieved by halving the window size. In our later developments, the window size is chosen, thus if the window is too large for a given number of terms accuracy suffers, if it is too small, we get the equivalent of wraparound problems which we interpret as reflections or the effect of image loads. To elaborate on this last point, in some of our later analyses we have a need to model a large space subjected to a relatively narrow distributed load (think of an infinite beam subjected to a point load). Figure 1.6a shows that a rectangular distribution is represented reasonably well with M = 40 when the window is W = 5a which implies that a window size of 800a requires M = 6400 which is a large number. Therefore, ways of doing this efficiently and rationally are useful. A Gaussian distribution has the representation e−(x/α) = 12 a0 + 2

M 1

am cos(ξ x) ,

am =

2 −(αmπ/W )2 e W

A function of somewhat similar shape is the sine-squared function of extent a and a reasonable connection sets a = 3.32α. The heavy and thin lines in Fig. 1.6b for M = 2048 compare the Gaussian and sine-squared functions, they differ only at their base where the latter has a definite delimination of its compact support. Staying with the example of W = 800a, Fig. 1.6b shows the effect of the number of terms in the summation. Superficially, there is a significant deterioration as the number of terms is decreased. What is surprising is that in each case shown the area under the curve is the same. Thus, if these plots represented a traction distribution, then the resultant force would be the same in each case. Consequently, especially if relatively remote responses are of interest, the resultant load is the significant factor. In other words, while the M = 256 case does not represent the actual distribution, it does represent the resultant effect and therefore could accurately predict remote behaviors. The example problems in Chaps. 6 and 7 use this extensively and we refer to the approach as periodically extended loads (pEL).

1.2 Applications Using the FFT Algorithm The following examples serve to show the basic procedures used in applying the FFT to transient signals. The FFT is a transform for which no information is gained or lost relative to the original signal. However, the information is presented in a different format that often enriches the understanding of the information. In the examples to follow, we try to present that duality of the given information.

1.2 Applications Using the FFT Algorithm

23

The examples are divided into two groups: one group explores the properties of the FFT, the other shows applications to experimental data.

1.2.1 Explorations of the Properties Reference [12] describes a few explorations of the frequency analysis of signals; it uses the program DiSPtool which is part of the QED package to accomplish the tasks. The following are abbreviated results. Consider the rectangular pulse already treated and that has the transform given by Eq. (1.2). The effect of different sample lengths and number of points used are shown in Fig. 1.7. Because of the discrete sampling, note that the vertical sides of the rectangle always have a rise time of T . In the examples, the jump is treated by using its half value as shown in the time plots. First, it is noticed that the transform is symmetric about the middle or Nyquist frequency. This is a consequence of the input signal being real-only—if it were complex, then the transform would fill the complete range. What this means is that N real points are transformed into N/2 complex points and no information is gained or lost. Therefore, the useful frequency range extends only up to the Nyquist, given by fNyquist =

1 2 T

FFT CFT

0.

160.

320.

480.

0.

50.

100.

150.

Fig. 1.7 FFT transform of a rectangular pulse of width a = 50 µs. (a) Time domain signals. Circles are sampled data. (b) Frequency domain transforms

24

1 Spectral Analysis of Wave Motion

This range is increased only by decreasing T . Thus, for a fixed number of points, fine resolution in the time domain (small T ) means course resolution in the frequency domain. Finer resolution in the frequency domain is achieved only by increasing the sample length T . It is also seen from the figure that the match between the FFT amplitude and the continuous Fourier transform is very good at low frequencies but gets worse at the higher frequencies. As mentioned before, the discrete and the continuous transforms match closely only if the highest significant frequency in the signal is less than the Nyquist. The rectangular pulse can be represented exactly only by using an infinite number of sinusoids but the contributing amplitudes get smaller as frequency increases and therefore a finite number of sinusoids can suffice. For a given sample rate T , the number of samples only determines the density of transform points. Thus the top two FFTs in Fig. 1.7 are numerically identical at the common frequencies. Increasing the sample rate for a given number of samples increases the Nyquist frequency, and therefore the range of comparison between the discrete and continuous transforms. It is seen that by the fourth plot, the amplitudes in the vicinity of the Nyquist are negligible. Using the discrete transform puts an upper limit on the maximum frequency available to characterize the signal. If the signal is not smooth, the amplitudes of the high-frequency sinusoids used to describe the signal are high. Any attempt at capping the high-frequency sinusoids introduces a distortion in the amplitudes of the lower-frequency sinusoids. This is called aliasing and is discussed in more detail presently. For future reference, we now summarize some of the inter-relationships among the various parameters. Consider a signal of duration T sampled as N points. The discretization rates in the two domains are T =

T , N

f =

1 T

Various forms for the Nyquist frequency are fNyquist =

1 N f N = = 2T 2 T 2

The complementarity of information between the time and frequency domains is illustrated in Fig. 1.8. On comparing the top plots, we notice that the longer duration pulse has a shorter main frequency range; however, both exhibit side lobes which extend significantly along the frequency range. The second trace shows a smoothed version of the second triangle compared to a rectangular pulse of the same duration; in the frequency domain it is almost identical to the shorter triangle, the only difference being the reduction in the high-frequency side lobes. Interestingly, however, the rectangular pulse has a shorter main significant frequency range but the side lobes are very significant extending beyond what is shown. Thus time domain smoothing (using moving averages, say) acts as a high band filter in the frequency

1.2 Applications Using the FFT Algorithm

0.

100.

200. .

300.

400.

25

0.

10.

20.

30.

40.

50.

Fig. 1.8 Comparison of some pulse-type signals. (a) Time domain signals. (b) Frequency domain transforms

domain. In the wave analyses of the later chapters, we need to control the frequency range of the inputs and we do this by using signals without sharp edges. As a rule of thumb, a triangular pulse has a frequency content of about 2/Tp where Tp is the duration of the pulse. This is confirmed in the figure for the smoothed triangular pulse. The remaining two plots are for modulated signals—in this case a sinusoid modulated (multiplied) by a triangle. The sinusoid function on its own would give a peak at 20 kHz, whereas the triangular modulations on their own are as shown for the top traces. The product of the two in the time domain is a convolution in the frequency domain. The convolution tends to distribute the effect of the pulse but it is worth noting that while both waveforms have the same time domain amplitudes, their transforms have different amplitudes indicating different energy levels. A point of interest here is that if a very narrow-banded signal in the frequency domain is desired, then it is necessary to extend the time domain signal as shown in the bottom example. The modulated waves are called wave packets or wave groups because they contain a narrow collection (group) of frequency components. It might have been thought that the triangular pulse is a more natural group because in the time domain it is highly localized, but what is also important to us is the frequency content of the wave and we see that the modulated wave is localized in both the space and time domains. We use this wave packet idea regularly throughout the remaining chapters to interrogate complex systems. The signals we deal with often have multiple reflections. A sense of their effect can be gaged from Fig. 1.9 where the traces are arranged so as to have a different number of reflections. In the time domain, we recognize the multiple reflections as localized disturbances in time; in fact, in this simple example we can cut the trace so as to isolate each of the reflections separately. No such separation is possible in the frequency domain because any small time-localized portion of signal contributes over the total frequency range. Consequently, we get an interference effect—the adding and

26

1 Spectral Analysis of Wave Motion

0.0

0.5

1.0

1.5

2.0

0.

5.

10.

15.

20.

Fig. 1.9 Effect of the superposition of multiple reflections. (a) Time domain signals. (b) Frequency domain transforms

subtracting of phases result in amplifying certain components. We can see from the figure the evolution of spectral peaks as the number of reflections is increased. In vibration analyses, which would correspond to an infinite number of reflections, the spectral peaks become very sharp and this situation would then be called resonance.

1.2.2 Experimental Aspects of Wave Signals It seems appropriate at this stage to summarize some of the experimental aspects of recording the wave signals in structures. By necessity, it is only cursory, but it is hoped that it gives a feel for the type and quality of data recorded. It is necessary to record an infinitely long trace in order to exactly characterize a general signal. This is obviously impractical, so the problem reduces to one of obtaining best estimates for the spectra from a finite trace. This section summarizes some of the necessary procedures used for spectral estimation, more details and citations are given in Ref. [11]. Some of these procedures are similar to those used for vibration signals. Some of the characteristics of the testing to record transient signals associated with impact and stress wave propagation in structures are: • • • •

High-frequency content (1 kHz–1 MHz), Only initial portion (∼ 5000 µs) of the signal is analyzed, Only a small number of “runs” are performed, The number of data recording channels is usually limited (2–8).

There is quite a range of measurement techniques available for studying wave propagation problems. Of course, some are more suitable for certain situations than others, but because the emphasis in this book is on structures, then only those techniques most commonly used for structural analysis are surveyed. A typical setup is shown in Fig. 1.10.

1.2 Applications Using the FFT Algorithm

27

Force Transducer

PCB 303A03 accelerometers

computer

Force Transducer

Power Supplies

Tek AM502 waveform recorder pre-amps

Fig. 1.10 Typical experimental setup for a plate (left) or frame (right)

Most types of electrical resistance strain gages are usually adequate and because the events are of short duration, temperature compensation is usually not a serious concern. The largest gage size allowable is dependent on the highest significant frequency (or shortest wavelength) of the signal, but generally gages equal to or less than 3 mm (0.125 in.) are good for most situations. A rule of thumb for estimating the maximum length is  1 L= 10f

E ρ

(1.12)

where L is the gage length, f is the highest significant frequency, and E and ρ are the Young’s modulus and density, respectively, of the structural material. A typical gage of length 3 mm (0.125 in.) is good to about 160 kHz. Other aspects of the use of strain gages in dynamic situations are covered in Refs. [2, 8, 18]. Either constantcurrent, potentiometer or Wheatstone bridge gage circuits can be used. Nulling of the Wheatstone bridge is not necessary for dynamic problems because only the ac portion of the signal is recorded. A wide variety of small accelerometers are available. There is a problem with accelerometers, however, in that if the frequency is high enough, ringing is observed in the signal. This can often occur for frequencies as low as 20 kHz. Generally speaking, accelerometers do not have the necessary frequency response of strain

28

1 Spectral Analysis of Wave Motion

gages, but they are movable, thus making them more suitable for larger, complex structures. Also, note that if the stress is of interest, then integration of the signal is required. It is impossible to be categorical about the minimum setup necessary for waveform recording but the following pieces of equipment definitely seem essential. Good preamplifiers are essential for dynamic work; they not only provide the gain for the low voltages from the bridge but also perform signal conditioning. The former gives the flexibility in choice of gage circuit as well as reducing the burden on the recorder for providing amplification. A separate preamplifier is needed for each strain gage circuit and a typical one should have up to 100K gain, dc to 1 MHz frequency response, and selectable band pass filtering. Generally, the analog filtering (which is always necessary) is performed at or below the Nyquist frequency associated with the digitizing. The most significant advance in recent years is the availability of digital recorders. A typical digital waveform recorder would have 4 channels, 2048 8bit words of memory, and a sample rate of 0.2 µs to 0.1 s. It is common for high-speed recorders to digitize to 8-bit words (although newer ones use 12-bit words). This means it has a resolution of 1/256 of the full scale. If the signal can be made to fill at least half full scale, then this resolution of about 1% of the maximum signal is adequate. This situation is best achieved by having good preamplifiers. Many of the newer digital recorders combine the features of an oscilloscope with those of a small stand-alone computer. The DASH-18 [1] is an example of the newer type of recorders; it is like a digital strip chart recorder and notebook computer rolled into one. The DASH-18 can record up to 18 channels and has a very flexible interface for manipulating the recorded data. There are also relatively inexpensive recorders available that can be installed in desktop computers. The Omega Instruments DAS-58 data acquisition card is an example; this is a 12-bit card capable of 1 Mhz sampling rate and storing up to 1M data points in the onboard memory. A total of 8 multiplexed channels can be sampled giving the fastest possible sampling rate of 1M/8 = 125 K samples per second. The measured signal can contain unwanted contributions from at least two sources. The first is the ever present electrical noise. Analog filters can be used to remove much of this, although care must be taken not to remove some of the signal itself. The second may arise from unwanted reflections. This is especially prevalent for dispersive signals as the recording is usually extended to capture as much as possible of the initial passage of the wave. There are many ways of smoothing digital data but the simplest is to use moving averages of various amounts and points of application. Figure 1.11 shows an example of smoothing a signal that has an exaggerated amount of noise and reflections present. The following sequence was used on data values sampled at every 1 μs:

1.2 Applications Using the FFT Algorithm

29

removed noise smoothed

as recorded

0.

200.

400.

600.

800.

1000.

Fig. 1.11 Effect of moving-average smoothing

5-point 33-point 77-point 201-point

starting at 0 μs starting at 220 μs starting at 450 μs starting at 600 μs

The last averaging brings the signal smoothly to zero in the vicinity of 1000 µs. Also shown in the figure is that part of the signal removed; this can be monitored to detect if too much of the signal is being filtered. When we compare the truncated measured signal to the (imagined) infinite trace, we say the former was “windowed”—only a small portion of the infinite trace is visible through the window. Examples of time windows commonly used are the rectangle and the Hanning window (which is like a rectangle with smoothed edges); see Refs. [3, 5] for more examples of windows. None of the windows usually used in vibration studies seem particularly appropriate for transient signals of the type obtained from propagating waves. The window used in most of the studies to follow is the rectangular window tapered in some way on the large time side. This window is thought to be adequate and is certainly the easiest to implement. A problem with rapid truncation is that the reconstructed signals exhibit significant disturbances at the point of truncation. If the sample size is large compared to the length of interest, then this is not a serious problem because the disturbances can be easily identified (they are localized in time) and thus not be confused with the true signal. By necessity, actual signal records are finite in length, and this can add some difficulties to the scheme of estimating its spectral content. This section reviews the major effects that should be considered. An actual signal of very long duration must be truncated to a reasonable size before the discrete Fourier transform is evaluated. The transform of the signal will therefore be dependent on the actual function F (t), say, and the way it is sampled. Note that if the sampling rate is every T , then the highest detectable frequency using the DFT is 1/(2 T ). Thus, if T is too large, the high frequency appears as

30

1 Spectral Analysis of Wave Motion

data sampled

0.

100.

200.

300.

400.

Fig. 1.12 Aliasing caused by coarse sampling

a lower frequency (its alias) as shown in Fig. 1.12. This phenomenon of aliasing can be avoided if the sampling rate is high enough. By “high enough” is implied a rate commensurate with the highest significant frequency in the signal. From a practical point of view, this is also avoided by recording with the analog filters set so as to remove significant frequencies above the Nyquist. The estimated spectrum is given only at discrete frequencies and this can cause a problem known as leakage. For example, suppose there is a spectral peak at 17 kHz and the sampling is such that the spectrum values are only at every 2 kHz. Then the energy associated with the 17 kHz peak “leaks” into the neighboring frequencies and distorts their spectral estimate. Leakage is most significant in instances where the signal has sharp spectral peaks and is least significant in cases where the signal is a flat, broad-banded signal. In the analysis of the impact of structures, the signals generated generally do not exhibit very sharp spectral peaks (as in the case of vibrations). However, it can become a problem if there are many reflections present in the signal. Because discrete Fourier analysis represents a finite sample of an infinite signal on a finite period, then schemes for increasing the apparent period must be used. The simplest means of doing this is by padding. There are various ways of padding the signal and the one most commonly used is that of simply adding zeros. Alternatively, Ref. [10] shows how a long term estimate of the signal behavior can also be used to pad the signal. The main consequence of padding is that because a larger time window is used then higher resolution in the frequency domain is achieved. As regards padding with zeros, it should be pointed out that it really does not matter if the zeros are added before the recorded signal or after. Thus even if the signal never decreases to zero, padding is justified because it represents the early quiescent part of the signal.

1.3 Spectral Analysis of Wave Motion

31

1.3 Spectral Analysis of Wave Motion The application of the continuous Fourier transform to the solution of wave propagation problems is quite standard as can be judged from some of the chapters of Ref. [20]. The application of discrete Fourier series, however, is quite limited with some of the early exceptions being Refs. [6, 9, 15]. It appears that the task of performing the summations is too formidable because a large number of terms are required to adequately describe the wave. With the advent of computers and the utilization of the FFT algorithm, the situation changed, and performing large summations rapidly is now feasible and economical. This section introduces, in a consistent manner, the application of the discrete Fourier transform (in the form of spectral analysis) to the solution of wave problems. The key to the spectral description of waves is to be able to express the phase changes incurred as the wave propagates from location to location. This is done conveniently through use of the governing differential equations for particular structural models—although in this chapter we try to make the results independent of any particular structural system.

1.3.1 General Functions of Space-Time and Spectrum Relations The solutions in wave problems are general functions of space and time. If the time variation of the solution is focused on at a particular point in space, then it has the spectral representation u(x1 , y1 , t) = F1 (t) =

 n

Cˆ 1n eiωn t

At another point, it behaves as a time function F2 (t) and is represented by the Fourier coefficients Cˆ 2n . That is, the coefficients are different at each spatial point. Thus, the solution at an arbitrary position has the spectral representation u(x, y, t) =

 n

uˆ n (x, y, ωn )eiωn t

(1.13)

where uˆ n are the spatially dependent Fourier coefficients. Notice that these coefficients are functions of frequency ωn and thus there is no reduction in the total number of independent variables. For shorthand, the summation and subscripts are often to be understood and the function is then given the representation u(x, y, t) ⇒ uˆ n (x, y, ωn ) Sometimes it is written simply as u. ˆ

or

u(x, ˆ y, ω)

32

1 Spectral Analysis of Wave Motion

The differential equations are given in terms of both space and time derivatives. Because they are linear, then it is possible to apply the spectral representation to each term appearing. Thus, the spectral representation for the time derivative is  ∂u ∂  iωn uˆ n eiωn t = uˆ n eiωn t = ∂t ∂t In shorthand, this becomes ∂u ⇒ iωn uˆ n ∂t

or

iω uˆ

In fact, time derivatives of general order have the representation ∂ mu ⇒ i m ωnm uˆ n ∂t m

i m ωm uˆ

or

(1.14)

Herein lies the advantage of the spectral approach to solving differential equations— algebraic expressions in the Fourier coefficients replace the time derivatives. That is, there is a reduction in the number of derivatives occurring. Similarly, the spatial derivatives are represented by  ∂ uˆ n ∂  ∂u = uˆ n eiωn t = eiωn t ∂x ∂x ∂x and in shorthand notation ∂ uˆ n ∂u ⇒ ∂x ∂x

or

∂ uˆ ∂x

(1.15)

In this case there appears to be no reduction, but as seen later, with the removal of time as an independent variable, these derivatives often become ordinary derivatives, and thus more amenable to integration. Consider the following general, linear (one-dimensional), homogeneous differential equation for the dependent variable u(x, t): u+a

∂u ∂ 2u ∂u ∂ 2u ∂ 2u +b +c 2 +d 2 +e + ··· = 0 ∂x ∂t ∂x∂t ∂x ∂t

The coefficients a, b, c, . . . are assumed not to depend on time, but could be functions of position. If, now, the solution is given the spectral representation u(x, t) =

N n

uˆ n (x, ωn )eiωn t

then on substitution into the differential equation get

1.3 Spectral Analysis of Wave Motion

33

   d uˆ n d 2 uˆ n 2 uˆ n + a + (iωn )buˆ n + c + (iω) d uˆ n + · · · eiωn t = 0 n dx dx 2 Because each eiωn t term is independent, then this equation must be satisfied for each n. That is, there are N simultaneous equations of the form    d uˆ  n + ··· = 0 1 + (iωn )b + (iωn )2 d + · · · uˆ n + a + (iωn )e + · · · dx These equations become, on grouping terms, A1 (x, ω)uˆ + A2 (x, ω)

d uˆ d 2 uˆ + A3 (x, ω) 2 + · · · = 0 dx dx

(1.16)

where A1 , A2 , . . . depend on frequency, possibly on position, and are complex. The subscript n is dropped, but it is understood that an equation of this form must be solved at each frequency. We see that the original partial differential equation becomes a set of ordinary linear differential equations in the Fourier coefficients uˆ n . Consider the special case when the coefficients a, b, c, . . . in the partial differential equation are independent of position and lead to linear differential equations with constant coefficients. Reference [16] considers in good detail the solution of equations of this form and should be consulted for the details not given here. These equations have solutions of the form eλx , where λ is obtained by solving the algebraic characteristic equation (obtained after substituting into Eq. (1.16)) A1 + A2 λ + A3 λ2 + · · · = 0

(1.17)

It is usual in wave analysis, however, to assume that λ is complex to begin with, that is, that the solutions are of the form u(x) ˆ = Ce−ikx

(1.18)

In this context, the exponent k is called the wavenumber. In general, the characteristic equation (for constant coefficients) becomes A1 + (−ik)A2 + (−ik)2 A3 + · · · = 0 and this has many values of k that satisfy it. That is, kmn = fm (A1 , A2 , A3 , · · · , ωn )

(1.19)

This relation between the wavenumber k and frequency ω is called the spectrum relation and is fundamental to the spectral analysis of waves. The different values of m correspond to the different modes. The solution is given as the superposition of modes in the form

34

1 Spectral Analysis of Wave Motion

u(x) ˆ = C1 e−ik1 x + C2 e−ik2 x + · · · + Cm e−ikm x There are as many modes (or solutions) as there are roots of the characteristic equation and these should not be confused with the number of solutions for each frequency. To reinforce this, the solution in total form is written as u(x, t) =

 N  C1n eik1n x + C2n eik2n x + · · · + Cmn eikmn x eiωn t n

(1.20)

The exponential form for each term is due to the coefficients of the differential equation being constant; however, the solution for any problem can always be expressed as u(x, t) =



ˆ mn x)eiωn t Pˆn G(k

(1.21)

where Pˆn is an amplitude spectrum (that is, the amplitude of each frequency ˆ (which may be a combination of modes) is the system transfer component) and G function. Analysis of the partial differential equations combined with the boundary conditions (BCs) (these are the conditions imposed at the edge of the model) ˆ and km (ω). Further, it is noted that the determines the particular forms for G wavenumber k acts as a scale factor on the position variable in the same way that the frequency ω acts on the time. The analysis of the scaling done by k(ω) provides a good deal of insight into the solution before the actual solution is obtained.

1.3.2 Some Wave Examples The following are some simple examples to demonstrate the effectiveness of the spectral approach to dynamic equations. Consider the following differential equation which is an example of the simple 1D wave equation 2 ∂ 2u 2∂ u = a ∂x 2 ∂t 2

(1.22)

where a is real and positive. Using the spectral representation gives d 2 uˆ + ω2 a 2 uˆ = 0 dx 2 ˆ −ikx is a solution which (on Because this has constant coefficients, assume Ae substitution) results in the characteristic equation ˆ =0 [−k 2 + ω2 a 2 ] A

1.3 Spectral Analysis of Wave Motion

35

Consequently, the two modes for k are k1 (ω) = +ωa ,

k2 (ω) = −ωa

and the solution is    ˆ 2 e+iωax }eiωt = ˆ 1 e−iωax + A ˆ 1 e−iω[ax−t] + ˆ 2 e+iω[ax+t] u(x, t) = {A A A ˆ 1 (ω), A ˆ 2 (ω) are the amplitude spectrums. The set of coefficients A This solution corresponds to sinusoids (infinite wave trains) moving to the left and right, respectively. A line in the x − t plane connecting a common point is a straight line. That is, points of common phase (position on the sinusoid) travel along straight lines in x − t as shown in Fig. 1.13a for a forward moving train. If the slope of this line is thought of as a speed, then it can be called the phase speed and is given by c=

x 1 ω = = t a k

The relation between phase speed and frequency is called the dispersion relation. We see that different frequency components travel at the same speed and that the superposition also travels at the same speed. When the phase speed is constant with respect to frequency, the signal is said to be nondispersive and it maintains its superposed shape. That is, at each position in space, the sinusoids have the same relative phase and so superpose to give the same shape except shifted in time.

-2.

0.

2.

4.

6.

8.

10.

12.

0.

2.

4.

6.

8.

10.

12.

14.

Fig. 1.13 Segments of an infinite wave train at different positions. Left: nondispersive system, individual sinusoids travel at the same speed. Right: dispersive system, individual sinusoids travel at different speed

36

1 Spectral Analysis of Wave Motion

Consider now the following differential equation (which actually represents flexural wave motion in beams) ∂ 4u ∂ 2u + a4 2 = 0 4 ∂x ∂t

(1.23)

where, again, a is real and positive. Because the coefficients are constant, assume ˆ −i[kx−ωt] is a solution giving the characteristic equation as Ae ˆ =0 [k 4 − ω2 a 4 ] A Note there are four modes because k 2 = ±ωa 2 or √ k1 (ω) = +a ω ,

√ k2 (ω) = −a ω ,

√ k3 (ω) = +ia ω ,

√ k4 (ω) = −ia ω

and the total solution becomes √ √   ˆ 2 e+i[a ωx+ωt] ˆ 1 e−i[a ωx−ωt] + A u(x, t) = A +



ˆ 3 e−a A

√

ωx+iωt

+



ˆ 4 ea A

√

ωx+iωt

The first two are wave terms, as in the previous example, and again travel with a constant phase speed given by c≡

1√ ω = ω k a

Different frequency components, however, travel at different speeds. As a result, the superposition of the components changes at the different locations and the wave then appears to change shape as it propagates. When the phase speed is not constant with respect to frequency, the signal is said to be dispersive. This is demonstrated for two wave train segments shown in Fig. 1.13b. Observe that the resultant signal travels at a speed different from the individual sinusoids. The last two terms in the solution are spatially damped vibrations. While not wave solutions per se (they are usually referred to as evanescent waves), they nonetheless play a crucial role in being able to satisfy the imposed boundary conditions when, say, a wave interacts with a joint. The two previous examples had the familiar double time derivative associated with inertia. Consider, as a different case, the following differential equation related to Fourier’s law of heat conduction ∂ 2u ∂u =a 2 ∂t ∂x

(1.24)

where a is real and positive. Again, because the coefficients are constant, assume ˆ −i[kx−ωt] is a solution giving the characteristic equation as Ae

1.3 Spectral Analysis of Wave Motion

37

ˆ =0 [−k 2 − iωa] A √ There are two modes having the spectrum relations k(ω) = ± −iaω or  k1 (ω) = −(1 − i) aω/2 ≡ −(1 − i)β,

 k2 (ω) = +(1 − i) aω/2 ≡ +(1 − i)β

The complete solution is u(x, t) =



ˆ 1 e−βx e−i[βx−ωt] + A



ˆ 2 e+βx e+i[βx+ωt] A

This is comprised of left and right propagating terms, but both are also spatially damped. Further, the solutions are dispersive in the sense that β is not a linear function of ω. Although this solution does not fall neatly into a simple wave description, the spectral approach solves it with ease. This is very fortunate, because as waves interact with boundaries and discontinuities, they lose their “wave nature” causing a more complicated dynamic motion and the spectral approach can still handle these cases.

1.3.3 Group Speed The dynamic solution is comprised of the superposition of many harmonics and it was shown that some are waves, some are damped waves, while some others are vibrations. All these combine to give the observed motion. It is of interest to know how to describe the propagating part of the disturbance because that is what arrives at a remote location. It must be reiterated, however, that for all practical problems, all types of solutions are present, and it may be unwarranted to attempt to isolate the so-called wave. In any local region of interest the solution (for a typical mode) can be written in the exponential form u(x, t) =



ˆ n (kn x)eiωn t = Pˆn G



Pˆn e−ikn x eiωn t

At a typical frequency, the wavenumber can be written in terms of its real and imaginary parts as k = kR + ikI giving the wave response in the form u(x, t) =



Pˆn e−kI x e−i[kR x−ωt]

38

1 Spectral Analysis of Wave Motion

This is recognized as having three parts: an amplitude spectrum Pˆn , a spatially decaying term e−kI x , and the propagating sinusoids e−i[kR x−ωt] . The phase speed of these sinusoids is given as c≡

ω kR

(1.25)

This is the speed at which the individual harmonics move as depicted in Fig. 1.13. Because it is the behavior of the superposition of all the sinusoids that forms the observed signal, then it is of interest to investigate how this group response differs from the individual sinusoids. Consider the interaction of two neighboring propagating components, that is, u(x, t) = Pˆn e−ikn x eiωn t + Pˆn+1 e−ikn+1 x eiωn+1 t Let these be written in terms of a central frequency ω∗ = (ωn + ωn+1 )/2, wavenumber k ∗ = (kn + kn+1 )/2, and the differences ω = ωn+1 − ωn , k = kn+1 − kn so that the discrete values are given by kn = k ∗ − 12 k,

kn+1 = k ∗ + 12 k;

ωn = ω∗ − 12 ω,

ωn+1 = ω∗ + 12 ω

Substitute into the displacement expression and simplify to get ∗ ∗ u(x, t) = Pˆ ∗ e−i[k x−ω t] 2 cos



1 2 kx

− 12 ωt

 (1.26)

(For simplicity, it is assumed that Pˆ ≈ 0.) This resulting wave is comprised of two parts besides the average amplitude spectrum. There is a sinusoid (called the carrier wave) of average frequency ω∗ and wavenumber k ∗ , and travels with average phase speed c∗ = ω∗ /k ∗ . But this is modulated by another wave (called the group wave) of wavenumber 12 k, frequency 12 ω, and travels at a phase speed of ω/ k. The phase velocity of the modulation is called the group speed. That is, ∂c ∂c ω ∂w −→ cg = = c/(1 − k ) = c + k k ∂k ∂ω ∂k In general, the group speed is numerically different from the phase speed. While the above analysis is for only two harmonics, it can be imagined that many harmonics interact in a similar manner giving rise to a carrier wave modulated by a group wave. This is what is depicted in Fig. 1.13b and also in Fig. 1.1; the group corresponds to the envelope of the disturbance and the phase to where the zeros occur. In general, however, it is not possible to “see” the individual sinusoids— only the group superposition is observed. It cannot be emphasized enough the importance of the group speed behavior because this is what is actually there. The ability to synthesize or assemble the group behavior is a unique feature of spectral

1.3 Spectral Analysis of Wave Motion

39

analysis coupled with the FFT algorithm. Put another way, the FFT allows both the assembling and disassembling of the group response. It is possible to have very complicated group behavior. For example, it is possible to have a solution for which the group speed is positive (the disturbance moves forward) and yet the phase speed is negative. According to Ref. [13], this is the situation as a caterpillar moves forward, the ripples in its body move from head to tail.

1.3.4 Summary of Wave Relations The wave for a particular propagating mode can be written in any of the following forms: ˆ −i 2πλ [x−ct] = Ae ˆ −i[kx−ωt] ˆ −ik[x−ct] = Ae u(x, t) = Ae ˆ can be complex and the above form is for a forward moving wave. The amplitude A The following is a collection of terms often used in wave analysis, the dimensions are in the square brackets: ω = angular frequency [radians/time] f = cyclic frequency [Hz, 1/time ] = T = period [time] =

ω 2π

2π 1 = f ω

ω 2π = λ c 2π 2π c = λ = wavelength [length] = ω k k = wavenumber [1/length] =

φ = phase of wave [radians] = [kx − ωt] =

2π ω [x − ct] = [x − ct] c λ

ω ωλ = k 2π ∂ω cg = group speed [length/time] = ∂k c = phase velocity [length/time] =

It should be kept in mind, however, that a general disturbance includes nonpropagating and attenuated components also.

40

1 Spectral Analysis of Wave Motion

1.4 Propagating and Reconstructing Waves The significance of the spectral approach to waves (coupled with the use of the differential equations) is that once the signal is characterized at one space position then it is known at all positions, and therefore propagating it becomes a fairly simple matter. This section illustrates the basic algorithm for doing this.

1.4.1 Basic Algorithm In its simplest terms, the solution to a waves problem is represented as u(x, t) =

 n

   ˆ 1 (k1n x) + G ˆ 2 (k2n x) + · · · eiωn t = ˆ mn x)eiωn t Pˆn G Pˆn G(k

ˆ is the analytically known transfer function of the problem. It is a function of where G position x and has different numerical values at each frequency. Pˆn is the amplitude spectrum; this is known from the input conditions or from some measurement. Thus ˆ is recognized as the Fourier transform of the solution. Of course, it is different Pˆn G at each position, but once it is evaluated at a particular position then its inverse immediately gives the time history of the solution at that point. Figure 1.14 is a flow diagram for the basic algorithm to propagate a wave. Briefly, the time input P (t) is converted to its spectrum Pˆn through a use of the forward FFT. The transformed solution is then obtained by evaluating the product

Fig. 1.14 Flow diagram for a wave reconstruction program

1.4 Propagating and Reconstructing Waves

41

ˆ mn x) uˆ n = Pˆn G(k at each frequency and some position. This is finally reconstructed in the time domain by use of the inverse FFT. In the process, it is necessary to realize (when using ˆ is evaluated only up to the Nyquist the FFT to perform the inversion) that Pn G frequency and the remainder is obtained by imposing that it must be the complex conjugate of the initial part. This ensures that the reconstructed time history is realonly. An additional feature of note is that when doing propagations the dc (zero frequency) component is undetermined because it does not propagate. It is advantageous to remove its arbitrariness by imposing that the first value of the reconstruction be zero, that is, Ny uˆ 0 = − uˆ n 1

where Ny is the Nyquist frequency. This is consistent with the idea that the system is quiescent before the arrival of the wave. Furthermore, it is good practice to give all signals a zero header (a beginning portion of zeros before the actual signal) to emphasize this point. Let the signal history at a particular position be a triangular pulse as shown in Fig. 1.15 and let uˆ n (xo ) be its transform. Also let the transfer function be that of the simple wave equation ˆ n = e−ikn x , G

kn = ωn /co

where co is a constant; here we use co = 5 km/s. The amplitude spectrum is Pˆn = u(x ˆ o )/e−ikn xo and the solution is then  u(x) ˆ = u(x ˆ o)

e−ikn x e−ikn xo



Figure 1.15 shows the reconstructions of the signal at different positions. It is obvious that the signal is propagating at a constant speed and that there is no change in shape. This is characteristic of nondispersive signals. To characterize the amplitude spectrum of the input, it was necessary to specify a time window, in the above a window of 5120 µs was used. With the wave propagating, however, it is possible for it to actually propagate out of the window. This occurs in the figure a little after x = 20 m, but because of the periodicity inherent in the spectral analysis, the wave from the neighboring window propagates into view from the left. The lesson here is that the window size must be chosen not only to allow proper characterization of the spectral content, but also to allow room for propagating the signal. Consequently, there is a connection between the size of the time window and the allowable distance the wave can move. A point of interest

42

1 Spectral Analysis of Wave Motion

x=0 x=5m x=10m x=15m x=20m x=25m

0.

1000.

2000.

3000.

4000.

5000.

Fig. 1.15 Reconstructions for a nondispersive wave

that may seem odd at first is that the window size is determined by the slowest wave component, not the fastest. Let the initial shape and transfer function be the same as in the previous example but choose the spectrum relation to be nonlinear in frequency, i.e., dispersive. That is, ˆ n = e−ikn x , G

kn =

√

ωn ωo /co

where ωo is a constant. (Strictly speaking the transfer function should also include the damped vibration terms, but the present form is adequate for far-field predictions.) The amplitude spectrum is obviously as given before but the reconstructions shown in Fig. 1.16 are very different. What starts out as a well-defined triangular shape disperses as it propagates. Not only does it change its shape, but it is no longer possible to identify the so-called pulse that is propagating. This is characteristic of dispersive signals and is due to the relation between k and ω being nonlinear. If the speed is slow for low frequencies, then it takes a very long time for the lowfrequency part of the signal to arrive. Thus it exhibits a long tail. The lower plots in Fig. 1.16 also show the results of propagating a dispersive signal a large distance. The curious feature is that it is the lower frequencies that propagate from the left into view first. To minimize the effect of this it is a good idea to pad the window with zeros. Thus the creeping components must first traverse the zeros before they come into view. When the higher frequencies arrive sooner than unwanted lower frequencies, then (depending on the circumstance) the unwanted components can often simply be filtered out. There are two ways of doing this. The first method is simply by band pass filtering, that is, every component below a certain frequency is blocked. The second scheme uses an n-point moving average but retains the difference and not the average. This latter scheme can do an excellent job in reconstructing the propagating wave. The most effective scheme, however, is

1.4 Propagating and Reconstructing Waves

43

x=0 x=0.5m x=1.0m x=1.5m x=2.0m x=2.5m

0.

1000.

2000.

3000.

4000.

5000.

Fig. 1.16 Reconstructions for a dispersive wave

.

0.

500.

1000.

. .

0.

500.

1000.

1500.

2000.

0.0

0.1

0.2

Fig. 1.17 SDoF system subjected to a rectangular pulse. (a) Time responses and load history. (b) Amplitude spectrum

simply to increase the window size at the price of adding more terms. This is feasible because of the efficiency of the FFT algorithm and its essentially linear dependence on N . To elaborate on this problem of wrap-around and in the process identify another source, or challenge, consider the simple spring-mass SDoF system described by Ku + C u˙ + M u¨ = P (t)

or

ˆ (1.27) uˆ = Pˆ /[K + iωC − ω2 ] = Pˆ G

and shown in Fig. 1.17. Let the force history be a rectangular pulse of the type examined in Fig. 1.2, although the history should always have a zero header so as to be able to detect any wrap-around problems.

44

1 Spectral Analysis of Wave Motion

The time window should be large enough to allow ring-down of response; this depends on the amount of damping. If there is no damping, no window is large enough to do a reconstruction. Thus, all modeled systems should have some damping which is true of real systems anyway. Another view of this large time window requirement is to recognize that the amplitude spectrum is very sharp and therefore requires a very fine f (i.e., large T = 1/ f = N T ) to characterize it as inferred from Fig. 1.17b.

1.4.2 Integration of Signals Inherent in the spectral analysis approach is the assumption that all functions of interest can be both integrated and differentiated. However, during the integration stage care must be taken to include the proper constants of integration. This is of special importance when, for example, obtaining displacement from strain because some unexpected side effects arise. To illustrate some of the challenges that arise due to the enforced periodicity of the results, Fig. 1.18 shows the displacement at the impact site of an infinitely long beam. The periodicity requires the beginning and final points of the reconstruction to be the same. This figure shows the profound effect of the window size. For this particular beam problem a nonzero value of displacement should persist for all time after the passage of the pulse. However, periodicity of the result on the finite window forces the end displacement to zero so that the trace is continuous from the end of one window to the beginning of the next. This problem is minimized when the output is at the same differentiation level as the input. For example, when obtaining strain at one location from strain at another, then padding with zeros can be used. But in the case of displacement from force it is seen that padding does not help. What is required is a window size large enough

corrected disp spectral disp

0.

1000.

2000.

3000.

Fig. 1.18 Integration of signals showing wrap-around effect

4000.

5000.

1.4 Propagating and Reconstructing Waves

45

so that the slope of the trailing edge of the history is not significant. For a given sample rate, the window can only be increased by increasing the number of points in the transform. Conversely, for a fixed number of points, the sample rate T must be increased, thus distorting the higher frequencies and running the risk of aliasing. To understand the effect in time, it is of interest to write the differential relations in terms of the frequency and wavenumber for both beams and rods. (More details on the origin of the rod and beam equations can be found in the next two chapters.) The simplest propagating solutions for rods and beams are, respectively, ROD:

u(x, t) =

BEAM:

v(x, t) =

 

An e−i[kx−ωt] ,

k = aω

Bn e−i[kx−ωt] ,

√ k=a ω

A table can now be constructed that shows the inter-relationship of all the derivatives: ROD

du du → ku → ωu → dx dt

BEAM dv → kv dx strain

d 2v dv → k 2 v → ωv → 2 dt dx d 3v → k3v dx 3

d 2u d 2u 2 2 → k u → ω u → dx 2 dt 2

loading

d 4v d 2v 4 2 → k v → ω v → dx 4 dt 2

Thus strain and velocity are at comparable levels of ω, and likewise for the loading and accelerations. This chart can be interpreted by noting, for example, that for beams integration of strain twice with respect to x to find displacement is equivalent to integration of velocity once with respect to time. The displacement is known, therefore, to within a linear function of time:   vˆ = eˆ dxdx + c1 t + c2 There are many ways of establishing these coefficients of integration. A simple method is to curve fit the first portion of the signal with a straight line and subtract it from the total signal. In fact, it is good practice to always include an initial zero header of the signal so as to be able to determine if there is an integration problem. Perhaps the simplest procedure is always to deal with signals that are nearly stationary. This is achieved by using time rates (such as velocity or acceleration) of the quantities of interest. Then, after reconstruction, the signal can be time integrated in the time domain. This is what was done to obtain the corrected displacements of Fig. 1.18.

46

1 Spectral Analysis of Wave Motion

Further Research 1.1 Show that by introducing the real coefficients an and bn such that (an − ibn ) ≡ Cn 2/T , we get the usual Fourier series representation F (t) =

 ∞     1 t t an cos 2π n ao + + bn sin 2π n 2 T T n=1

where the Fourier coefficients an and bn are obtained from an =

2 T

 0

T

 t F (t) cos 2π n dt , T

bn =

2 T



T 0

 t dt F (t) sin 2π n T

We have used ωn = 2π n/T and n = 0, 1, 2, . . .. —Ref. [17], p. 193. 1.2 Given the integral  I (t) =

+∞

−∞

f (ω)eih(ω,t) dω

Show that in the case when f (ω) is narrow banded about a frequency ωc , say, then the integral simply becomes I (t) ≈ f (ωc )eih(ωc ,t) —Ref. [14], p. 64. 1.3 Consider the other case when f (ω) is broad banded then the integral is most affected by the exponential term. Assume that most of the contribution to the integral comes from where the phase has a minimum value. Show that the integral then becomes  I (t) ≈

2π f (ωc )ei[h(ωc )+π/4] , h (ωc )

dh(ωc ) =0 dω

This approximation is the method of Stationary Phase. It is generally accurate in the large phase region which means asymptotically large time or asymptotically large distances. —Ref. [14], p. 67. 1.4 With t being the rms duration and ω being the rms bandwidth of a wave packet, establish the uncertainty relation t ω ≥

1 2

—Ref. [13], p. 465.

References

47

References 1. Astro-Med, Inc.: DASH-18, West Warwick, RI, USA 2. Bickle, L.W.: The response of strain gages to longitudinally sweeping strain pulses. Experimental Mechanics 10, 333–337 (1970) 3. Blackman, R.B., Tukey, J.W.: The Measurement of Power Spectra. Dover, New York (1958) 4. Brigham, E.O.: The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs, NJ (1973) 5. Chatfield, C.: The Analysis of Time Series: An Introduction. Chapman and Hall, London (1984) 6. Conway, H.D., Jakubowski, M.: Axial impact of short cylindrical bars. J. Appl. Mech. 36, 809–813 (1969) 7. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Mathematical Computations 19, 297–301 (1965) 8. Dally, J.W., Riley, W.F.: Experimental Stress Analysis, 3rd edn. McGraw-Hill, New York (1991) 9. Davies, R.M.: A critical study of the Hopkinson pressure bar. Philos. Trans. R. Soc. 240, 375– 457 (1948) 10. Doyle, J.F.: Further developments in determining the dynamic contact law. Experimental Mechanics 24, 265–270 (1984) 11. Doyle, J.F.: Modern Experimental Stress Analysis: Completing the Solution of Partially Specified Problems. Wiley, UK (2004) 12. Doyle, J.F.: Nonlinear Structural Dynamics Using FE Methods. Cambridge University Press, Cambridge (2015) 13. Elmore, W.C., Heald, M.A.: Physics of Waves. Dover, New York (1985) 14. Graff, K.F.: Wave Motion in Elastic Solids. Ohio State University Press, Columbus (1975) 15. Hsieh, D.Y., Kolsky, H.: An experimental study of pulse propagation in elastic cylinders. Proc. Phys. Soc. 71, 608–612 (1958) 16. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956) 17. Main, I.G.: Vibrations and Waves in Physics, 3rd edn. Cambridge University Press, Cambridge (1993) 18. Oi, K.: Transient response of bonded strain gages. Experimental Mechanics 6, 463–469 (1966) 19. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes. Cambridge University Press, Cambridge (1986); 2nd edn. (1992) 20. Sneddon, I.N.: Fourier Transforms. McGraw-Hill, New York (1951) 21. Speigel, M.R.: Fourier Analysis with Applications to Boundary Value Problems. McGraw-Hill, New York (1974)

Chapter 2

Longitudinal Waves in Rods

Rods and struts are important structural elements and form the basis of many truss and grid frameworks. Because their load bearing capability is axial, then as waveguides they conduct only longitudinal wave motion. This chapter begins by using elementary mechanics to obtain the equations of motion of the rod. As it happens, the waves are governed by the simple wave equation and so many of the results are easily interpreted. The richness and versatility of the spectral approach is illustrated by considering such usually complicating factors as viscoelasticity and elastic constraints. Particular emphasis is placed on the waves interacting with discontinuities such as boundaries and changes in cross section, as shown in Fig. 2.1. The basic waves in rods are described by a single mode; coupled thermoelasticity is introduced in this chapter as a first example of handling multi-mode problems where the coupling is between temperature and displacement (strain).

2.1 Elementary Rod Modeling There are various schemes for deriving the equations of motion for structural elements. The approach taken here is to begin with the simplest available model, and then as the need arises to append modifications to it. In this way, both the mechanics and the wave phenomena can be focused on without being unduly hindered by some cumbersome mathematics.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8_3

49

50

2 Longitudinal Waves in Rods

Fig. 2.1 Velocity response of a semi-infinite two-material rod free at one end and impacted at the junction. Note how the pulse is initially partially trapped in the upper material but eventually leaks away after multiple reflections

Fig. 2.2 Segment of rod with loads

2.1.1 Equation of Motion and Spectral Analysis The elementary model considers the rod to be long and slender and assumes it supports only 1D axial stress as shown in Fig. 2.2. It further assumes that the lateral contraction (or the Poisson’s ratio effect) can be neglected. Both of these restrictions are removed in Chap. 4 when the higher order rod models are developed. Following the assumption of only one displacement, u(x), the axial strain is given by xx =

∂u ∂x

Let the material behavior be linear elastic, then the 1D form of Hooke’s law gives the stress as σxx = Exx = E

∂u , ∂x

E = Young’s Modulus

This stress gives rise to a resultant axial force of

2.1 Elementary Rod Modeling

51

 F =

σxx dA = EA

∂u ∂x

Let qu (x, t) be the externally applied axial force per unit length. While not essential, we consider the cross section to be rectangular of depth b and height h. With reference to Fig. 2.2, the balance of forces gives −F + [F + F ] + qu x − ηA x u˙ = ρA x u¨ where ρA is the mass density per unit length of the rod, η is the damping (viscous) per unit volume, and the super dot indicates a time derivative. If the quantities are very small, then the equation of motion becomes (after dividing through by x) ∂F ∂ 2u ∂u = ρA 2 + ηA − qu ∂x ∂t ∂t The independent variables are x and t. Substituting for the force in terms of the displacement gives the equation of motion   ∂ ∂u ∂ 2u ∂u EA = ρA 2 + ηA − qu ∂x ∂x ∂t ∂t

(2.1)

In the special case of uniform properties and no damping, all dependent variables (stress, strain, etc.) have an equation of the form ∂ 2u co2 2 ∂x

∂ 2u − 2 = 0, ∂t

 co ≡

EA ρA

(2.2)

for the homogeneous part. In this case, the waves in the rod are governed by the simple wave equation and the general solution is that of D’Alembert, given by u(x, t) = f (x − co t) + F (x + co t)

(2.3)

This solution does not hold for the general rods of interest in this chapter and therefore is not pursued any further in its present form. Let the solution have the spectral representation developed in Chap. 1. The kernel solutions corresponding to this are obtained by considering the homogeneous equation   d uˆ d EA + ω2 ρAuˆ − iωηAuˆ = −qˆu dx dx

(2.4)

This is the basic equation used in the remainder of the chapter. Assume that both the modulus and the area do not vary with position, then the homogeneous differential equation for the Fourier coefficients becomes

52

2 Longitudinal Waves in Rods

EA

d 2 uˆ + (ω2 ρA − iωηA)uˆ = 0 dx 2

This is an ordinary differential equation with constant coefficients—the frequency is considered as a parameter. Because this equation has constant coefficients, then it has the exponential solutions e−ikx leading to  ˆ −ik1 x + Be ˆ +ik1 x , u(x) ˆ = Ae

k1 ≡

ω2 ρA − iωηA EA

(2.5)

ˆ and Bˆ are the undetermined amplitudes at each frequency. When combined where A with the time variation, this solution corresponds to two waves: a forward-moving wave and a backward-moving wave. That is, u(x, t) =



ˆ −i[k1 x−ωt] + Ae



ˆ +i[k1 x+ωt] Be

(2.6)

ˆ B, ˆ k, etc.) could depend It is understood that all quantities inside the summation (A, on the frequency ω. The spectrum relation for the undamped case is shown plotted in Fig. 2.3 as the continuous line; it is a straight line as seen from  k1 = ω

ρA EA

(2.7)

The comparison is with FE-generated data using a vibration eigenanalysis. To see how this is accomplished, consider a rod with fixed end conditions. The undamped solution is u(x) ˆ = c1 cos(kx) + c2 sin(kx) 3. 2. 1. 0. 0.

20.

40.

60.

80.

100.

Fig. 2.3 Spectrum relation for a rod. The circles are FE-generated data

2.1 Elementary Rod Modeling

53

where c1 and c2 are real only. The zero displacement BCs give 0 = c1 ,

0 = c1 cos(kL) + c2 sin(kL)

This leads to the eigensolution c1 = 0, c2 = ? ⇒ sin(kL) = 0. The latter is satisfied when k = nπ/L. Therefore, the spectrum relation is obtained by counting the number of half-waves and plotting against frequency. Confirmation is hardly necessary in this case but becomes crucially important in our later more complex cases, particularly when there are multiple modes present. The FE results show a slight deviation at the maximum frequency, which is explained in Sect. 9.4 and has to do with the discretization using elements (40 was used in the present case). We use the FE method in this way quite often and therefore refer to it as FE spectrum relation. The undamped spectrum relation gives constant phase and group speeds of  ω c= = k

 EA = co , ρA

dω = cg = dk

EA = co ρA

(2.8)

As a useful reference, the properties and speeds of some typical structural materials are given in Table 2.1. The units for density (and mass) are often a source of confusion in the common system of units; in the table, it is given as weight per unit volume divided by gravity, i.e., ρ = (W/V )/g.

2.1.2 Basic Solution for Waves in Rods We consider the basic problem for waves in a rod to be those arising from the point loading or point impact of a semi-infinite rod—this problem allows us to discuss the essential aspects of the waves.

Table 2.1 Typical properties for some structural materials Material Aluminum Brass Concrete Copper Epoxy Glass Rock Steel

Modulus psi 10.6×106 12.0×106 4.0×106 13.0×106 0.5×106 10.0×106 4.4×106 30.0×106

GPa 72.7 82.3 27.4 89.1 3.4 68.6 30.1 206

Density lb·s2 /in.4 0.26×10−3 0.79×10−3 0.23×10−3 0.83×10−3 0.10×10−3 0.23×10−3 0.24×10−3 0.73×10−3

kg/m3 2700 8100 2400 8500 1000 2300 2500 7500

Wave speed in./s 0.206×106 0.135×106 0.132×106 0.141×106 0.070×106 0.209×106 0.140×106 0.203×106

km/s 5.23 3.43 3.35 3.58 1.78 5.31 3.56 5.15

54

2 Longitudinal Waves in Rods

Fig. 2.4 Free body for the point impact of a rod

Let the end of the rod at x = 0 be subjected to a force history P (t); then according to the free body of Fig. 2.4, we have at x = 0 :

F = EA

∂u(x, t) = −P (t) ∂x

Let all functions of time have the spectral representation; the boundary condition becomes, in expanded form, EA

 d  uˆ n (x)eiωn t = − Pˆn eiωn t n n dx

This has to be true for all time; hence, the equality must be true on a term-by-term basis giving EA

d uˆ n = −Pˆn dx

Problems like this impact problem always reduce to establishing the relationships at a particular n (or frequency ωn ); hence, the subscript can usually be omitted (but understood). The condition at x = 0 now becomes (using the solution (2.5) and because there is only a forward-moving wave) ˆ = −Pˆ EA{−ik1 A}

or

ˆ = A

Pˆ ik1 EA

The complete solution for the forward-moving wave is therefore u(x, t) =



Pˆ e−i[k1 x−ωt] ik1 EA

or (to emphasize the frequency dependence) with every subscript written out explicitly u(x, t) =

1  Pˆn −i[k1n x−ωn t] e n ik1n EA

(2.9)

Knowing the spectrum of the force history, Pˆ , the displacement (and, consequently, the stress, etc.) can be determined at any location x. Examples of the resulting

2.1 Elementary Rod Modeling

55

R

R

ea

ea

l

l

Ima g

Fr eq

g

Ima

Fr eq

Fig. 2.5 Transfer functions for displacement at two positions

strain histories are shown in Fig. 1.15 for the undamped case.√As is apparent, the disturbance travels at the now familiar constant speed of co = EA/ρA. ˆ The transfer function, G(x, ω), for this case is ˆ u(x) ˆ = G(x, ω)Pˆ ,

ˆ G(x, ω) =

1 e−ik1 x ik1 EA

and is shown plotted in Fig. 2.5 for two x positions. We see that the position x contributes the phase change. Note that the amplitude varies as 1/k1 and therefore also as 1/ω, thus the reason the very large values at low frequency. The displacement is related to force by a single integration (that is why k1 occurs in the denominator). Its effect is to essentially amplify the low-frequency components and may cause windowing problems as discussed in Sect. 1.4. The transfer function for the velocity, on the contrary, is ˆ v (x, ω) = iωG(x, ˆ G ω) = 

ω (ρAω2

− iωηA)EA

e−ik1 x

and the amplitude is relatively insensitive to frequency because the damping is usually small. Consequently, the windowing parameters used for transforming the force are also appropriate for reconstructing the velocity. Using the subscripts i and r to represent the forward- and backward-moving waves, respectively, then the other mechanical quantities (at a particular frequency) can be obtained as Displacement: Velocity: Strain: Stress: Force:

ˆ −i[kx−ωt] ui = Ae u˙ i = iωui i = −ikui σi = −ikEui Fi = −ikEAui

ˆ +i[kx+ωt] ur = Be u˙ r = iωur r = ikur σr = ikEAur Fr = ikEAur

56

2 Longitudinal Waves in Rods

giving the following interesting inter-relationships for the undamped case (when k = ω/co ) u˙ i = ikco ui = −

co co σi = − Fi , E EA

u˙ r = ikco ur = +

co co σr = + Fr E EA

Thus the history profiles of the particle velocity, force, and stress are the same. Notice that a tensile stress moving forward causes a negative velocity (i.e., the particles move backward). Also, note that the convention for the sign of the forces, Fi and Fr , is that of the stress because they are internal forces.

2.2 Dissipation in Rods It is straightforward to generalize the descriptions of the rod without actually complicating the structure of the solution. In many cases, it is only the spectrum relation that changes, and otherwise the form of the transfer functions is unchanged. In this section, we consider ways in which the wave is dissipated; the types of dissipation elaborated on are distributed constraints and viscoelastic material behavior.

2.2.1 Distributed Constraint It can be imagined that as the wave moves down the rod, the response can be influenced by the surrounding medium. Specifically, let there be retarding forces (per unit length) proportional to displacement (−Ku u) and the velocity (−ηu); ˙ the latter we have already encountered as internal viscous damping, but it is realized that the same effect could arise from outside the rod. The differential equation of motion becomes EA

∂ 2u ∂ 2u ∂u = ρA 2 − Ku u − ηA 2 ∂t ∂x ∂t

(2.10)

The spectral representation gives EA

d 2 uˆ + [ω2 ρA − iωηA − Ku ]uˆ = 0 dx 2

(2.11)

Because the coefficients are constant, this has the same solution as Eq. (2.5) except that now the spectrum relation can be complex (even if the damping is zero) and given by

2.2 Dissipation in Rods

57

 k1 = ±

ω2 ρA − iωηA − Ku EA

(2.12)

The resulting waves, of course, are dispersive. First consider when only the velocity restraint is present, then the wavenumber is always complex indicating partial decay for all components. This is equivalent to damping. In the limit of small damping, the spectrum relation can be approximated as      ηA ω ρA ω−i =± − i η¯ k1 ≈ ± EA 2ρA co and the solution is ¯ −i[x−co t]ω/co ˆ −ηx e u = Ae

The wave behavior is damped in x. Hence, the generic solutions are damped proportional to ηx. ¯ Generally, we associate the complex part of the spectrum relation with damping or dissipation. As an aside, it is advisable to add numerical stability to all the calculations by incorporating some small damping in the system. The range of “small” should typically be specified on the order of η ≈ ωo × 10−3 ρ where ωo is a typical frequency range. Changes in the damping should then be done as order of magnitude changes on this value. Reference [6] gives an interesting experimental account of the effect of coulomb friction on the outside wall of the rod. Returning now to Eq. (2.12), an interesting case arises when only the elastic constraint Ku is present; Fig. 2.6 shows a plot of the spectrum relation. This may

3.

2.

1.

0. 0.

20.

40.

60.

80.

100.

Fig. 2.6 Spectrum relation for a rod. (a) Full spectrum showing imaginary contributions. (b) Realonly contributions. The circles are FE-generated data

58

2 Longitudinal Waves in Rods

seem like a rather complicated plot because of the occurrence of a cut-on frequency. That is, there is a nonzero frequency for which the wavenumber k is zero. This is given by  Ku ρA − =0 ω EA EA 2

Ku ρA

ωc =

or

(2.13)

Also shown is the FE-generated data for the real part of the spectrum relation. The model used 20 elements for the rod, fixed at both ends, attached to 19 springs (modeled as massless rods). The stiffness of each spring is EA/ l so that the stiffness per unit length is Ku = 19(EA/ l)/L. The agreement is quite good but deteriorates for high frequencies for the reason given in connection with Fig. 2.3. The idea of a cut-on (and cut-off) frequency occurs regularly in our later discussions and so it is of value now to consider this simpler case in a little more detail. Below the cut-on frequency, the wavenumber is imaginary-only and therefore the wave attenuates; for a spectrum wave the components below ωc do not propagate very far and a signal sampled at a large distance is rich in only the high-frequency components. This is illustrated in Fig. 2.7, which shows the significant filtering effect that is occurring. For illustrative purposes, the cut-on frequency was chosen as Fc = 6 kHz. The question must arise that if the rod is elastic and the constraint is elastic how can there be dissipation? For the system as a whole (the rod and springs together), there is no dissipation; what is happening is that the rod is transferring energy to the springs and it is the rod that is loosing energy and hence experiences the dissipation. When we have coupled elastic systems, we generally find cut-on and

0.0

0.5

1.0

1.5

2.0

0.

5.

10.

15.

20.

Fig. 2.7 Responses for a semi-infinite rod with elastic constraints, dashed lines are the nondispersive case. (a) Time-domain strain reconstructions. (b) Frequency-domain amplitude spectrums of the strains

2.2 Dissipation in Rods

59

cut-off frequencies indicating loss (or transfer) of energy from one system to the other. As we progress, we make this point clearer in terms of deformation modes.

2.2.2 Viscoelastic Rod Viscoelastic materials exhibit time-dependent material properties; a familiar example is creep straining under constant load. The purpose of this section is to show how waves in viscoelastic media may be posed in terms of spectral analysis. A good review of uniaxial wave propagation in viscoelastic rods (including some experimental results) can be found in Ref. [16]. The derivation of the equation of motion follows the same procedure as before except that the stress–strain behavior is time dependent. While there are many ways of expressing this relationship, the following form is adequate for present purposes:  p

ap

 d p σxx d q xx = b , q q dt p dt q

p, q = 0, 1, 2, . . .

(2.14)

where ap and bq are the material parameters. Viscoelastic materials require more parameters for their characterization than for simple elastic materials. Here the stress and strain are related through multiple derivatives in time. This can now be expressed in the spectral form as  p

   ap (iω)p σˆ xx = bq (iω)q ˆxx q

or simply σˆ xx

 q q bq (iω) ˆ E(ω) ≡  p p ap (iω)

ˆ = E(ω)ˆ xx ,

(2.15)

which resembles the linear elastic relation. The equation of motion also appears similar to the elastic case and is d uˆ ˆ + ω2 ρAuˆ = −qˆu E(ω)A dx 2 2

The only difference in the solution is that the spectrum relation is  k1 = ±ω

ρA ˆ E(ω)A

(2.16)

which is obviously dispersive. The nonzero imaginary component means there is some attenuation as we now show.

60

2 Longitudinal Waves in Rods

Consider the special case of the standard linear solid as visualized by a parallel spring (E2 ) and dashpot (η2 ) in series with another spring (E1 ) and described by dxx dσxx σxx E1 E2 [E1 + E2 ] = E1 xx + + dt η2 dt η2 The viscoelastic modulus can be written in the following forms: ˆ E(ω) = E1

E2 +iωη2 E0 +iωη2 α E1 −E0 = E1 = Eˆ R (ω)+i Eˆ I (ω) , α = E1 +E2 +iωη2 E1 +iωη2 α E1 (2.17)

A plot of this is shown in Fig. 2.8 using the PMMA properties E1 = 5.56 GPa ,

E2 = 21.3 GPa ,

η = 1.41 MPa s

These properties are taken from Ref. [24], which also gives parameters for the shear and bulk behavior as well as for a PVC material. The modulus has the very slow and very fast behavior limits of ˆ E(0) ≈

E1 E2 = E0 , E1 + E2

ˆ E(∞) ≈ E1

respectively. Note that both of these limits are time-independent elastic, and consequently, the viscoelastic energy dissipation occurs only in the middle frequency range as shown in Fig. 2.8a around 2.5 kHz. The dissipation is a maximum where ˆ the magnitude of I m{E(ω)} is a maximum and is given by

1.0

0.5

0.0 0.

5.

10.

15.

20. 0.

200.

400.

600.

800.

1000.

Fig. 2.8 Viscoelastic behavior of the standard linear solid. (a) Frequency behavior with PMMA properties. The fine line is the phase speed. (b) Effect of viscoelasticity on pulse propagation

2.3 Coupled Thermoelastic Waves

ωm =

61

E12 E1 = , η2 (E1 − E0 ) η2 α

EI m = 12 [E1 − E0 ] = 12 αE1

Reference [17] discusses this dissipation. Figure 2.8b shows some time reconstructions for the strain response in an impacted rod. The input force histories are sine-squared pulses of duration 100 µs (full line) and 50 µs (dashed line), which have a frequency content of about 20 kHz ˆ and 40 kHz, respectively. The nonzero imaginary component of E(ω) means that there is some attenuation in a propagated wave, and this can be seen in the time reconstructions. We also see a broadening of the response plus a kink in the tail end. The kink (which is clearly present in the experimental strains of Ref. [4]) is due to the viscoelastic recovery after the passage of the stress pulse, while the broadening is due to wave dispersion. The phase and group speeds, in general, are given as  ω c= = k

ˆ EA , ρA

 ∂ω cg = = ∂k

  ˆ ω d Eˆ EA / 1− ρA 2Eˆ dω

(2.18)

(Note that if the imaginary part of the wavenumber is large, then the concept of group speed has little meaning.) Both speeds are shown plotted in Fig. 2.8a (the phase speed is the thin line); both have the same limiting values of speed because the limits are elastic. It is observed that the group speed is a maximum where the damping is significant. Thus damping does not necessarily cause the wave to “slow down,” although its amplitude does decrease. In summary, the effect of viscoelasticity on the propagation of a pulse is to decrease the amplitude (because of the damping) and spread the pulse out (because of the spectrum of speeds) and because of recovery. Additional details and considerations of wave propagation in viscoelastic rods are given in Ref. [11]. It focuses on extracting material properties from wave propagation measurements.

2.3 Coupled Thermoelastic Waves Another generic type of loading that can give rise to stress waves is that of a thermal blast. While this is a very interesting problem in its own right (especially in the form of coupled thermoelastic waves), the treatment here is necessarily superficial; a thorough treatment is given in Ref. [5]. Its use here is primarily as the first example to show the spectral approach for coupled systems. That is, the problems treated so far are described by the single spectrum relation, k1 (ω); we now look at a problem that requires or gives rise to two spectrum relations.

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2 Longitudinal Waves in Rods

2.3.1 Governing Equations The derivation of the governing differential equation for the displacement is similar to that as shown earlier in the chapter, except that the constitutive relation must take into account the fact that the strain also changes because of the temperature. That is, 

F = σxx A = EA[xx

∂u − αT − αT ] = EA ∂x



where T is the change of temperature and α is the coefficient of thermal expansion. The governing equation becomes    ∂ ∂u ∂ 2u EA − αT = ρA 2 ∂x ∂x ∂t

(2.19)

To this, we must add a set of equations that govern the temperature. The change of heat flux q(x, t) is balanced by the change of temperature plus any dissipative mechanisms. In the case of coupled thermoelasticity, the dissipative mechanism is the mechanical straining. Hence, −

∂q ∂T ∂xx = ρCE + EαTo ∂x ∂t ∂t

where To is the absolute temperature and CE is the specific heat. The final piece of information is the Fourier law of heat conduction q = −K

∂T ∂x

where K is the thermal conductivity. The equation governing the temperature is then   ∂ ∂T ∂T ∂ 2u K = ρCE + EαTo ∂x ∂x ∂t ∂x∂t

(2.20)

Note that the coupling for u occurs through the constitutive relation, while for T it occurs through the balance relation. For simplicity, consider rods of uniform cross section with uniform properties, the coupled differential equations governing the axial displacement u(x, t) and temperature change T (x, t) are EA

∂ 2u ∂ 2u ∂T , = ρA + αEA 2 2 ∂x ∂x ∂t

K

∂ 2T ∂T ∂ 2u + EαT = ρC E o ∂t ∂x∂t ∂x 2

(2.21)

We want to know if this coupled system can exhibit wave-type solutions, that is, we ask if

2.3 Coupled Thermoelastic Waves

63

ˆ 1 e−i[kx−ωt] , u(x, t) = A

ˆ 2 e−i[kx−ωt] T (x, t) = A

are solutions. Note that the same k is used in both representations. On substituting into the differential equations, we find that the following system of equations must be satisfied  2   ˆ1 ikαEA −k EA + ρAω2 A =0 (2.22) 2 ˆ −Kk − iωρCE −kωETo α A2 ˆ 2 is if the determinant of the ˆ 1 and A The only way to have a nontrivial solution for A matrix is zero. The determinant is not zero in general; however, if k is considered undetermined and adjusted in such a way as to force the determinant to zero, then a nontrivial solution is possible. (The procedure is that used in finding the eigenvalues of a matrix.) Multiplying the determinant out and imposing that it be zero allows k to be determined from the characteristic equation [EAK]k 4 + [iωρCE EA − KρAω2 + E 2 Aα 2 iωTo ]k 2 − [iωρCE ρAω2 ] = 0 This can be solved as a quadratic equation in k 2 . Being quadratic means there are two solutions or modes—we amplify on this presently. Because solving quadratic equations is a regularly occurring requirement in spectral analysis, we take the opportunity now to state the preferred way (i.e., avoiding numerical round-off errors) of doing this. Following Ref. [21], consider the equation a2 z2 + a1 z + a0 = 0 and compute the quantity q = −a1 −



a12 − 4a2 a0

where the sign of the square root is chosen such that the imaginary part is negative. The two roots are then given by z1 =

q , 2a2

z2 =

2a0 q

√ The four roots, in our case, are then given as k = ± zi . In addition, it is imperative that double precision (e.g., Fortran real*8, complex*16) calculations be used. In all, there are four possibilities for the wavenumbers, but because they appear as ± pairs we say there are two modes of behavior. The complete spectral solution is written as ˆ 1 e−ik1 x + Bˆ 1 e−ik2 x + C ˆ 1 eik1 x + D ˆ 1 eik2 x u(x) ˆ =A

64

2 Longitudinal Waves in Rods

ˆ 2 eik1 x + D ˆ 2 eik2 x ˆ 2 e−ik1 x + Bˆ 2 e−ik2 x + C Tˆ (x) = A

(2.23)

But because uˆ and Tˆ are coupled according to Eq. (2.22), then the amplitudes for each mode (the corresponding barred and unbarred coefficients) are related by ( )2 =

2 EA − ρAω2 km ( )1 = Rm ( )1 ikm EAα

(2.24)

where km for the appropriate mode m must be used. Consequently, there are a total of four independent solution sets whose amplitude spectra are represented by the ˆ 1 , Bˆ 1 , C ˆ 1, D ˆ 1 . We emphasize this by writing the temperature solution coefficients A as ˆ 1 e−ik1 x + R2 B ˆ 1 e−ik2 x + R3 C ˆ 1 eik1 x + R4 D ˆ 1 eik2 x Tˆ (x) = R1 A Note that R3 and R4 are obtained by substituting −k1 and −k2 , respectively, into the expression for the amplitude ratio. The coefficients are determined through the BCs and the radiation condition. However, before solving an explicit problem, we first look in more detail at the spectrum relation.

2.3.2 The Spectrum Relation The third term in the second square bracket of the characteristic equation is the coupling, which is specifically due to the coefficient of thermal expansion α; if this is zero, then the uncoupled spectrum relations are  ρAω2 k1 = ± , EA

 k2 = ±

−iωρCE K

The first mode is obviously that of the longitudinal stress wave in the rod, and the other is associated with the temperature change according to Fourier’s Law. This second mode is highly dispersive and therefore does not contribute to the wave propagation. It is emphasized, however, that all modes are necessary to describe the displacement response and all modes are necessary to describe the temperature response because both variables are coupled through the BCs. That is, aside from coupling of the differential equations, there can also be coupling of the BCs, and all modes are necessary to satisfy these properly. We can get a better appreciation of the coupling effect by considering an approximate version of the spectrum relations. First rewrite the characteristic equation as

2.3 Coupled Thermoelastic Waves

65

[EAk 2 − ρAω2 ][Kk 2 + iωρCE ] + βEAiωρCE k 2 = 0 ,

β≡

Eα 2 To ρCE

where β is a measure of the strength of the coupling. Assuming that β is small, we then have as reasonable approximations that [EAk12 − ρAω2 ] ≈ −βEAiωρCE

ρAω2 /EA [KρAω2 /EA + iωρCE ]

[Kk22 + iωρCE ] ≈ −βEAiωρCE

−iωρCE /K [−EAiωρCE /K − ρAω2 ]

or more compactly  k1 = ±

ρAω2 (1−βγ ) , EA

 k2 = ±

−iωρCE (1+βγ ) , K

γ ≡

1 [1−iωK/CE E]

The coefficient γ is frequency dependent and ranges from 1 to 0. We could replace it with 1. It turns out that the spectrum relations are quite insensitive to the coupling parameter β—changes of many orders of magnitude are necessary before an effect is noticeable. The mode 1 behavior has a negligible imaginary component and a nearly linear relation between the frequency and the real part. Consequently, no attenuation (in space) is expected. Further, because the group speed is nearly constant, then this portion of the response propagates unchanged in shape. The mode 2 behavior is quite different in each of these regards. First, there is a sizable imaginary component indicating attenuation. Second, because the relation is nonlinear in frequency, that portion that is propagated changes its shape. Finally, in comparison to the first mode, the magnitudes of the respective wavenumbers are vastly different, indeed, nearly in a ratio of 100 to 1. Consequently, propagation speeds are very slow and attenuation is expected to be much more significant. In summary, it can be said that irrespective of the particular boundary value problem, the mode 2 contribution to the displacement and temperature is highly localized in space, whereas the other mode sends out a propagating nondispersive component. The relative magnitudes involved depend on the geometry and the particular imposed conditions.

2.3.3 Blast Loading of a Rod The physical parameters for this problem are motivated by the experimental work reported in Ref. [20]. In this work, a pulsed ruby laser is used to rapidly heat one end of an aluminum rod. A glass cap is used to capture the laser light and convert it to heat, and the rapid heating causes thermal expansion and hence stresses. Note

66

2 Longitudinal Waves in Rods

that the mechanical properties of the glass and aluminum are closely matched, and therefore the glass cap has little impedance mismatch effect. We take the nominal properties of the glass/aluminum rod as Density, ρ : 2700 kg/m3 Modulus, E : 70 GPa Specific heat, CE : 921 J/kg ◦ C

Diameter, 2a : 3 mm Expansion coefficient, α : 23 µ/◦ C Conductivity, K : 204 W/m ◦ C

The pulse is almost triangular lasting about 40 ns, and pumps about 0.1 J of energy; it is not clear, however, how much of this energy actually enters the rod system. In reality, this pulse generates a frequency content high enough that elementary rod modeling (as done here) is inadequate. To the credit of the cited paper, they consider some higher order models, and we explore them in Sects. 4.2 and 4.5. Here, our focus is on coupling of modes, and it is sufficient to use the elementary rod model for the longitudinal displacement mode. Let the rod be semi-infinite (x ≥ 0) in extent and experience a concentrated heat blast at the free end (x = 0). To model this problem, assume that the temperature is related to the prescribed heat input through the conduction law, and the stress (resultant force) is zero at the blast input end. The BCs for such a problem (following Ref. [2]) are ∂T K (0, t) = q(t) , ∂x

 ∂u(0, t) − αT (0, t) = 0 − αT ] = E ∂x 

σxx = E[xx

In terms of Eq. (2.23), these become ˆ 1 + ik2 R2 Bˆ 1 = −q/K ik1 R1 A ˆ ˆ 1 + (αR2 + ik2 )Bˆ 1 = 0 (αR1 + ik1 )A where only that portion of the solution satisfying the radiation condition in x > 0 is considered. (More specifically, as elaborated in the next chapter, the “radiation condition” requires that dissipation occur in the positive propagation direction, i.e., negative kI .) Solving for the coefficients and using Eq. (2.24) give ˆ 1 = qˆ [αR2 + ik2 ] , A K

ˆ 1 = −qˆ [αR1 + ik1 ] B K

where ≡ k1 k2 (R1 − R2 ) − iαR1 R2 (k1 − k2 ) A quantity of interest is the strain, and this is obtained simply as ˆxx =

∂ uˆ ˆ 1 e−ik1 x − ik2 B ˆ 1 e−ik2 x = G(x, ω)qˆ = −ik1 A ∂x

(2.25)

2.3 Coupled Thermoelastic Waves

67

1000.

500.

20. 0. 0.00

0.02

0.04

0.

0.06

500.

-20.

0. 0.0

2.0

4.0

6.0

8.0

0.0

0.5

1.0

1.5

2.0

Fig. 2.9 Responses due to blast loading of a rod. Positions are in mm. (a) Temperature (units of ◦ C), the x = 0 history is capped. The inset is an estimate of the initial temperature distribution. (b) Strain (units of µ). The x = 0 response is scaled by 10−4

The temperature and strain histories can now be reconstructed and some plots are shown in Fig. 2.9. It is noted that the reconstructions pose a severe wrap-around challenge, which was partially overcome by first obtaining the time derivatives and then integrating. Figure 2.9a shows the temperature histories at locations ranging over 0 ∼ .05 mm; the trend over time is toward a uniform distribution of temperature. We can get an approximate estimate of the initial nonuniform distribution by plotting the peak value at each location; this is shown as the inset plot. More or less, we see that the initial high temperature is localized to about 0.05 mm of the rod; if this were true, it would indicate the experimental impossibility of measuring the initial temperature. Figure 2.9b shows the strain response. What is remarkable is that a nondispersive wave escapes from the localized heated region; its profile is indicative that a portion of the input remains localized. What might be surprising is the very small magnitude of strain considering that at the end it is αT (0, t), which is on the order of 8000 µ. The explanation can be seen from the transfer function associated with the strain where it is noted that at x = 0, G = α, but for nonzero x the second mode attenuates leaving G(x, ω) ≈ −

αk12 k12 − k22

e

−ik1 x

 ≈ −α

k1 k2

2

e−ik1 x

That is, because k2 is so much larger than k1 , the overall amplitude diminishes considerably.

68

2 Longitudinal Waves in Rods

2.4 Reflection and Transmission of Waves Once a wave is initiated, it inevitably meets an obstruction in the form of a boundary or discontinuity because all structures are finite in extent. Thus to extend the wave analysis to structures, it is of great importance to know how the wave interacts with these obstructions. The previous section considered coupling on a differential level; this section deals with coupling at discrete points. Figure 2.10 shows the set of obstructions we consider. More complicated situations, such as waves in curved rods [7, 15] and helical springs [14, 25], and waves in noncollinear rods [1, 9, 18], generally create new wave modes (such as flexural waves) and so are left to Chaps. 3 and 5. The incident wave generates a reflected wave in such a way that the two waves superpose at the boundary to satisfy the BCs. The only waves that can be present are the two given by u(x, t) =



ˆ −i[k1 x−ωt] + Ae



ˆ +i[k1 x+ωt] Be

ˆ is associated with the known incident wave and Bˆ with the unknown where A reflected wave. This is an important point for later analyses: because the differential equations establish all the possible solutions, then the number of choices is automatically prescribed—there is no possibility of “leaving something out.” (This, of course, assumes that the choice of structural model is adequate for the problem.) The BCs are set in terms of Displacement:

u (or u, ˙ etc.)

Force:

EA

∂u ∂x

Fig. 2.10 Free bodies for typical boundaries and joints. (a) Elastic boundary. (b) Oscillator. (c) Joint with concentrated mass. (d) Distributed elastic joint.

2.4 Reflection and Transmission of Waves

69

Table 2.2 Some typical boundary conditions for rods BC type Fixed Free Spring Dashpot Mass

Time domain u(0, t) = 0 ∂u(0, t) =0 EA ∂x ∂u(0, t) EA = −K u(0, t) ∂x ∂u(0, t) ∂u(0, t) EA = −C ∂x ∂t ∂u(0, t) ∂ 2 u(0, t) EA = −M ∂x ∂t 2

Spectral domain uˆ = 0 d u(0) ˆ =0 EA dx d u(0) ˆ EA = −K u(0) ˆ dx d u(0) ˆ EA = −Ciωu(0) ˆ dx d u(0) ˆ EA ˆ = +M ω2 u(0) dx

A typical set of boundaries along with their equations is shown in Table 2.2. Note how the time-domain conditions are converted into conditions on the spectral components u. ˆ

2.4.1 Reflection from an Elastic Boundary Assume that at the end of the rod there is an elastic spring as shown in Fig. 2.10a. The resistive force is proportional to displacement; hence, at the boundary x = 0, we have F (t) = −K u(0, t) or ˆ + ik1 B} ˆ = −K{A ˆ + B} ˆ EA{−ik1 A giving the reflected wave amplitude as ik1 EA − K ˆ A Bˆ = ik1 EA + K

(2.26)

Because the wavenumber k1 is related to frequency, then it is apparent that each frequency component is affected differently and that, in general, the amplitude is complex. Figure 2.11 shows some reconstructions of the pulse after it is reflected from springs of different stiffness. The incident pulse is a compressive force triangle in each case. The behaviors are varied, and to best explain it, consider some of the extreme limits. The BC at a free end (x = 0) requires that the force be zero, that is, for each frequency component ˆ + ik1 B} ˆ =0 EA{−ik1 A ˆ = A, ˆ which says that the reflected displacement pulse is the same This gives B as the incident. By differentiating, it can be seen that the reflected stress pulse is

70

2 Longitudinal Waves in Rods

small stiffness

large stiffness

0.

500.

1000.

1500.

2000.

2500.

Fig. 2.11 Velocity responses for a rod attached to springs of various stiffnesses. The monitoring site is also the impact site

inverted. Thus for the no spring case ˆ , Bˆ = A

ur = ui ,

σr = −σi

On the contrary, if the boundary is fixed, the total displacement is zero giving ˆ, Bˆ = −A

ur = −ui ,

σr = σi

These limits are also achieved when K → 0, K → ∞, respectively. These limiting results are independent of frequency, and hence, the reflected pulse retains the same shape. This is shown as the two extreme cases of Fig. 2.11. For a given spring stiffness, the limiting behavior on the frequency gives rise to the same limits as above. That is, the high frequencies experience a free end and the low frequencies experience a fixed end. Consequently, for a pulse with a spectrum of frequencies, the reflected signal is distorted as shown for the spring of medium stiffness. (Parenthetically, medium in this situation means relative to the amplitude spectrum of the wave.) It is clear that for the medium stiffness spring there is a phase shift in the occurrence of the maximum velocity. This is a typical example of how a wave propagating in a nominally nondispersive medium behaves dispersively when interacting with discontinuities. To amplify on this, consider the relation between ˆ and Fig. 2.12a ˆ the coefficients to be a transfer function. That is, Bˆ = G(ω) A, ˆ shows the plot of G(ω) against frequency. What is striking is that there is a peak in the imaginary part, while the real part goes through zero. This is almost like resonance behavior (as discussed next in relation to Fig. 2.12b) except that the resonant frequency is zero.

2.4 Reflection and Transmission of Waves

71 Re

Re

al

al

g Ima

g Ima

Fre

Fre

q

q

Fig. 2.12 Transfer function for different end conditions. (a) Spring. (b) Oscillator

2.4.2 Reflection from an Oscillator We now consider a more complicated BC: assume that the end of the rod is attached to a concentrated mass via an elastic spring and dashpot as shown in Fig. 2.10b. The resistive force is proportional to both the displacement and the velocity; hence, the equilibrium conditions at the boundary x = 0 and for the mass are, respectively, −Fˆ −K[uˆ o − uˆ c ]−iωC[uˆ o − uˆ c ] = 0,

+K[uˆ o − uˆ c ]+iωC[uˆ o − uˆ c ] = −Mω2 uˆ c

where uo is the displacement of the rod and uc is that of the mass. To these, we add the relations obtained from the wave behavior in the rod ˆ + Bˆ , uˆ o = A

ˆ + ik B} ˆ Fˆ = EA{−ik A

giving the reflected wave amplitude as ikEA + α ˆ ˆ, ˆ Bˆ = A = G(ω) A ikEA − α

α≡

ω2 M[K + iωC] K + iωC − mc ω2

(2.27)

What is most interesting about the relation is the presence of the SDoF oscillator in the denominator of α. If there is no dashpot, this could become zero at the natural frequency, and therefore, α would dominate the response. In fact, at this frequency ˆ = −1 indicating the behavior of a fixed end. What is striking is that it has the form G of a damped resonance, i.e., a peak in one component while the other component goes through zero. This is clearly seen in Fig. 2.12b. To explain these results, consider some of the limiting cases. If there is no mass, i.e., mc = 0, then ˆ, Bˆ = A

ur = ui ,

σr = −σi

72

2 Longitudinal Waves in Rods

which, of course, is the free end condition because the spring has nothing to react against. On the contrary, as the mass is made very large, i.e., M → ∞, the resonance frequency shifts to zero and we recover Fig. 2.12a. If simultaneously, the stiffness is also made very large then ˆ, Bˆ = −A

ur = −ui ,

σr = σi

This gives the fixed end condition. An interesting case is when K = 0, M = ∞ and we take C = EAk/ω = co ρA, then we get that Bˆ = 0, that is, there are no reflections. This is one of the few cases where a single dashpot can absorb the complete wave spectrum.

2.4.3 Concentrated Mass Connecting Two Rods Any change in cross section or material properties causes the generation of new waves. While the actual 3D situation is very complicated, in the present 1D analysis, only a longitudinal transmitted wave and a longitudinal reflected wave are generated. That is, the displacements in the two sections of the rod are taken as ˆ 1 e−ik1 x + Bˆ 1 eik1 x , uˆ 1 = A

ˆ 2 e−ik2 x uˆ 2 = A

The analysis is performed by imposing equilibrium of forces and continuity of displacement across the joint. Consider two rod sections connected by a concentrated mass as shown in Fig. 2.10c. The equation of motion of the mass and the continuity condition gives −F1 + F2 = MJ u¨ J ,

u1 = u2 = uJ

where the subscript J refers to the joint. Because there is only a forward-moving wave in the second rod, these become ˆ 1 + Bˆ 1 }(ik1 ) + E2 A2 {−A ˆ 2 }(ik1 ) = MJ {A ˆ 2 }(iω)2 −E1 A1 {−A ˆ1 = A ˆ2 ˆ1 +B A Solving gives the amplitudes of the two generated waves as ik1 E1 A1 − ik2 E2 A2 + MJ ω2 ˆ Bˆ 1 = A1 ik1 E1 A1 + ik2 E2 A2 − MJ ω2 ˆ2 = A

2ik1 E1 A1 ˆ1 A ik1 E1 A1 + ik2 E2 A2 − MJ ω2

2.4 Reflection and Transmission of Waves

0.0

0.5

1.0

1.5

2.0

73

2.5

0.0

0.5

1.0

1.5

2.0

2.5

Fig. 2.13 Responses for a rod with various concentrated masses. Circles are FE-generated data. (a) Reflected side. The monitoring site is also the input site. (b) Transmitted side

These relationships are complex and frequency dependent; thus distortion of the wave can be expected. This is exhibited in the responses shown in Fig. 2.13. It is clear that the mass acts as a frequency filter. For the low frequencies and similar rods, for example, Bˆ 1 = 0 ,

ˆ2 = A ˆ1 A

showing the signal is unaffected. But at high frequencies ˆ1, ˆ 1 = −A B

ˆ2 = 0 A

showing that the mass acts as a fixed end and does not transmit any of the wave. Notice that the same result is effected by increasing the size of the mass. The case of transmission of waves between two dissimilar rods connected by a massless can be recovered by setting MJ = 0. This gives √ 1− rS rD ˆ ˆ A1 , B1 = √ 1+ rS rD

ˆ 2= A

2 ˆ 1; A √ 1+ rS rD

rS ≡

E2 A2 , E1 A1

rD ≡

ρ2 A2 ρ1 A1

The responses are independent of frequency. This is interesting in comparison to the case of the elastic spring; while the second rod and a spring behave the same statically, the mass distribution of the rod makes a profound difference dynamically. The expressions for the stresses are simply √ rS rD − 1 σi , σr = √ rS rD + 1

√

σt = 2 √

rS rD A1 σi rS rD + 1 A2

(2.28)

Reference [23] gives some experimental validation of these stepped rod results.

74

2 Longitudinal Waves in Rods

This is probably a good place to comment that the FE analysis works both ways: when necessary it can confirm our spectral analyses, and when necessary our spectral analyses can explain the FE results. Neither is set up to duplicate each other, and we view them as complementary to each other.

2.4.4 Interactions at a Distributed Elastic Joint Consider two long bars with an insert of a different material or cross section sandwiched between them as shown in Fig. 2.10d. When the two long portions are of similar material, then this is essentially a split Hopkinson pressure bar and Ref. [8] gives a thorough analysis of its use. Of interest here, however, is the role played by a finite rod in affecting the wave response and how that is indicated in the transfer function. The waves in each section are represented as ˆ 1 e−ik1 x + B ˆ 1 eik1 x uˆ 1 = A ˆ 2 e−ik2 x + B ˆ 2 eik2 x uˆ 2 = A ˆ 3 e−ik3 x uˆ 3 = A ˆ 1 , and there is only one wave in the third rod. Note that The incident wave is A the rod forming the joint has both forward- and backward-moving waves present simultaneously in the finite length, a form of reverberation. This is fundamentally different from the two waves in the semi-infinite first rod as shown here. The BCs at the joint interfaces, x = 0 and L, are at x = 0 :

u1 = u2 ,

F1 = F2

at x = L :

u2 = u3 ,

F2 = F3

Imposing these four conditions gives the system of equations ⎫ ⎤⎧ˆ ⎫ ⎧ ⎪ B1⎪ 1 ⎪ −1 1 1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎬ ⎥ A ⎢(kEA)1 ˆ (kEA)2 −(kEA)2 0 ⎥ 2 = (kEA)1 A ⎢ ˆ1 ⎣ 0 ⎪ eik2 L −e−ik3 L ⎦⎪ 0 ⎪ e−ik2 L Bˆ 2⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎭ ˆ 3⎭ 0 −(kEA)2 e−ik2 L(kEA)2 eik2 L(kEA)3 e−ik3 L ⎩A 0 ⎡

While this can be solved explicitly, it is best left as part of the algorithm for solving for the transfer function of the wave. If all the bars are of the same material, then this relation simplifies considerably, but leaving it in its present form allows, for example, a solution for the viscoelastic joint. Figure 2.14 shows some of the responses on both sides of the insert for an incident triangular pulse. It is apparent that a pulse is propagating backward and forward in the joint. Furthermore, each

2.4 Reflection and Transmission of Waves

75

incident side

reflected side

0.

500.

1000.

1500.

2000.

2500.

Fig. 2.14 Transmitted and reflected responses near a distributed elastic joint. The monitoring site is also the impact site Re

Re

al

al

g

g

Ima

Ima Fr

eq

Fr

eq

Fig. 2.15 Transfer function for the responses near an elastic joint. (a) Reflected side. (b) Transmitted side

reflected and transmitted pulse is similar in shape but diminishing in amplitude. It appears as if there are multiple copies of the pulse generated. In looking at the differences between the present system of equations and those of the previous cases considered, the presence of the eik2 L terms is most significant. These are periodic functions, and, consequently, the solution also expresses periodicity. Figure 2.15 shows plots of the transfer functions; its most striking feature is that as the real component goes through a maximum, the imaginary component goes through a zero. This is a resonance-like behavior. To summarize, the effect of the distributed elastic joint on the response of each term is similar to that of a resonator. In the frequency domain, this causes changes in phase, which result in spectral peaks in the transfer functions; in the time domain, it appears as repeated pulses. In this particular example, each transmitted pulse is similar in shape, perhaps leading to the conclusion that the response is nondispersive (i.e., “no change in shape”). But consider what happens when the joint is small—then the multiple transmissions overlap to give a shape that is definitely not triangular. The point is

76

2 Longitudinal Waves in Rods

that when the full response is compared to the incident wave, then on this basis the elastic joint has a dispersive effect. This raises another point. The effect of the joint is to increase the total duration of the response: a single incident pulse of duration 100 µs, say, may become ten smaller pulses over a duration of 2000 µs. As a consequence, a time window chosen to describe the incident pulse may turn out to be too small to adequately contain the response. The resulting reflections or transmissions from neighboring windows will then propagate into view. These negative effects can be minimized by choosing a very large window with the consequent expense of a large N. An alternative approach is to add a little damping to the system so as to force the multiple reflections to zero sooner. This recommendation holds, in any event, because it also adds stability to the calculations, and it makes the description more realistic since all physical systems have some amount of damping. These considerations re-appear later when we consider structures composed of multiple members.

2.5 Distributed Loading One of the themes to emerge in this book is the idea that complicated structures can be viewed as a collection of connected waveguides. We prepare this idea by analyzing the problem of a load distributed over a finite length of an infinite rod. The problem is analyzed in two ways: first as a distributed load periodically extended to infinity, and second as a segmented rod. These models are shown in Fig. 2.16, which we explain presently. The two models have very different implications.

2.5.1 Periodically Extended Load Model We start by letting the distributed load be represented by the form qu (x, t) = P (t)f (x)

⇐⇒

qˆu (x) = Pˆ f (x)

(2.29)

Fig. 2.16 Modeling of a rod with distributed load over the region 0 ≤ x ≤ L. (a) Actual problem with boundaries at infinity. (b) Periodically extended load. (c) Connected waveguide model

2.5 Distributed Loading

77

In this, the shape of the distribution, f (x), does not change over time—it is only the scaling, P (t), that changes. If the area under f (x) is unity, then P (t) can be interpreted as the resultant force history. We can represent the load distribution using the continuous Fourier transform  +∞  +∞ 1 −iξ x ˆ ˆ f (x) = dξ , f (x) = f (x)e+iξ x dx f (ξ )e 2π −∞ −∞ From a computational point of view, we do not want to deal with infinities and therefore replace the integral with summations. That is, we represent the load distribution shape with the Fourier series f (x) =

1  ˜ −iξm x , fm e m W

ξm ≡

2π m W

where W is the space window shown in Fig. 2.16b. Note that because f (x) is realonly, we could use the sine and cosine form of the Fourier series; in fact, that is how the analysis is implemented; however, for presentational purposes, the exponential form is easier to describe. A key point is that we have essentially assumed that the rod is periodically extended; thus, there are image loads in all the other windows. This is referred to as the wavenumber transform method or the periodically extended load (pEL) model. The usefulness of this solution method is that it can handle arbitrary load distributions such as multiple separate distributions. Now let the response have a similar representation so that the governing system, Eq. (2.4), becomes  1  2 −Pˆ  ˜ −iξ x −ξm EA + (ω2 ρA − iωηA) u˜ m e−iξm x = fm e W W This must be true for any m; hence, we conclude that the space transform components of the displacement are u˜ m =

−Pˆ f˜m [ξm2 EA − (ω2 ρA − iωηA)]

From this, we get the spectral components of the response as u(x) ˆ =

 f˜m −Pˆ   e−iξm x m ξ 2 EA − (ω2 ρA − iωηA) W m

(2.30)

This can then be incorporated in the inverse FFT for the space reconstructions. Generally, however, it is more efficient to compute the summation over m directly rather than use the FFT algorithm because the responses at only a few locations, x, are usually required.

78

2 Longitudinal Waves in Rods

waveguide transform x=0

force

velocity

x=L

400.

500.

600.

700.

800.

500.

600.

700.

800.

Fig. 2.17 Resultant axial force and velocity response for points under the distributed load

The remaining specification for the problem is the actual load distribution shape—for simplicity let us take it as being rectangular over a length L as shown in Fig. 2.16. This gives f˜m = [eiξm L − 1]/(iξm ) Responses are shown in Fig. 2.17 where the history is that of a smoothed pulse and the parameters M = 1200, W = 150 m (6000 in.) were used. A feature to note about the force responses is that at x = L/2, which is the middle of the distribution, the axial force is zero. This can be explained by considering material away from the distributed load site: to the left, the material is being pulled in tension, to the right, the material is being pushed in compression. Note, however, that the velocity of all points is positive. A point worth making about the pEL model is its connection to the Cauchy integral. Reverting to the integral transform, the solution is written as −Pˆ u(x)= ˆ 2π



∞ −∞



  f˜ −Pˆ ∞ f˜e−iξ x −iξ x dξ e dξ = 2π −∞ (ξ, ω) [ξ 2 EA−(ω2 ρA − iωηA)] (2.31)

We know that the value of the integral is simply the sum of the residues [3], that is, from the points where (ξ, ω) = 0. From the contours of Fig. 2.18, we see there are two poles corresponding to where the denominator is zero. These occur at  ξo = ± [ω2 ρA − iωηA]/EA

2.5 Distributed Loading

79

1.0 .8 .6 .4 .2 .0 -.2 -.4 -.6 -.8 -1.0 0

5

10

15

20

25

30

-1.0 -.8 -.6 -.4 -.2

.0

.2

.4

.6

.8 1.0

Fig. 2.18 Contours of the determinant (ξ, ω). On the left is a slice of Re(ξ ) against frequency for Im(ξ ) = 0, and on the right is Re(ξ ) against Im(ξ ) at 15 kHz

which, in fact, are the waveguide spectrum relations. Furthermore, the residues at each of these poles are proportional to e±iξo x which are the kernel function solutions for the waveguide. There is an intimate connection between what we call the waveguide solution and integral solutions of the form of Eq. (2.31). The significant difference lies in the size of the domains; as pointed out above, the waveguide domain can be finite, thus making it useful for structural analysis. We do not have a need for contour integration in the complex plane because, generally, we either do the integration (summations) numerically or replace the integrals with their waveguide form as shown next.

2.5.2 Connected Waveguide Solution In the connected waveguides approach, we divide the structure into segments corresponding to discontinuities in the sectional properties (at joints, for example) and changes of loading. In the present example, there are changes in the loading at x = 0 and x = L; we therefore divide the rod into three segments as indicated in Fig. 2.16c. The solutions in the three segments are x ξ the wavenumber is real and a component of a wave propagates. If we view each ξm as a waveguide, then all the lower frequencies are filtered in a manner analogous to that of a rod with elastic constraint. Hence for a spectrum of ξm , we expect the low-frequency content of the arriving signal to be dominated by the smaller wavenumbers or larger radii. Figure 6.11b shows the comparison of the experimental strains of Fig. 6.9 with those predicted by the above analysis. The results establish the equivalence of the Bessel and double series solutions away from the impact site. The specifications for the computations are time window T = 2048 µs, N = 1024, space window W = 20 m and M is as given in the figure. At the impact site, the singularity in strain is indicated by an increasing peak strain value as M is increased. Again, when dealing with experimentally measured strains, this singularity can be removed by integrating over the length of the strain gage. Although the solutions involving Bessel functions and the wavenumber summation are equivalent, the former is computationally more efficient because only a single series is involved. Unfortunately, the Bessel form of solution is not suitable for imposing BCs along a straight edge and so the double series form must be used. If the response is required at many instances of time and many locations in space, then a double FFT algorithm should be used for the inversion. In the present circumstances, however, the solution is required (almost) continuous in time but only at a limited number of space locations (because the experimental comparison is with a limited number of strain gage data sets). Consequently, the summations over m are performed long form, but those over n are done using the FFT.

6.3 Reflection and Scattering of Flexural Waves

263

6.3 Reflection and Scattering of Flexural Waves When a wave is incident normal to the boundary, then the plate generates a reflected wave and an attenuated wave localized to the boundary. In this respect, it is similar to that of a beam. When a wave is incident obliquely, however, a wave can propagate along the boundary. This is one of the situation of interest here. Reference [15] considers, quite extensively, the reflection of straight-crested flexural waves from the edge of a semi-infinite plate. Here we consider both planar and nonplanar incident wavefronts. For the 2D plate, it is possible for a boundary to be limited in extent—a hole or local distributed support, for example. This has the significant consequence that only some of the wave is affected by the boundary; the generated waves are referred to as scattered waves. This is a difficult subject usually covered in specialty books such as Ref. [24]; this section gives only a short introduction to some of the ideas involved.

6.3.1 Waves Reflected from a Straight Edge The three typical BCs of clamped, pinned, and free are used here so as to illustrate the nature of the reflected waves. We leave until Chap. 7 consideration of BCs involving the connection of multiple plates—while the mechanics are straightforward, the solutions are cumbersome and we opt to wait until we have developed a spectral element approach. The geometry under consideration is shown in Fig. 6.12a. The incident and reflected waves are described by ( ' −iβxC θ +Be ˆ −γ x e−iβySθ , ˆ wˆ i (x, y) = Ae

( ' iβxC θ +De ˆ γ x e−iβySθ ˆ wˆ r (x, y) = Ce

Fig. 6.12 Wave reflections near a straight edge. (a) Incident plane waves. (b) Incident arbitrary fronted waves

264

6 Waves in Plates and Cylinders

where, as before, Cθ ≡ cos θ and Sθ ≡ sin θ and the behavior in y enforces consistency of phase behavior along the boundary. The wavenumber of the reflected evanescent term is governed by the second of Eq. (6.15) and therefore must satisfy γ 2 − β 2 Sθ2 − β 2 = 0

 γ = β 1 + Sθ2

or

When θ = 0, we recover the beam-like behavior for the incident and reflected waves. Let the boundary be at x = 0 and take the total solution as    2 ˆ iβxCθ + De ˆ βx 1+Sθ e−iβySθ ˆ −iβxCθ + Ce w(x, ˆ y) = Ae

ˆ term being This gives a wave generally moving in the y direction, with the A ˆ term being reflected, and the D ˆ term being an x attenuated vibration. incident, the C For simplicity, we assume the incident evanescent Bˆ wave is negligible. Because they all have the common y term then the BCs are satisfied for any y. First consider a clamped boundary, the BCs at x = 0 give ˆ +C ˆ +D ˆ wˆ = 0 = A

 ∂ wˆ ˆ + iβCθ C ˆ + β 1 + S2D ˆ = 0 = −iβCθ A θ ∂x giving the coefficients of the generated waves as  ˆ = − C

1 + Sθ2 + iCθ

ˆ, A

1 + Sθ2 − iCθ

ˆ =  2iCθ ˆ D A 2 1 + Sθ − iCθ

(6.31)

When the angle of incidence, θ , is zero, then the reflected waves are phase shifted relative to the incident waves. On the other hand, when θ approaches π/2, then the ˆ term disappears and C ˆ = −A ˆ showing that the superposition is zero. D Consider now the situation where the edge of the plate is supported by springs such that the BCs become Mxx = −α

∂w , ∂x

Vxz = −Kw

where K and α are the linear and torsional spring constants, respectively. In terms of the wave coefficients these conditions become   ˆ + (1 + S 2 − νS 2 )D ˆ − (C 2 + νS 2 )C ˆ Dβ 2 − (Cθ2 + νSθ2 )A θ θ θ θ    ˆ + iCθ C ˆ + l + S2D ˆ = −αβ − iCθ A θ

6.3 Reflection and Scattering of Flexural Waves

265

   ˆ +C ˆ + D} ˆ ˆ + iCθ C ˆ + l + S2D ˆ [1 − (2 − ν)S 2 ] = −K{A Dβ 3 − iCθ A θ θ These are rearranged to give the amplitudes from ⎤   ˆ −1 + ν ∗ + iCθ α ∗ 1 + ν ∗ + 1 + Sθ2 α ∗ ⎦ C ⎣  ˆ 2 ∗ ∗ ∗ ∗ D iCθ (1 + ν ) − K 1 + Sθ (1 − ν ) + K   1 − ν ∗ + iCθ α ∗ ˆ = A iCθ (1 + ν ∗ ) + K ∗ ⎡

where ν ∗ = (1 − ν)Sθ2 ,

α ∗ = α/βD ,

K ∗ = K/β 3 D

Obviously, the special cases of K = 0 or ∞ and α = 0 or ∞ can easily be obtained. It is apparent that the general case is frequency dependent as was true for the analogous situation for the beam. An interesting point to note is that Poisson’s ratio appears only in association with sin θ ; this means that for normal incidence Poisson’s ratio does not play a role. In general, an arbitrary disturbance (think of a localized load) generates an arbitrarily crested wave, but we can synthesize this as a superposition of many plane waves. That is, we use the periodically extended model introduced in the previous section. The geometry is shown in Fig. 6.12b where the source is indicated as being a distance L from the boundary. The present formulation is set up to solve problems whose BCs are specified along x = constant. At the boundary x = L, the solution and the four conditions can be written as ˆ −ik1 x + Be ˆ −ik2 x + Ce ˆ −ik1 (L−x) + De ˆ −ik2 (L−x) w(x) ˜ = Ae ˆ ∗ + Bˆ ∗ + C ˆ +D ˆ w˜ = A ∂ w˜ ˆ ∗ − ik2 Bˆ ∗ + ik1 C ˆ + ik2 D ˆ = −ik1 A ∂x 1 ˜ 2 ˆ∗ 2 ˆ∗ 2 ˆ 2 ˆ A + kν2 B − kν1 C − kν2 D Mxx = −kν1 D −1 ˜ 2 ˆ∗ 2 ˆ∗ 2 ˆ 2 ˆ A − ik2 kν1 B + ik1 kν2 C + ik2 kν1 D Vxz = −ik1 kν2 D

(6.32)

where the abbreviations ˆ −ik1 L , ˆ ∗ ≡ Ae A

ˆ −ik2 L , Bˆ ∗ ≡ Be

2 kνi ≡ ki2 + νξ 2 = β 2 ∓ (1 − ν)ξ 2

266

6 Waves in Plates and Cylinders

have been used. The particular problems of interest are constructed from combinations of these. Assume the incident wave is characterized at a location a distance L from the ˆ and B ˆ are taken as known. boundary independent of the boundary. Thus, both A A simply supported edge condition requires that at the boundary the deflection and applied moment be zero. That is, w = 0,

Mxx = 0

giving the simple result ˆ = −Ae ˆ −ik1 L , C

ˆ = −Be ˆ −ik2 L D

(6.33)

and in full form ˆ −ik2 (2L−x) ]e−iξy ˆ −ik1 (2L−x) + Be wˆ ref = −[Ae Thus, the wave appears to originate from an image source a distance 2L away from the actual source. However, this is the only case where the image idea applies; in general, the new waves originate at the boundary itself—we develop this idea in the final section of this chapter. For a fixed edge condition both the deflection and slope are zero, ∂w =0 ∂x

w = 0, giving as the system to be solved 

1 1 ik1 ik2



ˆ C ˆ D





−1 −1 = ik1 ik2



ˆ∗ A Bˆ ∗



Using Cramer’s rule this gives      ∗ ˆ ˆ 1 C 2k2 k 1 + k2 A = ˆ ˆ k1 − k2 −2k1 −k1 − k2 B∗ D

(6.34)

The determinant for the system is = ik2 − ik1 = i

k22 − k12 −2β 2 =i k2 + k1 k2 + k1

This is zero only when the frequency is zero. The presence of both e−ik1 L and e−ik2 L ˆ and D ˆ shows that the reflected waves cannot be written simply in terms of in both C image waves.

6.3 Reflection and Scattering of Flexural Waves

267

A free-edge boundary requires the applied moment and shear to be zero, Mxx = 0 ,

Vxz = 0

The system of equations to be solved is 

2 2 −kν2 −kν1 2 ik k 2 ik1 kν2 2 ν1



ˆ C ˆ D





2 2 kν2 kν1 = 2 ik k 2 ik1 kν2 2 ν1



ˆ∗ A Bˆ ∗



where 2 kν1 = β 2 − (1 − ν)ξ 2 ,

2 kν2 = β 2 + (1 − ν)ξ 2

The solution is     ∗  4 + k k4 2 k2 ˆ ˆ −i k1 kν2 2k2 kν1 A C 2 ν1 ν2 = 2 2 4 4 ˆ −2k1 kν1 kν2 −k1 kν2 − k2 kν1 Bˆ ∗ D

(6.35)

The determinant is given by 4 4 = k1 kν2 − k2 kν1

(6.36)

ˆ and D ˆ have a very complicated dependence It is apparent that the coefficients C on both frequency ω and wavenumber ξ . With the present spectral formulation this actually poses no problems when obtaining the inverses (or time reconstructions). Figure 6.13 shows how this determinant depends on wavenumber and frequency. The horizontal axis shows the frequency from 0 to 40 kHz and the vertical axis is

8. 6. 4. 2. 0. -2. -4. -6. -8. 0.

5.

10. 15. 20. 25. 30. 35. 40.

-8.0 -6.0 -4.0 -2.0

.0

2.0 4.0 6.0 8.0

Fig. 6.13 Contours of the determinant. On the left is a slice of Re(ξ ) against frequency for Im(ξ ) = 0, and on the right is Re(ξ ) against Im(ξ ) at 20 kHz.

268

6 Waves in Plates and Cylinders

Re(ξ ) varying from −8.0 to 8.0. Even if the damping coefficient is nonzero, the determinant becomes so small that significant responses can be observed. Thus, in comparison to our previous double summation solution, this plot shows that, of the [N × M] matrix of components, only those along the locus of zero determinant contribute significantly. The free-edge BC is different from the others in that it is possible for the determinant to be zero. This is interesting enough that we consider it in greater detail next.

6.3.2 Free-Edge Waves In the absence of damping, it is possible for the determinant of Eq. (6.36) to equal ˆ D}. ˆ In other zero, therefore raising the possibility of a singular solution for {C, words, it is possible for a free wave to travel along the edge. This wave is analogous to a Rayleigh wave as detailed in Sect. 4.1, the difference is that the wave is dispersive. In its simplest form, we are interested in knowing if the wave ˆ −ik1 x + Fe ˆ −ik2 x ]e−i[ky−ωt] w(x, t) = [Ee where  k1 = ± β 2 − k 2 ,

 k2 = ±i β 2 + k 2 ,

β2 ≡



[ρhωn2 − icωn ]/D

can propagate along the free edge and satisfy the BCs. At present, the wavenumber k is as yet undetermined. By imposing the free-edge conditions, this gives 

2 2 −kν2 −kν1 2 ik k 2 ik1 kν2 2 ν1

  Eˆ =0 Fˆ

(6.37)

A nontrivial solution is possible only if the determinant 4 4 = k1 kν2 − k2 kν1 =0

(6.38)

(This expression for the determinant is obviously the same as occurred in the last section.) The wavenumbers where this happens are (after expanding and simplifying) (1 − ν)3 (3 + ν)k 8 − 2(1 − ν)(1 − 3ν)k 4 β 4 − β 8 = 0 This is quadratic in k 4 hence it is straightforward to solve and get

6.3 Reflection and Scattering of Flexural Waves

269



1/4 √ (1 − 3ν) ± 2 1 − 2ν + 2ν 2 k = ±(1, i)β (1 − ν)2 (3 + ν) These eight values coincide with the eight zeros of Fig. 6.13. We are interested in free waves, that is, when k is real-only, hence of the eight possible solutions, we are only interested in the following pair: 

1/4 √ (1 − 3ν) + 2 1 − 2ν + 2ν 2 k = ±β (1 − ν)2 (3 + ν) The quantity inside the bracket is close to unity for all values of Poisson’s ratio. For example, ν ν ν ν

= 0.00, = 0.25, = 0.30, = 0.35,

k k k k

= 1.00000β, = 1.00041β, = 1.00095β, = 1.00196β,

k1 k1 k1 k1

= −0.0000iβ, = −0.0287iβ, = −0.0436iβ, = −0.0626iβ,

k2 k2 k2 k2

= −1.4142iβ = −1.4145iβ = −1.4149iβ = −1.4156iβ

Our value of ν = 0.33 gives k = 1.00148β, k1 = −0.0544iβ, and k2 = −1.4152iβ. Using this result in Eq. (6.37) gives the amplitudes ratios for Eˆ and Fˆ as Fˆ =

2 kν1 2 kν2

1 − (1 − ν)1.00148 ˆ ν ˆ Eˆ = E≈ E 1 − (1 − ν)1.00148 2−ν

We can now rewrite the solution as a single summation over frequency as w=

 n

' Eˆ e−0.054βn x +

( ν e−1.415βn x e−i[1.00148βn y−ωn t] 2−ν

(6.39)

This wave is characterized by a speed that is slightly slower than the corresponding speed in the infinite sheet, and by an exponential decay as it penetrates into the interior of the plate. We refer to this wave as the “free-edge wave.” The free-edge wave is difficult to isolate because it is dispersive (specifically β is dispersive) and because it travels only marginally slower than the incident wave. To make the presence of this wave more discernible, we impact the plate with a force having a narrow-banded frequency range; the resulting waves disperse very slowly, and the group behavior of the edge wave is thus easier to observe. The force history of this pulse and its corresponding frequency spectrum can be inferred from Fig. 6.14. Figure 6.14 shows transverse velocity responses at various distances from the impact site along the edge. For comparison, we use the free-edge wave (with Eˆ characterized at y = 1500 mm) and use spectral analysis to propagate it both forward and backward as illustrated earlier in the previous section. The most striking

270

6 Waves in Plates and Cylinders

double summation free edge wave y=0.5m x=0

x=0

y=1.0m

x=50mm

y=1.5m

x=100mm

y=2.0m

x=150mm

y=2.5m

x=200mm

Time [µs]

0.

500.

1000.

1500.

2000.

2500.

3000.

2000.

2500.

3000.

3500.

Fig. 6.14 Edge behavior responses. Left, propagation along the edge. Right, propagation away from the edge

feature of this plot is that the amplitude of the edge wave diminishes very slowly over space compared to waves which propagate in the interior of the plate. In addition, our free-edge solution shows insignificant deviations from the double summation solution, thus confirming that the edge wave responses behave very similarly to Rayleigh waves. Figure 6.14 also shows how the edge wave penetrates the interior of the plate. The observation points are at y = 2500 mm, and various x locations inward from the edge. The magnitude of the response decreases rapidly, showing that the edge wave is localized to regions close to the edge. Furthermore, the close agreement of the single and double summation responses again helps confirm that the edge wave responses behave very similarly to Rayleigh waves.

6.3.3 Scattering of Flexural Waves In the case of waveguides, the end condition is total in the sense that the boundary covers the complete cross section—all of the wave experiences the end condition. A wave incident on a localized discontinuity—a circular inclusion as shown in Fig. 6.15, for example, has a significantly different response because only some of the wave is affected by the boundary. One of the basic challenges involved is that the coordinate system used to describe the incident wave is different from the coordinate system of the boundary geometry and consequently, different also from the coordinates used to describe the reflected waves. To illustrate the consequences of this, we consider a plane wave incident on a fixed circular boundary, the schematic of the event is shown

6.3 Reflection and Scattering of Flexural Waves

271

scattered wave

incident wave

shadow back scattered forward scattered

scattered wave

Fig. 6.15 Waves scattered from a fixed circular boundary

in Fig. 6.15. The incident plane wave extending indefinitely in the y direction and given by ˆ −ik2 (L+x) = A ˆ ∗ e−ik1 x + Bˆ ∗ e−ik2 x , ˆ −ik1 (L+x) + Be wˆ = Ae

k1 =

√ ω [ρh/D]1/4

where x = 0 is the origin of our new coordinate system describing the boundary and we have assumed negligible damping. For simplicity, we take the incident evanescent term to be negligible (Bˆ ≈ 0). The geometry of the boundary is circular as shown in Fig. 6.15 hence we need to represent the incident wave in this coordinate system. Let e−ik1 x = e−ik1 r cos θ =

 m

m (−i)m Jm (k1 r) cos mθ ,

0 = 1, m = 2

where Jm are Bessel functions of the first kind. This representation is actually a Fourier series with Jm as the Fourier coefficients; in that case we have the relation  1 Jm (k1 r) = e−ik1 r cos θ cos mθ dθ 2π We therefore have a representation of the incident wave as wˆ =

 m

ˆ ∗ Jm (k1 r) cos mθ m (−i)m A

Just as e−iξm y modified the plane waves in Eq. (6.28) to give circularly crested waves, so does cos mθ modify the circular waves to give the plane waves. The first question we need to resolve is the number of terms necessary in the expansion representing the incident wave. Figure 6.16 shows the reconstruction of the plane wave at x = 0 and y = 300 mm for different number of M. The symptom of not having enough terms in the expansion manifests itself as a sort of wrap-around problem. At the distance y = 300 mm, it is necessary to have about M = 60 to get

272

0.

6 Waves in Plates and Cylinders

500.

1000.

1500.

plane wave

y = 600mm

M=60

y = 300mm

M=40

y=0

M=20

plane wave

2000.

0.

500.

1000.

1500.

2000.

2500.

Fig. 6.16 Effect of number of terms on expansion

good convergence. The figure also shows that the representation deteriorates as the distance y is increased for a fixed number M. The governing equations for flexural waves in cylindrical coordinates are 

 ∂2 1 ∂2 1 ∂ wˆ ± β 2 wˆ = 0 + + r ∂r ∂r 2 r 2 ∂θ 2

Let the displacement be represented by a Fourier series in the hoop direction, that is,  w(r, ˆ θ) = w˜ m cos(mθ ) m

The equation for the amplitudes is  d2  m2 1 d 2 − w˜ m = 0 + ± β r dr dr 2 r2 These are Bessel equations with the solutions w˜ m (r) = AJm (βr) + BYm (βr) ,

w˜ m (r) = CKm (βr) + DIm (βr)

These Bessel functions are shown plotted in the appendix; note that both Ym and Km go to infinity at the origin r = 0, whereas Im goes to infinity at large argument. Hence, when we look at the exterior problem we take B = −iA so that the solution is a Hankel function that coincides with an outward propagating wave, and we take D = 0. A general wave emanating from a circular region is therefore represented by wˆ =

 m

m (−i)m [Am Hm (k1 r) + Bm Km (k1 r)] cos mθ

6.3 Reflection and Scattering of Flexural Waves

273

where Hm = Jm − iYm is the Hankel function. We have for the total response in the plate surrounding the boundary wˆ =

 m

ˆ ∗ Jm (k1 r) + Am Hm (k1 r) + Bm Km (k1 r)] cos mθ m (−i)m [A

The coefficients Am , Bm are now determined from the BCs. For simplicity, we consider only the fixed boundary case, Refs. [16, 24] are good sources for the treatment of some other cases. For the fixed edge condition at r = a, we have ∂w =0 ∂r

w(a) = 0 , giving as the system to be solved 

Hm Km  Hm Km



Am Bm



ˆ∗ = −A

  Jm Jm

where the prime indicates differentiation with respect to the argument z = k1 r. Using Cramer’s rule, this gives       Am Jm Km −Km ˆ∗ 1 = −A , Bm Jm −Hm Hm

 = Hm Km − Hm Km

For a fixed argument, this determinant approaches zero as the order m increases. To avoid numerical difficulties, we use ratios of the Bessel functions and write the solution as    Hm (k1 r) Km (k1 r) ∗ m ˆ ¯ ¯ + Bm cos mθ wˆ = A m (−i) Jm (k1 r) + Am m Hm (k1 a) Km (k1 a) where  (k a) Km Jm (k1 a) 1 − A¯ m K (k a) Jm (k1 a) ≡ m 1 , Km (k1 a) Hm (k1 a) Jm (k1 a) − Km (k1 a) Hm (k1 a)

Hm (k1 a) Jm (k1 a) − −B¯ m H (k a) Jm (k1 a) ≡ m 1 Km (k1 a) Hm (k1 a) Jm (k1 a) − Km (k1 a) Hm (k1 a)

Note that the derivatives are computed from the recurrence relations as given in the Appendix. Figure 6.17 shows some of the responses at various positions around the boundary. First, along the line x = −2a we see the effect of the back-scattered wave, that is, we see the wave reflecting off the object at an angle. Because there are responses in the shadow region, we could say there is a diffraction effect, that is, the wave has “bent around” the boundary. Generally, however, diffraction is associated with sharp edges and in the present case we say

274

6 Waves in Plates and Cylinders forward scattered only y = 8a

x = 2a

x = 2a

total response back scattered only

x = 4a y = 4a x = 6a

y=0

x = 8a x = 10a

0.

500.

1000.

1500.

2000.

0

500

1000

1500

2000

2500

Fig. 6.17 Scattered responses near the boundary. (a) Near side at x = −2a, (b) far side at y = 0

Fig. 6.18 Directivity plot of the power function |ψ(θ)|2 over a range of frequencies

there is a forward scattered wave. It should not be very surprising that there is a forward scattered wave because, in the absence of the boundary, the incident plane wave is present and therefore the scattered wave is necessary to cancel it. We expect the total response to be relatively small in the shadow region, hence the forward scattered wave must be large enough to cancel the incident wave. To further elaborate on the scattered waves, consider the far-field (k1 r  1) response. Replace the Bessel functions with their asymptotic forms  ˆ∗ wˆ s ≈ A

2 −ik1 r e ψs (θ ) π k1 r

where the angle dependence is in the term ψs (θ ) ≡

 m

m (−i)m

 A¯ (k a)  m 1 e−i(−π/4−mπ/2) cos mθ Hm (k1 a)

The scattered power is related to the quantity |ψ(θ )|2 and Fig. 6.18 shows the angle dependence of this quantity for a number of frequencies. The plots are normalized with respect to their areas (or total power). As the frequency increases, a greater

6.4 Waves in Cylinders and Curved Plates

275

proportion of the scattered power is directed in the forward direction; this field interacts with the incident wave to give a long shadow region.

6.4 Waves in Cylinders and Curved Plates There is considerable interest in shells because of such structural applications as arches, canopies, containment vessels, and fuselages to name a few. but, from a wave propagation point of view, they are treated quite differently. Specifically, for the former the waves propagate along the length while they propagate in the hoop direction for the latter. Consequently, we treat the cylinder as a waveguide and the curved plate as having a periodically extended load (pEL) along the length. Figure 6.19 shows the coordinate systems used. More detailed analysis of cylindrical models than what follows can be found in Refs. [17, 20]. Reference [9] goes deeper into the connection between spectral shapes and coupled deformations especially as applied to cylindrical shells and arches.

6.4.1 Deformation of Cylindrical Shells We looked at the response of curved beams in Sect. 3.4; the model derived neglected any variation of response with respect to the lengthwise z direction. Adding this variation is the main difference of this section. There are a variety of ways to derive the shell equations; we find it most expedient to continue using the approach established in the earlier chapters. That is, we first specify the deformation, obtain

Fig. 6.19 Deformation of cylinders. (a) Coordinate system for cylinders. (b) Curved plate segment

276

6 Waves in Plates and Cylinders

the strains, convert to the coordinates of Fig. 6.19, obtain the energies, and then use Hamilton’s principle to derive the equations of motion and the BCs. Following the procedure established in Sect. 3.4, we can approximate the deformation of the shell in the cylindrical coordinates shown in Fig. 3.17 as u¯ r (r, θ, z) ≈ ur (θ, z)

 ξ  ∂ur − uθ R ∂θ ∂ur u¯ z (r, θ, z) ≈ uz (θ, z) − ξ ∂z

u¯ θ (r, θ, z) ≈ uθ (θ, z) −

(6.40)

with ξ ≡ (r − R) and where the third equation allows for bending about the z axis. These give the nonzero strains as ¯θθ =

ur ∂uθ  1 ∂uθ ξ  ∂ 2 ur − + − 2 R R ∂θ ∂θ R ∂θ 2

¯zz =

∂ 2 ur ∂uz −ξ 2 ∂z ∂z

2¯θz =

1 ∂uz ∂uθ ξ  ∂ 2 ur ∂uθ  2 + − − R ∂θ ∂z R ∂z∂θ ∂z

(6.41)

In comparison to the curved beam equations, we have added the displacement in the axial direction, and retained any variations in that direction, that is, with respect to z. At this stage, it is worth our while to convert the above to a more usual form of notation. It is typical in shell analysis to have an axial coordinate x, a hoop coordinate θ , and a transverse coordinated ξ pointing away from the origin of the circle. That is, we have z −→ x ,

r −→ ξ ;

ur −→ w ,

uθ −→ v ,

uz −→ u

We now can write our approximate deformation relations as ∂w ∂x  ξ  ∂w −v v(x, ¯ θ, ξ ) ≈ u(x, θ ) − R ∂θ w(x, ¯ θ, ξ ) ≈ w(x, θ ) u(x, ¯ θ, ξ ) ≈ u(x, θ ) − ξ

The nonzero strains are then xx =

∂ 2w ∂u −ξ 2 ∂x ∂x

(6.42)

6.4 Waves in Cylinders and Curved Plates

w ξ  ∂ 2w 1 ∂v ∂v  + − 2 − R ∂θ R ∂θ R ∂θ 2  2 ξ ∂ w ∂v  1 ∂u ∂v + − 2 − = R ∂θ ∂x R ∂x∂θ ∂x

277

θθ = γθx

(6.43)

Other cylinder models have slightly different expressions for these strains; the present model is closest to that of Reisner [22, 25]. An excellent survey of the different models is given in Refs. [17, 20]. The one developed here is the shell equivalent of the classical plate and the Bernoulli–Euler beam. The strain energy for a small element of cylinder in-plane stress is  U=

1 2

V

' ∗ 2 ( 2 2 E [xx + θθ dV + 2νxx θθ ] + Gγxθ

Substitute for the strains and integrate through the thickness to get the total strain energy as U = UM + UF where  UM =

1 2

+

1 2

UF =

1 2

+

1 2

  

' ( E ∗ h u,2x +(w,θθ +w)2 /R 2 + 2νu,x (v,θ +w)/R dxRdθ ( ' Gh u,θ /R + v,x dsRdθ ( ' D w,2xx +(w,θθ −v,θ )2 /R 4 + 2νw,xx (w,θθ −v,θ )/R 2 dsRdθ ( ' D¯ (2w,xθ −v,x )2 /R 2 dsRdθ

(6.44)

where, as with flat plates, D ≡ Eh3 /12(1 − ν 2 ) and we also have D¯ = Gh3 /12 = D(1−ν)/2. The energy terms are grouped according to normal and shear membrane, and normal and shear flexural contributions. As developed in Ref. [9], this grouping makes clear the type of interaction effects occurring in a complex structure such as a shell. The total kinetic energy is  T =

1 2

V

 =

1 2

˙¯ ˙¯ y, z, t)2 + w(x, ˙¯ ρ[u(x, y, z, t)2 + v(x, y, z, t)2 ] dV   ρh u˙ 2 + v˙ 2 + w˙ 2 ds dy

where we have neglected the rotational inertia.

(6.45)

278

6 Waves in Plates and Cylinders

In this subsection we want to focus on the spectrum relations and therefore skip the load contributions to the potential energy. An application of Hamilton’s principle and taking the variations with respect to δu, δv, and δw leads to three governing equations ( ' ( ' E ∗ h − u,xx −νv,xθ /R−νw,x /R +Gh − u,θθ /R 2 −v,xθ +ρhu=0 ¨ ( ' ( ' (6.46) E ∗ h − νu,xθ /R−v,θθ /R 2 −w,θ /R 2 +Gh − u,xθ /R−v,xx ( ' ( ' ¨ D − v,θθ /R 4 −w,θθθ /R 4 +νw,xxθ /R 2 +D¯ − v,xx /R 2 +2w,xxθ /R 2 +ρhv=0 ( ' E ∗ h νu,x /R+v,θ /R 2 +w/R 2 ( ' D − v,θθθ /R 4 −νv,xxθ /R 2 +w,xxxx −w,θθθθ /R 4 +2νw,xxθθ /R 2 ( ' +D¯ − 2v,xxθ /R 2 +4w,xxθθ /R 2 +ρhw¨ = 0 We consider solutions that are periodic in θ , that is, for wave propagation in x, take the solution in the form u=uo cos(nθ )e−i[kx−ωt] ,

v=vo sin(nθ )e−i[kx−ωt] ,

w=wo cos(nθ )e−i[kx−ωt]

The assumed solution leads to an EVP; however, because the differential equations are complicated, the EVP system matrix is also complicated. It is therefore instructive to separate it into its different energy contributions. To make the patterns more apparent, we introduce the wavenumber ko ≡ 1/R; this makes all terms in the matrices comparable (that is, wavenumber of dimension 1/length). The membrane contributions are ⎡

⎤ k2 νikko n νikko E ∗ h ⎣ −νikko n ko2 n2 ko2 n ⎦ , ko2 −νikko ko2 n

⎡

⎤ ko2 n2 ikko n 0 Gh ⎣ −ikko n k 2 0⎦ 0 0 0

with the DoF arranged as {uo , vo , wo }T . It is noteworthy that the membrane shear affects only u and v, but the normal stresses affect all three displacement components. The flexural contributions are ⎡

⎤ 00 0 ⎦, D ⎣ 0 ko4 n2 ko4 n3 + νk 2 ko2 n 0 ko4 n3 + νk 2 ko2 n k 4 + ko4 n4 + 2νk 2 ko2 n2

⎡

⎤ 00 0 D¯ ⎣ 0 k 2 ko2 2k 2 ko2 n ⎦ 0 2k 2 ko2 n 4k 2 ko2 n2

The flexural behavior involves only the u and v displacements and the axial behavior is absent. In other words, the coupling is only that similar to the arch—if the cylinder had a curvature in the x-direction we would expect coupling in that direction also. As developed thoroughly in Ref. [9], each matrix is essentially a stiffness matrix. The inertia contribution is simply

6.4 Waves in Cylinders and Curved Plates

279

0.10 3. 2. 0.05 1. 0.

0.00 0.

5.

10.

0.0

15.

1.0

2.0

3.0

4.0

Fig. 6.20 Spectrum relations for a cylinder. (a) Showing the complicated interaction effects occurring. (b) Expanded view of (a) and comparison with FE generated data

⎡

⎤ 100 −ρhω2 ⎣ 0 1 0 ⎦ 001 The EVP becomes ' ∗ ( ¯ · ] − ω2 ρh[ · ] { u } = 0 (6.47) E h[ · ] + Gh[ · ] + D[ · ] + D[ The characteristic equation is quartic in k 2 which makes it difficult to solve in simple form. Reference [1] presents a solution scheme that involves solving two cubic equations. In the exact solution of waveguides, we encounter characteristic equations that are typically transcendental and therefore generally cannot be solved in closed form. Section 4.2 outlines a simple numerical scheme for obtaining complex roots; this is the scheme used to obtain the following plots. For each n, there are four mode pairs. Figure 6.20a shows the three real-only branches for n = 1 → 10, we consider the n = 0 case separately. The ring frequency is around 6.4 kHz and we see that below this the behavior is highly coupled. Above this frequency, there is decoupling with the dominant behaviors described by ' (1/2 flexural: k = (1 − 2n2 ) ± [1/4R 4 + kF4 ]1/2 , ' 2 1 ( 1/2 axial: k = kL − 2 (1 − ν)n2 /R 2 , ' 2 ( 1/2 torsional: k = kT − 2n2 /(1 − ν)R 2 ,

v,θ = −w v, w ≈ 0 u, w ≈ 0

where kF4 = ω2 ρh/D, kL2 = ω2 ρh/E ∗ h, kT2 = ω2 ρh/Gh. Each mode has a cut-on frequency that depends on n.

280

6 Waves in Plates and Cylinders

Fig. 6.21 Spectral shape for a cylinder radially constrained at the ends and free axially

Figure 6.21 shows an example spectral shape for a cylinder. While it looks complicated, thinking in terms of the cylinder being unfolded as a flat plate, we see the same pattern of half-waves in the length and transverse (hoop) directions. The FE shell global BCs imposed at each end are that {u, v, w; φx , φy , φz }g = {0, 0, 1; 1, 1, 1} Note that the axial DoF is unconstrained and warping can be observed in the side view of the figure. These conditions are equivalent to periodic BCs (pBCs) along the length, the BCs are already periodic in the hoop direction because the shape is continuous through 2π . The cylinder was meshed with 24 modules along the length and 64 in the hoop direction, using the MRT/DKT shell element [6]. Figure 6.20 shows a plot of the FE data; they show good agreement with the model results.

6.4.2 Wave Propagation Along a Cylinder To get a sense of the complexity involved, consider when the behavior is only axisymmetric. That is, there is no θ dependence. The governing equations reduce to ∂ 2u ν ∂w  + ρh =0 R ∂x ∂x 2 ∂t 2 ∂ 2v D¯ ∂ 2 v ∂ 2v −Gh 2 − 2 2 + ρh 2 = 0 ∂x R ∂x ∂t   ν ∂u 4 w ∂ w ∂ 2w + 2 + D 4 + ρh 2 = 0 E∗h R ∂x R ∂x ∂t − E∗h

 ∂ 2u

+

(6.48)

6.4 Waves in Cylinders and Curved Plates

281

¯ 2 = The second equation is a torsion equation (think of φx = v/R) where D/R 2 2 from the other Gh(h /12R ) can be neglected in comparison to Gh. It is uncoupled √ equations and gives a nondispersive spectrum relation k = ±ω Gh/ρh; this is the St. Venant torsion model. The first and third equations are coupled through Poisson’s ratio; if we set ν = 0, we recover the elementary rod and elementary beam on an elastic foundation models, respectively. We explore the nature of the coupling. Assume wave propagation solutions of the form u = uo e−i[kx−ωt] ,

w = wo e−i[kx−ωt]

which gives the corresponding EVP as      2     0 0 ω 0 uo k 2 iνkko ∗ E h +D − ρh =0 2 4 2 −iνkko ko wo 0 ω 0k The characteristic equation is [γ 2 ]k 6 − [γ 2 kL2 ]k 4 + [(1 − ν 2 )ko2 − kL2 ]k 2 + [kL4 − kL2 ko2 ] = 0 where γ 2 = D/E ∗ h and kL2 = ω2 ρh/E ∗ h. This is cubic in k 2 and readily solved. The real-only portions of the spectrum relation are shown in Fig. 6.22a; there is an abrupt change of behavior around 4 kHz which worth investigating but first consider some special cases. For ω → ∞  k = ±ω E ∗ h/ρh ,

√ k = ± ω[D/ρh]1/4 ,

√ k = ±i ω[D/ρh]1/4

The first is a longitudinal (axial) mode, the others correspond to a flexural (radial) mode. At ω = 0,

0.1

0.2

0.0

0.1

-0.1

0.0 0.

2.

4.

6.

8.

3.9

4.0

4.1

4.2

Fig. 6.22 Axisymmetric behavior of a cylinder. (a) Real-only portions of the spectrum relation. (b) Expanded view of the real and imaginary components of k in the cut-on region

282

6 Waves in Plates and Cylinders

[γ 2 k 4 −(1+ν)ko2 ]k 2 = 0

k = ±0 ,

or

k = ±(1±i)[3(1+ν)/R 2 h2 ]1/4

The axial mode begins at the origin, the others are complex and radially dominant. Thus the flexural behavior is indicative of a beam on an elastic foundation similar to Fig. 3.7, but the radial mode is inconsistent with the axial mode shown in Fig. 3.18b. Look for a cut-on frequency where k = 0; this gives kL2 [kL2 − ko2 ] = 0

or

ωc = 0 ,

ωc = co /R

Interestingly, this is the same cut-on frequency (the ring frequency) as obtained for the curved beam. Figure 6.22b shows an expanded view of the cut-on region; there is strong coupling between the modes. To the left, only the axial dominated mode is realonly, to the right, the axial dominated mode and one of the flexural modes are real-only. In between, approximately 4.0 ∼ 4.1 kHz, only one flexurally dominated mode is real-only, thus the axial mode has what is called a band gap, a frequency range where all components are attenuated. The band gap is explored using an FE analysis. Figure 6.23a shows the FE generated time domain responses for a cylinder axially impacted with a sine-squared forcing history of duration 200 µs. The cylinder has dimensions R = 203 mm (8 in), L = 2.03 m (80 in); 120 modules of the MRT/DKT [6] element are used along the length while 48 are used in the hoop direction. Although this is a fairly large mesh, it is only marginally converged for wave propagation studies. The impacting force has a significant frequency content out to about 10 kHz as shown by the dashed line in Fig. 6.23b. The two vertical lines in the figure indicate the frequency range of Fig. 6.22b; it is clear that a strong filtering action is happening. The amplitude spectrum is of the response at the end of

0.0

0.3

0.6

0.9

1.2

0.

5.

10.

15.

Fig. 6.23 FE generated axial responses for an √ axially loaded cylinder. (a) Velocity responses. Dashed lines are arrival times based on co = E ∗ /ρ. (b) Frequency domain response at x = L. The thin dashed lines delimit the plot region of Fig. 6.22b and the dashed line is the scaled force input

6.4 Waves in Cylinders and Curved Plates

283

the cylinder as shown in Fig. 6.23a. The cg plot in Fig. 6.23b, which is derived from the model, confirms the band gap at 4 kHz, but also shows that the nearby-frequency wave components are dispersive. Additional aspects of waves in cylindrical shells is covered in Refs. [11, 13]. The latter has some early experimental results while the former looks at reflection and transmission coefficients. Reference [19] illustrates how the spectrum relations can be obtained from a single finite element. Additional aspects of band gaps are covered in Chaps. 5 and 9.

6.4.3 Curved Plate Equations The biggest difference between this analysis and that just completed is that the wave propagation is in the hoop direction. Following the discussions associated with Fig. 3.17, we take Rθ −→ s ,

z −→ −y ,

r −→ −z

giving for the corresponding displacements uθ −→ u ,

uz −→ −v ,

ur −→ −w

The approximate deformation relations are u(s, ¯ y, z) ≈ u(s, y) − z

 ∂w

∂s ∂w v(s, ¯ y, z) ≈ v(s, y) − z ∂y

+

u R (6.49)

w(s, ¯ y, z) ≈ w(s, y) The nonzero strains are then ss = −

 ∂ 2w w ∂u 1 ∂u  + −z + R ∂s R ∂s ∂s 2

yy =

∂ 2w ∂v −z 2 ∂y ∂y

γsy =

 ∂ 2w ∂u 1 ∂u  ∂v + −z 2 + ∂s ∂y ∂s∂y R ∂y

(6.50)

Other shell models have slightly different expressions for these strains; the present model is closest to that of Reisner [22, 25]. An excellent survey of the different

284

6 Waves in Plates and Cylinders

models is given in Refs. [17, 20]. The one developed here is the shell equivalent of the classical plate and the Bernoulli–Euler beam. The strain energy for a small segment of shell in-plane stress is  U=

1 2

'

V

( 2 2 2 E ∗ [ss dV + yy + 2νss yy ] + Gγsy

where E ∗ ≡ E/(1 − ν 2 ). Substitute for the stresses and strains and integrate with respect to the thickness to get the total strain energy as U = UM + UF where 

) ∂u w * ∂v  w *2 ) ∂v *2 − dsdy + + 2ν ∂s R ∂y ∂s R ∂y  ) ∂u ∂v *2 1 + 2 Gh + dsdy ∂y ∂s  ) ) 1 ∂u ∂ 2 w * ∂ 2 w  1 ∂u ∂ 2 w *2 ) ∂ 2 w *2 1 + 2 + 2 dsdy UF = 2 D + + 2ν R ∂s R ∂s ∂s ∂y 2 ∂s ∂y 2  ) ∂ 2 w *2 1 ∂u 1 + 2 D¯ +2 dsdy (6.51) R ∂y ∂s∂y

UM =

1 2

E∗h

) ∂u

−

where, as with flat plates, D ≡ Eh3 /12(1 − ν 2 ) and we also have D¯ = Gh3 /12 = D(1 − ν)/2. The energy contributions are grouped according to normal and shear membrane, and normal and shear flexural. As developed in Ref. [9], this grouping makes clear the type of interaction effects occurring in a complex structure such as a shell. The total kinetic energy is  T =

1 2

V

 =

1 2

˙¯ ˙¯ y, z, t)2 + w(x, ˙¯ ρ[u(x, y, z, t)2 + v(x, y, z, t)2 ] dV   ρh u˙ 2 + v˙ 2 + w˙ 2 ds dy

(6.52)

where we have neglected the rotational inertia. Let the potential of the applied loads be  ∂w V = − [qu u+qv v+qw w] dsdy−Qu u−Qv v−Qw w−Qψ ψ−· · · , ψ≡ ∂s The ellipses indicate that additional loads could be applied either along the boundaries or a body loads.

6.4 Waves in Cylinders and Curved Plates

285

An application of Hamilton’s principle and taking the variations with respect to δu, δv, and δw leads to three governing equations  ∂ 2u 1 ∂w  ∂ 2v ∂ 2v  − + Gh + ∂s∂y R ∂s ∂s∂y ∂s 2 ∂y 2 D¯  ∂ 2 u ∂ 2u ∂ 3w ∂ 3w  ∂ 3w  D  ∂ 2u + = ρh + R + Rν + 2R + 2 R ∂s 2 ∂s 3 ∂s∂y 2 R 2 ∂y 2 ∂s∂y 2 ∂t 2  ∂ 2u  ∂ 2u ∂ 2v ∂ 2v  ∂ 2v ν ∂w  + 2− + Gh + 2 = ρh 2 E∗h ν ∂s∂y R ∂y ∂s∂y ∂y ∂s ∂t   1 ∂u ν ∂v w + − 2 (6.53) E∗h R ∂s R ∂y R  1 ∂ 3u ν ∂ 3u ∂ 4w ∂ 4w ∂ 4w  −D + + + 2ν + R ∂s 3 R ∂s∂y 2 ∂s 4 ∂s 2 ∂y 2 ∂y 4 ∂ 2w ∂ 4w  D¯  ∂ 3 u 2 = ρh + 4R − R ∂s∂y 2 ∂s 2 ∂y 2 ∂t 2

E∗h

 ∂ 2u

+ν

This rather complicated collection of equations are a combination of the flat membrane, flat plate, and curved beam equations. The associated BCs on the side s =constant are specified in terms of one each of the following pairs:  ∂u w ∂v  D  ∂ 2 w ∂ 2w  1 ∂u − +ν + + ν + u or Qu = E¯ ∂s R ∂y R ∂s 2 R ∂s ∂y 2   ∂v ∂u + v or Qv = 12 (1 − ν)E¯ ∂s ∂y   ∂ 3w 2 D ∂ u ∂ 2u  ∂ 3w  w or Qw = − − D + (1 − ν) + (2 − ν) R ∂s 2 ∂y 2 ∂s 3 ∂s∂y 2   2 ∂w ∂ w ∂ 2w 1 ∂u or Qm = D (6.54) + ν + 2 2 ∂s R ∂s ∂s ∂y It remains now to interpret the resultants and relate them to these BCs. Referring to Fig. 6.19b, we can form the resultants per unit length as  Nss ≡

 σss dz ,

Nsy ≡

σsy dz

After substituting for the stresses and strains in terms of our approximations leads to Nss = E ∗ h

 ∂u ∂s

−

w ∂v  +ν , R ∂y

Nsy = 12 (1 − ν)E ∗ h

 ∂v ∂s

+

∂u  ∂y

(6.55)

286

6 Waves in Plates and Cylinders

We can also form the resultant moments per unit length  Mss ≡ −

 Msy ≡ −

σss z dz ,

σsy z dz

Again, after substituting for the stresses and strains in terms of our approximations leads to Mss = D

 ∂ 2w ∂s 2

+ν

∂ 2w 1 ∂u  , + R ∂s ∂y 2

Msy = 12 D(1 − ν)

∂w  ∂ u +2 ∂y R ∂s

Comparing these expressions to those for the BCs, we see that the natural BCs are equivalent to specifying Qu = Nss +

1 Mss R

Qv = Nsy Qw = −

∂Msy ∂Mss −2 = Vsz ∂s ∂y

Qm = Mss

(6.56)

The first of these resembles the resultant load expression used for curved beams, while the third resembles the Kirchhoff shear stress relation.

6.4.4 Spectrum Relation for Propagation in the Hoop Direction The loading, and hence the response, is periodic in y. For wave propagation in s a suitable solution form is u=uo cos(ξy)e−i[ks−ωt] ,

v=vo sin(ξy)e−i[ks−ωt] ,

w=wo cos(ξy)e−i[ks−ωt]

As usual, these assumed solution forms lead to an EVP. The membrane contributions are ⎡

⎤ −k 2 −νikξ ikko E ∗ h ⎣ νikξ −ξ 2 νko ξ ⎦ , −ikko νko ξ −ko2

⎡

⎤ −ξ 2 −ikξ 0 Gh ⎣ ikξ −k 2 0 ⎦ 0 0 0

with the DoF arranged as {uo , vo , wo }T . The flexural contributions are

6.4 Waves in Cylinders and Curved Plates

287

⎡

⎤ −k 2 ko2 0 ik 3 ko + νikko ξ 2 ⎦, D ⎣0 00 3 2 4 2 2 4 −ik ko − νikko ξ 0 −k − 2νk ξ ξ

⎡

⎤ −ko2 ξ 2 0 2ikko ξ 2 ⎦ D¯ ⎣ 0 00 2 2 2 −2ikko ξ 0 −4k ξ

The inertia contribution is simply ⎡

⎤ 100 ρhω2 ⎣ 0 1 0 ⎦ 001 These matrices have a lot in common with those of Sect. 6.4, the differences stem from the choice of coordinates. The EVP becomes ' ∗ ( ¯ · ] + ω2 ρh[ · ] { u } = 0 (6.57) E h[ · ] + Gh[ · ] + D[ · ] + D[ The characteristic equation is quartic in k 2 and therefore we use the numerical introduced in Sect. 4.2 to obtain the following plots. An idea of the variety of behaviors is shown in Fig. 6.24 for a value of ξm = 2π m/W with m = 50. Not surprising, there are many branch points in the spectrum relation. Most of the complicated behavior occurs in the vicinity of low frequency as was also the case for the curved beam in Sect. 3.4. We know this region is where the thickness is relevant and to exaggerate its effects, the plots are for the case when h/R = 0.1 and R = 250 mm for an aluminum plate. The figure only shows those modes that have negative imaginary parts, that is, the branches associated with forward-moving waves. Fig. 6.24 Spectrum relations for m = 50 K1 K2 K3 K4

Re

al

Fre

q

Ima g

288

6 Waves in Plates and Cylinders

6.4.5 Donnell Shell Equations Equations (6.53) are too complicated for our present purpose and therefore we would like to affect some simplifications. For the curved beam, the effect of curvature on the flexural behavior was restricted to the very low-frequency range as witnessed by Fig. 3.18. This is not true of plates because of the complicating role played by the axial wavenumber ξm = 2mπ/W . Nonetheless, the results in Fig. 6.20 show that the spectral shapes themselves are predominantly transverse at low frequency and we utilize this in our approximation. Consider the third of Eqs. (6.53), the leading term can be interpreted as E ∗ h[u,s +νv,y −w/R]/R = E ∗ h[ss + νyy ]/R = σss h/R This is a membrane stress contribution. Assume that the in-plane displacements otherwise have negligible contributions (i.e., no inertia effects) and with D¯ = 12 (1 − ν)D, the equation becomes σss h/R − D∇ 2 ∇ 2 w = ρhw¨ ,

∇2 =

∂2 ∂2 + ∂s 2 ∂y 2

It is worth noting in Eq. (3.34) for the curved beam that the middle term in the second equation is an axial force EA[v − Ru,s ] = EAss = F . Thus, for a beam and plate, the effect of curvature is that of a prestress. However, because we do not know the stress, the problem remains a coupled membrane-flexure problem. Donnell [2] introduced an elegant solution for the membrane part of the problem which we now illustrate. For a flat plate in-plane stress, the three nonzero strains are not independent of each other. To see this, consider the collection of derivatives xx ,yy +yy ,xx −γxy ,xy = u,xyy +v,yxx −u,yxy −v,xxy = 0 This is called the compatibility equation. For the curved plate, a similar collection of derivatives gives ss ,yy +yy ,ss −γsy ,sy = u,syy −w,yy /R + v,yss −u,ysy −v,ssy = −w,yy /R This shows the connection between the membrane strains and the out-of-plane deflection. At this stage, we replace the strains with stresses (e.g., ss = [σss − νσyy ]/E), and using the Airy stress function where σss = ψ,yy ,

σxx = ψ,ss ,

the compatibility equation becomes

σsy = −ψ,sy

6.4 Waves in Cylinders and Curved Plates

289

∇ 2 ∇ 2 ψ/E = −w,yy /R When stresses are derived from an Airy stress function, they automatically satisfy the equilibrium equations when body forces are absent. When body forces are present, as, for example, when inertia is present, the equations become more complicated. We make the assumption that, for the purpose of computing the flexural behavior, the in-plane inertia effects can be neglected. This is equivalent to saying that we are considering frequencies below the ring frequency. Consequently, the complete formulation for the flexural behavior is given by −D∇ 2 ∇ 2 w + ψ,yy h/R = ρhw¨ ,

∇ 2 ∇ 2 ψ = −w,yy E/R

These are the equations due to Donnell [2] which elegantly shows the coupling in the curved plate problem. For wave propagation in s, we can assume solutions of the form w = wo cos(ξy)e−i[ks−ωt] ,

φ = φo cos(ξy)e−i[ks−ωt]

or, more generally, using eiξy . These give the coupled system    ρhω2 − D[ξ 2 + k 2 ]2 −ξ 2 h/R wo =0 2 2 2 2 [ξ + k ] φo −ξ E/R The characteristic equation is D[ξ 2 + k 2 ]4 − ρhω2 [ξ 2 + k 2 ]2 + ξ 4 Eh/R 2 = 0 which is quadratic in [ξ 2 + k 2 ]2 and therefore easily solved; this is the simplification we sought. The four root pairs are given by k = ±[±β 2 −ξ 2 ]1/2 ,

( ' β 4 ≡ [ξ 2 +k 2 ]2 = ρhω2 ±[(ρhω2 )2 −4DEh/R 2 ]1/2 /2D

These are shown plotted in Fig. 6.25a as the lines for two small values of m. The squares are the selected comparisons with the more exact equations and the circles are a replot of the FE data from Fig. 6.20. Clearly the new model captures much of the complex interactions between the modes. The outer plots are the flexurally dominated modes, these seem to be well represented at all frequencies. The inner plots deteriorate as the frequency increases. This is because membrane inertia effects are beginning to make a contribution. We correct for this presently. Figure 6.26 shows the spectrums for larger values of m. The comparisons with the more exact model are uniformly good. This is because increasing m shifts the cut-on frequencies further out. To get improved membrane modeling, let us return to the governing system of Eqs. (6.53) and assume that membrane actions involve very little flexing. That is, in

290

6 Waves in Plates and Cylinders

40.

20.

30. 20.

10.

10. 0.

0. -10.

-10.

-20. -30.

-20.

-40. 0.

5.

10.

15.

0.

10.

20.

30.

40.

Fig. 6.25 Spectrum relations for wave propagating in the hoop direction. Dashed lines are imaginary components. (a) Low order m. Circles are FE data, squares are more exact model. (b) High-frequency membrane dominated behavior for m = 5. Squares are the more exact model results

40. 20. 0. -20. -40. 0.

3.

6.

9.

12.

15.

0.

3.

6.

9.

12.

15.

Fig. 6.26 Spectrum relations for wave propagating in the hoop direction. Dashed lines are imaginary components, symbols are the more exact model.

the first two equations, we assume φy = u/R + w,s ≈ 0 ,

φs = w,y ≈ 0

The two equations become E ∗ h[u,ss +u/R 2 − νv,sy ] + Gh[u,yy +v,sy ] = ρhu¨ E ∗ h[νu,sy +v,yy ] + Gh[u,sy +v,ss ] = ρhv¨ For wave propagation in s, we can assume solutions of the form

6.4 Waves in Cylinders and Curved Plates

u = uo cos(ξy)e−i[ks−ωt] ,

291

v = vo sin(ξy)e−i[ks−ωt]

This gives the coupled EVP system  2   kL − k 2 + ko2 − γ ξ 2 −ikξ(ν + γ ) uo =0 2 2 2 vo ikξ(ν + γ ) kL − γ k − ξ where kL2 = ω2 ρ/E ∗ , ko = 1/R, and γ = G/E ∗ . The characteristic equation is [γ ] k 4 +[ξ 2 −kL2 −γ (kL2 +ko2 −γ ξ 2 )−ξ 2 (ν+γ )2 ] k 2 +[(kL2 +ko2 −γ ξ 2 )(kL2 −ξ 2 )] = 0 This is quadratic in k 2 and easily solved. Figure 6.26b shows the high-frequency behavior for m = 5; the simplified model captures the essentials of the behaviors. By separating the flexurally dominated behaviors from the membrane dominated behaviors we could write the spectrum relations in fairly simple analytical forms. The validity of these forms depends on the frequency region and the space wavenumber region. To summarize: the flexurally dominated behaviors are accurately described by the Donnell model for all frequencies and length wavenumber ξ , the membrane dominated behaviors are accurately described by splicing the Donnell model with the simplified membrane model. It is worth remembering that from a wave propagation point of view, the branches (modes) of the spectrum relation with imaginary components are usually not significant because they evanesce. Thus, in general, we can be more relaxed about accurately modeling these complex components.

Further Research 6.1 Show that the governing equation for a membrane pre-loaded plate on an elastic foundation is ∂ 2w ∂ 2w ∂ 2w ∂ 2w + N¯ yy 2 = −ρh 2 + q(x, y) D∇ 2 ∇ 2 w + N¯ xx 2 + 2N¯ xy ∂x∂y ∂x ∂y ∂t where N¯ xx , N¯ yy , N¯ xy are the in-plane loads per unit length. —Ref. [26], p. 431 6.2 Consider an infinite . Show that the solution for the initial value problem 2 2 w(r, t) = fo e−r /a , w(r, ˙ t) = 0 is given by w(r, t) =

   −¯r 2   −¯r 2  −¯r 2 /(1+τ 2 ) fo cos + τ sin e 1 + τ2 1 + τ2 1 + τ2

where τ 2 = 16Dt 2 /(ρha 4 ) and r¯ = r/a. —Ref. [12], p. 239

292

6 Waves in Plates and Cylinders

6.3 For the infinite plate, show that the response to a rectangular pulse of duration τ is     r2    r2 P tH − (t − τ )H , b = D/ρh w(r, t) = 4π bρh 4bt 4b(t − τ ) π H (x) ≡ − Si(x) − sin(x) + xCi(x) 2 where Si and Ci are the sine and cosine integrals, respectively. —Ref. [12], p. 243

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965) 2. Donnell, L.H.: A new theory for the buckling of thin cylinders under axial compression and bending. ASME Aeronaut. Eng. 56, 795–806 (1934) 3. Doyle, J.F., Kamle, S.: An experimental study of the reflection and transmission of flexural waves at an arbitrary T-joint. J. Appl. Mech. 54, 136–140 (1987) 4. Doyle, J.F.: An experimental method for determining the dynamic contact law. Experimental Mechanics 24, 10–16 (1984) 5. Doyle, J.F.: Further developments in determining the dynamic contact law. Experimental Mechanics 24, 265–270 (1984) 6. Doyle, J.F.: Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability. Springer, New York (2001) 7. Doyle, J.F.: Guided Explorations of the Mechanics of Solids and Structures: Strategies for Solving Unfamiliar Problems. Cambridge University Press, Cambridge (2009) 8. Doyle, J.F.: Nonlinear Structural Dynamics Using FE Methods. Cambridge University Press, Cambridge (2015) 9. Doyle, J.F.: Spectral Analysis of Nonlinear Elastic Shapes. Springer, New York (2017) 10. Dyer, I.: Moment impedance of plates. J. Acoust. Soc. Am. 32, 1290–1297 (1960) 11. Fuller, C.R.: The effect of wall discontinuities on the propagation of flexural waves in cylindrical shells. J. Sound Vib. 75(2), 207–228 (1981) 12. Graff, K.F.: Wave Motion in Elastic Solids. Ohio State University Press, Columbus (1975) 13. Heimann, J.H., Kolsky, H.: The propagation of elastic waves in thin cylindrical shells. J. Mech. Phys. Solids 14, 121–130 (1966) 14. Kalnins, A.: On fundamental solutions and Green’s functions in the theory of elastic plates. J. Appl. Mech. 33, 31–38 (1966) 15. Kane, T.R.: Reflection of flexural waves at the edge of a plate. J. Appl. Mech. 21–22, 213–220 (1954) 16. Konenkov, Y.K.: Diffraction of a flexural wave by a circular obstacle in a plate. Sov. Phys. Acoust. 10(2), 153–156 (1964) 17. Leissa, A.W.: Vibration of Shells. NASA SP-288 (1973) 18. Ljunggren, S.: Generation of waves in an elastic plate by a vertical force and by a moment in the vertical plane. J. Sound Vib. 90, 559–584 (1983) 19. Manconi, E., Mace, B.R.: Wave characterization of cylindrical and curved panels using a finite element method. J. Acoust. Soc. Am. 125(1), 154–163 (2009) 20. Markus, S.: Mechanics of Vibrations of Cylindrical Shells. Elsevier, New York (1988) 21. Medick, M.A.: On classical plate theory and wave propagation. J. Appl. Mech. 28, 223–228 (1961)

References

293

22. Naghdi, P.M., Berry, J.G.: On the equations of motion of cylindrical shells. J. Appl. Mech. 21(2), 160–166 (1964) 23. Oden, J.T.: Mechanics of Elastic Structures. McGraw-Hill, New York (1967) 24. Pao, Y.H., Mow, C.C.: The Diffraction of Elastic Waves and Dynamic Stress Concentrations. Crane, Russak and Company, New York (1973) 25. Reissner, E.: Stress and displacement of shallow spherical shells. J. Math. Phys. 25(1), 80–85 (1946) 26. Shames, I.H., Dym, C.L.: Energy and Finite Element Methods in Structural Analysis. Hemisphere, Washington (1985) 27. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1963) 28. Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, New York (1968)

Chapter 7

Thin-Walled Structures

The thin-walled structures of interest in this chapter are generally enclosed regions, as typified by an aircraft fuselage. They are modeled as combinations of folded plates (flat plate segments assembled along their edges) and cylindrical shell segments as shown in Fig. 7.1. Other examples of structures that fall into this category include plates with stringers, corrugations, channels, ducts, troughs, and open and closed thin-walled tubes with one or more cells. A characteristic feature is that the axial length is large relative to the hoop dimensions. As pointed out in the previous chapter, the only way to efficiently handle problems with complicated boundaries and discontinuities is to develop a matrix methodology for use on a computer. The approach we use here is similar to that of the spectral element method developed earlier for the waveguide analysis; however, for the present extended structures, the stiffness is established in the frequency/wavenumber domain. That is, the individual elements are considered to be infinite along the length, but the loads are periodically extended. The focus is on propagation in the circumferential direction. The method to be developed has some features in common with the finite strip methods of Cheung [1] and Eterovic and Godoy [7]. The significant difference, however, is that the current approach uses the exact dynamic solution and, hence, has the advantage that there is no restriction on size or frequency range—they are not strip elements. The limitations are those inherited from the use of periodic extended loads (pELs).

7.1 Spectral Elements for Flat Plates A flat plate generally loaded has both in-plane (membrane) and out-of-plane (flexural) actions. We develop separate elements for each of these actions. The

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8_8

295

296

7 Thin-Walled Structures

Fig. 7.1 Complex multicelled thin-walled structure modeled as a collection of flat and curved extended elements

Fig. 7.2 Flat plate spectral element. (a) DoFs and nodal loads on a positive face. (b) Periodic loads in y

geometry for both elements is shown in Fig. 7.2. Note that the elements are infinitely long, but the loads are restricted to be periodic in y.

7.1.1 Membrane Spectral Elements The in-plane or membrane response of the thin plate is developed in Sect. 4.2. Much of the element developments follow by analogy with Chap. 5, hence we emphasize only those aspects that are different. More details are found in Refs. [10, 11]. The Helmholtz potentials for both symmetric and antisymmetric loadings are given as    ˆ nm e−ik1nm x + Bˆ nm e−ik1nm (L−x) {cos ξm y, sin ξm y} ˆ nm (x, y)=  A m

7.1 Spectral Elements for Flat Plates

297

   ˆ nm e−ik2nm (L−x) {sin ξm y, cos ξm y} ˆ nm e−ik2nm x + D Hˆ znm (x, y)= C m   2 2 k2 = kP2 − ξm2 (7.1) k1 = kP − ξ m , where L is the length of the element as shown in Fig. 7.2, and kP and kS are the wavenumbers as given in Chap. 6. The displacements u and v, therefore, have spectral representations with the kernels   ˆ nm e−ik1nm x + Bˆ nm e−ik1nm (L−x) ik1nm u˜ nm (x, ηm , ωn ) = + −A   ˆ nm e−ik2nm x + D ˆ nm e−ik2nm (L−x) ξm ± +C   ˆ nm e−ik1nm x − Bˆ nm e−ik1nm (L−x) ξm v˜nm (x, ηm , ωn ) = ± −A   ˆ nm e−ik2nm x − D ˆ nm e−ik2nm (L−x) ik2nm + +C

(7.2)

where ± corresponds to choosing either the symmetric or antisymmetric terms, respectively. It is noticed that these signs are associated with the ξ term, and, hence, we drop the ± and it need only be necessary to make the change ξ → −ξ as appropriate. The subscripts n and m are cumbersome to carry along; therefore, in the subsequent developments, these are also dropped and their presence can be inferred from the overhead tilde. To get the shape functions, we replace the above coefficients with the frequency and wavenumber dependent generalized nodal DoFs u˜ 1 , v˜1 , u˜ 2 , and v˜2 . Although we have only four coefficients, we have two functions and this results in two sets of shape functions. These can be arranged as u(x) ˜ = g˜ 1u (x)u˜ 1 + g˜ 2u (x)v˜1 + g˜ 3u (x)u˜ 2 + g˜ 4u (x)v˜2 v(x) ˜ = g˜ 1v (x)u˜ 1 + g˜ 2v (x)v˜1 + g˜ 3v (x)u˜ 2 + g˜ 4v (x)v˜2

(7.3)

The frequency and wavenumber dependent shape functions for the two-noded element are significantly more complicated than those we have looked at before and are given as ˜− g˜ iu (x) = ik1 h˜ − i1 (x) + ξ hi2 (x) , where the functions h˜ ± ij are

˜+ g˜ iv (x) = ξ h˜ + i1 (x) + ik2 hi2 (x)

(7.4)

298

7 Thin-Walled Structures

h˜ ± 11 (x) ˜h± (x) 21 h˜ ± 31 (x) h˜ ± 41 (x) h˜ ± 12 (x) ˜h± (x) 22 h˜ ± 32 (x) h˜ ± 42 (x)

= = = = = = = =

+ik2 +ξ −ik2 −ξ −ξ −ik1 +ξ −ik1

[(e2 f1 + f2 )e−ik1 x [(e2 f1 − f2 )e−ik1 x [(e2 f2 + f1 )e−ik1 x [(e2 f2 − f1 )e−ik1 x [(e1 f1 + f2 )e−ik2 x [(e1 f1 − f2 )e−ik2 x [(e1 f2 + f1 )e−ik2 x [(e1 f2 − f1 )e−ik2 x

± (e2 f2 + f1 )e−ik1 (L−x) ]/ ± (e2 f2 − f1 )e−ik1 (L−x) ]/ ± (e2 f1 + f2 )e−ik1 (L−x) ]/ ± (e2 f1 − f2 )e−ik1 (L−x) ]/ ± (e1 f2 + f1 )e−ik2 (L−x) ]/ ± (e1 f2 − f1 )e−ik2 (L−x) ]/ ± (e1 f1 + f2 )e−ik2 (L−x) ]/ ± (e1 f1 − f2 )e−ik2 (L−x) ]/

(7.5)

and f1 = (k1 k2 − ξ 2 )(e1 − e2 ), ≡ f12 − f22 ,

f2 = (k1 k2 + ξ 2 )(1 − e1 e2 )

e1 = e−ik1 L ,

e2 = e−ik2 L

Figure 7.3 illustrates the wavenumber dependence of the shape functions where ξ = 2π m/W and W = 4 m. The shape functions occur in pairs where g˜ 3 and g˜ 4 are the mirror images of g˜ 1 and g˜ 2 , respectively. Note that as m increases each shape function exhibits behavior localized to the node. This occurs because both k1 and k2 are complex for large m as illustrated in Fig. 4.4. Working in the analogous way, we get for the semi-infinite plate u(x) ˜ = g˜ 1u (x)u˜ 1 + g˜ 2u (x)v˜1 ,

v(x) ˜ = g˜ 1v (x)u˜ 1 + g˜ 2v (x)v˜1

(7.6)

1. m=0 m=100 m=200 m=300

0.

.0

.2

.4

.6

.8

Fig. 7.3 Membrane shape functions at 5 kHz

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

7.1 Spectral Elements for Flat Plates

299

where g˜ 1u (x)=[k1 k2 e−ik1 x + ξ 2 e−ik2 x ]/ ,

g˜ 2u (x) = ik1 ξ [e−ik1 x − e−ik2 x ]/

g˜ 2v (x)=[k1 k2 e−ik2 x + ξ 2 e−ik1 x ]/ ,

g˜ 1v (x) = ik2 ξ [e−ik2 x − e−ik1 x ]/

≡k1 k2 + ξ 2

(7.7)

We reiterate that the significance of the shape functions is that they allow the complete dynamic description of the element to be captured entirely in terms of the nodal DoFs—the functions themselves are used to interpolate (exactly) between the nodes. It is now a straightforward task of differentiation in order to get the stresses and nodal tractions. We begin with the two noded element. The spectral representation of the normal and shear stresses (σ˜ xx , σ˜ yy , and σ˜ xy ) is found from the displacements, the strain–displacement relations, and the constitutive relations. The unit normals at node 1 are nx = −1, ny = 0 and at node 2 are nx = 1, ny = 0, thereby giving the relation between the nodal forces per unit length and stresses as N˜ xx1 = −σ˜ xx1 h ,

N˜ xx2 = σ˜ xx2 h

N˜ xy1 = −σ˜ xy1 h ,

N˜ xy2 = σ˜ xy2 h

These play the role of nodal loads. The nodal displacements are related to the generalized nodal forces through the frequency and wavenumber dependent dynamic element stiffness matrix {F˜nm } = [ k˜nm ]{u˜ nm }

(7.8)

where now the dynamic stiffness matrix [k˜nm ] is of size [4 × 4], is complex and symmetric, and can be written as ⎤ k11nm k12nm k13nm k14nm   ⎢ k22nm −k14nm k24nm ⎥ ⎥ k˜nm = ⎢ ⎣ k11nm −k12nm ⎦ sym k22nm ⎡

where the individual stiffness terms are explicitly   μh (−ik2 kS2 ) k1 k2 z12 z21 + ξm2 z11 z22 nm   μh = (−ik1 kS2 ) k1 k2 z11 z22 + ξm2 z12 z21 nm   μh = ξm k1 k2 (−kS2 + 4ξm2 ) 4e−ik1 L e−ik2 L − z12 z22 nm

k11 = k22 k12

(7.9)

300

7 Thin-Walled Structures

10.

1. m=0 m=50 m=100 m=150 m=200

0.

5.

10.

0.

15.

5.

10.

15.

20.

Fig. 7.4 Dynamic stiffnesses normalized as k˜11 /[E ∗ h/L] and k˜22 /[Gh/L], respectively

  μh ξm ξm4 − ξm2 k22 + 2k12 k22 z11 z21 nm   μh = 2ξm k1 k2 kS2 e−ik1 L z22 − e−ik2 L z22 nm   μh = 2ik2 kS2 k1 k2 e−ik2 L z21 + ξm2 e−ik2 L z11 nm   μh = 2ik1 kS2 k1 k2 e−ik2 L z11 + ξm2 e−ik1 L z21 nm −

k14 k13 k24

The determinant quantity nm is nm = 2ξm2 k1 k2 [4e−ik1 L e−ik2 L − z12 z22 ] − [k12 k22 + ξm4 ]z11 z21 zj 1 = 1 − e−2ikj L ,

zj 2 = 1 + e−2ikj L

A very interesting aspect of this matrix is that it has a form (in terms of repetition of entries) similar to that of the spectrally formulated beam element of Sect. 5.3. This comes about essentially because both the beam and the 2-D membrane have two DoFs per node. Figure 7.4 shows the frequency and wavenumber dependences of the diagonal stiffnesses. Note that the higher the m the larger the stiffnesses and that significant “resonant” behavior is only seen at the lower m values. This is consistent with the behavior of the shape functions. The one-noded or throw-off element is treated in the same way. The unit normals at node 1 are nx = −1, ny = 0, thereby giving the relation between the nodal loads (per unit length) and stresses as N˜ xx1 = −σ˜ xx1 h ,

N˜ xy1 = −σ˜ xy1 h

(7.10)

7.1 Spectral Elements for Flat Plates

301

Substituting for the stresses (using the expressions of Sect. 4.1), we eventually get the relationship between the nodal loads and the nodal displacements as      μh N˜ xx1 ik2 kS2 ξm [2ξm2 + 2k1 k2 − kS2 ] u˜ 1 = v˜1 N˜ xy1 ik1 kS2 k1 k2 + ξm2 sym

(7.11)

This is written in shorthand as {F˜nm } = [ k˜nm ]{u˜ nm } where [ k˜nm ] is a [2 × 2] dynamic stiffness matrix. It is symmetric and complex. Each stiffness shows a significant imaginary contribution. Furthermore, the throwoff element does not exhibit a resonant behavior. This element recovers the solution for the half-plane [10]; the Rayleigh wave behavior, which might have been thought to be in the above, enters into the relation only after solving for the displacements. The BCs are implemented as done for the waveguide elements. The normal load BC, for example, is Nxx1 = S(y)P (t)   N M M = 12 ao + am cos(ξm y) + bm sin(ξm y) Pˆn eiωt m=1

Nxy1 = 0

m=1

n=1

(7.12)

The last portion of Nxx1 corresponds to an arbitrary normal loading with S(y) representing a spatial variation of the load having the Fourier coefficients am , bm , and P (t), the time variation of the load with the Fourier coefficients Pˆn .

7.1.2 Flexure Spectral Elements We summarize the development of flexural plate elements. Unlike the case of the membrane behavior where there are two displacement functions, here we use a single function to represent the transverse displacement. Nonetheless, there are two DoFs at each edge, and therefore, the results are quite similar to those of the membrane case. More details can be found in Refs. [2–5]. The general spectral solution for the dynamic response of the plate in vacuum can be written as  ˆ mn e−ik1 x + B ˆ mn e−ik2 x w(x, ˆ y) = m[A

(7.13)

ˆ mn e−ik1 (L−x) + D ˆ mn e−ik2 (L−x) ]{cos(ξm y), sin(ξm y)} +C where the wavenumbers are given by

302

7 Thin-Walled Structures

k1 =



 k2 = −i β 2 + ξm2 ,

β 2 − ξm2 ,

β 4 = (ρhω2 − iηhω)/D

The subscripts m and n are understood on the wavenumbers k1 and k2 , and L is introduced to associate the reflections coming from a boundary at x = L. For convenience, most of the subsequent expressions omit the summation and the m and n subscripts; both of these are to be implied. ˆ =D ˆ = 0. The end conditions require First consider the throw-off element with C that ˆ + Bˆ , w(0) ˜ = w˜ 1 = A

ˆ + −ik2 Bˆ ψ˜ x (0) = ψ˜ 1 = −ik1 A

ˆ and B ˆ can be rewritten in terms of the nodal displacement w˜ 1 and The constants A ˜ rotation ψx1 . Doing this, we can therefore rewrite the displacements in the form w(x) ˜ = g˜ 1 (x)w˜ 1 + g˜ 2 (x)ψ˜ 1

(7.14)

The functions g˜ i (x) are the frequency dependent plate shape functions and are given by   g˜ 1 (x) = −ik2 e−ik1 x + ik1 e−ik2 x /   g˜ 2 (x) = −e−ik1 x + e−ik2 x / ,

≡ i(k1 − k2 )

(7.15)

The complete description of this element has now been captured in the two nodal degrees of freedom w˜ 1 and ψ˜ 1 . The process of obtaining the stiffness relation is simply that of differentiation to obtain the bending moments and shear forces on the edges. First, moment and shear expressions as given by the BC expressions are expanded by replacing w(x, ˜ y) with the spectral representation of Eq. (7.17). This leads to 

V˜1 M˜ 1





−ik1 k2 (k1 + k2 ) −k1 k2 + νξ 2 =D −k1 k2 + νξ 2 −i(k1 + k2 )



w˜ 1 ψ˜ 1

 (7.16)

This is a dynamic stiffness relationship between the loads applied at the edge of a semi-infinite plate, and the deflections along this edge. The development of the two-noded element, shown in Fig. 7.2, follows along the same lines. The end conditions on the element are w(0) ˜ = w˜ 1 ,

d w(0) ˜ = ψ˜ 1 , dx

w(L) ˜ = w˜ 2 ,

d w(L) ˜ = ψ˜ 2 dx

Solving for the coefficients in terms of the nodal degrees of freedom allows the transverse displacement to be written in the form

7.1 Spectral Elements for Flat Plates

303

w(x) ˜ = g˜ 1 (x)w˜ 1 + g˜ 2 (x)Lψ˜ 1 + g˜ 3 (x)w˜ 2 + g˜ 4 (x)Lψ˜ 2

(7.17)

The frequency and wavenumber dependent shape functions for the two-noded element are significantly more complicated than those for the throw-off element and are given as g˜ 1 (x) g˜ 2 (x) g˜ 3 (x) g˜ 4 (x)

= [r1 h˜ 1 (x) + r2 h˜ 2 (x)]/ = [r1 h˜ 3 (x) + r2 h˜ 4 (x)]/ = [r1 h˜ 2 (x) + r2 h˜ 1 (x)]/ =−[r1 h˜ 4 (x) + r2 h˜ 3 (x)]/ ,

(7.18) = r12 − r22

where h˜ 1 (x) h˜ 2 (x) h˜ 3 (x) h˜ 4 (x)

= ik2 [e−ik1 x − e2 e−ik1 (L−x) ] − ik1 [e−ik2 x − e1 e−ik2 (L−x) ] = −ik2 [e2 e−ik1 x − e−ik1 (L−x) ] + ik1 [e1 e−ik2 x − e−ik2 (L−x) ] = [e−ik1 x + e2 e−ik1 (L−x) ] − [e−ik2 x + e1 e−ik2 (L−x) ] = [e2 e−ik1 x + e−ik1 (L−x) ] − [e1 e−ik2 x + e−ik2 (L−x) ]

and r1 = i(k2 − k1 )[1 − e1 e2 ], e1 = e−ik1 x , e2 = e−ik2 x

r2 = i(k2 + k1 )[e1 − e2 ]

Functionally, these are the same shape functions used in describing the beam waveguide behavior; the difference is that the wavenumbers k1 and k2 have a dependence on ξ . Figure 7.5 illustrates the wavenumber dependence of the shape functions. The shape functions occur in pairs where g˜ 3 and g˜ 4 are the mirror images of g˜ 1 and g˜ 2 , m=0

1

m=100 m=200 m=300

0

0.

2.

4.

6.

8.

0.

2.

4.

6.

Fig. 7.5 Shape functions g˜ i (x) and their derivatives for a plate in flexure at 1 kHz

8.

10.

304

7 Thin-Walled Structures

respectively. Note that the higher m terms tend to give distributions localized to the node. The process to obtain the stiffness relation for the double-noded element is the same as for the throw-off element and leads to ⎫ ⎡ ⎧ ⎤⎧ ⎫ w˜ 1 ⎪ k11nm k12nm k13nm k14nm ⎪ V˜1 ⎪ ⎪ ⎪ ⎪ ⎨ ˜ ⎪ ⎬ ⎢ ⎬ ⎨ ˜ ⎪ ⎥ k22nm −k14nm k24nm ⎥ ψ1 M1 ⎢ = ⎣ ⎪ k11nm −k12nm ⎦ ⎪ w˜ ⎪ ⎪ ⎪ ⎪ V˜2 ⎪ ⎩ 2⎪ ⎭ ⎭ ⎩ ψ˜ 2 symm k22nm M˜ 2

(7.19)

˜ is the dynamic plate element stiffness matrix; its individual terms are The matrix [k]   k˜11 = D −k1 k2 [k12 − k22 ][r1 z22 + r2 z21 ] /   2 2 [r1 z11 + r2 z12 ] + ik2 kν2 [r1 z11 − r2 z12 ] / k˜12 = D −ik1 kν1   k˜13 = D −k1 k2 [k12 − k22 ][r1 z21 + r2 z22 ] /   2 2 [r1 z12 + r2 z11 ] − ik2 kν2 [r1 z12 − r2 z11 ] / k˜14 = D −ik1 kν1   k˜22 = D −[k12 − k22 ][r1 z22 − r2 z21 ] /   k˜24 = D −[k12 − k22 ][r1 z21 − r2 z12 ] / k˜23 = −k˜14 ,

k˜33 = k˜11 ,

k˜34 = −k˜12 ,

k˜44 = k˜22

= r22 − r12 r1 = i(k2 − k1 )[1 − e−ik1 L e−ik2 L ],

r2 = i(k2 + k1 )[e−ik1 L − e−ik2 L ]

and z11 = 1 − e−ik1 L e−ik2 L ,

z12 = e−ik1 L − e−ik2 L

z22 = 1 + e−ik1 L e−ik2 L ,

z21 = e−ik1 L + e−ik2 L

The wavenumber combinations dependent on Poisson’s ratio are given by 2 ≡ k12 + (2 − ν)ξ 2 , kν1

2 kν2 ≡ k22 + (2 − ν)ξ 2

˜ is a symmetric complex [4 × 4] matrix. This matrix relates applied Notice that [k] forces and moments along the edges to transverse displacement and rotation there. Figure 7.6 shows the variation of the two main diagonal terms as a function of frequency. The main point to note is that the number of spectral peaks diminish as m is increased. This is in keeping with the idea that the plate stiffens as m increases.

7.1 Spectral Elements for Flat Plates

305

m=0 m=50 m=100 m=150 m=200

10 1

0.

5.

10.

15.

0.

5.

10.

15.

20.

Fig. 7.6 Dynamic stiffness for k˜11 /(12D/L3 ) and k˜22 /(4D/L) for different values of m

7.1.3 A Simple Example As a simple example, we consider the response of an infinite sheet subjected to a concentrated load. This is a problem whose solution can be represented in many forms—it is instructive to see how it relates to the element approach. We conceive of the infinite sheet as made of two semi-infinite elements sharing common u1 and v1 displacements along the axis x = 0. To help in understanding the assemblage process, consider a node 2 at x = +∞ and a Node 3 at x = −∞. The stiffness relations for each element is 12     μh ik2 kS2 ξm [2ξm2 + 2k1 k2 − kS2 ] u˜ 1 N˜ xx1 = v˜1 N˜ xy1 ik1 kS2 sym 13      μh ik2 kS2 −ξm [2ξm2 + 2k1 k2 − kS2 ] u˜ 1 N˜ xx1 = v˜1 N˜ xy1 ik1 kS2 sym The negative off-diagonal term comes about because the second element is rotated 180◦ about the y-axis. (The effect of orientation is developed more formally later in this chapter.) The assemblage process is simply a matter of imposing nodal equilibrium. In this case, of course, we are dealing with a generalized node, but the process is completely equivalent to what was done in Chap. 5. We need balance of both the x and y loads, that is,  −

N˜ xx1 N˜ xy1

12

 −

N˜ xx1 N˜ xy1

13 +

Substituting for the stiffness relations, we get

  N˜ xx = P˜ =0 P˜y = 0

306

7 Thin-Walled Structures

     μh 2ik2 kS2 0 u˜ 1 P˜ = 2 v˜1 sym 2ik1 kS 0 This gives the solution u˜ 1 =

P˜ , 2μhik2 kS2

v˜1 = 0

This is a result obtained by Lamb [8]. It is surprising that the v displacement along x = 0 is zero. The actual responses are reconstructed by using the double summation. Rather than do that, consider the response at an arbitrary position. Using the shape functions and nodal solution, we get  uˆ 1 [k1 k2 e−ik1 x + ξ 2 e−ik2 x ] cos ξm y m  uˆ 1 v(x, ˆ y) = [−ik2 ξ e−ik1 x + ik2 ξ e−ik2 x ] sin ξm y m

u(x, ˆ y) =

If we now replace the functions of y with their exponential form, then we can write the radial displacement for the first mode as  uˆ r = uˆ cos θ + vˆ sin θ =

∞

−∞

uˆ 1 [k1 k2 cos θ + k2 ξ sin θ ]e−ik1 x e−iξy dξ

We can put this in a form suitable for approximating the far-field by the method of stationary phase (this approximation is developed in Chap. 8) as  uˆ r (x, z) =

+∞ −∞

(ξ )e−i(ξ ) dξ

Specializing the method to our case, we have =k1 x + ξy = =

'

( kp2 − ξ 2 sin θ + ξ sin θ r

Pˆ uˆ 1 [k1 k2 cos θ + k2 ξ sin θ ] = [k1 k2 cos θ + k2 ξ sin θ ] 2(2μ + λ)ik2 hkP2

where we have introduced cylindrical coordinates. Setting the first derivative of the phase to zero gives ξo = kP sin θ , and the integral approximation is then

o = k P r

7.2 Folded Plate Structures

−i Pˆ cos θ uˆ r (r, θ ) ≈ 2(2μ + λ)h

307

 2π −i[kP r+π/4] e kP r

√ This shows that the displacement diminishes as 1/ r in the far-field—a conclusion we also drew for the flexural behavior of an infinite sheet subjected to a point load.

7.2 Folded Plate Structures This section reviews the manner in which a 3D plate structure is treated globally. Because much of the development in this section is taken, by analogy, from that in Chap. 5, we concentrate mainly on establishing the notation and meaning of the participating terms. Again, a point to bear in mind is that the spectral analysis approach essentially replaces the dynamic problem with a series of pseudo-static problems and, from this point of view, the assemblage process is therefore identical to that of the static case. The folded plate structures of interest are joined along common edges as illustrated in Fig. 7.7.

7.2.1 Structural Stiffness Matrix Plates of unequal thickness, stiffness, and viscous damping may be joined along common edges by merely summing together the dynamic stiffness matrices [ k˜ ] ˜ To join the membrane and for different elements into a global stiffness matrix [K]. flexural elements, we introduce nodal load and DoFs vectors with respect to local coordinates as {F˜¯ }T ≡ {N˜¯ xx1 , N˜¯ xy1 , V˜¯xz1 , M˜¯ xx1 ; N˜¯ xx2 , N˜¯ xy2 , V˜¯z2 , M˜¯ x2 } Fig. 7.7 Assemblage of 3D folded plate elements. The complete connection edge is a generalized node

308

7 Thin-Walled Structures

{ u˜¯ }T ≡ {u˜¯ 1 , v˜¯1 , w˜¯ 1 , ψ˜¯ x1 ; u˜¯ 2 , v˜¯2 , w˜¯ 2 , ψ˜¯ x2 } We then have the combined element stiffness relation as {F˜¯ } = [ k˜¯ ]{ u˜¯ } where [ k˜¯ ] is an [8 × 8] matrix. The terms of [ k˜¯ ] representing in-plane DoFs (u, v) and for out-of-plane DoFs (w, ψx ) are already given. It is now necessary to relate these two sets of global quantities. The global y-axis coincides with the local y-axis, and each plate element can be at an arbitrary orientation about the y-axis as shown in Fig. 7.7. During a rotation, v˜ and ψ˜ are left unaffected. The rotation matrix is ⎡

cos θ ⎢ 0 [ R ]=⎢ ⎣ sin θ 0

⎤ 0 − sin θ 0 1 0 0⎥ ⎥ 0 cos θ 0 ⎦ 0 0 1

As the translational displacements are independent of rotational displacements, the transformation of a nodal vector from the local to the global member coordinate axes system uses the matrix [ T ] as { u } = [ T ]{ u¯ } ,

{F } = [ T ]{F¯ } ,



[ R ] 0 [ T ]≡ 0 [ R ]



where [ T ] is the 8 × 8 rotation matrix. As a consequence, the stiffness matrix in global coordinates can be written as [ k˜ ] = [ T ]T [ k˜¯ ][ T ] The resulting general plate element models both in-plane and out-of-plane waves. There are many ways to establish the structural stiffness matrix from the element stiffness matrices. The procedure chosen here (consistent with viewing the problem as a pseudo-static one) imposes nodal (dynamic) equilibrium at each node in turn. At each node, there are eight quantities of interest: P˜x , P˜y , P˜z T˜y u, ˜ v, ˜ w˜ φ˜ y

are the applied force components, is the applied moment, are the resulting translational displacement components, is the resulting rotational displacement.

Note that the global rotation about the y-axis, φ˜ y , is equal to the negative local ψ˜ x . To illustrate the procedure for constructing the structural stiffness matrix,

7.2 Folded Plate Structures

309

assume that the structure is in a state of (dynamic) equilibrium, and thus, each node (joint) must also be in a state of (dynamic) equilibrium. Consider a free body containing a node under the action of externally applied loads and internal nodal loads (N˜ xx , N˜ xy , . . . , M˜ x ). Replace the member nodal forces in terms of the (augmented) stiffness matrices to get equilibrium expressed as {P˜ } = [K˜ (12) ]{ u˜ } + [K˜ (13) ]{ u˜ } + · · · = [ K˜ ]{ u˜ }

(7.20)

where [ K˜ ] is called the dynamic structural stiffness matrix. A more detailed description of this assembling procedure for static problems is given in Ref. [6]. The applied loads are taken in the form S(y)P (t) ⇐⇒ S˜m Pˆn where S(y) is the distribution shape along a node. We then have the structural stiffness relation [ K˜ ]{ u˜ } = {S˜ Pˆ } where {S˜ Pˆ } is a vector indicating the DoFs that are loaded. For a single impact, say, { S˜ } has zeros at all indices excluding the term representing the load point and becomes { S˜ }Pˆ .

7.2.2 Computer Program Structure The dynamic structural stiffness matrix is assembled using techniques analogous to those used for spectral elements of Chap. 5. The major difference is that here the Do-Loops are over all of the frequency and wavenumber components. This results in the assemblage of a dynamic stiffness matrix at each frequency and wavenumber. The nonzero nodal DoFs of the resulting system are found using a similar equation solver. The process of generating the global stiffness matrix [K˜ nm ] involves the following operations for each element: Step 1: Step 2: Step 3: Step 4: Step 5: Step 6:

Compute the in-plane stiffness matrix in local coordinates. Compute the flexural stiffness matrix in local coordinates. Augment and sum the two matrices into an [8 × 8] matrix, [k˜nm ]. Rotate the element to match the global coordinate system. Delete the zero global DoFs. Assemble the reduced global stiffness matrix, [K˜ nm ].

310

7 Thin-Walled Structures

The program then solves the complex, banded, linear system of equations [K˜ nm ]{u˜ nm } = { S˜ } to get the unit response or system response functions. Thus, the problem can be thought of as a sequence of N × M pseudo-static problems. In common with the waveguide spectral elements, a big advantage of the spectral formulation for folded plates is that there is a small number of nodes, and hence, the in-core memory requirements are very small. For example, a folded plate with three members can have, say, four nodes, giving a system size of 16 DoFs, and the memory required (excluding code) is about 16 × 8 × 2 = 256 bytes. This is very modest. However, if we wish to store the complete solution, then the information stored per node is very large. The storage requirements based on a 1024-point FFT transform, a 512-space transform, and 8-byte words are about 16 × 512 × 512 × 8 × 2 = 67 MB This is a relatively large memory requirement; however, it does mean that it is now feasible to add a post-processing capability that allows the reading in of the nodal solution and obtaining the member loads or distributions at any desired point without having to redo the complete solution. The spectral method, like other techniques for dynamic analysis of plate structures, is computationally intensive; it requires several hundred thousand, or even millions, of solutions to a small system of linear equations. Unlike other techniques however, the spectral method gives high-fidelity results over large regions and automatically produces the frequency response function. Another benefit of the spectral element method of finding dynamic behavior of structures is that it is ideally suited for multiprocessor computers of the single instruction multiple data (SIMD) type. The many solutions to the small dynamic stiffness matrix equations cited above are computations, which are completely independent of each other, and can therefore be performed simultaneously. Although the efficiency of most parallel algorithms usually decreases as the number of processors increases, the spectral element method maintains maximum efficiency on the largest of multiprocessor computers [2–4]. As computer hardware architectures become increasingly parallel, the methods underlying the approaches shown here will gain tremendous computational advantages.

7.2.3 Structural Applications Joints connecting out-of-plane folds change the nature of the reflected waves because the joints couple in-plane and flexural behavior. We look at one example, the cylindrical box shown in Fig. 7.8, that illustrates the interactions nicely. When a structure is impacted, there are generally two aspects of the response we are interested in. First is the dynamic response local to the impact site because the interaction between the structure and the impactor determines the actual force

7.2 Folded Plate Structures

311

history. Second is the response a distance away from the impact site because this determines how energy propagates into the structure. The cylindrical box allows us to study both of these aspects. To help put the results in perspective, we have also included results from a Timoshenko beam analysis. Our test structure is a thin-walled cylindrical box with a square cross section having sides of length 102 mm and thickness h = 2.5 mm as shown in Fig. 7.8. The box is modeled as a collection of folded plates with the folds located at the nodal dots. Later, to make the example more interesting, we increase the thickness of the plates so as to make the box have properties (in the context of our applied force history) of both a folded plate structure and a simple beam; the relationship between these two structures gives an interesting insight into the behavior of complex structures. The top face of the box is impacted at its center with the smoothed force history of Fig. 1.8. Velocity responses at three locations on an x − z plane through the impact site are shown in Fig. 7.9. The sets of curves labelled “w” are flexural responses. The “u” curve is an in-plane velocity response in the side panel, which is induced

Fig. 7.8 Box cylinder. (a) Geometry. (b) Sequence of folded plate geometries analyzed. The structures are symmetric about the line of action of the force

w bottom

u side x10

w side x10 w top

0.

1000.

2000.

Fig. 7.9 Cylinder box velocity responses

3000.

4000.

5000.

312

7 Thin-Walled Structures

folds 0

1

2

3

full

0.

1000.

2000.

3000.

4000.

0.

2.

4.

6.

8.

10.

12.

14.

Fig. 7.10 The evolution of structural resonance at the impact site. (a) Time responses, w(t). ˙ ˆ (b) System frequency response |G|

by the original flexural wave interacting with the folds. In-plane waves travel faster than flexural waves and their effects are therefore seen much earlier. Apart from the arrival times, there is very little insight immediately obvious from these plots. We now show the effect of the various folds on the reflections and in doing so illustrate some of the unique features of the spectral element method. An interesting study is to look at the effect of adding folds to the plate. Figure 7.8b shows the sequence of models of the box beam where an additional corner is added at each step. For clarity, only the right half of the symmetric sections are shown. The four models in this figure have the same properties as the complete 102 mm × 102 mm structure. Throw-off elements (denoted by elements ending with a squiggle, ) absorb all waves transmitting through the last corner in each model. The sequence helps to explain the responses in the lowest curves of Fig. 7.9. Each fold adds a new set of reflections. These reflections are discernible in Fig. 7.10a. A deeper insight is gained by looking at the system transfer functions for the same cases, where in the frequency domain we have ˆ Pˆ uˆ = G What we see is the formation of spectral peaks. This is essentially the vibration resonance behavior. It is interesting to note that while the time domain responses have established themselves by the third fold, the frequency plots show that the spectral peaks continue to get sharper. We see this because the spectral element method can show the system response not modified by the spectrum of the input loading.

7.2 Folded Plate Structures

313

folds: 0

1

2

3

full

0.

1000. 2000. 3000. 4000.

0.

2.

4.

6.

8.

10. 12. 14.

Fig. 7.11 The evolution of structural resonance at y = 200 mm. (a) Time responses, w(t). ˙ ˆ (b) System frequency response |G| top middle bottom beam

x=0

x=700mm

0.

1000.

2000.

3000.

4000.

5000.

Fig. 7.12 Box beam displacements

Figure 7.11 shows the same information but at a location 200 mm down the cylinder. Again, we see the evolution of the resonances, but we also see a very important filtering effect. It is noticed that the low-frequency components have very ˆ is almost flat up to about 400 Hz. This would have small magnitudes and that G significant effect if a force deconvolution is attempted. That is, no low-frequency information is propagated significant distances from the impact site. The following analyses use a slightly larger plate thickness of 3 mm and a smaller cross-sectional dimension of 51 mm × 51 mm. These were chosen so as to make the separate trends of plate-like and beam-like behavior easier to identify. We call this structure a box beam.

314

7 Thin-Walled Structures

top

side beam resonances

0.

2.

4.

bottom

6.

8.

10.

12.

14.

16.

Fig. 7.13 System response functions at x = 700 mm

Figure 7.12 shows displacement histories in the global z direction at three locations on the cross section through the impact site. Although the dimensions of the box beam are relatively small, the three locations show quite different behaviors. The top and bottom faces show significant vibratory behaviors that are out of phase with each other; the middle point has very little of this. In fact, the middle point resembles more the character of the Timoshenko solution although the latter overestimates the total displacement. The curve at the impact site shows that the box beam is much softer than the Timoshenko beam, that is, the box beam gives an initial displacement that is significantly larger. We now examine responses of the box beam and Timoshenko beam in the frequency domain. Frequency domain acceleration responses at a cross section 700 mm from the impact site are shown in Fig. 7.13. While not obvious from the plots, the characteristics of the box beam and Timoshenko beam responses are very similar at low frequencies but differ markedly after 2.8 kHz when the box beam begins to exhibit resonant peaks. Above this frequency, both the centerline of the box beam side panels and Timoshenko results show a pronounced attenuation, while the top and bottom faces show enhanced response. The transition frequency occurs at the first natural frequency of the box beam’s cross section; to illustrate this, we compare in Fig. 7.13 the resonance frequencies of a square frame having the same properties as a unit length of the box beam cross section (acting under plane strain conditions). The multiple resonances of the frame coincide, approximately, with the beginning of the peaks in the box beam response. The pertinent question arises: can the box beam be approximated by a Timoshenko beam model? The significant difference between the two models is that the folded plate model, being 3D, has many more modes than the simpler beam model. The issue is not so much whether or not the modes are excited but whether they appear in the monitored DoFs. For example, the vertical (global z direction) behavior of the centerline of the box beam side panel is similar to that of the

7.3 Spectral Elements for Curved Plates

315

Timoshenko beam; however, its out-of-plane behavior bears no resemblance. An understanding of when the Timoshenko model is not applicable can be obtained by considering the vibrations of the cross section. The first resonance occurs at about 2.8 kHz (which is inside the input range of 0 to 12 kHz) and therefore is significant. If, however, the cross section were reduced to 25 mm × 51 mm, say, while keeping the same amount of material, the first resonance would increase to about 15 kHz and the cross-sectional behavior would not be significant; a Timoshenko model would then suffice.

7.3 Spectral Elements for Curved Plates Our final extension of the spectral element approach is to curved extended structures. This was initiated in the previous section, but it is only after implementing an appropriate matrix approach that any reasonable problems can be tackled. We motivate the matrix approach by first solving the problem of the point impact of an infinitely long curved plate. We saw that the membrane behavior of a flat plate required four generalized DoFs, while the out-of-plane flexural behavior also required four. A folded plate structure, which exhibits both membrane and flexural behaviors, then required eight DoFs. A segment of a curved cylindrical shell, as shown in Fig. 7.14, exhibits behavior similar to that of the folded plate and therefore has eight DoFs. For the folded plate, we assembled the [8 × 8] stiffness as a combination of two [4 × 4] elements plus a rotation matrix. But the curved element has all displacements coupled, and hence, we have to tackle the stiffness matrix directly as an [8 × 8] system. This is too cumbersome to do explicitly; consequently, we lay out a computer based method for establishing the shape functions and, subsequently, the stiffness matrix. This, of course, adds to the computational burden, but in return we Fig. 7.14 Segment of cylindrical shell using spectral element notation

316

7 Thin-Walled Structures

get an approach that is conceptually simpler and helps to unify the special results established earlier in this chapter and in Chap. 5.

7.3.1 Impact of an Infinite Curved Plate Consider an infinite curved plate of radius R; Physically, this would mean that the plate is in the form of a coiled helix. Our approach to the solution parallels that of the earlier impact problems, what makes this one interesting is that we now have four coupled modes. That is, we wish to illustrate the level of complexity introduced by the multiple modes and justify the use of a matrix methodology. What we are dealing with is a throw-off element, but initially we do not use the element terminology. The solution for the forward propagating terms is written as ˆ 1 e−ik1 s + Bˆ 1 e−ik2 s + C ˆ 1 e−ik3 s + D ˆ 1 e−ik4 s u(s) ˆ =A ˆ 2 e−ik1 s + Bˆ 2 e−ik2 s + C ˆ 2 e−ik3 s + D ˆ 2 e−ik4 s v(s) ˆ =A ˆ 3 e−ik1 s + Bˆ 3 e−ik2 s + C ˆ 3 e−ik3 s + D ˆ 3 e−ik4 s w(s) ˆ =A The 12 coefficients are not independent but are related through the amplitude ratios. Let us write the solutions as ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎨ φu ⎬ ⎨u⎬ ⎨1⎬ −ik1 s ˆ ˆ e + B e−ik2 s (s) = A v φ 1 ⎩ ⎭ ⎩ ⎭ ⎩ v⎭ φw 1 w φw 2 ⎧ ⎫ ⎧ ⎫ ⎨φu ⎬ ⎨φu ⎬ ˆ φv e−ik3 s + D ˆ φv e−ik4 s +C ⎩ ⎭ ⎩ ⎭ 1 3 1 4 where the symbol φ indicates an amplitude ratio. The solution is arranged so that we can set φ = 0 to recover the familiar form for the uncoupled cases. We can regroup this solution into the somewhat more compact form ⎧ ⎫ ⎡ ⎫ ⎤ ⎧ ˆ −ik1 s ⎫ ⎧ ⎪ φ11 φ12 φ14 ⎪ ⎨ Ae ⎬ ⎨ φ13 ⎬ ⎨u⎬ ˆ −ik3 s ˆ −ik2 s + φ23 Ce v = ⎣ φ21 φ22 φ24 ⎦ Be ⎪ −ik s ⎪ ⎩ ⎩ ⎭ ⎭ ⎩ ⎭ 4 ˆ w φ31 φ32 φ34 φ33 De The free body for the impact region looks similar to that of Fig. 3.4 except that there is also an axial force. At the impact site s = 0, we impose u(0, t) = 0 ,

∂w =0 ∂s

7.3 Spectral Elements for Curved Plates

317

and we specify that the in-plane shear traction is zero:  ∂u ∂v  + =0 σsx = G ∂y ∂s ˆ as the This gives three equations for the four unknown coefficients. Let us choose C reference unknown, then the system of three equations can be arranged as ⎧ ⎫ ⎤⎧ ˆ ⎫ ⎪ φ11 φ12 φ14 ⎨A⎪ ⎬ ⎨ φ13 k3 ⎬ ˆ ⎣ φ21 k1 φ22 k2 φ24 k4 ⎦ Bˆ = − φ23 k3 C ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ˆ φ31 k1 φ32 k2 φ34 k4 φ33 k3 D ⎡

While a [3 × 3] system can be solved in closed form, no advantage is gained by doing so. We opt instead to solve it numerically and write the solution for the three coefficients in the symbolic form ˆ B, ˆ D} ˆ T = {gA , gB , gD }T C ˆ {A, This system is solved at each value of frequency ω, and each value of wavenumber ξ . We are now in a position to write all of the displacements in terms of the single ˆ First introduce the subscript notation unknown C. g1 (s) ≡ gA e−ik1 s ,

g2 (s) ≡ gB e−ik2 s ,

g3 (s) ≡ e−ik3 s ,

g4 (s) ≡ gD e−ik4 s

The matrix form is ⎫ ⎧ ⎫ ⎤⎧ ⎫ ⎧ ⎡ φ11 φ12 φ14 ⎨ g1 ⎬ ⎨ φ13 ⎬ ⎨u⎬ v (s) = ⎣ φ21 φ22 φ24 ⎦ g2 + φ23 g3 ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ φ31 φ32 φ34 g4 φ33 w The final piece of information comes from the impacting force. We have 1 2P

= −Vsz=

 ∂ 3w D  ∂ 2u ∂ 2u  ∂ 3w  + D + (1 − ν) + (2 − ν) R ∂s 2 ∂y 2 ∂s 3 ∂y 2 ∂s

or 1 ˜ 2P

=

     D  ˆ kj2 + 2ξ 2 (1 − ν) φ1j gj + Dikj kj2 + ξ 2 (2 − ν) φ3j gj C j R

ˆ in terms of where the summations are over the modes. This allows us to solve for C ˆ P. Some velocity reconstructions are shown in Fig. 7.15. Note that the response at the impact site is scaled down by 10, which indicates the very rapid decay of

318

0.

7 Thin-Walled Structures

2000.

4000.

6000.

360

360

270

270

180

180

90

90

0/10

0/10

8000.

0.

2.

4.

6.

8.

10.

12.

14.

Fig. 7.15 Out-of-plane velocity responses for an impacted curved plate. Numbers indicate the angle in degrees from the impact site. (a) Time responses w. ˙ (b) Frequency response |w| ˆ

amplitude. The most significant point of interest is the oscillatory behavior of the transverse velocity. This we observed also in the curved beam example. What is interesting here is that while there is not a single cut-on frequency (there is one for each m mode), yet a single dominant spectral peak establishes itself. Reference [9] elaborates on the point excitation of curved plates. A key notion is that for finite cylinders this is equivalent to having image sources at multiples of 2π R. This is not the multiple images discussed in this chapter (i.e., the periodically extended load case), which are in the axial direction.

7.3.2 General Shape Functions Because the system size is large, we need a more organized way to handle establishing the shape functions and, subsequently, the element stiffness. In developing our matrix scheme, we see that the key new ingredient is the matrix of amplitude ratios. For each wavenumber kj , Eq. (6.57) gives the relation among the amplitudes. The characteristic system can be represented as ⎤⎧ ⎫ ⎨ uo ⎬ ⎣(k, ξ, ω) ⎦ vo = 0 ⎩ ⎭ wo ⎡

7.3 Spectral Elements for Curved Plates

319

This is homogeneous, and therefore, at best, we can only get amplitude ratios. For example, we can solve for the remaining two terms as a function of uo . In anticipation of a later need, we augment the system with ∂w −ψ =0 ∂s

or

ψo = −ikwo

and write the solutions at a particular wavenumber kj as ⎫ ⎧ ⎫ ⎧ uo ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎬ ⎨ v vo = uo = {}j uo ⎪ ⎪ w ⎪  ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ o⎪ ⎩ w⎪ ψo j ψ j where the symbol  indicates an amplitude ratio. Although the vector {} shown is normalized with respect to uo , it is possible for other modal vectors to be normalized differently. This must be done for each mode kj , and hence, there are eight vectors. We choose to represent these as [  ] = [[ A ], [B ]] where ⎡⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎤ 1 ⎪ u ⎪ u ⎪ u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎥ ⎢⎨ 1 v v ⎢ v ⎥ [ A ] ≡ ⎢ ⎥ ⎪ ⎪ ⎪ w ⎪ 1 ⎪ 1 ⎪ w ⎪ ⎣⎪ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ψ 1 ψ 2  ψ 3  ψ 4 and [B ] is the same but evaluated at the wavenumbers −kj . That is, the [4 × 4] partitions [ A ] and [B ] are evaluated at +kj and −kj , respectively; they are fully populated and typically are not symmetric. The normalizations are arranged so that the uncoupled flat plate solutions are easily recovered. For each mode, the corresponding amplitude uoj is undetermined; to make the ˆ B, ˆ . . .. notation resemble what we have already used, we label each of the these A, Consider a segment of shell of length L in the hoop direction. We begin by expressing the displacements as ˆ 12 e−ik2 s + · · · + G17 e−ik3 (L−s) + H18 e−ik4 (L−s) ˆ 11 e−ik1 s + B u(s) ˜ = A ˜ but involving the amplitude ratios 2j , with similar expressions for v, ˜ w, ˜ and ψ, 3j , and 4j , respectively. The length is introduced to include reflections coming from a boundary located at s = L. This displacement solution can be rewritten as T ˜ {u(s), ˜ v(s), ˜ w(s), ˜ ψ(s)} = {U }(s)

ˆ −ik1 s + · · · + {}8 He−ik4 (L−s) = {}1 Ae

320

7 Thin-Walled Structures

We write this in an even more compact matrix form; so as to make the matrix notation more accessible, we take the developments of the rod in Sect. 5.2 as the archetype and use its notation (except changed to matrices). The displacement for the shell segment is written as ˆ + [B ]e(L − s){ B} ˆ {U }(s) = [ A ]e(s){ A} where the diagonal matrix of exponentials and the vector of amplitudes are given by ⎤ 0 e−ik1 s ⎢ 0 0 ⎥ ⎥ e(s) ≡ ⎢ ⎣ 0 ··· 0 ⎦ , 0 e−ik4 s ⎡

⎧ ⎫ ˆ⎪ ⎪ A ⎪ ⎪ ⎪ ⎨ˆ⎪ ⎬ B ˆ { A} ≡ ˆ , ⎪C⎪ ⎪ ⎪ ⎪ ⎩ˆ⎪ ⎭ D

⎧ ⎫ Eˆ ⎪ ⎪ ⎪ ⎬ ⎨ˆ⎪ ˆ ≡ F { B} ⎪ ⎪G⎪ ⎪ ⎩ ⎭ H

ˆ and { B} ˆ in terms of the nodal displacements at We wish to replace the vectors { A} s = 0 and s = L. That is, we introduce u(0) ˜ = u˜ 1 ,

v(0) ˜ = v˜1 ,

w(0) ˜ = w˜ 1 ,

˜ ψ(0) = ψ˜ 1

with similar terms at s = L. We write this in matrix notation as ˆ − [B ]e(L){ B} ˆ {u˜ 1 , v˜1 , w˜ 1 , ψ˜ 1 }T = { u˜ }1 = {U }(0) = [ A ]e(0){ A} ˆ − [B ]e(0){ B} ˆ {u˜ 2 , v˜2 , w˜ 2 , ψ˜ 2 }T = { u˜ }2 = {U }(L) = [ A ]e(L){ A} Solving for the coefficients gives 

      ˆ { A} { u˜ }1 [G11 ] [G12 ] { u˜ }1 =[ G ] = [G21 ] [G22 ] { u˜ }2 { u˜ }2 ˆ { B}

where each partition of [ G ] is of size [4 × 4]. We are now in a position to write the displacements in terms of the shape functions. They are {U }(s) = [g(s)]1 { u˜ }1 + [g(s)]2 { u˜ }2

(7.21)

where the [4 × 4] matrix of shape functions are defined as [g(s)]1 = [ A ]e(s)[G11 ] + [B ]e(L − s)[G21 ] [g(s)]2 = [ A ]e(s)[G12 ] + [B ]e(L − s)[G22 ]

(7.22)

There are a total of 4 × 4 × 2 = 32 shape functions. While not obvious from the above, it turns out that, even in this general case, the collection of shape functions

7.3 Spectral Elements for Curved Plates

321

Fig. 7.16 Sample of shape functions for a 270◦ shell segment

2 kHz, m=0

2 kHz, m=40

4 kHz, m=0

4 kHz, m=40

associated with the DoFs at the second node is the mirror image of those associated with the DoFs at the first node. The results for the throw-off element are simply those associated with [G11 ]. These formulas can be used to recover all the shape functions already derived in this chapter and in Chap. 5. By way of example, Fig. 7.16 shows the g33 (s) cos(ξm y) shape function of a 270◦ shell segment; this shape function is associated with the w˜ 1 DoF and thus can be plotted as a radial displacement of the original shape. The other shape functions behave in a similar manner. It is clear from this figure and from the above developments that once the nodal DoFs are determined then the shape functions can be used to compute the responses at any intermediate locations. This is a crucial attribute because the segments can be very large if desired.

7.3.3 Dynamic Stiffness Relation for a Curved Shell Element The next step in the element development is to derive a stiffness relation for the shell segment. The process is simply that of expressing the resultant forces and moments in terms of the displacement solutions given in Eq. (7.21). We write these resultants in matrix form as  T 1 {F }(s) = Nss + Mss , Nsy , Vsz , Mss (s) = [ ∂ ]{U }(s) R

322

7 Thin-Walled Structures

where [ ∂ ] is the matrix collection of differential operators of size [4 × 4]. After substituting for {U }(s) in terms of the shape functions, we get {F }(s) = [ ∂ ][g(s)]1 { u˜ }1 + [ ∂ ][g(s)]2 { u˜ }2 ≡ [∂g(s)]1 { u˜ }1 + [∂g(s)]2 { u˜ }2

(7.23)

Relating the member resultants at s = 0 and s = L to the nodal loads at the same locations leads to the stiffness relation      {F˜ }1 [−∂g(0)]1 [−∂g(0)]2 { u˜ }1 = (7.24) [+∂g(L)]1 [+∂g(L)]2 { u˜ }2 {F˜ }2 or simply ˜ {F˜ } = [k(ω, ξ )]{ u˜ } where {F˜ } ≡ {N˜ 1 , F˜y 1 , V˜1 , M˜ 1 ; N˜ 2 , F˜y 2 , V˜2 , M˜ 2 }T { u˜ } ≡ {u˜ 1 , v˜1 , w˜ 1 , ψ˜ 1 ; u˜ 2 , v˜2 , w˜ 2 , ψ˜ 2 }T The matrix [ k˜ ] is the [8 × 8] dynamic element stiffness matrix that is frequency and wavenumber dependent, complex, and symmetric. The stiffness relation for the throw-off or one-noded element is simply ˜ ξ )]{ u˜ }1 {F˜ }1 = [−∂g(0)]1 { u˜ }1 = [k(ω,

(7.25)

The stiffness matrix in this case is of size [4 × 4]. These stiffness relations can be used to recover the results already given for flat elements. For illustrative purposes, Fig. 7.17 shows the normalized k˜11 , k˜22 , k˜33 , and k˜44 diagonal terms for the same shell used to illustrate the shape functions. As is typical with spectral elements, they exhibit a very large dynamic (numerical) range. The normalizations are with respect to the stiffnesses for straight thin beams [6] but modified for plates. That is, they are presented as k˜11 /(Eh/L) ,

k˜22 /(Gh/L) ,

k˜33 /(12D/L3 ) ,

k˜44 /(4D/L)

Note that the k˜11 stiffness is substantially less than the static values, and it is only after the cut-on frequency (≈ 3 kHz for m = 0) does it become greater than unity. On the other hand, k˜33 is always significantly larger than the static straight value; this is because the L3 in the denominator predicts an inordinately small static value.

7.3 Spectral Elements for Curved Plates

323

100 1

unity reference m=0 m=40

.0

2.0

4.0

6.0

8.0

10.0

Fig. 7.17 Normalized stiffnesses k˜11 , k˜22 , k˜33 , and k˜44 , for a 270◦ shell segment. The vertical axis is the logarithm of the magnitudes

Fig. 7.18 Geometry for the point-loaded circular cylinder

7.3.4 Point Loading of a Complete Cylinder The advantage of the curved element is that it can be combined with other elements—either flat or curved—to form significantly more complex structures; an example is given in Fig. 7.1. The variety of possibilities is too great to pursue here, so we are content with the single example of a closed cylinder because that is the geometry specifically excluded from our analysis. The geometry of our problem is shown in Fig. 7.18.

324

7 Thin-Walled Structures

We form the complete cylinder by combining two curved elements of the same material properties each of which is a half-circle. Because of symmetry, we could model the shell with a single half-circle element in which case, half of the applied load must be placed at each node and the extra conditions of u˜ 1 = 0, ψ˜ 1 = 0; u˜ 2 = 0, ψ˜ 2 = 0 be imposed. The two-element model was chosen so as to verify the assemblage process. The elements are joined at the common nodes by merely summing together the appropriate dynamic stiffness matrix components to form a global stiffness matrix [ Kˆ ]. We must therefore first rotate each element stiffness to this global system. The transformation requires the use of a simple [6 × 6] rotation matrix [ T ] that is of the form 

[R(α)] 0 [ T ]= 0 [R(α + 2β)]



where R(θ ) is the [3 × 3] rotation matrix [6]. This takes into account that the ends of the curved element are oriented differently to each other.  In our case, if the nodal , z ) and (x , z ), then with 2p = (x2 − z1 )2 + (y2 − z1 )2 coordinates are (x 1 1 2 2  2 2 and q = R − p , the angles are given by −1

α = tan



(x2 − x1 )p − (z2 − z1 )q (x2 − x1 )q + (z2 − z1 )p

 ,

β = tan

−1

  p q

Because only two elements are present, we assign the coordinate system of one to be the global coordinate system and rotate the other. The global stiffness matrix is then simply formed by adding the two global element stiffness matrices together. Typical velocity reconstructions for three points, θ = 0◦ , 90◦ , 180◦ , along the circumference are shown in Fig. 7.19. For comparison purposes, the same shell was modeled using 64 flat plate spectral elements. The two sets of results are indistinguishable even though there are very many reflections. It is worth pointing out that when fewer flat elements were used the results deteriorated as the time increased. This shows the significant computational savings in using the curved element. The waves travel around the circumference multiple times and interact with each other. We can get an insight into this by looking at the system response function ˆ where uˆ = G ˆ Pˆ and Pˆ is the input load. Note that this facility is an integral G attribute of the spectral element formulation. From Fig. 7.19b, we can see that many resonance frequencies are in the process of being established. These would be the frequencies of interest if an impulse/modal analysis vibration experiment were being performed. It is interesting to note that the w responses at 0◦ and 180◦ exhibit similar resonant behavior, but the response at 90◦ is missing many of the intermediate resonances. A significant peak appears at about 3.3 kHz. An analysis of the amplitude ratios shows that this peak is coming entirely from the first mode even though the response overall is dominated by the third mode. This is the same peak that is observed in

References

325

curved elements 64 flat elements

0.0

2.0

4.0

6.0

0.0

2.0

4.0

6.0

8.0

Fig. 7.19 Point excitation of a closed cylinder. The lengthwise position is y = 0. (a) Velocity responses. (b) System response functions. Dotted line is the zero reference

Fig. 7.15 where the curvature acts effectively as a continuous boundary and sets up a standing wave.

Further Research 7.1 Show that for a plane wave traveling in a flat plate the in-plane stiffness uncouples the u and v DoFs. —Ref. [11], p. 325 7.2 Show that, in respect to Fig. 7.2, at large wavenumber, the behavior at node 1 is uncoupled from that at node 2. —Ref. [11], p. 325

References 1. Cheung, Y.K.: Finite Strip Method in Structural Analysis. Pergamon Press, New York (1976) 2. Danial, A.N.: Inverse solutions in folded plate dynamics. Ph.D. Thesis, Purdue University (1994) 3. Danial, A.N., Doyle, J.F.: A massively parallel implementation of the spectral element method for impact problems in plate structures. Comput. Syst. Eng. 5, 375–388 (1994) 4. Danial, A.N., Doyle, J.F.: Dynamic response of folded plate structures on a massively parallel computer. J. Comput. Struct. 54, 521–529 (1995) 5. Danial, A.N., Rizzi, S.A., Doyle, J.F.: Dynamic analysis of folded plate structures. J. Vib. Acoust. 118, 591–598 (1996) 6. Doyle, J.F.: Static and Dynamic Analysis of Structures. Kluwer, The Netherlands (1991)

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7. Eterovic, A.L., Godoy, L.A.: An exact strip method for folded plate structures. Comput. Struct. 32, 263–276 (1989) 8. Lamb, H.: On the propagation of tremors over the surfaces of an elastic solid. Philos. Trans. R. Soc. Lond. A 203, 1–42 (1904) 9. Pierce, A.D., Hyun-Gwon K.: Elastic wave propagation from point excitations on thin-walled cylindrical shells. J. Vib. Acoust. 112, 399–406 (1990) 10. Rizzi, S.A.: A spectral analysis approach to wave propagation in layered solids. Ph.D. Thesis, Purdue University (1989) 11. Rizzi, S.A., Doyle, J.F.: A spectral element approach to wave motion in layered solids. J. Vib. Acoust. 114, 569–577 (1992)

Chapter 8

Structure–Fluid Interactions

There are many practical situations where the interaction between the dynamics of a structure and a surrounding fluid is of great importance. The most obvious is noise; noise is the propagation of structurally generated acoustic energy through the fluid. In return, the interaction can also influence the response of the structure itself; examples include dams, chimney stacks, ships, fuselages, propellers, and transmission cables. A structure immersed in a fluid can experience three types of loading. The first is the structure-borne excitation caused by the propagating structural waves. The second is a pressure loading arising from some source within the fluid; this acts as a distributed external loading on the structure. The third is the “self-loading” or fluid loading caused by the moving structure interacting with the fluid. This also acts as a distributed loading, but because the magnitude depends on the motion of the structure, it has the effect of coupling the structure and fluid motions. Our primary interest in this chapter is of a problem where a typically finite structure is coupled to a relatively extended second medium as shown in Fig. 8.1. We briefly review some of the basics of acoustic waves, but more thorough treatments can be found in Refs. [5, 7, 9]; the finer points of structure–fluid interaction are covered in the excellent summary paper by Crighton [2].

8.1 Plate–Fluid Interactions The modeling of plates as done in Chap. 6 is unaffected by the presence of the fluid, and we need to only account for the distributed applied pressure effect. Wave motion in fluids is a special case of fluid dynamics; however, to effect practical solutions, it is usual to make some assumptions and approximations. This section establishes the background of these assumptions and develops the relevant equations.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8_9

327

328

8 Structure–Fluid Interactions

baffle

plate

baffle

Fig. 8.1 Moir´e of the real and imaginary parts of the fluid pressure near a finite panel vibrating at 1 kHz. Note the nonzero pressure near the immobile baffle Fig. 8.2 Element of fluid with stresses

8.1.1 Linearized Acoustic Wave Equations We restrict ourselves to motion only in the x-z plane as shown in Fig. 8.2. The fluid and any structural components extend uniformly in the y direction. The time variation of the fluid density ρ¯a and pressure p¯ a , due to the passage of an acoustic wave, can be written as ρ¯a (r, t) = ρa (r) + ρ(r, t) ,

p¯ a (r, t) = pa (r) + p(r, t)

(8.1)

where ρa and pa are the mean (ambient) values, and ρ(r, t) and p(r, t) are the fluctuations. (We use the subscript “a” to stand for “acoustic medium.”) Substituting these into the equations for fluid motion and treating the fluctuations as small quantities lead to the continuity and equilibrium equations, respectively, ∂ u˙ ∂ w˙ ∂ρ + ρa + ρa =0 ∂t ∂x ∂z ∂σxz ∂σzz ∂σxz ∂σxx + = ρa u¨ , + = ρa w¨ ∂x ∂z ∂x ∂z

(8.2)

where σij are the stress components. The constitutive behavior of the fluid is described by

8.1 Plate–Fluid Interactions

329

∂ u˙ ∂ w˙ , σzz = −p + 2μ , ∂x ∂z ) ∂u ∂w * ρ + p=B = −B ρa ∂x ∂z

σxx = −p + 2μ

) ∂ u˙ ∂ w˙ * σxz = μ + ∂z ∂x (8.3)

where B is the bulk modulus and μ is the viscosity. Note that the stress is related to both the volumetric behavior as well as the strain rate. These equations are more easily manipulated in the frequency domain, and in their spectral form, they appear as ρˆ + ρa

∂ uˆ ∂ wˆ + ρa =0 ∂x ∂z

∂ σˆ xz ∂ σˆ xx + = −ρa ω2 uˆ , ∂x ∂z σˆ xx = −pˆ + 2μiω σˆ zz = −pˆ + 2μiω

∂ uˆ , ∂x

∂ wˆ , ∂z

∂ σˆ zz ∂ σˆ xz + = −ρa ω2 wˆ ∂x ∂z ) ∂ uˆ ∂ wˆ * σˆ xz = μiω + ∂z ∂x ) ∂ uˆ ρˆ ∂ wˆ * pˆ = B + = −B ρa ∂x ∂z

(8.4)

Rearranging these equations, the spectral form for the displacement becomes (B + μiω)∇∇ · uˆ + μiω∇ 2 uˆ + ρa ω2 uˆ = 0

(8.5)

which is similar to the Navier’s equations presented in Sect. 4.1. Therefore, a Helmholtz decomposition representation can be taken as uˆ =

∂ Hˆ ∂ φˆ + , ∂x ∂z

wˆ =

∂ φˆ ∂ Hˆ − ∂z ∂x

which leads to (B + 2μiω)∇ 2 φˆ + ω2 ρa φˆ = 0 ,

(μiω)∇ 2 Hˆ + ω2 ρa Hˆ = 0

(8.6)

These equations are equivalent to those presented in Sect. 4.1. It is clear that there are two modes of behavior. In order to look at the wave behavior of these modes, consider plane waves propagating in the fluid. As done before, consider a plane wave propagating in an arbitrary direction θ and described by ˆ −ika (xCθ +ySθ ) wˆ = Ae where we have set Cθ ≡ cos(θ ) and Sθ ≡ sin(θ ). The wavenumbers are

330

8 Structure–Fluid Interactions





ρa , ka ≡ ω B + 2μiω

kas ≡ ω

ρa μiω

2 is entirely imaginary and depends linearly on the frequency— Note that kas this means that the response is similar to diffusion (or Fourier heat conduction), and consequently, the shear response is localized to the boundaries or where the disturbance is initiated. The other mode exhibits dissipation due to the viscosity, but when the viscosity is negligible, the acoustic wave speed is nondispersive and given by

 ω ca = = ka

B ρa

Table 8.1 shows some typical property values for air and water. Later, we consider the interaction of acoustic waves with plates; for comparison, Fig. 8.3 shows the phase speeds of waves in a thick steel plate and a thin aluminum plate. A common assumption for most fluids is that they are inviscid; this leads to certain simplifications. The constitutive relations become

Table 8.1 Nominal properties for air and water Material

Modulus psi

GPa

Density lb· s2 /in.4

kg/m3

Wave speed in./s

km/s

Air

0.02×103

0.14×10−3

0.11×10−6

1.20

13.5×103

0.30

Water

0.42×106

2.90

0.12×10−3

1100

60.0×103

1.50

.4 25mm steel

.3

water

.2 coincidence 2.5mm aluminum

.1 air

.0

0.

2.

4.

6.

8.

10.

12.

14.

16.

Fig. 8.3 Phase speeds for waves in air and water and their comparison with the flexural phase speeds in aluminum and steel plates

8.1 Plate–Fluid Interactions

331

σˆ xx = −pˆ = −B

ρˆ , ρa

σˆ zz = −pˆ = −B

ρˆ , ρa

σˆ xz = 0

leading to the momentum relations ∂ pˆ = ρa ω2 uˆ , ∂x

∂ pˆ = ρa ω2 wˆ ∂z

Combining these with the expression for ρ/ρ ˆ a gives the Helmholtz equation governing the pressure field as  B∇ pˆ + ρa ω pˆ = 0 2

2

or

∇ pˆ 2

+ ka2 pˆ

= 0,

ka ≡ ±ω

ρa B

(8.7)

plus the associated BCs 

 ∂ pˆ = ρa ω2 uˆ , ∂x

pˆ ;

 pˆ ;

 ∂ pˆ = ρa ω2 wˆ ∂z

This set of equations is our primary equations governing the dynamic response of the fluid. Note that it exhibits only one mode of wave behavior. In the subsequent sections, we consider structures composed of multiple panels modeled as collections of plane surfaces or plates. There a Cartesian coordinate system local to the plane is used. Under these circumstances, for example, a distributed pressure source can be represented by p(x, ˆ z) =

 m

p(ξ ˜ m , ωn )e−ikz z e−iξm x ,

kz ≡



ka2 − ξm2

(8.8)

The wavenumber behavior of kz is similar to the first plate mode in Fig. 6.3. In the case when p˜ is unity, this corresponds to a concentrated pressure source; the representation, however, is valid for more general pressure distributions. A similar representation for the components of the motion gives the displacements in the fluid as w˜ =

−ikz p˜ , ω2 ρa

u˜ =

−iξ p˜ ω2 ρa

(8.9)

Because of our interest in fluids contacting plates, consider the fluid to be in contact with a surface located at z = 0 and whose normal displacement is w, ˆ and then the first of the above equations indicates that the displacement and the pressure are related through convolution integrals because, at z = 0, w(x, ˆ 0) =

1  ω2 ρa

m

− ikz p˜ m e−iξm x

or

p(x, ˆ 0) = ω2 ρa



w˜ m −iξm x e m −ikz

332

8 Structure–Fluid Interactions

Fig. 8.4 Geometry of the plate with fluid loading

In other words, the pressure in the fluid at any point is related to the displacement response at every point on the surface. It is this convolution relation that is the source of many difficulties in the structure–fluid interaction problem.

8.1.2 Incident Plane Wave on a Plate Because the panels investigated in most interior noise applications are thin, it seems adequate to use the classical plate model for the panel systems. Our interest here is on aspects of the fluid interaction; therefore, for simplicity, we take the plate as acting uniformly in the y direction. Figure 8.4 depicts the typical loadings experienced by the plate. Following the developments of Chap. 6, the equation of motion for the plate reduces to D

∂ 2w ∂ 4w ∂w + ρh + ηh = qw (x, t) − p(x, z = 0, t) ∂t ∂x 4 ∂t 2

(8.10)

The loading on the plate has two parts: qw (x, t) is the applied distributed transverse loading and p(x, z = 0, t) is the distributed fluid pressure loading. These equations are complemented by the fluid equations ∇ 2 pˆ + ka2 pˆ = 0

(8.11)

and the interface condition between the fluid and plate wˆ plate (x) = wˆ fluid (x, z = 0) ,

∂ p(x, ˆ z = 0) = ω2 ρa wˆ plate (x) ∂z

(8.12)

We apply these equations to a number of situations so as to elucidate the effects of the fluid. As a first example, consider an infinite plane wave impinging on a single infinite sheet completely immersed in fluid. The total pressure variation is a combination of waves moving in x and ±z making an angle θ with the normal to the plate. That is,

8.1 Plate–Fluid Interactions

333

pˆ + (x, z) = p˜ i e−ika (xSθ −zCθ ) + p˜ r e−ika (xSθ +zCθ ) , pˆ − (x, z) = p˜ t e−ika (xSθ −zCθ ) where Cθ ≡ cos θ and Sθ ≡ sin θ . The incident pressure amplitude p˜ i is known, but we do not know the reflected pressure amplitude p˜ r nor the transmitted pressure amplitude p˜ t . If the plate separates two dissimilar fluids, then there would be a refraction effect and the transmitted angle would be different from the incident angle, and thus introducing an additional unknown. (The extra relation would then come from matching the phases.) The pressure induces a response in the plate given by wˆ = we ˜ −ikx which must satisfy the plate equation  d4  D 4 − ρhω2 + iωηh wˆ = −p(x, ˆ z = 0) dx At this stage, we do not know the wavenumber k. Substituting for the displacement and pressures into this plate equation gives [Dk 4 − ρhω2 + iωηh]we ˜ −ikx = −p˜ i e−ika xSθ − p˜ r e−ika xSθ + p˜ t e−ika xSθ There are three unknown amplitudes w, ˜ p˜ r , and p˜ t and one unknown wavenumber k; we need another equation. We get this from the BCs at z = 0, namely, ∂ pˆ + = ρa ω2 wˆ ∂z

or

  ika Cθ p˜ i e−ika xSθ − p˜ r e−ika xSθ = ρa ω2 we ˜ −ikx

and ∂ pˆ − = ρa ω2 wˆ ∂z

or

  ika Cθ p˜ t e−ika xSθ = ρa ω2 we ˜ −ikx

where continuity of displacement between the plate and fluid is assumed. When we combine these equations together, they must all have a common phase behavior with respect to x; hence, we conclude that the wavenumbers are related by k = ka Sθ ,

k a Cθ =

 ka2 − k 2

Canceling the common phases, we find the amplitudes from [Dk 4 − ρhω2 + iωηh]w˜ = −p˜ i − p˜ r + p˜ t ,

p˜ i − p˜ r =

ρa ω2 w˜ , ika Cθ

p˜ t =

ρa ω2 w˜ ika Cθ

334

8 Structure–Fluid Interactions

Re

transmitted reflected

al

Ima g Fre

q

Fig. 8.5 Transmitted and reflected pressures for an incident angle of θ = 40◦

This leads to the displacement solution −2p˜ i

w˜ = Dka4 Sθ4

− ρhω2

2ρa ω2 + iωηh + −ika Cθ

(8.13)

and the reflected and transmitted pressures obtained from p˜ r = p˜ i +

ρa ω2 w˜ , −ika Cθ

p˜ t =

ρa ω2 w˜ ika Cθ

Note that if the fluid is on one side of the plate only, then the factor of 2 in the denominator of w˜ is changed to unity. Figure 8.5 shows the variation of transmitted and reflected pressures for an incident angle of θ = 40◦ . These responses are surprisingly complicated. To help understand them, we consider some special cases. If the plate is very stiff, then w˜ ≈ 0 and we get p˜ r = p˜ i and p˜ t = 0, and the total pressure acting on the plate is 2p˜ i . This is true for any angle of incidence. This is the so-called blocked pressure. For flexible plates in the limit of ω = ∞, we also have w˜ ≈ 0 and p˜ r = p˜ i , p˜ t = 0. That is, the plate is behaving as if it is very stiff. Conversely, in the limit of ω = 0, we have p˜ r = 0 and p˜ t = p˜ i ; the plate is transparent to the wave. Figure 8.5 shows that at a certain frequency, there is a significant change in behavior. To explain this, first, consider when the incident angle is normal, then k = 0 and the displacement is w˜ =

−ρhω2

−2p˜ i , + iωηh + 2iωρa ca

p˜ r = p˜ i + iωρa ca w˜ ,

p˜ t = −iωρa ca w˜

because ω/ka = ca . The responses are dominated by the inertia of the plate and its stiffness has no influence. The reason for this is that the plate wavelength λ = 2π/k

8.1 Plate–Fluid Interactions

335

is infinite, and hence the plate is not in flexure. Consequently, the plate moves as a rigid body. Also, note that the inertia of the fluid appears like a viscous damping term. For arbitrary incident angles, the mass and stiffness contributions are opposite in sign, and hence there are combinations when it is possible for the reactive component to be zero, that is, when Dka4 Sθ4

2 − ρhωco

=0

or

ωco

c2 = a2 Sθ



ρh D

The frequency, ωco , at which this occurs is called the coincidence frequency. Thus, for a given angle of incidence, there is a unique coincidence frequency ωco ; however, because Sθ cannot exceed unity, there is a lower limiting frequency for the coincidence phenomenon given by  ωc =

ca2

c2  ρh = a 12(1 − ν 2 ) D co h

(8.14)

where ωc is known as the critical frequency or lowest coincidence frequency. We refer to ωc simply as the coincidence frequency. The coincidence frequency occurs when the phase speeds of the plate bending wave and of the acoustic wave in the fluid are equal as is indicated in Fig. 8.3. It might seem odd that having emphasized the group speed in the previous chapters that it is the phase speed that is significant here; at coincidence, it is the phase of the waves rather than their speed that are matched. The significance of the coincidence frequency is that wave components in the plate above this frequency are radiated easily into the fluid. That is, at coincidence, the transmitted and incident waves are the same, and there is no reflected wave. This is indicated in Fig. 8.5 by noting how the trend to zero for the transmitted wave was interrupted at coincidence. This is another example of the energy transfer between two coupled elastic systems appearing as increased dissipation in one of them. The phenomena discussed above depend on the extent of fluid loading; thus, it is useful to have a measure of this loading. Following Ref. [2], fluid loading is characterized by two independent parameters: a mass ratio ρa ca /ρhω and a speed ratio M = c/ca = ka /k; we have M = 1 at coincidence. Both parameters are frequency dependent; we can arrange for only one parameter, which is varied with ω by introducing ≡

ρa ca ρa co  = / 12(1 − ν 2 ) , ρhωc ρ ca

M=

c ka = = ca k



√ ω =  ωc

The parameter , called the intrinsic fluid loading parameter, is the same for all plates of a given material embedded in a given fluid. This is typically small with values such as

336

8 Structure–Fluid Interactions

steel/water: 0.130 ,

aluminum/air: 0.002 ,

aluminum/water: 0.460

In the examples that follow, we take aluminum/air as the case of light fluid loading and steel/water as the case of heavy fluid loading.

8.2 Panel Excitations A common situation is for the waves to arise from the impact or machinery loading of the panel. In addition to the wave propagating in the structure as discussed in the other chapters, we also have radiation into the fluid. We consider two cases: one is a semi-infinite fluid, and the other is a two-panel system.

8.2.1 Line Loading of an Infinite Plate in a Fluid Consider a single infinite sheet with fluid only on one side and impacted along a narrow line. Spectral analysis assumes solutions in the form wˆ =

 m

w˜ m e−iξm x ,

qˆ =

 m

q˜m e−iξm x ,

ξm ≡ m2π/W

(8.15)

This is an application of our periodically extended load (pEL) representation. The pressure has a similar representation, except that we must realize it is twodimensional:   pˆ = p˜ m e−iξm x e−ikz z , kz ≡ ka2 − ξm2 (8.16) m

Combining these with the plate equations, we get [Dξm4 − ρhωn2 + iωn ηh]w˜ m = q˜m − p˜ m We need one more equation. This comes from the conditions at z = 0: ∂ pˆ = ρa ω2 wˆ , ∂z

−ikz p˜ m = ρa ω2 w˜ m

where continuity of displacement between the plate and fluid is assumed. The displacement of the plate is determined to be w(x) ˆ =

 m

w˜ m e−iξm x =

 m

q˜m e−iξm x Dξm4

− ρhωn2

ρa ω2 + iωn ηh + −ikz

(8.17)

8.2 Panel Excitations

337

z=100mm x=0

x=0

x=0.5m

x=0.5m x=1.0m

x=1.0m

0.

2000.

4000.

6000.

x=1.5m

x=1.5m

x=2.0m

x=2.0m

8000.

0.

2000.

4000.

6000.

8000. 10000.

Fig. 8.6 Time domain responses for the line loading of an infinite aluminum plate in air. (a) Plate responses w(x, ˙ t). (b) Fluid pressures p(x, z = L, t)

As with the case of a plate in a vacuum, q˜m is chosen to be 1.0 to represent a point load at x = 0. Figure 8.6 compares the w-velocity responses for an aluminum plate in air and the resulting pressures 100 mm from the plate. The plate responses look similar to those of the beam and plate encountered in Chaps. 3 and 6, respectively. The pressures, however, exhibit an unusual trailing behavior. In addition, the period of the zero crossings is almost constant, whereas for the plate velocity it increases. Figure 8.7 compares the same responses but in the frequency domain. The most striking feature of the pressure is the spectral peak in the vicinity of 5 kHz—this corresponds to the coincidence frequency as seen from Fig. 8.3. The peak gets sharper further away from the impact site. What we also notice is that the amplitudes in the vicinity below coincidence are diminishing rapidly. This is seen through the exponential term e−ikz z = e−i

√

2z ka2 −ξm

For a given wavenumber ξ , low frequencies are evanescent. This becomes even more severe for the higher wavenumbers. Hence, at any given location, the pressure response lacks in low-frequency content and lacks information about the shape of the applied load. From an inverse problems point of view, this makes the acoustics problem very difficult because it indicates that remote measurements are incapable of inferring the source of the disturbance. This has given rise to an experimental area called Near-Field Holography [8], where an array of sensors is placed very close to the vibrating surface so as to minimize the loss of information due to the evanescing of the waves.

338

8 Structure–Fluid Interactions

z = 100mm

0.

2.

4.

6.

8.

x=0

x=0

x=0.5m

x=0.5m

x=1.0m

x=1.0m

x=1.5m

x=1.5m

x=2.0m

x=2.0m

10. 12.

0.

2.

4.

6.

8.

10. 12. 14. 16.

Fig. 8.7 Magnitude of the frequency domain responses. (a) Plate responses |iωw(x)|. ˆ (b) Fluid pressure |p(x, ˆ z = L)| Fig. 8.8 Geometry of the double plate system

8.2.2 Double Panel Systems Noise radiation is often reduced by the addition of a “trim panel,” that is, a second panel parallel to the source panel. We consider a simple version of the problem where the only coupling is through the fluid as indicated in Fig. 8.8. This is the analogous problem to that solved in Sect. 3.2, where the beams were connected by distributed springs. The difference here is that the connecting medium has inertia as well as stiffness. For simplicity, we take both plates as having identical properties. The equations of motion for the two plates are, respectively, D

d 4 wˆ 1 − ør hwˆ 1 = qˆ1 − p(z ˆ = 0) dx 4

D

d 4 wˆ 2 − ør hwˆ 2 = qˆ2 + p(z ˆ = L) dx 4

(8.18)

The pressure in the cavity has waves propagating in both z directions; hence, we represent it by p(x, ˆ z) =

 m

[A˜ m e−ikz z + B˜ m e−ikz (L−z) ]e−iξm x ,

kz ≡



ka2 − ξm2

8.2 Panel Excitations

339

Concentrate on the plates for the moment, so remove the pressure in the following manner. Assume there is continuity of displacement between plate and fluid, the momentum relation becomes ∂ pˆ = ρa ω2 wˆ ∂z

or

ρa ω2 w˜ = −ikz [A˜ m e−ikz z − B˜ m e−ikz (L−z) ]

This can be particularized to both interfaces so as to solve for the coefficients in terms of the displacements. That is,      ρa ω2 /ikz 1 −e w˜ 1 A˜ , = B˜ −1 + e2 e −1 w˜ 2

e ≡ e−ikz L

We are now in a position to write the relation between the pressure on the two plates in terms of the displacements 

       ρa ω2 /ikz −1 − e2 2e −p(0) ˜ w˜ 1 −1 −e A˜ = = 2e −1 − e2 w˜ 2 +p(L) ˜ e 1 B˜ −1 + e2

This, in fact, is an acoustic spectral element in the sense that the pressures at the generalized nodes (the location of the plates) are related only to the displacements at these nodes. Also, the form of the matrix is identical to that of the spectral element for the rod, see Eq. (5.6). We complete the solution by adding this pressure to the equations of motion for the plates to get  4   0 Dξm − ør h w˜ 1 0 Dξm4 − ør h w˜ 2      ρa ω2 /ikz 1 + e2 −2e w˜ 1 q˜ = 1 + 2 2 w˜ 2 q˜2 −2e 1 + e −1 + e

(8.19)

These can now be solved to get responses as shown in Fig. 8.9. Figure 8.9 compares the w-velocity responses for two aluminum plates separated by an air gap of 100 mm (4 in.). The response of the first plate looks typical and is almost identical to Fig. 8.6a. The response of the second plate is much more interesting. The response is much smaller (which is why double panels are often used for noise suppression), but we see two quite separate effects. The first is an almost nondispersive group that seems to get bigger the further it propagates. The other is a low-frequency undulation. The nondispersive part is obviously related to the pressure histories of Fig. 8.6b. The undulation is due to the development of a coupled mode between the two plates quite similar to that of the antisymmetric mode of the spring coupled beams in Sect. 3.2. Figure 8.10 compares the frequency domain response, and again we see the spectral peaks in the vicinity of 5 kHz for the trim panel. These peaks are not as crisp

340

8 Structure–Fluid Interactions

z = 100 mm x=0 x=0 x=0.5m x=0.5m x=1.0m

x=1.0m

x=1.5m x=1.5m x=2.0m

x=2.0m

0.

2000.

4000.

6000.

8000.

0.

5000.

10000.

15000.

20000.

Fig. 8.9 Time domain responses for the double panel. (a) Impacted plate. (b) Trim plate magnified by 15 z = 100 mm

0.

2.

4.

6.

8.

10. 12.

x=0

x=0

x=0.5m

x=0.5m

x=1.0m

x=1.0m

x=1.5m

x=1.5m

x=2.0m

x=2.0m

0.

2.

4.

6.

8.

10. 12. 14.

Fig. 8.10 Frequency domain responses for the double panel. (a) Impacted plate, (b) Trim plate magnified by 10

as in the single plate case; it seems that the cavity resonances may be contributing to this. To help bracket this effect, consider the case of rigid walls and only plane waves propagating in the cavity. The resonance frequencies are given by fN =

Nca ≈ 1.7, 3.4, 5.1, 6.8, . . . 2L

where the frequency is in kHz. These resonances occur in the vicinity of the frequency range of interest as well as cover the coincidence frequency; hence, we now consider the effect of these resonances a little further.

8.2 Panel Excitations

341

The thrust of the reverberation analysis is to decompose the result of Eq. (8.19) and thereby help explain the features of Figs. 8.9 and 8.10. We begin by noting that, without approximation, we can write       1 2e 1 + e2 −2e 10 e −1 = − + 01 −1 + e2 −2e 1 + e2 −1 + e2 −1 e The plate equations become 

       (ρa ω2 /ikz )2e e −1 w˜ 1 q˜ w˜ 1 = + 1 w˜ 2 w˜ 2 q˜2 −1 e −1 + e2

0 0

where = Dξm4 − ør h − ρa ω2 /ikz We recognize the first matrix as describing uncoupled single plates with fluid loading. The second matrix causes the reverberation where the acoustic wave reflects multiple times inside the cavity. Note that 1 eN +2 = 1 + e2 + e4 + · · · + ≈ 1 + e2 + e4 + · · · 2 1−e 1 − e2 where the approximation is most valid if there is some dissipative mechanism. We can now write the reverberant pressures as   2    ρa ω2 −p(0) ˜ e −e w˜ 1 2 4 (1 + e = 2 + e + · · · ) 2 w ˜2 −e e +p(L) ˜ ik z rev



Look closer at the pressure on the first plate, for example, −p(0) ˜ = +2

ρa ω2 −ikz 1L [e + e−ikz 3L + e−ikz 5L + · · · ]w˜ 2 ikz

−2

ρa ω2 −ikz 2L [e + e−ikz 4L + e−ikz 6L + · · · ]w˜ 1 ikz

The interpretation is that w2 , for example, causes a pressure Wave, which impinges on the first plate after it travels a distance L; this is reflected and travels a further distance of 2L before it impinges again. And the process continues. The leading factor of “2” is important because it is the pressure if the plate were rigid, this is the blocked pressure mentioned previously. A similar interpretation is given for the pressure on the second plate.

342

8 Structure–Fluid Interactions

A final point worth noting is that there are two attenuation effects occurring in the solution. Because all materials have some damping, the waves associated with the longer reflection paths eventually disappear. Second, any initially localized disturbance expands to occupy a greater volume over the reflection paths and therefore diminishes. A useful approximation, then, is to assume that only the very first incident wave causes a fluid loading, that is, for our double panel, we have 

−p(0) ˜ +p(L) ˜

 rev

   ρa ω2 0 −e w˜ 1 ≈ 2 w˜ 2 −e 0 ikz

This is the blocked pressure approximation. It must be realized that this approximation is separate from the self-loading effect, which is already included. The blocked pressure is a useful approximation of the fluid loading when the fluid is light or the structure is relatively stiff.

8.2.3 Cylindrical Cavity Our final acoustic example looks at the case where the fluid is completely surrounded by the structure. The simplest such structure is a circular cylinder; we look at it because it also gives another illustration of the use of Bessel functions. The geometry of our problem is shown in Fig. 8.11. With reference to Fig. 8.11, we expect a variation of pressure in the hoop (or θ ) direction when a general load is applied. The Helmholtz equation in cylindrical coordinates is  ∂2 1 ∂2  1 ∂ + 2 2 pˆ + ka2 pˆ = 0 + 2 r ∂r ∂r r ∂θ Let the pressure be represented by a Fourier series in the hoop direction, that is, p(r, ˆ θ) =

Fig. 8.11 Geometry of a circular cylinder containing a fluid

 m

p˜ m (r) cos(mθ )

8.2 Panel Excitations

343

This is an example of pEL where the periodicity is inherent in the cylinder. The amplitudes are given by p˜ m (r) = AJm (ka r) + BYm (ka r) The Bessel functions of the second kind, Ym , approach infinity at the origin r = 0; hence, for this particular problem, we take B = 0. Note, however, that if we were looking at the exterior problem, then we would take B = −iA, so that the solution is a Hankel function that coincides with an outward propagating wave. The pressure and displacements in the fluid are represented by p(r, ˆ θ) =

 m

Am Jm (ka r) cos(mθ ) ,

uˆ r (r, θ ) =

 m

u˜ r (r) cos(mθ )

As we have done before, we wish to express the coefficient Am in terms of the boundary displacement. In general, we have ∂ pˆ = ρa ω2 uˆ r ∂r

or

 m

Am ka Jm (ka r) cos(mθ ) = ρa ω2

 m

u˜ r (r) cos(mθ )

Let the displacements be specified at r = R and be indicated by u˜ R r , and then we have Am = u˜ R r

ρa ω2 ka Jm

The pressure and displacements are now represented by p(r, ˆ θ)=

 m

u˜ R r

  ρa ω2 Jm (ka r) R Jm (ka r) cos(mθ ), u ˆ cos(mθ ) (r, θ ) = u ˜ r r m ka Jm (ka R) ka Jm (ka R)

In these, the pressure and displacement are written in terms of the single function u˜ R r , which is the radial displacement of the outer shell. We can therefore interpret the displacement expression as a shape function relation. The contours of some of these shape functions are shown in Fig. 8.12. These contours help visualize the standing wave that is established inside the cylindrical cavity. We looked at the response of curved plates in Sect. 6.4; we now adapt these equations to the complete cylindrical shell. This will be straight-forward since we are assuming that there is no variation in the (global) axial y direction. Replacing s = Rθ , we get 2 3 ¯ d uˆ − E¯ d wˆ + D¯ d wˆ + ρhω2 uˆ = qˆθ (E¯ + D) dθ dθ 2 dθ 3 3 d 4 wˆ ¯ wˆ + D¯ d uˆ − E¯ d uˆ − ρhω2 wˆ = qˆr − p(r + E ˆ = R) D¯ dθ dθ 4 dθ 3

(8.20)

344

8 Structure–Fluid Interactions

Fig. 8.12 Contours of displacement u(r, ˜ θ) at 3 kHz

with the definitions E¯ ≡

Eh , (1 − ν 2 )R 2

D¯ ≡

Eh3 12(1 − ν 2 )R 4

The cylinder radius R appears explicitly only in these material properties. Because we want to match the displacement around the boundary of the structure with that of the fluid, represent the displacement as a Fourier series in the same form as done for the pressure. That is, let u(r, ˆ θ) =

 m

u˜ m sin(mθ ) ,

w(r, ˆ θ) =

 m

w˜ m cos(mθ )

The system of equations to be solved is then 

¯ 2 + ρhω2 ¯ + Dm ¯ 3 −(E¯ + D)m Em 3 4 ¯ ¯ ¯ ¯ −Em − Dm Dm + E − ρhω2



   u˜ m q˜θ = w˜ m q˜r − p˜ m

Substituting for the pressure, we get ⎡

⎤     ¯ + Dm ¯ 3 Em q˜ u˜ m 2 ⎦ = θ ¯ 4 + E¯ − ρhω2 − ρa ω J (ka R) w˜ m q˜r Dm ka J  (ka R)

¯ 2 + ρhω2 −(E¯ + D)m

⎣

¯ − Dm ¯ 3 −Em

Consider the case where there is only a normal applied load qˆr = q, ˆ qˆθ = 0, the solution is obtained directly by use of Cramer’s rule. The two displacements are  (Em ¯ + Dm ¯ 3) q˜m sin(mθ ) u(r, ˆ θ) = − m  −(E¯ + D)m ¯ 2 + ρhω2 q˜m cos(mθ ) w(r, ˆ θ) = m where the system determinant is

8.2 Panel Excitations

345

¯ 2 + ρhω2 ][Dm ¯ 4 + E¯ − ρhω2 ] + [Em ¯ + Dm ¯ 3 ]2 = [−(E¯ + D)m ¯ 2 − ρhω2 ]ρa ω2 +[(E¯ + D)m

Jm (ka R) ka Jm (ka R)

These displacements can be reconstructed in the usual manner. What is of more direct interest here is the role of the fluid loading as it affects the cylinder in comparison to its effect on the infinite plate and the two parallel plates.

8.2.4 Comparison of Fluid Loadings So far, we have considered the fluid loading on a single plate, the parallel double plate, and the case of a cylinder surrounding the fluid [7]. The relevant equation in each case is ' ρa ω2 ( single: Dξ 4 − ρhω2 − w˜ = q˜ ikz ' ρa ω2 ) 1 + e−i2kz L *( parallel: Dξ 4 − ρhω2 − w˜ 1 = q˜1 ikz 1 − e−i2kz L ' 1 ρa ω2 J (ka R) ( w˜ = q˜r cylinder: Dξ 4 − ρhω2 + E ∗ h 2 − ka J  (ka R) R Some reinterpretation of parameters needed to be done in order to make the comparisons with the infinite plate. For the two parallel plates, we take w2 = 0 and W = 2π R, L = 2R. For the cylinder, we take u = uθ = 0 and ξ = m/R. All three have the common [Dξ 4 − ρhω2 ] term. The shell has the added elastic constraint effect of E ∗ h/R 2 , but this is unrelated to the fluid. In fact, if there is no fluid, then the three equations are very similar. We expect the cylinder to exhibit resonances, but where does this enter these equations? We must keep in mind that the infinite plates are represented over a space window W that is very large, but the cylinder has W = 2π R that is relatively small. The waves coming from the neighboring windows cause the resonances. We have a hierarchy of pressure contributions given by single: ρa ω2

1 ikz

parallel: ρa ω2

(1 + e−i2kz L ) ikz (1 − e−i2kz L )

cylinder: ρa ω2

J (ka R) ka J  (ka R)

346

8 Structure–Fluid Interactions

infinite parallel cylinder m=0 10

m=2

1

m=4

0.

5.

10.

15.

20.

25.

30.

Fig. 8.13 Comparison of pressure contributions from infinite plate, two parallel plates, and circular cylinder

 In each case, kz = ka2 − ξm2 , ξm = m2π/W . The variation of these pressure contributions against frequency is shown in Fig. 8.13 for a few values of wavenumber ξm . To further elaborate on the plots, look just at the portion of the pressure term that is specifically different. We have  ) m *2 single: ikz ←→ ka −1 ka R¯  √   2 ) m *2 1 − e−i2kz L 1 − e−i2 1−(m/ka L) ka L parallel: ikz ←→ −1 √ 2 ka L 1 + e−i2kz L 1 + e−i2 1−(m/ka L) ka L cylinder:

m ka Jm+1 (ka R) ka Jm (ka R) ←→ − Jm (ka R) ka R Jm (ka R)

The second form in each case corresponds to the appropriate expansion of kz . For each m, a frequency spectrum of kz goes through zero once. For the parallel plates, this occurs where  ω −i2kz L 1−e = 0 or ka = = (N π /L)2 + (2mπ /W )2 ca When m = 0, this corresponds to the characteristic equation for the free vibrations of the fluid with free boundaries. Similarly, the zeros occur for the cylinder when m ka Jm+1 (ka R) − =0 R Jm (ka R)

8.3 Waveguide Modeling of Distributed Pressures

347

When m = 0, this corresponds to the characteristic equation for the free vibrations of the fluid with free boundaries, and as m → ∞ it corresponds to the free vibrations of the fluid with fixed boundaries. Note that there are no spectral peaks for frequencies such that ka < ξ

or

ω < 2π mca /W

where W is the space window. At these frequencies, the pressure waves are evanescent. Although we only focused on the pressure contribution of the fluid loading, this example serves to demonstrate the rather complicated interaction between the wave phenomena and the structural dynamics that can occur in coupled structural systems.

8.3 Waveguide Modeling of Distributed Pressures We saw in the other chapters that if we can replace the wavenumber transform solution (our pEL representation) with a waveguide solution, then that allows us to terminate the waveguide, and we are thus in a position to assemble complex structures. Our goal here is to formulate the acoustics problem in terms of a waveguide; but while the effect of the fluid loading is that of a distributed pressure, its nonlocal character makes it different in many respects from the pressure caused by a distributed spring, say, and hence we must invoke special procedures.

8.3.1 Free Wave Response To begin our construction of a plate waveguide with fluid loading, we ask if it is possible for free waves to propagate in the plate immersed in a fluid. That is, are there wave solutions of the form w(x, t) = we ˜ −i[kx−ωt] ,

p(x, z, t) = pe ˜ −ikz z e−i[kx−ωt]

The spectral form of the equations of motion for the plate and the fluid can be written as, respectively, D

ˆ d 4 w(x) − [ρhω2 − iωηh]w(x) ˆ = −q(x, ˆ z = 0) , dx 4

B∇ 2 pˆ + ρa ω2 pˆ = 0

Because of the assumed wave solutions, the pressure response is given by pˆ = p˜ e−ikx e−ikz z ,

p˜ =

ρa ω2 w˜ , −ikz

kz ≡

 ka2 − k 2

348

8 Structure–Fluid Interactions

2.0 1.5 1.0 .5 .0 -.5 -1.0 -1.5 -2.0 -2.0 -1.5 -1.0 -.5

.0

.5

1.0 1.5

-2.0 -1.5 -1.0 -.5

.0

.5

1.0 1.5 2.0

Fig. 8.14 Contours of the characteristic equation at 5 kHz (left) and 15 kHz (right) for a steel plate in water

It must be borne in mind that as long as kz is chosen as above then, irrespective of the value of k, the fluid equations are satisfied. This leads to the characteristic equation Dk 4 − ør h +

ω2 ρa = 0, −ikz

kz ≡



ka2 − k 2 ,

ka2 ≡

ω2 ca2

(8.21)

The third term in the first equation describes the effect of the fluid loading. A detailed explanation of the significance of each term in Eq. (8.21) is given by Crighton [2], and the roots of this characteristic equation have been studied extensively in Refs. [1, 3, 4]. With the presence of the fluid loading, an analytical solution to this characteristic equation is not obvious. Figure 8.14 shows contours of the characteristic equation below and above the coincidence frequency. We see that it is dominated by four roots. It is desired to identify the roots that correspond to the plate flexural and evanescent waves. When the response of Eq. (8.17) is viewed as a contour integral in the complex planes of Fig. 8.14, it has contributions from the poles and a contribution, which arises from the branch cut associated with kz . This latter contribution is most significant at impact points and corners [2]. By neglecting this contribution, we would be in a position to replace the responses with just the pole contributions. That is, we would have a waveguide representation. To quantify this contribution, we consider the relatively large fluid loading case of the response of a 25 mm thick steel plate in water. Figure 8.15 shows the responses with and without the branch cut contribution. The agreement of the two solutions is good with some deviations occurring at the load site that are primarily in the low frequency range. This is confirmed at the large x locations. We therefore conclude that in cases of moderate

8.3 Waveguide Modeling of Distributed Pressures

349

transform waveguide no fluid x=0

x=1m x=2m

0.

1000.

2000.

3000.

4000.

5000.

Fig. 8.15 Comparison of the wavenumber transform solution to the waveguide solution that neglects the branch cut contribution

fluid loading (and especially for aluminum plates in air), the branch cut contribution can be neglected. Furthermore, the approximation is reasonable even for steel in water if the responses away from the impact site are of interest. It would be possible to include an approximate model for the branch cut contribution, but that is not be pursued here. Considering Eq. (8.21), it can be noticed that if the fluid loading term is set to zero, then the characteristic equation of the plate in a vacuum is recovered. For the vacuum response, the plate can be described in terms of two modes with the corresponding spectrum relations given by 1/4  2 ¯k1 = ρhω − iωηh , D

1/4  2 ¯k2 = −i ρhω − iωηh D

(8.22)

where k¯1 corresponds to the propagating flexural wave and k¯2 corresponds to an evanescent flexural wave as we have seen in Chap. 3. The plot of these wavenumbers for plates of 2.5 mm aluminum in air and 25.4 mm steel submerged in water can be found as the thin dashed lines in Figs. 8.16 and 8.17, respectively. A steel plate submerged in water is seen to exaggerate the effect of fluid loading on the structure. Solutions to the characteristic equation, found numerically as illustrated in Sect. 4.2, are also shown in Figs. 8.16 and 8.17 as circles. These figures indicate that the fluid loading primarily has the effect of altering the imaginary part of the first mode. We see that at coincidence and beyond, the effect is of increased viscous damping—this is consistent with the observation that the fluid is receiving more

350

8 Structure–Fluid Interactions

4. 3.

exact 2.

real

approx’n

1.

no fluid

0. -1.

real x400

-2.

imaginary x400

imaginary

-3. -4. 0.

1.

2.

3.

4.

5.

6.

7.

8.

9. 10. 0.

1.

2.

3.

4.

5.

6.

7.

8.

9. 10.

Fig. 8.16 Wavenumber behavior for an aluminum plate in air. (a) k1 (ω) and (b) k2 (ω) 2.0 1.5

exact 1.0

real

.5

approx’n no fluid

.0 -.5

real x10

-1.0

imaginary

imaginary x10

-1.5 -2.0 0.

2.

4.

6.

8. 10. 12. 14. 16. 18. 20. 0.

2.

4.

6.

8. 10. 12. 14. 16. 18. 20.

Fig. 8.17 Wavenumber behavior for a steel plate in water. (a) k1 (ω) and (b) k2 (ω)

of the energy at these frequencies. But otherwise the behavior is very similar to the in-vacuum behavior. That is, the pole contributions are associated with root k1 , which corresponds predominantly to the propagating flexural wave, and with root k2 , which corresponds predominantly to an evanescent flexural wave. There exists a third root to the characteristic equation that appears only above the coincidence frequency, which is not seen in either of these plots. The possibility of this root causing a structural wave is of interest and requires a more detailed interrogation of the structure. The wavenumber transform solution for the impact of a plate, as described in the previous section, is now used as a computer experiment to examine the effect of this third root. To make the wave more discernible, we impact the plate with a force having a narrow-banded frequency range as shown in

8.3 Waveguide Modeling of Distributed Pressures

351

x=1m x4

x=2m

x16

x=3m

x64

x=4m

x256

x=5m

0.

1000.

2000.

3000.

4000.

5000.

Fig. 8.18 Response at large distances on the structure showing the presence of a wave traveling at the sonic speed ca ≈ 1.5 km/s. Note the magnifications used

Fig. 1.8. The resulting waves disperse very slowly, and the group behavior of the waves is easier to observe. By inputting a force centered about a certain frequency, a wave traveling at the corresponding group speed should be identifiable if it exists. The velocity responses for an input force centered about 15 kHz are shown in Fig. 8.18. This figure is very interesting in that the presence of two waves can be identified, one wave corresponds to the plate flexural wave mode and the other corresponds to a wave traveling at the speed of sound in the fluid. Although the fluid loading on the structure causes a third structural wave to be present, it can be seen that this wave can only be distinguished at large distances with very large magnification and only over a very narrow frequency range. It therefore seems reasonable to neglect this wave in the modeling of connected structures. That is, we assume that there are two dominant structural waves resembling those for a structure in a vacuum.

8.3.2 Modified Spectrum Relations Making the assumption that there are only two dominant structural waves allows us to obtain approximate analytical expressions for the roots of Eq. (8.21). The approximations can be determined by rewriting the characteristic equation in the form (k 2 − β 2 )(k 2 + β 2 ) −

ρa ω2 = 0, iDkz

β2 ≡

 ør h/D ,

kz ≡



ka2 − k 2

352

8 Structure–Fluid Interactions

Notice that if the fluid loading term is set to zero, then k¯1 and k¯2 expressed in Eq. (8.22) can be found from the terms inside the first and second set of brackets, respectively. By assuming that the appropriate roots also come from these terms and using k¯1 and k¯2 as the first guess, approximations for the exact roots can be written as   ρa ω2 , k ≡ ka2 − β 2 k1 ≈ ± β 2 + z1 2Dikz1 β 2   ρa ω2 k2 ≈ ±i β 2 + , k ≡ ka2 + β 2 (8.23) z2 2Dikz2 β 2 These approximate solutions for k1 and k2 are also plotted in Figs. 8.16 and 8.17, respectively. There is very good agreement between the approximations and the exact numerical roots over the entire frequency range including the region near coincidence. By incorporating the fluid loading term directly into the modified spectrum relations, we are now in a position to replace the double summation wavenumber transform solution by a single summation over frequency. This is useful when an enclosed structure is viewed as having two distinct regions: an interior region and an exterior region. The interior region has loadings due to both the fluid and the radiation from the vibration of other plates. However, the exterior region only experiences loadings from the fluid. Using the modified spectrum relations, the fluid loading for the exterior problem can be accounted for without considering the fluid response. In this way, the solid–fluid interaction problem is partially decoupled. We conclude this section by illustrating the difference the waveguide formulation makes by reconsidering the example of the impact of an infinite plate. Although we have solved essentially the same problem before in Sect. 3.1, we do it again in the new variables. The transverse displacement of an infinite plate with two forward propagating waves is expressed as −iωt ˆ −i[k1 x−ωt] + Be ˆ −i[k2 x−ωt] w(x)e w(x, t) = Ae ˆ

where k1 and k2 are the modified spectrum relations presented in Eq. (8.23). The BCs for this problem are that at x = 0, the slope of the plate is zero (∂ w/∂x ˆ = 0) and the applied load is related to the shear by − 12 Pˆ = Vˆ = −D

∂ 3 wˆ ∂x 3

The response of the plate is then determined as w(x) ˆ =

 k1 −ik2 x  Pˆ −ik1 x e − e k2 2Dik1 (k12 − k22 )

(8.24)

8.4 Radiation from Finite Plates

353

This solution is to be compared to that of Eq. (8.17). The biggest difference is that the present solution does not require the summation over m. The waveguide responses of Fig. 8.15 were computed using this solution. It is clear that it is now possible to solve for the response of impacted finite plates with a surrounding fluid.

8.4 Radiation from Finite Plates When investigating the interior noise of a cabin, say, the response of the fluid, in addition to that of the structure, is required. We are also interested in the fluid response because it can be the source of an excitation on a different part of the structure. Radiation of energy from the dynamically loaded plate was implicit in the other sections, but here we wish to concentrate on the radiation from finite plates. The geometry of our problem is shown in Fig. 8.19.

8.4.1 Finite Plate Response We write the general transverse displacement for the plate in the form ˆ −ik2 x + Ce ˆ −ik1 (L−x) + De ˆ −ik2 (L−x) ˆ −ik1 x + Be w(x) ˆ = Ae ˆ B, ˆ C, ˆ and D ˆ are constants determined from the BCs on the segment. Also, where A, by using the modified spectrum relations, we claim that this adequately represents the behavior of a plate immersed in a fluid. It is advantageous, when dealing with finite or multiply connected structures, to use a solution formulation that already incorporates the connectivities—we made use of this in Chap. 5 when developing the spectral element method. The end conditions on the plate segment are w(0) ˆ = wˆ 1 ,

d w(0) ˆ = ψˆ 1 , dx

w(L) ˆ = wˆ 2 ,

d w(L) ˆ = ψˆ 2 dx

Solving for the coefficients in terms of the nodal DoFs allows the transverse displacement of the plate to be written in the form Fig. 8.19 Geometry of the finite plate of length 2L baffled to infinity

354

8 Structure–Fluid Interactions

w(x) ˆ = gˆ 1 (x)wˆ 1 + gˆ 2 (x)Lψˆ 1 + gˆ 3 (x)wˆ 2 + gˆ 4 (x)Lψˆ 2

(8.25)

The frequency dependent shape functions are given as gˆ 1 (x) gˆ 2 (x) gˆ 3 (x) gˆ 4 (x)

= = = = =

[ r1 hˆ 1 (x) + r2 hˆ 2 (x)]/ [ r1 hˆ 3 (x) + r2 hˆ 4 (x)]/ [ r1 hˆ 2 (x) + r2 hˆ 1 (x)]/ , [−r1 hˆ 4 (x) − r2 hˆ 3 (x)]/ , e1 = e−ik1 L , −r12 − r22 ,

r1 = i(k1 − k2 )[1 − e1 e2 ] r2 = i(k1 + k2 )[1 − e1 e2 ] e2 = e−ik2 L

(8.26)

where hˆ 1 (x) = +ik2 [e−ik1 x − e−ik2 L e−ik1 (L−x) ] − ik1 [e−ik2 x − e−ik1 L e−ik2 (L−x) ] hˆ 2 (x) = −ik2 [e−ik2 L e−ik1 x − e−ik1 (L−x) ] + ik1 [e−ik1 L e−ik2 x − e−ik2 (L−x) ] hˆ 3 (x) = [e−ik1 x + e−ik2 L e−ik1 (L−x) ] − [e−ik2 x + e−ik1 L e−ik2 (L−x) ] hˆ 4 (x) = [e−ik2 L e−ik1 x + e−ik1 (L−x) ] − [e−ik1 L e−ik2 x + e−ik2 (L−x) ] These shape functions, when plotted, are similar to those as depicted in Fig. 5.6 for the beam. Indeed, these become the shape functions for the Bernoulli–Euler beam model when we consider thin plates in vacuum. Again, the shape functions occur in pairs, where gˆ 3 and gˆ 4 are the mirror images of gˆ 1 and gˆ 2 , respectively. The deflection of a semi-infinite plate can be written in a similar manner as w(x) ˆ = gˆ 1 (x)wˆ 1 + gˆ 2 (x)ψˆ 1

(8.27)

with   gˆ 1 (x) = −ik2 e−ik1 x + ik1 e−ik2 x /   gˆ 2 (x) = −e−ik1 x + e−ik2 x / ,

≡ i(k1 − k2 )

The complete description of this plate segment has now been captured in the two nodal degrees of freedom wˆ 1 and φˆ 1 . As an example of using this procedure, consider the response due to the impact of a finite aluminum plate in air. Treating only half of the plate, we have that w1 = wo , ψ1 = 0, w2 = 0, ψ2 = 0, and the shear relation 12 Pˆ = −Vˆ = D wˆ o  . This gives the solution w(x) ˆ = wˆ o gˆ 1 (x) ,

wˆ o =

Pˆ 2D gˆ 1  (0)

These responses are shown in Fig. 8.20a; the presence of multiple reflections is obvious. Figure 8.21a shows the corresponding frequency domain behavior where the reflections give rise to multiple spectral peaks. Also shown are the resonance

8.4 Radiation from Finite Plates

0.

2000.

4000.

6000.

355

x=0

x=0 z=L

x=0.25L

x=L z=L

x=0.50L

x=2L z=L

x=0.75L

x=2L z=0

8000.

0.

2000.

4000.

6000.

8000. 10000.

Fig. 8.20 Time domain responses for an impacted aluminum plate of length 2L = 500 mm. (a) Plate responses w(x, ˙ t). (b) Fluid pressures p(x, z = L, t) resonance, sym resonance, anti-sym

0.

2.

4.

6.

8.

10.

x=0

x=0 z=L

x=0.25L

x=L z=L

x=0.50L

x=2L z=L

x=0.75L

x=2L z=0

12.

0.

2.

4.

6.

8.

10.

12.

14.

Fig. 8.21 Frequency domain responses for an impacted plate of length 2L = 500 mm. (a) Plate responses |iωw(x)|. ˆ (b) Fluid pressure |p(x, ˆ z = L)|

frequencies for a vibrating plate, it is clear that the impact has excited many of the symmetric modes of vibration.

8.4.2 Fluid Response from a Finite Plate The challenge we have here is to match the motion of the finite plate to that of the fluid. But the domain for the fluid is (at least) the half space z > 0 and −∞ < x < ∞, which is considerably larger than the length of the plate. Therefore, to match the

356

8 Structure–Fluid Interactions

plate and fluid boundaries, we must extend the plate boundary in the x direction. If the finite plate is baffled, that is, extended on both sides with very stiff material, then the displacements can be matched by imposing w = 0 outside of the finite plate. We assume that this can always be done even if the plate is not physically baffled but is attached to other plate segments. At the surface of the plate, z = 0, the fluid displacement must be equal to the plate displacement. This response in the fluid has the spectral representation w(x, ˆ z) =

 m

w˜ m e−ikz z e−iξm x ,

kz ≡

 ka2 − ξm2

By extending the plate deflection over the full space window of the fluid, we can then give it the spectral representation w(x) ˆ =

 m

w˜ m e−iξm x

Applying this to the shape functions gives w˜ m = wˆ 1 g˜ 1m + Lψˆ 1 g˜ 2m + wˆ 2 g˜ 3m + Lψˆ 2 g˜ 4m ,



gˆ j (x)e−iξm x dx

g˜ j m = W

(8.28) These integrals are easily evaluated; indeed, we need to only replace the exponential terms of hˆ j (x) in Eq. (8.26) with  I1 =

L

e−ik1 x e−iξm x dx = i[e−ik1 L e−iξm L − 1]/(ξm + k1 )

0

 I2 =

L

e−ik2 x e−iξm x dx = i[e−ik2 L e−iξm L − 1]/(ξm + k2 )

0

 I3 =

L

e−ik1 (L−x) e−iξm x dx = i[e−iξm L − e−ik1 L ]/(ξm − k1 )

0

 I4 =

L

e−ik2 (L−x) e−iξm x dx = i[e−iξm L − e−ik2 L ]/(ξm − k2 )

(8.29)

0

The continuity of wi and ψi between plate segments ensures the continuity of the fluid field. In the case of the semi-infinite plate segment, the displacement only consists of the terms wˆ 1 and ψˆ 1 , then only the first two integrals I1 and I2 need to be evaluated. Furthermore, letting L → ∞ gives the integrals as Ij = i/(ξm + kj ). In piecing together the solution, we treat the plate as having two segments with the BCs Seg 1:

w1 , ψ1 , ψ2 = 0, w2 = wo ;

Seg 2:

ψ1 , w2 , ψ2 = 0, w1 = wo

8.4 Radiation from Finite Plates

357

This leads to the solution for the fluid displacement written in terms of the central deflection of the plate as w(x, ˆ z) = wˆ o

   g˜ 3m e−iξm (x+L) + g˜ 1m e−iξm x e−ikz z m

Note that the representation for the first plate segment must be shifted an amount L. The pressure is given by p(x, ˆ z) = wˆ o

  ρa ω2  g˜ 3m e−iξm (x+L) + g˜ 1m e−iξm x e−ikz z m −ikz

The pressure responses for the fluid can be seen in Fig. 8.20b for a baffled aluminum plate in air. Similar to the plate response, these responses indicate the presence of multiple reflections occurring in the plate. This figure also indicates that the baffled finite plate excites the fluid not only directly in front of it but also at a distance along the baffle. The pressure at x = 2Landz = 0 is nonzero even though the plate is baffled at that location—this is further indication that, in fluids, the relation between the pressure and displacement is nonlocal. This, perhaps, is seen even more clearly in Fig. 8.1, which shows the 2D pressure distribution at 1 kHz. The frequency domain depiction of these responses is shown in Fig. 8.21b. The structural resonances again appear as spectral peaks. It can be seen that the frequencies at which the structural resonances were present, shown in Fig. 8.21b, are readily transmitted into the fluid. Intuitively, we might have thought that the response should be largest along a line normal to the plate. The responses shown in Fig. 8.20a indicate that this is not so. Furthermore, Fig. 8.21b shows that it is even frequency dependent. To help explain the situation, we now look at a far-field approximation to the previous results.

8.4.3 Far-Field Approximation We take this opportunity to introduce the method of stationary phase as a means of approximating the integrals often occurring in spectral analysis. The pressure response can be written in the form  p(x, ˆ z) =

+∞

−∞

ρa ω2 −iξ x −ikz z w˜ e e dξ = −ikz



+∞ −∞

(ξ )e−i(ξ ) dξ ,  = ξ x + kz z

Consider the situation when (ξ ) is somewhat broad banded, then the integral is most affected by the exponential term. Because the exponential term is oscillatory, most of the contribution to the integral comes from where the phase has a minimum value because at large phase they tend to self-cancel. Approximate the phase in that vicinity as

358

8 Structure–Fluid Interactions

(ξ ) ≈ (ξo ) +

do d 2 o 1 (ξ − ξo ) + (ξ − ξo )2 + · · · dξ dξ 2 2

We use the assumption that the first derivative is zero to determine the wavenumber ξo , where the phase has a stationary value. The integral then becomes  p(x, ˆ z) ≈



+∞

−∞

(ξ )e

−i(o +o ξ 2 /2)

dξ =

2π (ξo )e−i[(ξo )+π/4] o

with the stipulation that  (ξo ) = 0. This approximation is the method of Stationary Phase. It is generally accurate in the large phase region, which means asymptotically large time or large distances. See Refs. [6, 7] for more details on its application. Specializing the method to our case, we have  ' (  = ξ x + kz z = ξ cos φ + ka2 − ξ 2 sin φ r ( ∂ ' −ξ = cos φ +  sin φ r ∂ξ ka2 − ξ 2 −ka2 r ∂ 2 = sin φ 3 ∂ξ 2 (ka2 − ξ 2 ) /2 where we have introduced cylindrical coordinates. Setting the first derivative to zero gives ξo = ka cos φ ,

o = k a r ,

o =

−r 2

ka sin φ

,

˜ o) o = w(ξ

ρa ω2 −ika sin φ

These are the values of wavenumber that tend to make the phase a minimum value and hence give the main contribution of the integral. Substituting these into the integral approximation then gives 2πρa ω2 −i[ka r+π/4] , e p(x, ˆ z) ≈ w(ξ ˜ o) √ − 2π ka r

ξo = ka cos φ

This interesting result shows that the far-field pressure depends on angle only insofar as w(ξ ˜ o ) depends on the angle. For example, suppose that the displacement shape is that of a rectangle, then we have the angle dependence of the pressure given by w(ξ ˜ o) =

sin(ka L cos φ) sin(ξo ) = ξo L ka L cos φ

Realizing that ka = ω/ca , we see that the angle contribution would change dramatically as the frequency is changed.

8.4 Radiation from Finite Plates

359

10kHz 5kHz 1kHz 500Hz 100Hz Fig. 8.22 Directivity patterns for impacted finite plate. Also shown are the plate shapes at each frequency

Figure 8.22 shows the directivity patterns at a number of frequencies; the nearfield behavior was computed from the full solution with r = 2L. It is clear that these patterns are very sensitive to direction when the frequency gets close to coincidence. Also shown are the deflected shapes of the plate at each frequency. These shapes indicate an almost sinusoidal plate deflection except at the center and edges—these are the points of significant radiation.

Further Research 8.1 Show that the far-field pressure radiated from a piston of radius a is pˆ = ρa ω2

wJ ˜ 1 (ka a sin θ ) e−ika r ka a sin θ r

—Ref. [7], p. 97 8.2 Show that the far-field pressure radiated from a point loaded infinite plate is p(r, ˆ θ ) = ρa ω2

w˜ cos θ 1 − ika h(ρ/ρa ) cos θ [1 − (ω/ωc )2 sin4 θ ]

—Ref. [7], p. 245 8.3 Consider the 1D problem of a spring mounted piston sliding in a tube of radius a and terminated by a rigid closure. Show that the relation between a force acting on the piston and the displacement is Pˆ = iωu[π ˆ a]2

'

i(ω2 M − K) ( −iρa ca + 2 π a tan ka L ω(π a 2 )2 —Ref. [5], p. 137

360

8 Structure–Fluid Interactions

8.4 Show that the asymptotic response of a line loaded plate in a fluid has the branch cut contribution wˆ B (x) ≈

Pˆ √

ρa ca2 2π

e−i[ka x−π/4] (ka x)

3/2

—Ref. [1], p. 229 8.5 Consider a plane wave incident on a rigid cylinder of radius R. Show that the scattered pressure is p(r, ˆ θ ) = pˆ i



m (i)m

Jm (ka R)Hm (ka r) cos mθ Hm (ka R)

—Ref. [7], p. 322 8.6 Consider the effect of shell elasticity on the scattering of plane waves. —Ref. [7], p. 370

References 1. Crighton, D.G.: The free and forced waves on a fluid-loaded elastic plate. J. Sound Vib. 63(2), 225–235 (1979) 2. Crighton, D.G.: The 1988 Rayleigh medal lecture: fluid loading—the interaction between sound and vibration. J. Sound Vib. 133(1), 1–27 (1989) 3. Crighton, D.G., Innes, D.: Low frequency acoustic radiation and vibration response of locally excited fluid-loaded structures. J. Sound Vib. 91(2), 293–314 (1983) 4. Crighton, D.G., Dowling, A.P., Ffowcs Williams, J.E., Heckl, M, Leppington, F.G.: Modern Methods in Analytical Acoustics: Lecture Notes. Springer, Berlin (1992) 5. Fahy, F.J.: Sound and Structural Vibration: Radiation, Transmission and Response. Academic Press, New York (1985) 6. Graff, K.F.: Wave Motion in Elastic Solids. Ohio State University Press, Columbus (1975) 7. Junger, M.C., Feit, D.: Sound, Structures, and Their Interaction. MIT Press, Cambridge (1986) 8. Maynard, J.D., Williams, E.G., Lee, Y.: Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH. J. Acoust. Soc. Am. 78(4), 1395–1413 (1985) 9. Norton, M.P.: Fundamentals of Noise and Vibration Analysis for Engineers. Cambridge University Press, Cambridge (1989)

Chapter 9

Discrete and Discretized Structures

Consider a long continuous rod (Fig. 9.1a) modeled using FE methods so that the stiffness and mass matrices are given by a representation as shown in Fig. 9.1b. This representation is said to be the discretized version of the continuous rod and leads to a set of N linear equations of motion (EoM). Typically, in structural dynamics we are concerned with the conditions under which the discretized model accurately reproduces the continuous results. Contrast this with the chain of atoms shown in Fig. 9.1c; the interatomic forces (and hence stiffness relations) are different but for small disturbances, the wave motion is linear. This is a naturally discrete system and a question of interest is: under what circumstances can it be replaced with a continuous system? We therefore see that both systems (the continuous and naturally discrete) have much in common. More explicitly, we have the four states: continuous systems (state 1) modeled in discrete form (state 2), and naturally discrete systems (state 3) modeled in continuous form (state 4). We refer to state 2 as a discretized state whereas state 4 is referred to as a homogenized state. The analyses to follow are divided along these lines but we begin with the discrete state because the previous chapters covered the continuous state.

9.1 Wave Propagation in 1D Discrete Systems Some systems are naturally modeled as discrete systems. A beaded string is an example: while the string has a distributed mass, it is considered negligible in comparison to the beads so that the model divides as: beads have mass, no stiffness; the string has stiffness, no mass. Atomic systems are a more complicated example because of the distributed mass of the electrons resulting in distributed stiffness. The Born-Oppenheimer approximation that “the nuclei move in accordance with an effective potential function, and the electrons move as if the nuclei are fixed” [4] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8_10

361

362

9 Discrete and Discretized Structures

Fig. 9.1 Four modelings for longitudinal wave propagation in a rod-type structure. (a) Continuous system. (b) Long chain of springs and masses approximating the continuous system. (c) Long chain of atoms with interatomic bonds that possibly could be approximated as a continuous system (d)

means the two mass systems are uncoupled and the molecular dynamics are associated only with the nuclei. We begin with the classical example of a massless pretensioned string with equispaced massive beads to establish our basic procedures. We then complicate the systems in a number of ways that include adding a different periodic mass and adding springs with different connectivities.

9.1.1 Beaded String Consider the beaded string in Fig. 9.2; the strain and kinetic energies are, respectively, U = · · · + 12 K[vn−1 − vn ]2 + 12 K[vn − vn+1 ]2 + · · · , T = · · · + 12 M v˙n2 + · · · , K = F¯o /a

where F¯o is the pretension in the string. In contrast to our previous systems, the energies are functions of discrete unknowns. Nonetheless, Hamilton’s principle of Eq. (4.46) is applicable. To see this, rewrite it as 

t2

[δ U − δ T ] dt = 0 ,

t1

δU =

 ∂U δuI , I ∂uI

δT =

 ∂U δ u˙ I I ∂u ˙I

Using integration by parts (with respect to time) on the kinetic energy term then gives 

t2

t1

 ∂ U d ∂T  δuI dt = 0 + I ∂vI dt ∂ v˙I

Realizing that the δuI can be varied independently, then leads to ∂U d ∂T + =0 ∂vI dt ∂ v˙I

I = 1, 2, · · ·

(9.1)

9.1 Wave Propagation in 1D Discrete Systems

363

Fig. 9.2 Pretensioned string with massive beads. (a) Geometry and free body of a single bead. (b) spectrum relations

This is a form of Lagrange’s equation [8]. To apply Lagrange’s equation to our case, focus on the mass at position xn , then the differentiations give the equation of motion K[−vn−1 + 2vn − vn+1 ] + M v¨n = 0 We recognize the bracketed term as the second space derivative of the displacement. This is an ordinary differential equation (ODE) in time, the space dependence is implicit in the subscripts because the DoFs vn−1 , vn , vn+1 are associated with the space positions xn−1 , xn , xn+1 . We enquire if it has wave solutions of the form ˆ −i[kxn −ωt] v(xn , t) = Ae where k is an as yet unspecified wavenumber. The node positions are not continuous but at discrete points xn = na associated with the masses. We therefore have, for example, e−ikxn−1 = e−ikna e+ika . Substitute into the differential equation, cancel common terms, and regroup to get ,

ˆ =0 K[−e+ika + 2 − e−ika ] − ω2 M A

(9.2)

The condition for the assumed form to be a wave solution is that the braced term be zero. Replace the complex exponentials in terms of sines and cosines to get 2K[1 − cos(ka)] − ω2 M = 0

(9.3)

Knowing that 1 − cos(ka) = 2 sin2 (ka/2), then the condition becomes ω = ωc sin(ka/2)

or

k = (2/a) sin−1 (ω/ωc ) ,

ωc =

 4K/M

364

9 Discrete and Discretized Structures

This is the spectrum relation for the beaded string and is shown plotted in Fig. 9.2b. There are a number of interesting features about this spectrum relation. First note that the relation is not linear in ω, and therefore the system is dispersive; but for low frequencies (ω  ωc ) we get ω ≈ ωc ka/2

ω k= a

or



 M M =ω K F¯o a

(9.4)

Associating M → ρAa this is then the spectrum relation for a continuous pretensioned cable as discussed in Sect. 3.1. Thus the discretized system differs from the continuous system only at high frequencies. Also worth noting is the multiplicity of wavenumbers for a given frequency indicating multiple modes. However, the higher modes are different by km = k 1 + m

2π , a

λm =

λ1 2π 2π = = km k1 + m2π /a 1 + mλ1 /a

Because λ1 > a, we conclude that all the higher mode wavelengths are smaller than the spacing between the masses. While we can imagine the string deforming into such shapes, the fact that the masses are unaffected means that the dynamics are unaffected. Reference [14] has a very nice pictorial representation of this. The wavenumber at the critical frequency is kc = π/a, this corresponds to a sinusoid that as having its zero crossings at the mid-length of each spring in Fig. 9.1b. Each mass is √ then in between two springs of stiffness 2K giving a resonance frequency of ω = 2 × 2K/M = ωc . The most interesting feature of our result for wave propagating problems is that for frequencies above ωc , the wavenumbers are complex implying spatial dissipation. That is, there is exponential decay of the response as the wave propagates along the string. The phase and group speeds are, respectively,

c=

ω ω = , k [2/a] sin−1 [ω/ωc ]

cg =

 ∂ω = (ωc a/2) cos(ka/2) = (ωc a/2) 1 − (ω/ωc ) ∂k

These are shown plotted in Fig. 9.3 as the inset (c) and indicates that the group speed decreases to zero as ωc is approached. The implication is that frequencies higher than ωc are filtered; that is, there is dissipation. In a finite element context, this means that the discretization process puts an upper limit on the frequency content that can be propagated. This is the converse of what was observed for the constrained rod in Sect. 2.2 and beam on an elastic foundation in Sect. 3.2 where the lower frequencies were filtered. The region greater than ωc is called a stop band. Figure 9.3a shows the velocity responses due to a 15 µs sine-squared pulse; there is a good deal of dispersion even though the propagation distance is not large. Figure 9.3b shows the amplitude spectrum and it is clear how frequency

9.1 Wave Propagation in 1D Discrete Systems

365

1.0

0.5

0.0 0.0

0.0

0.2

0.4

0.5

0.6

1.0

0.

50.

100.

150.

Fig. 9.3 Pretensioned string with massive beads. (a) Responses for input velocity. (b) Transform of responses. (c) Dispersion relations

components above the cut-off are filtered. The parameters used are K = 1.8 GN/m, M = .044 kg, and a = .025 m, which gives a cut-off frequency of about 63 kHz. The responses show odd behavior for nonzero x in that there are responses before the pulse is initiated. Usually, this can be explained in terms of a wrap-around problem but that is not a complete explanation here because the magnitude gets larger as the pulse origin is approached from t = 0. Reference [2] refers to points such as ωc as resonant frequencies. By way of contrast, we can also apply these results to an understanding of crystals. With an atomic mass on the order of m 3.3 × 10−27 kg where m is the atomic number (number of protons in an atom) and bond stiffness √ on the order of K ≈ 500 N/m, this gives a cut-off frequency in the range of 4/ m × 1014 Hz. This is a very high frequency, it is on the order of the frequency of visible light. Later examples delve more deeply into the atomistic relevance of these results.

9.1.2 Two-Mass String A diatomic crystal such as NaCl (sodium chloride, common salt) has alternating masses in a regular pattern like that shown simplistically in Fig. 9.4a but extended in three dimensions. Figure 9.4b shows the FE generated vibration frequencies for 401 such masses (the vertical axis is simply an enumeration of the modes). What is striking is that there is a definite gap in the frequencies between 64 kHz and 90 kHz which we need to explain. The FE model is chosen to represent a mechanical system and not the diatomic system; it consists of a 1D string of alternating masses M and 12 M with M = ρAa connected with springs of stiffness K = 2Fo /a = 2EA/a. The spacing a is between similar masses and set as a = 25.4 mm (1.0 in). The properties of Fo ,

366

9 Discrete and Discretized Structures 400 300 200 100 0 0.

25.

50.

75.

100.

125.

Fig. 9.4 Diatomic crystal. (a) 1D lattice representation. (b) Spectrum relation showing a gap around 75 kHz

0. 200. 400. 600. 800.

0. 200. 400. 600. 800.

0. 200. 400. 600. 800.

Fig. 9.5 Velocity responses due to narrow-banded pulses. Dashed lines correspond to model speeds at the central frequency

A, and ρ are set as those of aluminum. For this analysis the first and last DoFs were fixed (so as to have periodic BCs) giving 399 DoFs. An effective way to interrogate the frequency behavior of a structure is through narrow-banded wave packets. To that end, three packets were constructed and are shown as the dashed lines in Fig. 9.4b. They are centered before the gap, in the gap, and after the gap. For this wave propagation analysis the last mass was fixed and the load applied to the first mass. Velocity responses for the three pulses are shown in Fig. 9.5. The first point to note is that the wave does not propagate in the frequency gap; that is, the gap coincides with significant dissipation. The waves propagate with well-defined fronts outside the gap but the higher frequency pulse shows an extending duration which indicates that some of the frequency components are travelling very slow. This is in spite of the fact that the pulse itself is relatively narrow-banded. To analyze this problem, we follow the basic procedure of the previous example. Let the masses of size M1 have displacements ξn−1 , ξn , ξn+1 , · · · , and the masses

9.1 Wave Propagation in 1D Discrete Systems

367

of size M2 have displacements ηn−1 , ηn , ηn+1 , · · · as indicated in Fig. 9.4a. We consider the masses to displace transversely as beads on a stretched string with stiffness K = 2F¯o /a. Consider the two typical masses indicated in Fig. 9.4; the strain and kinetic energies are U = 12 K[ηn−1 − ξn ]2 + 12 K[ξn − ηn ]2 + 12 K[ηn − ξn+1 ]2 T = 12 M1 ξ˙n2 + 12 M1 η˙ n2 There are two equations of motion given by the Lagrange’s equations d ∂T ∂U + = 0, ˙ dt ∂ ξn ∂ξn

d ∂T ∂U + =0 dt ∂ η˙ n ∂ηn

leading to M1 ξ¨n + K[−ηn−1 + 2ξn − ηn ] = 0 ,

M2 η¨n + K[−ξn + 2ηn − ξn+1 ] = 0

These are coupled equations. Assume wave responses of the forms ξ(xn , t) = ξo e−i[kna−ωt] ,

η(xn , t) = ηo e−i[kna−ωt]

On substitution, these lead to the eigenvalue problem 

−ω2 M1 + 2K −K[e+ika − 1] −K[e−ika − 1] −ω2 M2 + 2K

  ξo =0 ηo

(9.5)

The characteristic equation is M1 M2 ω4 − 2K[M1 + M2 ]ω2 + 4K 2 sin2 (ka/2) = 0 Solving for the frequency gives    1  1 1 1 2 4 sin2 (ka/2) ±K ω2 = K + + − M1 M2 M1 M2 M1 M2 The spectrum relation indicates that the waves behave dispersively. Because it is a coupled system, there are two separate modes. The real-only behavior is shown in Fig. 9.6b; this figure is to be compared to Fig. 9.2b except only the purely real wavenumbers are shown. The plots repeat themselves with respect to ka similar to Fig. 9.2b; the region shown is referred to as the first Brillouin zone. Consider the case when ka is small (i.e., close to the horizontal arrow in Fig. 9.6a), then

368

9 Discrete and Discretized Structures

3.

1.0

2. .

.

0.5

1. 0.

0.0 0.

25.

50.

75.

100.

125.

0.

25.

50.

75.

100.

125.

Fig. 9.6 One-dimensional two-mass system. (a) Spectrum relation. (b) Dispersion relation

 1  1 1  1  2(ka/2)2  ±K − + + M1 M2 M1 M2 M 1 + M2   2 2K(ka) 2K(ka/2)2 1 1 = 0+ − , 2K + M 1 + M2 M1 M2 M 1 + M2

ω2 ≈ K

The frequencies for the two modes are 



K/2 ω ≈ ka , M 1 + M2

ω≈

2K

 1 1  − α(ka/2)2 + M1 M2

For the first mode k ∝ ω and therefore this wave is nondispersive with a speed related to the average mass in the unit cell. The second mode is dispersive with a zero group speed. Now consider the case when ka/2 = π/2 at the border of the first Brillouin zone, then  2K ω = , M1 2

2K M2

or

ωa =

 2K , M1

ωo =

2K M2

and both modes are dispersive with zero group speed. The mode of the lower frequency is called the acoustic mode (ωa ) because it has the speed for which sound travels in the crystal; the mode of the higher frequency is called the optical mode (ωo ) because it is the one that makes the crystal optically active (interacts with electromagnetic waves). To elaborate on this, substitute for ωa2 and ωo2 at small ka into Eq. (9.5) to get the amplitude ratios acoustic:

η = +1 , ξ

optic:

η M1 =− ξ M2

For the acoustical mode, the neighboring atoms vibrate in unison; for the optical mode, neighboring atoms vibrate against each other thus creating an alternating dipole moment. If the two atoms have opposite charges (as is the case for NaCl) this vibration can be excited by the electric field of an incident light wave. To

9.1 Wave Propagation in 1D Discrete Systems

0.

50.

100.

150.

200.

369

0.

50.

100.

150.

200.

Fig. 9.7 Mode shapes obtained from the FE vibration eigenanalysis (a) Low frequencies. (b) High frequencies

further elaborate on the meaning of these amplitude ratios, we return to the vibration eigenanalysis discussed earlier. Figure 9.7 shows the mode shapes for the very low and very high frequencies. The plots are for the positions of each mass (treated as a transverse deflection for clearer viewing), the masses are not connected by straight lines that would represent the connecting string or spring. It is clear for the lower modes that neighboring masses move together. For the higher modes, the two sets of masses plot oppositely with the smaller mass having the larger displacement that is close to −M1 /M2 = −2. The actual mode shape has a typical high-frequency look in that the connecting string has many zero crossings, but the mass positions themselves show a much simpler shape. With the length of the string being L = 200 a, the wavenumbers in the plots (going from bottom to top) are k=

2π 2π 4π 10π 20π 2π = , , , , , λ 2L L L L L

A final point is that the frequency gap is given by 

 2K 0. Table 9.2 shows the parameters for some fcc metals, the original Finnis–Sinclair paper [11] considered bcc metals, and Ref. [12] considers extension to the hcp metals. It is not required that the exponents be integers but they usually are and determined such that n > m. The force on atom α is Piα = −

' 1 ∂V    ' ro (n+2 1 1 (' ro (m+2  β  n [xi − xiα ] = − − Cm + √ 2 ∂xiα r βα ro2 ρα ρ β r βα β=α

This has some commonality with the L-J force expression but the presence of the density makes a big difference. That is, when there are other atoms present, their interactions with even some other atoms again also affect the diatomic interaction. Consequently, this is something we cannot observe just by looking at the diatomic behavior. Further differentiations give the elasticity tensor as [5] Dklmn = 12 

βα βα βα βα    ' 1 ' 1 ( 1 ( r r rm rn F  − F  − C √ α f  − f  k l 2 r r r ρ α β=α

βα βα βα βα 1 ' 1 (3/2    rk rl    rm rn  (9.13) + C α f f 2 ρ r r β=α

β=α

where F αβ (r βα ) = 

 ro  n , r βα

f αβ (r βα ) =

 ro m r βα

9.3 Atomic Lattice Structures

393

and the prime means differentiation with respect to the argument. This stiffness has some similarities to Eq. (9.12), but again the presence of the densities ρ α makes the dependence on the atomic configuration quite complicated. Let us consider the bulk properties of silver. We first use a series of free standing blocks and then contrast the result with the use of pBCs. Silver in its more stable form is fcc. Figure 9.20a shows a side view of a [2 × 2 × 2] fcc lattice, a 3D view of a single cell is shown in Fig. 9.15c. The empty circles represent the face-centered atoms. This is the configuration extended for the free boundary cases. Blocks of silver were constructed in the fcc configuration and the single lattice parameter ao adjusted to give a minimum energy configuration. The energy and elasticities are shown in Table 9.3. The physical components of the elasticities are obtained from the energy version as Dij =

Dij∗ V

× 1.602 × 10−19

where V = ao3 and this gives Dij in units of [GPa]. There is, however, an ambiguity in the appropriate volume that we discuss later.

Fig. 9.20 Block representations for the bulk behavior of fcc lattices. (a) [2 × 2 × 2] with free boundaries; empty circles are the face-centered atoms. (b) [1 × 1 × 1] with periodic boundaries Table 9.3 Convergence properties for silver blocks, the bottom half are extrapolated estimates. The spacings ao and ao are in units of [nm], the other entities are in units of [eV]. The number of atoms for each block is given in parentheses in the first column Size 1 × 1 × 1 (14) 2 × 2 × 2 (63) 4 × 4 × 4 (365) 8 × 8 × 8 (2457) 16 × 16 × 16 32 × 32 × 32 64 × 64 × 64 128 × 128 × 128

V /N 3

−2.0249 −2.3812 −2.6317 −2.7834

ao .3938 .4009 .4049 .4069 .4079 .4084 .4086 .4087

ao .0071 .0040 .0020 .0010 .0005 .0002 .0001

∗ D11 144.7 95.6 76.2 68.0 63.9 61.9 60.8 60.3

∗ D12 97.7 64.7 51.8 46.2

∗ D44 63.9 40.5 32.0 28.5

∗ D11

49.1 19.4 8.4 4.1 2.1 1.0 0.5

394

9 Discrete and Discretized Structures

∗ show that the difference between successive numbers is The ao and D11 approximately halved. We can use this to estimate what the numbers would be if the convergence study were continued. It is clear that a good deal of computational effort must be expended to get small changes in the numbers. The estimated modulus is

D11 =

60.3 × 1.602 × 10−19 = 142.7 GPa .40873 × 10−27

This is close to the value of 141 GPa reported in Ref. [5] but different from the experimental value of 131 GPa also reported. Both are significantly different from the nominal bulk value of D11 = E(1 − ν)/(1 + ν)(1 − 2ν) = 98 GPa. Optimistically, the extrapolated value represents the bulk volume and therefore there is no ambiguity in volume. However, for any finite sized block estimating the size as V = N 3 ao3 seems like an underestimate. For example, treating Fig. 9.20a as a 2D lattice, we see that the center atom occupies an area b2 where b is the distance to the nearest neighbor. If this idea is applied to the fcc block, then the volume is estimated as V = [b/2 + Nao + b/2]3 = [1 + b/Nao ]3 N 3 a 3 = αN 3 a 3 Suppose b/aa = 0.5, then the premultiplier in square brackets becomes N = 1, 2, 4, 8

α = 3.375, 1.95, 1.42, 1.19

Without claiming that this is a good or accurate estimate of the “true” volume, what it does show is the level of ambiguity present. It is only for N > 64 does the premultiplier become unity. But, already for N = 8 there are nearly 2500 atoms and we conclude there is a need for another method to simulate a larger bulk volume. This is achieved using periodic boundary conditions (pBC) where a basic block, [8 × 8 × 8] say, is replicated in the three directions simply by imposing special BCs on the basic block. This is the same use of pBCs first introduced in Sect. 6.4. We illustrate its use. When preparing lattices for use in conjunction with pBCs, two changes are necessary relative to the free lattices. First, the basic lattice must be constructed so that it has the proper extension; we use the face-centered cubic arrangement to illustrate this point. After the energy of the basic lattice is minimized under free BCs and placed in the periodic box, a significant compressive stress can be generated which must be removed. For the pBC case, the basic lattice is terminated at the face-centered atoms as shown in Fig. 9.20b. The size of the periodic box is indicated by the dotted lines. We use the box as the measure of the volume. For illustrative purpose, we use the N = 4 block which has 500 atoms, slightly more than the free case because of the extra face-centered atoms. When the energy of this block is minimized under free BCs it has a lattice constant of ao = 0.40536 nm giving a box size of L = 2.0268 nm.

9.3 Atomic Lattice Structures

395

200.

1. 0.

150.

-1. .

.

100.

-2. 50.

-3. -4.

2.02

2.07

0.

2.02

2.07

Fig. 9.21 Lattice analysis using periodic boundary conditions (pBCs). (a) How the stress is affected by the box size. (b) How the moduli are affected by the box size. The arrows indicate the zero stress results

The presence of the image boxes give rise to a very large compressive hydrostatic stress on the order of 3.4 GPa. This could be removed using an MD simulation in conjunction with the Berendsen barostat [3]; because we are dealing with zero temperature, it is simpler to manually change the periodic box. Figure 9.21a shows the change in stress as the box size is increased. Clearly, it is easy to identify the box size for zero stress. The change in size seems very small (about 0.03 nm) but this corresponds to a strain of about 1.5%, which is quite large for a metal lattice. Figure 9.21b shows the effect that the box size has on the moduli. The zero stress value is ⎡ ⎤ 136.42 93.08 93.08 0.00 0.00 0.00

⎢ 93.08 136.42 93.08 0.00 0.00 0.00 ⎥ ⎢ ⎥ ⎢ 93.08 93.08 136.42 0.00 0.00 0.00 ⎥ [ D ]=⎢ ⎥ GPa ⎢ 0.00 0.00 0.00 57.24 0.00 0.00 ⎥ ⎣ 0.00 0.00 0.00 0.00 57.24 0.00 ⎦ 0.00

0.00

0.00 0.00 0.00 57.24

We see that the material is isotropic and that D12 /D44 = 1.63 which is near the desired value of 1.9.

9.3.4 Continuum Models for Discrete Systems We finish this section with a discussion of how a naturally discrete system can be replaced by a continuous system. The motivation for doing this is that in some cases a computational efficiency arises and in others a conceptual simplicity is the result. This is sometimes referred to as “continualization” but more widely as straingradient elasticity; see Refs. [1, 10, 15] and the citations therein for a sampling of

396

9 Discrete and Discretized Structures

the current literature. We use the example of a nanowire to illustrate some aspects of this interesting area. Let us start with the simplest of our discrete models: concentrated masses with a nearest neighbor interaction. Before the square root is taken, we have the spectrum relation from Eq. (9.3) that cos(ka) − 1 + Mω2 /2K = 0 This is the true relation and we want to approximate it over a certain frequency range. Do a Taylor series expansion of the cosine term to get the polynomial approximation 1 − 12 (ka)2 +

4 1 24 (ka)

+ · · · − 1 + Mω2 /2K = 0

In the continuum modeling, k appears through a space derivative while ω appears through a time derivative. Substituting the continuum operators (and canceling the unity terms), we get + 12

∂ 2u 2 a + ∂x 2

1 24

∂ 4u 4 a + · · · − M u/2K ¨ =0 ∂x 4

Substituting for K = EA/a and M = ρAa, we get a recognizable continuum model EA

∂ 2u + ∂x 2

4 2∂ u 1 12 EAa ∂x 4

+ · · · = ρAu¨

The leading terms on both sides of the equation are simply the elementary rod model. The additional term is interesting because it contains a, the length between the original discrete masses. Traditionally, continuum models do not have a length scale associated with the atomic structure and therefore to recreate the dynamic behavior reasonably well, we must include such a scale. Keep in mind, however, that the true system is the discrete system (for which there is a known scale length) and we adjust the parameters accordingly. Figure 9.22 shows a comparison of the exact (discrete) and approximate (continuum) spectrum relations. Clearly, there is a range where both are very close, by the same token, there is a range where both models are vastly different, the relevance of this is to be assessed in a particular context. The EoM for the continuum rod arises from the equation ∂σxx /∂x = ρ u; ¨ this leads to the interpretation of the stress as σxx = E

∂ 3u ∂ 2 xx ∂u + E  3 = Exx + E  ∂x ∂x ∂x 2

9.3 Atomic Lattice Structures

397

Fig. 9.22 Comparison of discrete (full line) and continuum (dashed line) spectrum relations

0.

20.

40.

60.

where E  = Ea 2 /12. This is a constitutive relation and the presence of the space derivatives of the strain is why it is called strain-gradient elasticity. The problem we face is that the origin of the dispersion is different in the two models. Look at the model in Fig. 9.11, the discrete masses cause a stop band, springs K2 cause higher space derivative dependence, the resulting spectrum relation of Fig. 9.12 is not the same as in Fig. 9.22. To elaborate, consider the Love rod model given by Eq. (4.60), an interpretation of the constitutive relation is σxx = E

∂u ∂ u¨ + ν 2 ρ(I /A) = Exx + ν 2 ρ(I /A)¨xx ∂x ∂x

This, too, causes a concave up dispersion behavior as seen in Fig. 4.19 but the physical origin of it is the transverse kinetic energy. Reference [15] draws an interesting conclusion from their studies of many material systems: “. . . our results appear to indicate that strain-gradient elasticity is irrelevant for most crystalline metals and ceramics.” We interpret that as saying the wave propagation behavior of metals is sufficiently described by just the elasticity matrix [ D ]. We examine the wave behavior of a nanowire in this light. Figure 9.23a shows a simple aluminum nanowire. It is made of 32 planes each containing three atoms forming an equilateral triangle. Each subsequent triangle is rotated 60◦ relative to the previous one. The nominal lattice parameters are ao = 0.2671 nm for the triangles, al = 0.2084 nm between planes so that the total length is L = 6.46 nm. Our first test is a quasi-static axial loading. The nanowire is very sensitive to temperature because all atoms are actually surface atoms and want to collapse into a cluster therefore the temperature was set to a low 2◦ K. For the first plane of atoms, one atom was fixed and the other two constrained to move in the plane. The load was applied to the second last plane of atoms, one of these atoms was constrained to move only axially. This conformation was equilibrate at 2◦ K changing the average lattice parameters slightly. The integration was performed with a time step of 5 fs. Figure 9.23b shows the load-deflection curve; it is nearly linear and the stiffness readily estimated as

398

9 Discrete and Discretized Structures

0.6

0.4

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.5

Fig. 9.23 Nanowire. (a) Sample nanowire; isometric and end views. (b) Force deflection curve

K=

P 0.6 × 10−9 = 39.88 N/m = L/N 0.45114 × 10−9 /30

where N is the total number of loaded segments. Our estimated stiffness is objective in the sense that the MD simulation provided the forces and the displacements, no continuum concepts such as modulus were involved. We shortly discuss these continuum concepts but for now Fig. 9.24a shows the actual recorded displacement histories of the three loaded atoms, the inset plot shows the history of the applied load. The applied load is a smoothed ramp designed to minimize dynamic (inertia) effects. The slight oscillation at the end shows some inertia effects were initiated, but the resultant final deflection can be accurately estimated. The alternative is to use a slower ramp with the additional computational cost; that does not seem warranted in this case. Parenthetically, not much difference is observed when the initial temperature of 2◦ K is uncontrolled during the experiment, it has a maximum excursion up to 3◦ K. From a simplified model point of view, let us initially think of the nanowire as a lumped mass system similar to those in Fig. 9.1. The mass is objective and given by M = 3 × 27au = 3 × 27 × 1.67265 × 10−27 = 135.49 × 10−27 kg where 27 is the atomic mass number for aluminum. With this mass and our previous estimated stiffness we can now estimate two important parameters for the nanowire: the cut-off frequency and the low-frequency speed. These are, respectively, fc = ωc /(2π ) =



4K/M/(2π ) = 5.46 × 1012 Hz ,

cg = ωc al /2 = 3577 m/s

The bulk speed for aluminum is on the order of 5000 m/s which is a sizable difference. Figure 9.24b shows the velocity responses due to a sine-squared pulse

9.3 Atomic Lattice Structures

399

0.5 0.4 0.3 0.2 0.1 0.0 0.

5.

10.

15.

20.

0.

2.

4.

.

6.

8.

Fig. 9.24 Time histories for a nanowire. (a) Displacement responses for quasi-static loading. The inset is the applied load history. (b) Velocity responses due to a sine-squared pulse of duration 2 ps

of duration 2 ps with an applied load level the same as in the quasi-static test. The frequency content of the loading extends to about 1.0 × 1012 Hz which puts it in the low-frequency range. The propagation definitely seems to follow our simplified model. Something interesting occurs at the end of the shown traces: the initial wave is tensile, it reflects from the fixed end as a compressive wave, this reaches the free end, and the negative velocity is doubled. This results is a crushing mode of failure of the nanowire. Of additional interest, if the initial load level is doubled the crushing actually occurs near the fixed end. Clearly, this is fascinating wave behavior but it is beyond the scope of this book to investigate. Instead, we opt to focus on the connection between the nanowire discrete model and some possible continuum level models of it. It is obvious that for small objects made of discrete particles, that concepts such as volume and area are highly ambiguous and unlike other cases such as lattices there is not the opportunity to use a periodic box to help define effective size. If we consider the centers of the atoms as forming a cylindrical shell, then the diameter, area, and volume are given by, respectively, D = 0.2843 nm ,

A = π D 2 /4 = 0.0636 nm2 ,

V = AL = 0.4107 nm3

This gives a mass density of ρ = 96 × 27 au/V = 10556 kg/m3 which is about four times larger than the nominal bulk value of 2800 kg/m3 . The effective diameter needs to be increased by a factor of two in order to have the same density. When an atom is part of a lattice structure its effective volume is estimated based on the nearest neighbor atoms. When this approach is used for the nanowire we get a diameter of D = 0.426 nm. This is shown as the dotted circle in

400

9 Discrete and Discretized Structures

the end view of Fig. 9.23a. The new estimate of volume is obviously larger but still not sufficient to give the low bulk value. On the other hand, if we use the smallest rectangular box surrounding the atoms, we get A = 0.0824 nm2 ,

V = AL = 0.5326 nm3 ,

ρ = 8141 kg/m3

This density is also too large. Although Eq. (9.13) is not quite appropriate for a nanowire (because it does not satisfy the ALMS condition), it is nonetheless interesting to use it so. The computed moduli are ⎤ ⎡ 156 116 156 203 0 0 0 0

⎢ 116 ⎢ ⎢ 203 [ D ]=⎢ ⎢ 0 ⎣ 0

203 203 404 0 0 0

0 0 0 20 0 0

0 0 0 0 40 0

0 0 ⎥ ⎥ 0 ⎥ ⎥ GPa 0 ⎥ 0 ⎦ 40

where the volume used is simply that of the surrounding box. The material symmetries are essentially hexagonal or transversely isotropic. Assuming true uniaxial stress, we have for the Young’s modulus and Poisson’s ratio E = D33 −

2 2D13 = 101 GPa , D11 + D12

ν=

D13 = 0.75 D11 + D12

This is not too far off the nominal value of 70 GN. We can use the modulus to estimate the discrete spring stiffness according to K=

101 × 109 × 0.533 × 10−18 EA = = 39.93 N/m al 0.208 × 10−9

We recover our earlier estimated value. Finally, consider the speed given by cg =



E/ρ = 3521 m/s

which is essentially the same as previously obtained. We see that the low speed value is associated with the greater density.

9.4 Spectral Analyses of FE Discretized Systems Section 9.1 began the discussion of systems that are inherently discrete. As part of the numerical modeling of structures, particularly when using FE methods, the continuous system is typically replaced with a discrete system. Thus, the new system

9.4 Spectral Analyses of FE Discretized Systems

401

is an approximate version of the continuous system and we use a spectral analysis to elucidate the differences. The two aspects we focus on are: establishing the spectrum relations, and modeling periodic systems. We illustrate schemes for obtaining spectral relations directly from the single-element information or from the structure as a whole with periodic BCs.

9.4.1 Spectrum Relations from Element Stiffness The spectrum relations are associated with the governing differential equations and not with any particular boundary value problem. The linear elastic element stiffness relation is the governing equation but in space-discretized form. Therefore, it should be possible to derive the complete spectrum relation from single-element information. We show how this can be done. Consider a uniform waveguide modeled with two ended elements. Example elements are frame, Hex8, and MRT/DKT shell. To fix ideas, consider a C-channel as shown in Fig. 9.25 modeled with the triangular MRT/DKT shell element. The assembled stiffness of the indicated segment can be partitioned as   Kˆ Kˆ [ Kˆ ] = ˆ 11 ˆ 12 , K21 K22

[ Kˆ ]{ u } = [K − ω2 M]{ uˆ }

where the subscripts 1 and 2 refer to the first and second face, respectively. The assemblage process of each segment is done according to standard procedures and the dynamic equilibrium equation involving just the indicated nodes is

.

.

Fig. 9.25 Modeling a uniform waveguide. (a) FE mesh showing the extracted segment participating in the modeling. (b) Schematic of the stiffness assemblage process. (c) Alternative assemblage process

402

9 Discrete and Discretized Structures

[Kˆ 21 ]{ uˆ }n−1 + [Kˆ 22 + Kˆ 11 ]{ uˆ }n + [Kˆ 12 ]{ uˆ }n+1 = 0 Because the waveguide is uniform, as done earlier, we can relate the DoF by { uˆ }n−1 = e+ika { uˆ }n ,

{ uˆ }n+1 = e−ika { uˆ }n

where a is the length of the segment (i.e., element length). The EVP for the segment becomes ( ' +ika [Kˆ 21 ] + [Kˆ 22 + Kˆ 11 ] + e−ika [Kˆ 12 ] { uˆ }n = 0 e This general result can be used to recover the 1DoF beaded string result of Eq. (9.2) and 2DoF beam result of Eq. (9.15). It is a straightforward demonstration to show how the spectrum relations for the C-channel are obtained from the EVP of the extracted segment shown in Fig. 9.25a. Some of the interesting side questions revolve around the choice of what waveguide DoF to use—remember that the number of modes obtained is proportional to the number of nodes (and their DoF) representing the cross section. We opt not to explore this topic here and refer back to Fig. 5.15b and its discussions which addresses the same questions; additionally, a later example problem also considers the same questions. Instead, we first consider the curved beam (modeled as a collection of straight frame elements) because it has an interesting aspect not anticipated from dealing with the governing equations such Eq. (3.34). Figure 9.26a shows a curved beam (arch) modeled as a collection of straight frame segments. The stiffness and mass of each segment is its stiffness and mass in local coordinates shown in Fig. 9.26b rotated to the global coordinates shown in

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.

2.

4.

6.

8.

Fig. 9.26 FE modeling of a curved beam. (a) Curved beam modeled as a collection of straight segments. (b) Alternative schematic of the assemblage process. (c) Spectrum relation. Circles are FE data, continuous lines are model results

9.4 Spectral Analyses of FE Discretized Systems

403

Fig. 9.26a. The stiffness and mass in local coordinates of a segment are assembled as the sum of that of the straight rod and beam to give from Eqs. (5.19) ⎡

1 ⎢ 0 EA ⎢ ⎢ 0 [ k¯ ] = ⎢ a ⎢ −1 ⎣ 0 0

0 0 0 0 0 0

⎤

⎡

0 −1 0 0 0 0 0 ⎢ 0 12 6L 0 0 0 0⎥ ⎥ EI ⎢ ⎢ 0 6L 4L2 0 0 0 0⎥ ⎥+ 3 ⎢ 0 1 0 0⎥ 0 a ⎢0 0 ⎣ 0 −12 −6L 0 0 0 0⎦ 0 000 0 6L 2L2

⎤

0 0 0 0 −12 6L ⎥ ⎥ 0 −6L 2L2 ⎥ ⎥ 0 0 0⎥ 0 12 −6L ⎦ 0 −6L 4L2

(9.14)

¯ T . For simplicity, we use a lumped where the DoFs at each node are { u¯ } = {u, ¯ v, ¯ φ} mass model where  M  = (ρAa/2)1, 1, αa 2 /39; 1, 1, αa 2 /39T . In the FE modeling of the curved beam, each straight frame element is rotated; we achieve the same effect for the single element by rotating the second group of DoF relative to the first. Construct a [6 × 6] transformation [ T ] matrix as { u } = [ T ]{ u¯ } ,

   I [ 0 ] [ T ]= , [ 0 ] [ R ]

⎡

⎤ cos θ − sin θ 0 [ R ] = ⎣ sin θ cos θ 0⎦ 0 0 0

where  I  is the unit matrix and [ 0 ] is zeros. The strain energy of the straight element with rotated end DoFs is U = 12 { u }T [ k ]{ u } = 12 { u¯ }T [ T ]T [ k¯ ] [ T ]{ u¯ } The stiffness of the approximately rotated element is the inner triple product. In general, the transformation also gets applied to the mass matrix, in the present case, however, it is invariant. Figure 9.26c shows the results for a curved beam with the parameters associated with Fig. 3.18. Without doubt, the comparisons between the two figures are comparable. The significant difference, however, is that the imaginary parts of the spectrum relations are also generated from the FE modeling. This is an important point because it is the strength of the dissipation that sometimes is more important than the energy that is propagated. An alternative way to get the stiffness is to form the two element structure shown in Fig. 9.26b with free BCs. This is of size [9 × 9] and the relevant substiffness is the [3 × 9] central portion. As another example, consider an FE model of a flat plate in flexure. Figure 9.27a shows a portion of the mesh (using the DKT element) surrounding a typical node, in this case it is labelled node 5. As we have shown earlier, the EoM of the mass at node 5 depends on the stiffness of the other connecting nodes. A simple way to get this stiffness is to form the [4 × 4] module shown in Fig. 9.27a with free BCs. The DoFs at each node are {w, φx , φy }T so that the total stiffness is of size [27 × 27].

404

9 Discrete and Discretized Structures

0.2 0.1 0.0 -0.1

0.

1.

2.

3.

4.

5.

Fig. 9.27 FE generated spectrum relations for a plate. (a) Mesh used to generate stiffness and mass matrices. (b) Results. Circles are FE data, lines are for model

As illustrated in Fig. 9.25b we only need the central band of stiffness to establish the EoMs. The central band is of size [3 × 27] ranging from row 13 to row 15. Parse it into [3 × 3] submatrices associated sequentially with the nine nodes and label them [ Kn ]. The wave solution is of the form ˆ −iξ mb e−i[kna−ωt] ˆ −iξym e−i[kxn −ωt] = Ae w(x, y, z) = Ae where a and b are the dimensions of a single element. For the infinite plate, ξ is a continuous specified wavenumber. The central band can then be written as [ Kˆ ] =

[ K1 ]e+iξ b e+ika + [ K2 ]e+iξ b e+ik0 + [ K3 ]e+iξ b e−ika +[ K4 ]e+iξ 0 e+ika + [ K5 ]e+iξ 0 e+ik0 + [ K6 ]e+iξ 0 e−ika +[ K7 ]e−iξ b e+ika + [ K8 ]e−iξ b e+ik0 + [ K9 ]e−iξ b e−ika − ω2 [ M5 ]

where the subscripts on [ K ] and [ M ] refer to the node and for convenience, the lumped mass model is used. The EVP consists in setting the determinant of [ Kˆ ] to zero; note that it is only of size [3 × 3]. Results are shown in Fig. 9.27b. The comparisons with the model from Sect. 6.1 are quite good except at the very low frequencies.

9.4.2 Real-Only Spectrum Relations and Spectral Shapes Structural vibrations and wave propagation in structures have much in common because, as pointed out before, they have the same governing equations. Vibrations

9.4 Spectral Analyses of FE Discretized Systems

405

15 12 9 6 3 0 0.

2.

4.

6.

8.

10.

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 9.28 Vibration of a fixed-fixed rod. Circles are FE results, lines are model results. (a) Estimated frequencies for different discretizations. The straight line is the continuum result. (b) Symmetric mode shapes for the N = 8 elements. (c) Antisymmetric mode shapes for the N = 8 elements

occur in finite structures, waves propagate in infinite structures. In order to connect the two, we conceive of the vibrational system as being extended to infinity by use of periodic BCs. We first consider a simple rod, and then the more complicated case of a beam; for simplicity of concept, we assume no damping. Consider a rod of finite length L = 2.54 m (100. in), fixed at both ends, and modeled with seven free masses and eight axial springs. The circles in Fig. 9.28 are the FE generated vibration eigenvalue results. The mode shapes as reported by the eigensolver and shown in Figs. 9.28b and c are simply the axial nodal displacements for the case N = 8 which has nine nodes and seven DoFs. These figures show the displacements connected by straight lines because the rod element uses linear interpolation functions. They exhibit an increasing jaggedness with n. Superposed on the plots are the sinusoids sin(mπ x/L) and what is quite curious, is that in each case the discretized mode shapes align perfectly well with the sinusoid. We explain this presently. When these frequencies are plotted simply as frequency against mode number as in Fig. 9.28a, we see another pattern. For the largest number of elements (N = 16), the initial portion of the plot is linear in frequency (i.e., nondispersive) with nonlinear (i.e., dispersive) behavior toward the maximum frequency. This is replicated in the lower discretizations but it is not so obvious. This, of course, is the same behavior as observed in Fig. 9.2 but our interpretation is different because we want to see the connection to continuous systems indicated by the straight line in the figure. The model for the rod is the same as for the beaded string of Fig. 9.2 except that K = EA/a and M = ρAa. The fine discretization moves the critical frequency higher so that a greater range coincides with the continuous model. We see that the sine functions are solutions of the (discretized) differential equation and not of a particular boundary value problem of continuous systems. That they are solutions for the fixed-fixed rod problem is because that problem actually has periodic BCs. That is, if a second such rod is attached to the first to form a rod of total length 2L, then the mode shapes are unaffected although the frequency

406

9 Discrete and Discretized Structures

80 60 40 20 0 0.

50.

100.

150.

Fig. 9.29 FE generated vibration data for a simply supported beam. (a) Eigenvalues. (b) Mode shapes; open circles are displacement, full circles are rotations

discretization is essentially halved. The consequences of this are made clearer in the next case where the system has two deformation modes. Now consider the case of the dynamics of a discretized beam. Figure 9.29a shows the FE results for the eigenvalues of a vibrating simply supported beam using 40 elements. The modeling uses the beam stiffness of Eq. (5.19) but with a lumped mass [8]; the DoFs are the transverse displacement v and rotation φ. All the modes were computed and for this purpose the Jacobi rotations algorithm seems quite appropriate because we want all modes to be computed to comparable accuracy. The vertical axis is simply the mode number as reported when the modes are sorted in ascending order according to frequency. There are two aspects to the plot that need to be explained: first is the gap between 50 kHz and 90 kHz, and second is the interpretation of the modes above the gap. Figure 9.29b shows the first four and (c) the last four mode shapes. The m and M − 1 − m modes have the same shapes but the ratio of displacement to rotation is vastly different. For the first four modes, the nodal data are plotted as is, but for the last four modes, the displacements needed to be multiplied by 250 in order to be observed. That is, the last four modes are rotationally dominant. To help explain some of the results, consider a very long beam modeled with elements of length a and let the mass matrix be diagonal; the stiffness matrix is banded spanning the DoF (vn−1 , φn−1 ), (vn , φn ), (vn+1 , φn+1 ). (Figure 9.25 illustrates the general procedure for assemblage.) The EoMs for these DoFs are K[−12vn−1 − 6aφn−1 + 24vn − 12vn+1 + 6aφn+1 ] + M1 v¨n = 0 K[6avn−1 + 2a 2 φn−1 + 8a 2 φn − 6avn+1 + 2a 2 φn+1 ] + a 2 M2 φ¨n = 0 where K = EI /a 3 , M1 = ρAa and M2 = αρAa/39. These are obtained by assembling two elements with stiffnesses given by Eq. (5.19) and extracting the two middle rows. As done for the rod, let us represent the wave solutions in the form v(x, t) = vo e−i[kx−ωt] and φ(x, t) = φo e−i[kx−ωt] . After substituting into the

9.4 Spectral Analyses of FE Discretized Systems

407

differential equations we get  K

−6ae+ + 6ae− −12e+ + 24 − 12e− + − 2 6ae − 6ae 2a e+ + 8a 2 + 2a 2 e−



    vo M1 0 vo −ω2 =0 φo 0 a 2 M2 φ o

where e± = e±ika . This is an EVP which can be simplified to 

−12a sin ka 24[1 − cos ka] − ω2 /ω12 +12a sin ka 4a 2 [2 + cos ka] − a 2 ω2 /ω22



 vo =0 φo

(9.15)

√ √ √ √ where ω1 = K/M1 = EI /ρA/a 2 and ω2 = K/M2 = 39EI /αρA/a 2 . In comparison to Eq. (9.2), we see that for the beam this is a [2 × 2] EVP and as such, we expect two modes. In fact, the system has much in common with Eq. (9.5). As usual, to have a nontrivial solution requires that the determinant be zero. This gives the characteristic equation ( ' ω4 − 24[1 − cos ka]ω12 + 4[2 + cos ka]ω22 ω2 ' ( + 96[1 − cos ka][2 + cos ka] − 144 sin2 ka ω12 ω22 = 0 This is quadratic in ω2 and therefore easily solved for real values of k. These are shown plotted in Fig. 9.29a as the continuous line where # = kL/π . The model captures the data exactly. It also gives an additional interpretation of the high-frequency FE data: if these data are flipped horizontally about mode #40, the wavenumbers are seen to decrease (and wavelength increase) as shown for the mode shapes in Fig. 9.29b. The gap occurs where ka = π giving the two frequencies as  ω1g =

√ 1 48K = 48 2 M1 a



 EI , ρA

ω2g =

 4K 2 = 2 M2 a

EI 39 ρA α

The frequency position of the gap is dictated by the size of the discretization, essentially as depicted in Fig. 9.28a for the rod. The size of the gap is dictated by the lumped modeling of the rotational inertia through the parameter 39/α. The gap can be set to zero by choosing α = 13/4. Effectively what happens is that all high frequencies are shifted down which is to be expected because increasing α increases the rotational inertia. But does removing the gap improve the useful range of the FE modeling for the given discretization? To answer this question, we need to make clear what it is we want the FE modeling to achieve. For example, if we want it to accurately model a continuous beam, then we must compare to the exact solution for a continuous beam. Chapter 3 shows that the exact spectrum relation for a continuous Bernoulli–Euler beam is

408

9 Discrete and Discretized Structures

k=

√  ρA 1/4 ω EI

and this is labelled BE in Fig. 9.29. Shortening the gap does not increase the useful frequency range. As for the rod, the discretized beam differs from the continuous beam only at high frequencies; based on Fig. 9.29, we can take the useful frequency range as some proportion of ωg1 , say two-thirds. When a broad-banded excitation is input to a FE modeled rod, the high frequencies are filtered as seen in Fig. 9.3. For a beam, however, some of the high frequencies can propagate and this can give misleading results. To elaborate, Fig. 9.30a shows some FE generated responses for a long beam similar to Fig. 9.29. The input is a sine-squared pulse of duration 15 µs with frequency content about 150 kHz. There are two significant differences in comparison to traces like in Fig. 3.5. First, there is a high-frequency rider on the latter part of the trace that are not reflections—the beam is of length 2032 mm (80 in) whereas the monitoring site is at 762 mm (30 in). Second, the rotational response φ˙ has a high-frequency leader that is not present in the displacement response v. ˙ We can gain insight into this behavior by looking at the responses in the frequency domain. The top plots in Fig. 9.30b are the amplitude spectrums and are somewhat like Fig. 9.3b for a rod; they show the filtering action but the beam exhibits a sharp resonance-like peak in the displacement. Furthermore, the rotational amplitude spectrum exhibits activity beyond 90 kHz. The bottom plot in Fig. 9.30b shows the dispersion behavior based on Fig. 9.29a; it is clear that on average the higher frequencies have a greater speed. However, it is the comment in relation to Fig. 9.29b that the displacement (for large m) needed to be multiplied by 250 in order to be observed that explains why the high-frequency activity is not observed in the displacement responses.

0.5

0.0

0.5

1.0

0.0 0.

50.

100.

150.

Fig. 9.30 FE analysis of an impacted long beam. (a) Responses. (b) Amplitude spectrums and dispersion relation. The normalizing factor is cf = 508 m/s (20 000 in/s)

References

409

Further Research 9.1 Consider a 2D lattice similar to Fig. 9.14a but let the springs be replaced by pretensioned strings with distributed mass. Show that the characteristic equation is sin(ωa/co )[2 cos(ωa/co ) − cos(kx a) − cos(ky a)] = 0

—Ref. [16], p. 55. 9.2 If the beaded string is taken to represent a 1D array of atoms, then the interaction of all the other atoms should also be taken into account. Show that the resultant EoMs are ∞ ' ( M u¨ α − K αβ uβ−1 − 2uβ + uβ+1 = 0 β=1

and that the dispersion relation is ∞ ' ( Mω2 = 2 K αβ 1 − cos nka β=1

Recognizing the dispersion relation as a Fourier series, show that K

αβ

Ma =− 2π



+π/a

−π/a

ω2 cos nak dk

How might this relation be used to investigate the interatomic forces? —Ref. [14], p. 235

References 1. Askes, H., Bennett, T., Aifantis, E.C.: A new formulation and C 0 -implementation of dynamically consistent gradient elasticity. Int. J. Numer. Methods Eng. 72, 111–126 (2007) 2. Ayzenberg-Stepanenko, M.V., Slepyan, L.I.: Resonant-frequency primative waveforms and star waves in lattices. J. Sound Vib. 313, 812–821 (2008) 3. Berendsen, H.J.C., Postma, J.P.M., Gunsteren, W.F., DiNola, A., Haak, J.R.: Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81(8), 3684–3690 (1984) 4. Born, M., Haung, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, Oxford (1954) 5. Cagin, T., Dereli, G., Uludogan, M., Tomak, M.: Thermal and mechanical properties of some FCC transition metals. Phys. Rev. B 59, 3468–3473 (1999) 6. Delph, T.J.: Local stresses and elastic constants at the atomic scale. Proc. R. Soc. A 461, 1869– 1888 (2005) 7. Doyle, J.F.: Mechanics of structural materials: from atoms to continua. Supplemental Class Notes, Purdue University, 2010

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8. Doyle, J.F.: Nonlinear Structural Dynamics using FE Methods. Cambridge University Press, Cambridge (2015) 9. Doyle, J.F.: Spectral Analysis of Nonlinear Elastic Shapes. Springer, New York (2017) 10. Engelbrecht, J., Berezovski, A.: Mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Syst. 3(1), 43–82 (2015) 11. Finnis, M.W., Sinclair, J.E.: A simple empirical N-body potential for transition metals. Philos. Mag. A 50(1), 45–55 (1984) 12. Igarashi, M., Khantha, M., Vitek, V.: N-body interatomic potentials for hexagonal close-packed metals. Philos. Mag. B 63(3), 603–627 (1991) 13. Jones, R.M.: Mechanics of Composite Materials. McGraw-Hill, New York (1975) 14. Main, I.G.: Vibrations and Waves in Physics, 3rd edn. Cambridge University Press, Cambridge (1993) 15. Maranganti, P., Sharma, P.: A novel atomistic approach to determine strain-gradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies. J. Mech. Phys. Solids 55, 1823–1852 (2007) 16. Martinsson, P.G., Movchan, A.B.: Vibrations of lattice structures and phononic band gaps. Q. J. Mech. Appl. Math. 56(1), 45–64 (2003) 17. Sutton, A.P., Chen, J.: Long-range Finnis-Sinclair potential. Philos. Mag. Lett. 61, 139–146 (1990)

Afterword

One moves forward from this mystic landing to Fourier analysis, periodogram analysis, Fourier analysis over groups, and one comes rapidly to great technological applications, and always a sense of the actual and potential unities that lurk in the corners of the universe. P.J. Davis and R. Hersh [2]

It is opportune now to assess the accomplishments of this book and (perhaps more important) to point out what is missing. This book has emphasized the analysis of structures in situations that facilitate their treatment as connected waveguides. Thus, the primary effort is devoted to developing the spectral analysis method for solving wave propagation problems in rods, beams, and plates. It is shown that within the spectral framework, the incorporation of damping, coupling effects, and higher order theories, is a straightforward and simple matter. The introduction of the spectral element approach formalized the connection procedure for use on a computer and accordingly extended the range of applications of these methods to more general structures. Furthermore, the inherent parallelism of the spectral methods makes them eminently suitable for hosting on the new multi-processor computer architectures. Additionally, the incorporation of experimental data in many of the examples demonstrates the affinity of the spectral methods for experimental methods. In this latest edition, this affinity also extends to the computer experiments primarily performed by FE methods. What emerges is an approach that exhibits a striking unity between the analytical, computational, and experimental methodologies. A topic introduced but not developed is that of a global approach to waves in structures. The spectral element approach demonstrated in Chaps. 4 and 6 is very exciting because it offers the possibility of performing wave analysis in complicated structures without having to resort to expensive, time-consuming cascading methods. The efficiency of the approach resides in the fact that the only unknowns are those associated with the connectivities at the joints and therefore are few in number. The beauty of such an approach is that the structure can be © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8

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412

Afterword

analyzed as a whole without worry of the detailed history of each wave in each member—only if necessary is the detailed wave behavior reconstructed, and this can be done quite efficiently as part of the post-processing. The additional material on periodic structures reinforces this view. A further side benefit is that it brings wave propagation and vibration analyses closer together and makes such global methods as modal analysis possible, but all the time it still has the possibility of recovering the detailed behavior. It seems that what is really needed in dynamic structural analyses is some global or higher level evaluation of the results. The spectral analysis approach has the potential to offer this—the frequency domain energy measures introduced in Chap. 4 could play a role—but it is not developed as yet. The development of a global approach to wave propagation in complex structures must be considered the next challenge. Only barely treated in this book is the whole series of problems dealing with waves in extended media interacting with discontinuities (such as holes or cracks) causing dynamic stress concentrations, diffraction, and scattering. These problems can be treated by spectral analysis, but they require the development of some new tools invariably involving multiple series (or integral) representations. These topics are really part of the more general one of the exact formulations of wave problems and their subsequent asymptotic approximation. While many such solutions can be found in the literature, they generally are too cumbersome to be of direct value within our connected structures context. An exciting possibility is to extend the spectral super-element concept to encompass these problems. Some conjectures for achieving this were given at the end of Chap. 6, but very little work in the area has been done as yet. The compelling aspect of the spectral super-element approach is its suitability for hosting on multi-processor computers. Without doubt, the most significant development over the last two/three decades is that of nanotechnology. Again, as shown in Chap. 8, spectral analysis unifies our approaches originating from quite disparate sources. The nanotechnology problems also highlight the inherent nonlinearity of the systems. The role of spectral analysis for nonlinear systems is very complicated and still an open question. These three problems represent different departures from the contents of this book: the first is to a level of integration, the second to a level of more local detail, and the third to the incorporation of nonlinearities. The spectral methodologies presented in this book provide a useful middle ground for beginning all three studies.

Appendix A

A.1 Bessel Functions Bessel functions, and the other special functions, arise as solutions to many boundary value problems, and their use can greatly extend the range of the spectral methods. The major properties of the Bessel functions are summarized here so as to facilitate their use. Additional sources of information are Refs. [1, 4, 5]; the book by Press et al. [3] has Fortran coding of these functions.

A.1.1 Bessel Equations and Solutions An equation of the form z2

d 2w dw + (z2 − n2 )w = 0 , +z 2 dz dz

n≥0

is called Bessel’s equation and has the two independent solutions w(z) = C1 Jn (z) + C2 Yn (z) where Jn , Yn are Bessel functions of the first and second kinds, respectively. These are shown plotted in Fig. A.1. Observe that Yn (z) is singular as z → 0 and that both Jn and Yn tend to zero as z → ∞. Actually, both exhibit a damped (in space) oscillation, and for this reason, the solutions are often written in terms of the Hankel functions of the first and second kinds given by Hn1 (z) ≡ Jn (z) + iYn (z) ,

Hn2 (z) ≡ Jn (z) − iYn (z)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8

413

414

Appendix A 1.0 .8 .6 .4 .2 .0 -.2 -.4 -.6 -.8 -1.0

Jn 0 1 2 10

z 0.

1.0 .8 .6 .4 .2 .0 -.2 -.4 -.6 -.8 -1.0

2.

4.

6.

8.

10.

12.

14.

16.

18.

20.

Yn 0 1 2 10

z 0.

2.

4.

6.

8.

10.

12.

14.

16.

18.

20.

Fig. A.1 Bessel functions Jn (z) and Yn (z)

As shown presently, these have far-field behaviors similar to the trigonometric functions. A related Bessel equation (which can be obtained by replacing z with iz) is z2

d 2w dw − (z2 + n2 )w = 0 +z dz dz2

and has the two independent solutions w(z) = C3 In (z) + C4 Kn (z) where In , Kn are called modified Bessel functions of the first and second kinds, respectively. These are shown plotted in Fig. A.2 as e−z I (z) and e+z K(z), respectively. Many equations are transformable into Bessel equations, and a very useful general form is z2

d 2w dw + (β 2 z2γ + δ 2 )w = 0 + (1 + 2α)z 2 dz dz

This has the solution [4]

A.1

Bessel Functions

415

Kn e z

1.0 .8 .6

0 1

.4

2 10

.2 .0

z 0.

2.

4.

6.

8.

10.

12.

14.

16.

18.

20.

In e – z

1.0 .8

0 1 2 10

.6 .4 .2 .0

z 0.

2.

4.

6.

8.

10.

12.

14.

16.

18.

20.

Fig. A.2 Modified Bessel functions e+z Kn (z) and e−z In (z)

w=

) βzγ * ) βzγ * 1 + C2 Y ν , C1 Jν α z γ γ

ν=



α 2 − δ 2 /γ

A similar general form can be set up for the modified functions.

A.1.2 Limiting Behavior When manipulating Bessel functions, the following recurrence relations are very useful: Mn+1 =

2n Mn − Mn−1 , z

dMn = Mn+1 − Mn−1 dz 2n Ln+1 = − Ln + Ln−1 , z dLn = Ln+1 + Ln−1 2 dz

M−1 = −M1

−2

L−1 = +L1

416

Appendix A

where Mn represents any of the functions Jn , Yn , Hn1 , Hn2 , and Ln represents either In or einπ Kn . Bessel functions have the small argument expansions Jn (z) = ( 12 z)n

∞ 

1 m!(n + m + 1)

(− 14 z2 )m

m=0

 (n − m − 1)! 1 2 ( 14 z2 )m + ln( 12 z)Jn (z) Yn (z) = − ( 12 z)−n π m! π n−1

m=0

1 − ( 12 z)n π In (z) = ( 12 z)n

∞ 

∞ 

{ψ(m + 1) + ψ(n + m + 1)}(− 14 z2 )m

m=0

( 14 z2 )m

m=0

Kn (z) = 12 ( 12 z)−n

1 m!(n + m)!

1 m!(n + m + 1)

n−1  (n − m − 1)! 1 2 m (− 4 z ) + (−1)n+1 ln( 12 z)In (z) m!

m=0

+(−1)n 12 ( 12 z)n

∞ 

{ψ(m + 1) + ψ(n + m + 1)}( 14 z2 )m

m=0

where  is the gamma function and ψ(n) = arguments, in particular, J0 ≈ 1 − 14 z2 , Y0 ≈



1 m!(n + m)!

1/n − .5772. For very small

2 2 {ln( 12 z) + γ }Jo + 14 z2 π π

I0 ≈ 1 + 14 z2 , K0 ≈ −{ln( 12 z) + γ }Io + 14 z2 It is apparent that the functions Y0 , I0 exhibit singularities z = 0. When the argument is increased, then these functions have the asymptotic behavior of   ) ) π nπ * π nπ * 2 2 Jn ⇒ cos z − − , Yn ⇒ sin z − − πz 4 2 πz 4 2  π −z 1 In ⇒ √ e e z , Kn ⇒ 2z 2π z It is apparent that Jn , Yn exhibit a (spaced) damped oscillation and In , Kn behave as exponentials. The Hankel functions have the limiting behaviors

References

417

 Hn1

⇒

2 i(z− π − nπ ) 4 2 , e πz

 Hn2

⇒

2 −i(z− π − nπ ) 4 2 e πz

That is, they have the exponential form of the trigonometric functions with the difference that they decrease in amplitude for large argument. Consequently, the use of the Bessel functions in spectral analysis of waves is similar to that of the trigonometric functions. One moves forward from this mystic landing to Fourier analysis, periodogram analysis, Fourier analysis over groups, and one comes rapidly to great technological applications, and always a sense of the actual and potential unities that lurk in the corners of the universe. P.J. Davis and R. Hersh [2]

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965) 2. Davis, P.J., Hersh, R.: The Mathematical Experience. Birkhäuser, Boston (1981) 3. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes. Cambridge University Press, Cambridge (1986); 2nd edn. (1992) 4. Relton, F.E.: Applied Bessel Functions. Dover, New York (1965) 5. Tranter, C.J.: Bessel Functions. Hart, New York (1968)

Index

A Accelerometer, 27, 227 Acoustic mode, 368 Adjustable parameters, 158, 161, 167, 241 Airy stress function, 288 Aliasing, 24, 30 Amplitude ratio, 64, 102, 119, 161, 195, 316, 368 Amplitude spectrum, 34 Angular frequency, 8 Anisotropy, 374 Assemblage, 204, 217, 309, 324 Atomistically large, macroscopically small (ALMS) approximation, 389

B Baffled plate, 356 Banded matrix, 406 Band gap, 209, 211, 225, 282 Beam, 85 Bernoulli-Euler beam, 407 Bessel functions, 255, 272, 413 Blocked pressure, 334, 341 Born-Oppenheimer approximation, 361 Boundary condition (BC), 69, 152, 158, 168, 240, 331, 352 Branch cut, 348, 360 Brillouin zone, 367 Broad banded, 46, 357 Buckling, 94

C Cable, 94 Carrier wave, 38 Cauchy integral, 78, 99, 348 Cavity resonance, 340 Characteristic equation, 63, 115, 159, 168 Classical plate model, 237, 241 Coincidence frequency, 335 Compatibility equation, 288 Conservative system, 375 Continuous Fourier transform (CFT), 8 Convolution, 12, 25, 226 Coupled system, 61, 98, 289, 367 Crack, 223 Cubic material, 377 Cut-on/off frequency, 58, 95, 116, 144, 160, 168, 207, 247 Cylindrical coordinates, 113, 271, 276, 342

D D’Alembert’s principle, 150 D’Alembert solution, 51, 80, 88 Damping, viscous, 51, 57, 155, 215, 243, 335, 349 Degree of freedom (DoF), 188 Delta function, 13 Diffraction, 273 Diffusion, 330 Directivity pattern, 359 Dispersion relation, 35, 53, 166, 364

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. F. Doyle, Wave Propagation in Structures, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-59679-8

419

420 Dispersive, 36, 57, 70, 75 Distributed loads, 76, 134, 140, 194 Dynamic stiffness, 190, 204, 309, 322 Dynamic stiffness, reduced, 205

E Elastic constraint, 57, 95 Elastic material, 375 Energy, 147, 150, 239, 362 Essential BC, 152 Evanescence, 108 Evanescent waves, 36, 89, 91, 131, 136, 161

F Far-field, 274, 357 Fast Fourier transform (FFT), 18 Filter, 58, 109, 206, 282, 313, 408 Flexural motion, 85 Flexural stiffness, 87 Fluid loading parameter, 335 Folded plate, 295, 307, 312 Forward transform, 8, 18 Fourier heat conduction, 62, 64 Fourier series, 14, 342 Fourier transform, discrete, 16 Free edge wave, 135, 268 Free wave, 268, 347

G Global stiffness, 204, 309, 324 Group speed, 38, 65, 166 Group wave, 38

H Hamilton’s principle, 151 Hankel functions, 272, 343, 413 Heat conduction, 36, 62, 330 Helmholtz decomposition, 126, 296, 329 Helmholtz equation, 331, 342 Hexagonal material, 377 Holography, near field, 337 Hooke’s law, 50, 125, 239

I Impact, 90, 119, 161, 255, 260 Inertia, 150, 155 Integration by parts, 153 Interference, 25, 229 Inverse transform, 8, 11

Index Inviscid fluids, 330 Isotropic material, 377 J Jacobi method, 406 Joints, 74, 90, 107, 109, 110, 112, 212, 221, 231 K Kinematically admissible, 148 Kirchhoff shear, 243 L Lagrange’s equation, 363 Lamb wave, 141 Lamé constants, 125, 377 Lateral contraction, 50, 164 Leakage, 30 Linearity of transforms, 10 Line loading, 132 Local stiffness, 185 Love waves, 181 M Mass matrix, 190, 201 Material properties, 53, 66, 330 Matrix, modal, 197 Mindlin-Herrmann rod, 167 Mindlin plate, 237 Modal matrix, 197 Mode conversion, 107 Mode shape, 369 Modes, propagation, 33, 144, 145 Modulated signal, 25 Moire, 327 Moment-deflection, 87 Monoclinic material, 376 N Narrow banded, 46, 103, 139 Natural BC, 151, 152 Navier’s equations, 126, 329 Noise, 230, 327 Nondispersive, 35 Nyquist frequency, 17, 23, 24, 41 O Optical mode, 368 Orthotropic material, 376, 377

Index P Padding, 30 Pair-potential, 384 Parallel computation, 206, 310 Periodically extended load (pEL), 22, 77, 336 Periodic BC (pBC), 280, 394, 405 Periodic structure, 208 Phase, 10, 14 Phase shift, 31, 106 Phase speed, 35, 38 Phononic material, 208 Plane strain, 126, 128, 129, 314 Plane stress, 126, 128, 148, 239 Plate, Mindlin, 240 Plate stiffness, 240 Pochhammer–Chree theory, 181 Polarization, 127, 379 Power, scattered, 274 Principle of virtual work (PoVW), 149 Propagation equation, 211 P-wave, 123, 127

R Radiation, 353 Rayleigh wave, 124, 136, 268 Residue, 78 Resonance, 26, 71 Resultant, 114, 285 Reverberation, 74, 112, 341 Ring frequency, 116, 282, 289 Ritz semidirect method, 152 Root solver, 63, 116, 142

S Scaling, of argument, 10 Scattered wave, 263, 273, 274 Shape function, 187, 189, 196, 297, 302, 320, 343, 354 Shear distribution, 139 SH-wave, 130 Side lobe, 24 Spectral analysis, 7 Spectral element, 219, 296, 301, 339 Spectrum, 7 Spectrum relation, 33, 52, 56, 59, 64, 260, 286 St. Venant torsion model, 202, 253, 281

421 Standing wave, 80, 97, 141, 247, 343 Stationary phase, 46, 306, 358 Stonely wave, 182 Stop band, 364, 374 Strain-gradient elasticity, 395 Strain energy, 148 Strain gage, 27, 258, 262 Stress intensity factor, 223 String under tension, 81, 94, 362 Strong form, 152 SV-wave, 129 S-wave, 123, 127 System transfer function, 34, 205

T Taylor series, 375 Thin plate, 241 Throw-off element, 187, 199, 300 Time shifting, 10 Timoshenko beam, 155, 158 Traction, 246 Transfer function, 40 Transform real function, 11 Transversely isotropic, 377 Trim panel, 338

V Van der Waal bond, 383 Variation, 150 Variational principle, 152 Velocity restraint, 57 Virtual displacement, 149 Virtual work, 148, 149 Viscous damping, 155 Voight notation, 376

W Warping, 280 Wave group, 25 Wavenumber, 33 Weak form of equilibrium, 146 Windowed signal, 29 Winkler foundation, 95 Wrap-around, 22, 43, 90, 194, 271, 365