Mechanical Wave Vibrations: Analysis and Control [1 ed.] 1119135044, 9781119135043

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Mechanical Wave Vibrations

Mechanical Wave Vibrations Analysis and Control

Chunhui Mei

University of Michigan-Dearborn Dearborn, MI, USA

This edition first published 2023 © 2023 Chunhui Mei All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Chunhui Mei to be identified as the author of this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty 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 work’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. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Mei, Chunhui, author. Title: Mechanical wave vibrations : analysis and control / Chunhui Mei. Description: Chichester, West Sussex, UK : John Wiley & Sons, 2023. | Includes bibliographical references and index. | Summary: “In this book titled Mechanical Wave Vibrations, vibrations in solid structures are viewed as waves that propagate along uniform waveguides, and are reflected and transmitted incident upon discontinuities similar to light and sound waves. The wave vibration description is particularly useful for structures consisting of onedimensional structural elements where a finite number of waves with given directions of propagation exist. In conventional textbooks on mechanical vibrations vibration problems in distributed structures are solved as boundary value problems. The coverage is typically limited to the analysis of a single type of vibration in a simple beam element due to the complexity in boundaries imposed by builtup structures.”-- Provided by publisher. Identifiers: LCCN 2022060202 (print) | LCCN 2022060203 (ebook) | ISBN 9781119135043 (hardback) | ISBN 9781119135067 (pdf) | ISBN 9781119135050 (epub) | ISBN 9781119135074 (ebook) Subjects: LCSH: Vibration. | Waves. Classification: LCC QC136 .M45 2023 (print) | LCC QC136 (ebook) | DDC 531/.32--dc23/eng20230429 LC record available at https://lccn.loc.gov/2022060202 LC ebook record available at https://lccn.loc.gov/2022060203 Cover Image: Courtesy of the Author Cover Design: Wiley Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India

To my mother Yaoguang Zeng who still takes care of me the same way she did before I left home for college, and my late father Fuchu Mei who was always proud whatever my pursuit was.

vii

Contents Preface  xi Acknowledgement  xiii About the Companion Website  xv

1 1.1 1.1.1 1.1.2 1.2 1.3

Sign Conventions and Equations of Motion Derivations  1 Derivation of the Bending Equations of Motion by Various Sign Conventions  1 According to Euler–Bernoulli Bending Vibration Theory  2 According to Timoshenko Bending Vibration Theory  7 Derivation of the Elementary Longitudinal Equation of Motion by Various Sign Conventions  10 Derivation of the Elementary Torsional Equation of Motion by Various Sign Conventions  12

2 2.1 2.2 2.3 2.4 2.5 2.6

Longitudinal Waves in Beams  15 The Governing Equation and the Propagation Relationships  15 Wave Reflection at Classical and Non-Classical Boundaries  16 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes  20 Force Generated Waves and Forced Vibration Analysis of Finite Beams  24 Numerical Examples and Experimental Studies  27 MATLAB Scripts  32

3 3.1 3.2 3.3 3.4 3.5 3.6

Bending Waves in Beams  39 The Governing Equation and the Propagation Relationships  39 Wave Reflection at Classical and Non-Classical Boundaries  40 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes  46 Force Generated Waves and Forced Vibration Analysis of Finite Beams  50 Numerical Examples and Experimental Studies  55 MATLAB Scripts  59

4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5

Waves in Beams on a Winkler Elastic Foundation  69 Longitudinal Waves in Beams  69 The Governing Equation and the Propagation Relationships  69 Wave Reflection at Boundaries  70 Free Wave Vibration Analysis  71 Force Generated Waves and Forced Vibration Analysis of Finite Beams  72 Numerical Examples  76 Bending Waves in Beams  79 The Governing Equation and the Propagation Relationships  79 Wave Reflection at Classical Boundaries  82 Free Wave Vibration Analysis  84 Force Generated Waves and Forced Wave Vibration Analysis  84 Numerical Examples  89

viii

Contents

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Coupled Waves in Composite Beams  97 The Governing Equations and the Propagation Relationships  97 Wave Reflection at Classical and Non-Classical Boundaries  100 Wave Reflection and Transmission at a Point Attachment  102 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes  104 Force Generated Waves and Forced Vibration Analysis of Finite Beams  105 Numerical Examples  108 MATLAB Script  114

6 Coupled Waves in Curved Beams  119 6.1 The Governing Equations and the Propagation Relationships  119 6.2 Wave Reflection at Classical and Non-Classical Boundaries  121 6.3 Free Vibration Analysis in a Finite Curved Beam – Natural Frequencies and Modeshapes  127 6.4 Force Generated Waves and Forced Vibration Analysis of Finite Curved Beams  128 6.5 Numerical Examples  134 6.6 MATLAB Scripts  143 7 7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.5 8 8.1 8.2 8.3 8.3.1 8.3.2 8.4 8.4.1 8.4.2 8.5 8.5.1 8.5.2 8.5.3 8.6 8.6.1 8.6.2 8.7

Flexural/Bending Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges Simply-supported  151 The Governing Equations of Motion  151 Closed-form Solutions  152 Wave Reflection, Propagation, and Wave Vibration Analysis along the Simply-supported x Direction  154 Wave Reflection, Propagation, and Wave Vibration Analysis Along the y Direction  156 Wave Reflection at a Classical Boundary along the y Direction  157 Wave Propagation and Wave Vibration Analysis along the y Direction  159 Numerical Examples  159 In-Plane Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges Simply-supported  189 The Governing Equations of Motion  189 Closed-form Solutions  190 Wave Reflection, Propagation, and Wave Vibration Analysis along the Simply-supported x Direction  192 Wave Reflection at a Simply-supported Boundary along the x Direction  192 Wave Propagation and Wave Vibration Analysis along the x Direction  195 Wave Reflection, Propagation, and Wave Vibration Analysis along the y Direction  197 Wave Reflection at a Classical Boundary along the y Direction  198 Wave Propagation and Wave Vibration Analysis along the y Direction  201 Special Situation of k0 = 0: Wave Reflection, Propagation, and Wave Vibration Analysis along the y Direction  201 Wave Reflection at a Classical Boundary along the y Direction Corresponding to a Pair of Type I Simple Supports along the x Direction When k0 = 0  202 Wave Reflection at a Classical Boundary along the y Direction Corresponding to a Pair of Type II Simple Supports along the x Direction When k0 = 0  203 Wave Propagation and Wave Vibration Analysis along the y Direction When k0 = 0  205 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along the y Direction When k0 ≠ 0  207 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along the y Direction When k0 ≠ 0, k1 ≠ 0, and k2 ≠ 0  207 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along the y Direction When k0 = 0, and either k1 = 0 or k2 = 0  209 Numerical Examples  212

Contents

8.7.1 8.7.2 8.7.3

Example 1: Two Pairs of the Same Type of Simple Supports along the x and y Directions  212 Example 2: One Pair of the Same Type Simple Supports along the x Direction, Various Combinations of Classical Boundaries on Opposite Edges along the y Direction  217 Example 3: One Pair of Mixed Type Simple Supports along the x Direction, Various Combinations of Classical Boundaries on Opposite Edges along the y Direction  223

9 9.1 9.1.1 9.1.2 9.1.3 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.4 9.4.1 9.4.2 9.5 9.5.1 9.5.2 9.6 9.7

Bending Waves in Beams Based on Advanced Vibration Theories  227 The Governing Equations and the Propagation Relationships  227 Rayleigh Bending Vibration Theory  227 Shear Bending Vibration Theory  228 Timoshenko Bending Vibration Theory  230 Wave Reflection at Classical and Non-Classical Boundaries  232 Rayleigh Bending Vibration Theory  232 Shear and Timoshenko Bending Vibration Theories  238 Waves Generated by Externally Applied Point Force and Moment on the Span  244 Rayleigh Bending Vibration Theory  245 Shear and Timoshenko Bending Vibration Theories  246 Waves Generated by Externally Applied Point Force and Moment at a Free End  247 Rayleigh Bending Vibration Theory  248 Shear and Timoshenko Bending Vibration Theories  249 Free and Forced Vibration Analysis  250 Free Vibration Analysis  250 Forced Vibration Analysis  250 Numerical Examples and Experimental Studies  252 MATLAB Scripts  257

10 10.1 10.1.1 10.1.2 10.1.3 10.2 10.2.1 10.2.2 10.2.3 10.3 10.3.1 10.3.2 10.3.3 10.4 10.4.1 10.4.2 10.4.3 10.5 10.5.1 10.5.2 10.6

Longitudinal Waves in Beams Based on Various Vibration Theories  263 The Governing Equations and the Propagation Relationships  263 Love Longitudinal Vibration Theory  263 Mindlin–Herrmann Longitudinal Vibration Theory  264 Three-mode Longitudinal Vibration Theory  265 Wave Reflection at Classical Boundaries  267 Love Longitudinal Vibration Theory  267 Mindlin–Herrmann Longitudinal Vibration Theory  268 Three-mode Longitudinal Vibration Theory  269 Waves Generated by External Excitations on the Span  271 Love Longitudinal Vibration Theory  271 Mindlin–Herrmann Longitudinal Vibration Theory  272 Three-mode Longitudinal Vibration Theory  273 Waves Generated by External Excitations at a Free End  275 Love Longitudinal Vibration Theory  275 Mindlin–Herrmann Longitudinal Vibration Theory  276 Three-mode Longitudinal Vibration Theory  276 Free and Forced Vibration Analysis  277 Free Vibration Analysis  278 Forced Vibration Analysis  278 Numerical Examples and Experimental Studies  281

11 11.1 11.2 11.3

Bending and Longitudinal Waves in Built-up Planar Frames  287 The Governing Equations and the Propagation Relationships  287 Wave Reflection at Classical Boundaries  289 Force Generated Waves  291

ix

x

Contents

11.4 11.5 11.5.1 11.5.2 11.5.3

Free and Forced Vibration Analysis of a Multi-story Multi-bay Planar Frame  292 Reflection and Transmission of Waves in a Multi-story Multi-bay Planar Frame  304 Wave Reflection and Transmission at an L-shaped Joint  304 Wave Reflection and Transmission at a T-shaped Joint  308 Wave Reflection and Transmission at a Cross Joint  315

12 12.1 12.2 12.3 12.4 12.5 12.5.1 12.5.2

Bending, Longitudinal, and Torsional Waves in Built-up Space Frames  329 The Governing Equations and the Propagation Relationships  329 Wave Reflection at Classical Boundaries  333 Force Generated Waves  336 Free and Forced Vibration Analysis of a Multi-story Space Frame  338 Reflection and Transmission of Waves in a Multi-story Space Frame  341 Wave Reflection and Transmission at a Y-shaped Spatial Joint  343 Wave Reflection and Transmission at a K-shaped Spatial Joint  353

13 13.1 13.1.1 13.1.2 13.2 13.2.1 13.2.2 13.3 13.4 13.5 13.6 13.7

Passive Wave Vibration Control  369 Change in Cross Section or Material  369 Wave Reflection and Transmission at a Step Change by Euler–Bernoulli Bending Vibration Theory  371 Wave Reflection and Transmission at a Step Change by Timoshenko Bending Vibration Theory  372 Point Attachment  373 Wave Reflection and Transmission at a Point Attachment by Euler–Bernoulli Bending Vibration Theory  374 Wave Reflection and Transmission at a Point Attachment by Timoshenko Bending Vibration Theory  375 Beam with a Single Degree of Freedom Attachment  376 Beam with a Two Degrees of Freedom Attachment  378 Vibration Analysis of a Beam with Intermediate Discontinuities  380 Numerical Examples  381 MATLAB Scripts  390

14 14.1 14.1.1 14.1.2 14.2 14.2.1 14.2.2 14.3 14.4

Active Wave Vibration Control  401 Wave Control of Longitudinal Vibrations  401 Feedback Longitudinal Wave Control on the Span  401 Feedback Longitudinal Wave Control at a Free Boundary  405 Wave Control of Bending Vibrations  407 Feedback Bending Wave Control on the Span  407 Feedback Bending Wave Control at a Free Boundary  410 Numerical Examples  413 MATLAB Scripts  416 Index  421

xi

Preface In this book titled Mechanical Wave Vibrations, vibrations in solid structures are viewed as waves that propagate along uniform waveguides and are reflected and transmitted incident upon discontinuities, similar to light and sound waves. The wave vibration description is particularly useful for structures consisting of one-dimensional structural elements where a finite number of waves with given directions of propagation exist. In conventional textbooks on mechanical vibrations, vibration problems in distributed structures are solved as boundary value problems. The coverage is typically limited to the analysis of a single type of vibration in a simple beam element because of the complexity in boundaries of each structural element imposed by built-up structures. From the wave standpoint, however, a structure, regardless of its complexity, consists of only two components, namely, structural elements and structural joints. Vibrations propagate along uniform structural elements, and are reflected and/ or transmitted at structural discontinuities such as joints and boundaries. Assembling these propagation, reflection, and transmission relationships provides a concise and systematic approach for vibration analysis of a complex structure. Unlike the conventional modal vibration analysis approach that has been taught in standard vibration courses for decades, the wave vibration analysis approach is seldom taught to students and the related knowledge is limited to the research community through journal or conference publications. This textbook is written with both undergraduate and graduate students in mind. The author hopes to see this wavebased vibration analysis approach incorporated into the engineering curriculum to allow engineering students a better understanding of mechanical vibrations and to equip them with additional tools for solving practical vibration problems. In addition, researchers and educators in the vibration and control field will find this book helpful. This textbook is written in such a way that there is no prior knowledge on vibrations needed, although it requires knowledge on mechanics of materials and dynamics at an undergraduate level. As a result, courses based on this textbook can be offered prior to, concurrent with, or after any conventional courses on mechanical vibrations. Students are expected to either be familiar with or be willing to learn MATLAB technical computing language. Sample MATALB scripts for numerical simulations are provided at the end of most chapters. This book is organized as follows. Chapter 1 is devoted to the coverage of sign conventions and the derivation of equations of motion using the Newtonian approach. Sign conventions, which are often a source of error for engineering analysis, play important roles not only in the derivation of governing equations of motion for bending, longitudinal, and torsional vibrations, but also in wave vibration analysis. In Chapters 2 and 3, longitudinal and bending vibrations in beams are studied, both based on elementary vibration theories. Fundamental concepts related to wave vibration analysis are introduced, such as the propagation of vibration waves along uniform structural elements (the waveguides) and the reflection of vibration waves at classical and non-classical boundaries (the discontinuities). Free and forced longitudinal and bending vibrations are analyzed from the wave vibration standpoint. Natural frequencies, modeshapes, as well as steady state frequency responses are obtained and compared with experimental results. Chapter 4 studies both longitudinal and bending waves in beams on a Winkler elastic foundation. The concepts of cut-off frequency and wave mode transition are introduced. The analysis is presented in non-dimensional form, a different form than the previous chapters. Chapter 5 studies vibration waves in composite beams, in which the concept of coupled waves caused by material coupling in a composite beam is introduced.

xii

Preface

In Chapter 6, coupled vibration motions along the radial and tangential directions in a thin curved beam are analyzed based on Love’s vibration theory. Cut-off frequencies, wave mode transitions, and dispersion relationships are studied. Wave reflections at classical and non-classical boundaries are derived. Natural frequencies, modeshapes, as well as steady state frequency responses are obtained from the wave vibration standpoint. Chapters 7 and 8 cover out-of-plane and in-plane vibrations in rectangular plates with at least one pair of opposite edges simply supported, which is required for closed form solutions to exist in plates. Chapters 9 and 10 advance the coverage of bending and longitudinal vibrations of Chapters 3 and 2 by taking into account effects that are neglected in the elementary theories. For example, the effect of rotary inertia and transverse shear deformation for bending vibration neglected in the Euler–Bernoulli bending vibration theory, are included in part or in full by the Rayleigh, Shear, and Timoshenko bending vibration theories. Free and forced longitudinal and bending vibrations are analyzed with comparison to experimental results. In Chapters 11 and 12, vibrations in built-up planar and space frames are studied. An angle joint, in general, introduces wave mode conversion. As a result, multiple wave types co-exist in built-up frame structures. For example, in a planar frame that undergoes in-plane vibrations, in-plane bending and longitudinal waves co-exist. In built-up space frames, inand out-of-plane bending, longitudinal, and torsional vibrations co-exist. Solving vibration problems of such complexity has proven to be challenging by the conventional modal analysis approach, however, the wave-based vibration analysis approach is seen to offer a concise assembly approach for systematically analyzing complex vibrations in built-up planar as well as space frames. The final two chapters of this book, Chapters 13 and 14, are devoted to vibration control from the wave standpoint, either by adding discontinuities to the path of wave propagation for the purpose of altering vibration characteristics of a structure or by minimizing the transmitted and/or reflected vibration energy in a structure. It is recommended to cover Chapters 1, 2, 3, 4, 5, 9, 11, 13, and 14 in an introductory course on Mechanical Wave Vibration at undergraduate level. The remaining chapters can be selected and added at an instructor’s discretion for a similar course at graduate level.

xiii

Acknowledgement First and foremost, I wish to thank my Ph.D. advisor and lifelong role model, Brian Mace. Brian brought me into the field of mechanical wave vibrations and trustfully handed me this book project. He has always been there whenever I needed guidance and encouragement. I am grateful to my academic brother Neil Harland. As the first reader of this book manuscript, Neil has provided much valuable and constructive feedback. His time and effort are greatly appreciated. Most importantly, I wish to thank my mom Yaoguang, my husband Chundao, my sons Yonglu and Yongwei, and my daughter-in-law Yujiang, for their love, support, and patience. Last but not least, I would like to acknowledge the Mechanical Engineering Department and the College of Engineering and Computer Science at the University of Michigan-Dearborn for jointly purchasing a laptop computer for this book project.

xv

About the Companion Website This book is accompanied by a companion website: www.wiley.com/go/Mei/MechanicalWaveVibrations This website includes ● ●

Instructor Solutions Manual Instructor PPT Slides

1

1 Sign Conventions and Equations of Motion Derivations Sign conventions and coordinate systems play important roles in wave vibration analysis and in the derivation of governing equations of motion for bending, longitudinal, and torsional vibrations. In this book, Cartesian coordinate system is adopted. For a planar structure, the x- and y-axis of a two-dimensional Cartesian coordinate system are chosen to be in the plane of the structure. The x-axis is always chosen to be along the longitudinal axis of a member. The axial and shear force are parallel to the x- and y-axis, respectively. Angle ϕ is defined by the right hand rule rotation from the x-axis to the y-axis. For an in depth understanding of sign conventions, which are often a source of error for engineering analysis, the governing equations of motion are derived using the Newtonian approach following various sign conventions.

1.1  Derivation of the Bending Equations of Motion by Various Sign Conventions Figure 1.1 shows the positive sign directions for internal resistant shear force V and bending moment M of four possible sets of conventions. In the figure, subscripts L and R denote parameters related to the left and right side of the cut section, respectively. The set shown in Figure 1.1a is a convention that has been popularly adopted by many authors in textbooks and research papers, while the remaining sets presented in Figures 1.1b to 1.1d are less often adopted. The best way to interpret a sign convention is to look at how the internal resistant forces and moments deform or rotate the corresponding element. In Sets (a) and (b) shown in Figure 1.1, the shear force is positive when it rotates its element along the positive direction of angle ϕ. The convention for the bending moment is defined differently. In Set (a) the bending moment is positive when it bends its element concave towards the positive y-axis direction; however, in Set (b) the positive bending moment is when it bends its element convex towards the positive y-axis direction. In Sets (c) and (d) in Figure 1.1, the shear force is positive when it rotates its element along the negative direction of angle ϕ . The convention for the bending moment is defined differently, in Set (c) the bending moment is positive when it bends its element concave towards the positive y-axis direction; while in Set (d) the positive bending moment is when it bends its element convex towards the positive y-axis direction. This deformation and rotation based interpretation holds regardless of the orientation of the beam; one only needs to be consistent with the choice of the coordinate system and the definition of positive sign directions. Consider now, as shown in Figure 1.2, a beam of length L that is subjected to an external distributed transverse load of f ( x , t ) per unit length. The x-axis is chosen to be along the neutral axis of the beam, t is the time, and y ( x , t ) is the transverse deflection of the beam. In the absence of axial loading, the bending equations of motion of the beam derived using the four sets of sign conventions shown in Figure 1.1 are presented below. Figures 1.3a to 1.3d are the free body diagrams of a differential element of the beam according to the four sets of sign conventions of Figures 1.1a to 1.1d, respectively. The bending moments and shear forces on both sides of the differential element are with positive sign directions by the corresponding sign conventions.

Mechanical Wave Vibrations: Analysis and Control, First Edition. Chunhui Mei. © 2023 Chunhui Mei. Published 2023 by John Wiley & Sons Ltd. Companion Website: www.wiley.com/go/Mei/MechanicalWaveVibrations

2

1  Sign Conventions and Equations of Motion Derivations

(a)

(b)

(c)

(d)

Figure 1.1  Definitions of positive sign directions for internal resistant shear force and bending moment by various sign conventions: (a) sign convention 1, (b) sign convention 2, (c) sign convention 3, and (d) sign convention 4.

Figure 1.2  A beam in bending vibration.

1.1.1  According to Euler–Bernoulli Bending Vibration Theory The bending equations of motion in the Euler–Bernoulli (or thin beam) theory are derived based on the following three assumptions. First, the neutral axis does not experience any change in length. Second, all cross sections remain planar and perpendicular to the neutral axis. Third, deformation at the cross section within its own plane is negligibly small. In other words, the rotation of cross sections of the beam is neglected compared to the translation, and the angular distortion due to shear is neglected compared to the bending deformation. The concept of curvature of a beam is central to the understanding of beam bending. Mathematically, the radius of curvature R of a curve y ( x , t ) can be found using the following formula

R=

3/2   dy ( x , t )2     1 +    dx    

(1.1)

d y( x,t ) 2

dx 2 For a beam element in a practical engineering structure that undergoes bending vibration, the transverse deflection of the centerline y ( x , t ) normally forms a shallow curve because of limitations set forth by engineering design codes on dy ( x , t ) is normally allowable deflection of engineering structures. Consequently, the slope of the deformation curve dx

1.1  Derivation of the Bending Equations of Motion by Various Sign Conventions

very small, and its square is negligible when compared to unity. Therefore, the radius of curvature as defined above can be approximated by 1 R≈ 2 d y( x, t) dx 2

(1.2)

By definition, the neutral axis, which lies on the x-axis, does not experience any change in length. Consequently, the lengths of the neutral axis of the differential element remain the same amount of dx before and after deformation, as shown in Figure 1.4. In the absence of axial loading, the longitudinal strain in the beam is produced only from bending and by definition of strain, ε =

ds − dx dx

(1.3)

(a)

(b)

(c)

(d)

Figure 1.3  Free body diagram of a beam element in bending vibration by the various sign conventions defined in Figure 1.1: (a) sign convention 1, (b) sign convention 2, (c) sign convention 3, and (d) sign convention 4.

3

4

1  Sign Conventions and Equations of Motion Derivations

Neutral Axis

(a)

(b)

(c)

Figure 1.4  Strain and radius of curvature: (a) before deformation, (b) after concave bending deformation, and (c) after convex bending deformation.

There are two types of bending deformations with reference to the positive y-axis, concave and convex, because of internal bending moments M1  or M3 and M2  or M 4 , respectively. The normal strains ε1,3 and ε2,4 on the differential element that is a distance z above the neutral axis correspond to the concave and convex deformations shown in Figures 1.4b and 1.4c are ε1,3 =

ds1,3 − dx ( R − z )dψ − Rdψ z =− = R dx Rdψ

(1.4a)

ε2,4 =

ds2,4 − dx ( R + z )dψ − Rdψ z = = R dx Rdψ

(1.4b)

where R is the radius of curvature of the transverse deflection of the centerline y ( x , t ), and ψ ( x , t ) is the angle of rotation of the cross section due to bending. Subscripts 1, 2, 3, and 4 denote parameters related to sign conventions 1, 2, 3, and 4 defined in Figure 1.1. For concave deformation, strains ε1,3 are negative above the neutral axis (where z is positive) because of compressive normal stress in the region caused by the internal resistant bending moment at the given direction. Strains ε1,3 are positive below the neutral axis (where z is negative) because of tensile normal stress in the region caused by the internal resistant bending moment at the given direction. This explains the negative sign in Eq. (1.4a). For convex deformation, the region above the neutral axis (where z is positive) is subject to positive strain, and below the neutral axis (where z is negative) is subject to negative strain; hence, strains ε2,4 carry the same sign as z, as reflected in Eq. (1.4b). For homogeneous materials behaving in a linear elastic manner, the stress σ and strain ε are related by the Young’s modulus E, σ = E ε

(1.5)

From Eqs. (1.4a), (1.4b), and (1.5), σ = Eε = −E z 1,3 1,3 R

(1.6a)

σ = Eε = E z 2, 4 2, 4 R

(1.6b)

1.1  Derivation of the Bending Equations of Motion by Various Sign Conventions

Balancing the internal normal stress σ and the internal bending moment M requires M = − ∫ z (σdA)

(1.7)

A

Substituting Eqs. (1.6a) and (1.6b) into Eq. (1.7) gives   E z M1,3 = −∫ A z −E dA = R  R 

∫ A z dA = R I ( x )

 z  E M 2,4 = −∫ z  E dA = − A  R  R 

∫ A z dA = − R I ( x )

E

2

(1.8a)

E

2

(1.8b)

In Eqs. (1.7), (1.8a), and (1.8b), A is the area of the cross section and I ( x ) = ∫ z 2dA is the area moment of inertia about the A centroidal axis that is normal to the plane of bending. Substituting Eq. (1.2) into Eqs. (1.8a) and (1.8b) gives M

1,3 = EI ( x )

d2 y ( x , t )

M2,4 = −EI ( x )

(1.9a)

dx 2

d2 y ( x , t ) dx 2

(1.9b)

From the free body diagrams of Figure 1.3, the force equations along the y-axis direction obtained by sign conventions 1, 2, 3, and 4 are  ∂2 y ( x , t ) ∂V ( x , t )  V ( x , t ) + , dx  − V ( x , t ) + f ( x , t )dx = ρ A( x )dx ∂x ∂t 2  

0