Elastic Wave Propagation in Structures and Materials Elastic Wave Propagation in Structures and Materials initiates with a brief in- troduction to wave propagation, different wave equations, integral transforms including fundamentals of Fourier Transform, and its numerical implementa- tion. The concept of spectral analysis and procedure to compute the wave pa- rameters, wave propagation in 1-D isotropic waveguides, and wave dispersion in 2-D waveguides are explained. Wave propagation in different media such as laminated composites, functionally graded structures, granular soils, and non- local elasticity models is addressed. Separate chapters on signal processing and wave propagation in viscoelastic waveguides are included in this book. The entire book is written in modular form for easy understanding of the basic concepts. Features: • Brings out the idea of wave dispersion and its utility in the dynamic responses • Introduces concepts such as Negative Group Speeds, Einstein’s Causality, and escape frequencies using solid mathematical framework • Discusses the propagation of waves in materials such as laminated composites and functionally graded materials • Proposes spectral finite element as analysis tool for wave propagation • Supports each concept/chapter with homework problems and MATLAB® codes This book is aimed at senior undergraduates and advanced graduates in all fields of engineering, especially Mechanical and Aerospace Engineering. Elastic Wave Propagation in Structures and Materials Srinivasan Gopalakrishnan MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. First edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Srinivasan Gopalakrishnan Reasonable efforts have been made to publish reliable data and information, but the author and pub- lisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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ISBN: 9780367637576 (hbk) ISBN: 9780367637637 (pbk) ISBN: 9781003120568 (ebk) DOI: 10.1201/9781003120568 Publisher’s note: This book has been prepared from camera-ready copy provided by the authors. This book is dedicated to my Late mother Saraswathi, my father Srinivasan, my wife Anu and my Children Karthik and Keerthana 5 Contents Preface xv Author Biography xvii hapter C 1(cid:4) Introduction to Wave Propagation 1 1.1 ESSENTIALCOMPONENTSOFAWAVE 2 1.2 INTERFERENCEOFWAVES 5 1.2.1 Interference of Two Similar Waves Propagating in the Same Direction 6 1.2.2 Standing Waves 7 1.3 NEEDFORWAVEPROPAGATIONANALYSISIN STRUCTURESANDMATERIALS 9 1.4 SOME WAVE PROPAGATION PROBLEMS IN SCIENCE ANDENGINEERING 13 1.5 ORGANIZATIONANDSCOPEOFTHEBOOK 16 hapter C 2(cid:4) Introduction to Fourier Transforms 21 2.1 FOURIERTRANSFORMS 22 2.1.1 Continuous Fourier Transforms (CFT) 22 2.1.2 Fourier Series 27 2.1.3 Discrete Fourier Transform 29 2.2 COMPARATIVEMERITSANDDEMERITSOFFFT 31 ® NOTEONMATLAB SCRIPTSPROVIDEDINTHIS CHAPTER 33 SUMMARY 33 EXERCISES 33 vii viii (cid:4) Contents hapter C 3(cid:4) Introduction to Wave Propagation in Structures 37 3.1 CONCEPT OF WAVENUMBER, GROUP SPEEDS AND PHASESPEEDS 38 3.2 WAVEPROPAGATIONTERMINOLOGIES 41 3.3 SPECTRALANALYSISOFMOTION 43 3.3.1 Second-Order System 43 3.3.2 Fourth-Order System 46 3.4 GENERAL FORM OF WAVE EQUATION AND THEIR CHARACTERISTICS 48 3.4.1 General Form of Wave Equations 49 3.4.2 Characteristics of Waves in Anisotropic Media 49 3.4.3 Characteristics of Waves in Inhomogeneous Media 50 3.4.4 Characteristics of Waves in Non-local Waveguides 51 3.5 DIFFERENTMETHODSOFCOMPUTING WAVENUMBERSANDWAVEAMPLITUDES 52 3.5.1 Method - 1 : The Companion Matrix & the SVD Technique 53 3.5.2 Method - 2 : Linearization of Polynomial Eigenvalue Problem (PEP) 54 SUMMARY 55 EXERCISES 56 hapter C 4(cid:4) WavePropagationinOne-DimensionalIsotropic Structural Waveguides 59 4.1 WAVEPROPAGATIONIN1-DELEMENTARY WAVEGUIDES 60 4.2 LONGITUDINALWAVEPROPAGATIONINRODS 61 4.2.1 D’Alemberts Solution 63 4.2.2 Spectral Analysis 63 4.2.3 PropagationofWavesinanInfiniteLongitudinal Waveguide 64 4.2.4 Interaction of Waves with Fixed and Free Boundaries 66 4.2.5 Reflection from an Elastic Boundary 67 4.2.6 ReflectionandTransmissionfromaJointHaving Concentrated Mass and Stepped Rod 67 Contents (cid:4) ix 4.3 FLEXURALWAVEPROPAGATIONINBEAMS 73 4.3.1 Wave Propagation Analysis 74 4.3.2 Propagation of Waves in an Infinite Beam 76 4.3.3 Reflection from Boundaries 77 4.3.4 Reflection from Elastic Boundary 79 4.3.5 Reflection and Transmission from a Stepped Beam and a Joint with Concentrated Mass 81 4.3.6 WavePropagationinBeamswithPre-Tensionor Pre-Compression 85 4.3.7 Wave Propagation in a Beam on Elastic Foundation 89 4.3.8 Wave Propagation in a Framed Structure 90 4.4 WAVEPROPAGATIONINHIGHER-ORDERWAVEGUIDES 97 4.4.1 Wave Propagation in Timoshenko Beam 97 4.4.2 Wave Propagation in Mindlin-Herrmann Rod 103 4.5 WAVEPROPAGATIONINROTATINGBEAMS 110 4.6 WAVEPROPAGATIONINTAPEREDWAVEGUIDES 113 4.6.1 Wave Propagation in Tapered Rod Having Exponential Depth Variation 114 4.6.2 Wave Propagation in Tapered Rod Having Polynomial Depth Variation 116 4.6.3 Wave Propagation in Tapered Beam 119 ® NOTEONMATLAB SCRIPTSPROVIDEDINTHIS CHAPTER 121 SUMMARY 122 EXERCISES 123 hapter C 5(cid:4) Wave Propagation in Viscoelastic Waveguides 129 5.1 CONSTITUTIVEMODELSFORVISCOELASTIC WAVEGUIDES 131 5.1.1 Two-Parameter Models 132 5.1.2 Three-Parameter Models 133 5.2 WAVEPROPAGATIONINVISCOELASTICROD 137 5.2.1 Wave Propagation in Two-Parameter Viscoelastic Rod 139 5.2.2 Wave Propagation in Three-Parameter Viscoelastic Rod 141