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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Mechanical Wave Vibrations
An elegant and accessible exploration of the fundamentals of the analysis and control of vibration in structures from a wave standpoint
In Mechanical Wave Vibrations: Analysis and Control, Professor Chunhui Mei delivers an expert discussion of the wave analysis approach (as opposed to the modal-based approach) to mechanical vibrations in structures. The book begins with deriving the equations of motion using the Newtonian approach based on various sign conventions before comprehensively covering the wave vibration analysis approach. It concludes by exploring passive and active feedback control of mechanical vibration waves in structures.
The author discusses vibration analysis and control strategies from a wave standpoint and examines the applications of the presented wave vibration techniques to structures of various complexity. Readers will find in the book:
- A thorough introduction to mechanical wave vibration analysis, including the governing equations of various types of vibrations
- Comprehensive explorations of waves in simple rods and beams, including advanced vibration theories
- Practical discussions of coupled waves in composite and curved beams
- Extensive coverage of wave mode conversions in built-up planar and spatial frames and networks
- Complete treatments of passive and active feedback wave vibration control
- MATLAB® scripts both in the book and in a companion solutions manual for instructors
Mechanical Wave Vibrations: Analysis and Control is written as a textbook for both under-graduate and graduate students studying mechanical, aerospace, automotive, and civil engineering. It will also benefit researchers and educators working in the areas of vibrations and waves.
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Mechanical Wave Vibrations
Analysis and Control
Chunhui Mei
University of Michigan-DearbornDearborn, MI, USA
This edition first published 2023
© 2023 Chunhui Mei
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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.
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 wave-based 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.
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, in- and 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.
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.
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 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 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.
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.
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 . 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 . 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 per unit length. The x-axis is chosen to be along the neutral axis of the beam, t is the time, and is the transverse deflection of the beam.
Figure 1.2 A beam in bending vibration.
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.
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.
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 of a curve can be found using the following formula
For a beam element in a practical engineering structure that undergoes bending vibration, the transverse deflection of the centerline normally forms a shallow curve because of limitations set forth by engineering design codes on allowable deflection of engineering structures. Consequently, the slope of the deformation curve is normally very small, and its square is negligible when compared to unity. Therefore, the radius of curvature as defined above can be approximated by
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.
Figure 1.4 Strain and radius of curvature: (a) before deformation, (b) after concave bending deformation, and (c) after convex bending deformation.
In the absence of axial loading, the longitudinal strain in the beam is produced only from bending and by definition of strain,
There are two types of bending deformations with reference to the positive y-axis, concave and convex, because of internal bending moments and , respectively. The normal strains and 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
where is the radius of curvature of the transverse deflection of the centerline , and 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 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 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 carry the same sign as z, as reflected in Eq. (1.4b).
For homogeneous materials behaving in a linear elastic manner, the stress and strain are related by the Young’s modulus E,
From Eqs. (1.4a), (1.4b), and (1.5),
Balancing the internal normal stress and the internal bending moment M requires
Substituting Eqs. (1.6a) and (1.6b) into Eq. (1.7) gives
In Eqs. (1.7), (1.8a), and (1.8b), A is the area of the cross section and is the area moment of inertia about the centroidal axis that is normal to the plane of bending.
Substituting Eq. (1.2) into Eqs. (1.8a) and (1.8b) gives
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
where is the volume mass density of the beam. Note that Eqs. (1.10a) and (1.10b) are identical, so are Eqs. (1.10c) and (1.10d).
Simplifying the above equations gives
From the free body diagrams of Figure 1.3, the moment equations by sign conventions 1, 2, 3, and 4, under the assumption that the rotary inertia is negligibly small, are
Neglecting second-order terms in dx and canceling appropriate terms, Eqs. (1.12a-d) reduce to
From Eqs. (1.9a), (1.9b), and (1.13a-d),
From Eqs. (1.14a), (1.14b), and (1.11a-d), a unique fourth order partial differential governing equation of a beam in bending, regardless of the sign conventions, is obtained,
It is important to note that although the equations of motion are not sign convention dependent, the relationships between moments/forces and deflections are. Their relationships in the Euler–Bernoulli bending vibration theory, corresponding to the various sign conventions illustrated in Figure 1.1, are as given in Eqs. (1.9a), (1.9b), (1.14a), and (1.14b).
1.1.2 According to Timoshenko Bending Vibration Theory
The Timoshenko bending vibration theory is applied in the derivation of the governing equations of motion of a beam in bending vibration. It is still assumed that the neutral axis does not experience any change in length and the cross section remains a plane after deformation. However, the restrictions imposed by the Euler–Bernoulli bending vibration theory on the rotation of the differential element and angular distortion due to shear are eliminated. In other words, the rotation of the differential element does not have to be negligible compared to the translation, and the angular distortion due to shear does not have to be small in relation to the bending deformation. Consequently, the analysis based on the Timoshenko bending vibration theory provides greater accuracy at higher frequencies than that of the Euler–Bernoulli bending vibration theory.
The normal strains and on the differential element at a distance z above the neutral axis corresponding to the concave and convex deformation shown in Figures 1.4b and 1.4c are
where is the radius of curvature of the transverse deflection of the centerline and is the angle of rotation of the cross section due to bending. Subscripts 1, 2, 3, and 4 denote that the parameters are related to sign conventions 1, 2, 3, and 4, as defined in Figure 1.1, respectively.
From Eqs. (1.5), (1.16a), and (1.16b),
Substituting Eqs. (1.17a) and (1.17b) into Eq. (1.7) gives
where A is the area of the cross section and is the area moment of inertia about the centroidal axis that is normal to the plane of bending.
The bending moment causes the cross section to rotate an angle . The net shear force acting on the section also contributes a shear deformation . As a result, the total deformation of the neutral line of the differential element of a beam comprises two components, namely bending deformation and shear deformation . Figure 1.5 shows the deformations corresponding to the four sets of sign conventions. They are related by the following algebraic summation,
Figure 1.5 Deformation of a beam in bending subject to positive moment and shear force 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.
Equation (1.19) gives
is a positive value corresponding to sign conventions 1 and 2, and it is negative for sign conventions 3 and 4.
Recall that in the Euler–Bernoulli bending vibration theory, the shear deformation is assumed negligibly small, from Eq. (1.20), one has .
For homogeneous materials behaving in a linear elastic manner, the shear stress and shear strain are related by Hooke’s law,
where G is the shear modulus.
The internal resistant shear forces corresponding to the positive bending moment and shear force defined in Figure 1.1 are
where is the shear coefficient and has been tabulated in (Cowper 1966).
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
Note that Eqs. (1.23a) and (1.23b) are the same, so are Eqs. (1.23c) and (1.23d). Simplifying the above equations gives
From Eqs. (1.22a), (1.22b), and (1.24a-d), a unique force-based equation of motion, regardless of the sign conventions, is obtained,
From the free body diagrams of Figure 1.3, the moment equations by sign conventions 1, 2, 3, and 4 are
Neglecting second-order terms in dx and canceling appropriate terms, Eqs. (1.26a-d) reduce to
From Eqs. (1.18a), (1.1.8b), (1.22a), (1.22b), and (1.27a-d), a unique moment-based equation of motion, regardless of the sign conventions, is obtained,
It is important to note that although the equations of motion are not sign convention dependent, the relationships between moments/forces and deflections are. Their relationships by the Timoshenko bending vibration theory, corresponding to the various sign conventions illustrated in Figure 1.1, are as given in Eqs. (1.18a), (1.18b), (1.22a), and (1.22b).
1.2 Derivation of the Elementary Longitudinal Equation of Motion by Various Sign Conventions
Figure 1.6 shows positive sign directions of internal resistant axial force F for two types of sign conventions. In the figure, subscripts “L” and “R” denote parameters related to the left and right side of the cut section, respectively.
Figure 1.6 Definitions of positive sign directions for internal resistant axial force by various sign conventions: (a) sign convention 1 and (b) sign convention 2.
In Figure 1.6a, the internal resistant axial force is defined positive when it stretches the element, which is a convention that has been popularly adopted by many authors in textbooks and research papers. In Figure 1.6b, the internal resistant axial force is defined positive when it compresses the element, which is less often adopted.
Consider a beam that is subjected to an external distributed axial load of per unit length, as shown in Figure 1.7, where the x-axis is chosen to be along the centerline of the beam and t is the time.
Figure 1.7 A beam in longitudinal vibration.
First, consider the sign convention of Figure 1.6a. From the free body diagram of Figure 1.8a,
Figure 1.8 Free body diagram of a beam element in longitudinal vibration by the various sign conventions defined in Figure 1.6: (a) sign convention 1 and (b) sign convention 2.
Simplifying Eq. (1.29a),
Applying Newton’s second law,
where is the longitudinal deflection of the beam, is the volume mass density, and A(x) is the cross-sectional area of the beam.
Simplifying Eq. (1.30) gives
From the mechanics of materials and the sign convention of Figure 1.6a (Hibbeler 2017), the internal resistant force is related to the axial deflection by
Substituting Eq. (1.32) into Eq. (1.31) gives the equation of motion for longitudinal vibration,
Next, consider the sign convention of Figure 1.6b. From the free body diagram of Figure 1.8b,
Simplifying Eq. (1.34a),
Applying Newton’s second law,
Simplifying Eq. (1.35) gives
From the mechanics of materials and the sign convention of Figure 1.6b, the internal resistant force is related to the axial deflection by
Substituting Eq. (1.37) into Eq. (1.36) gives the same equation of motion for longitudinal vibration obtained in Eq. (1.33).
It is important to note that although the equations of motion are not sign convention dependent, the relationships between forces and deflections by sign conventions 1 and 2 are described in Eqs. (1.32) and (1.37), respectively.
1.3 Derivation of the Elementary Torsional Equation of Motion by Various Sign Conventions
Figure 1.9 shows positive sign directions of internal resistant torque T for two types of sign conventions. In the figure, subscripts L and R denote parameters related to the left and right side of the cut section, respectively. The double arrow notation by the right hand rule is adopted for torque and torsional deflection, where the double arrow shows the direction the thumb points.
Figure 1.9 Definitions of positive sign directions for internal resistant torque by various sign conventions: (a) sign convention 1 and (b) sign convention 2.
In Figure 1.9a, the internal resistant torque is defined positive when the thumb directs outwards from the beam element by following the right hand rule, which is a convention that has been popularly adopted by many authors in textbooks and research papers. In Figure 1.9b, the internal resistant torque is defined positive when the thumb directs inwards to the beam element by following the right hand rule, which is less often adopted.
Consider a shaft that is subjected to an external distributed torque load of per unit length, as shown in Figure 1.10, where the x-axis is chosen to be along the centerline of the shaft and t is the time.
Figure 1.10 A shaft in torsional vibration.
First, consider the sign convention of Figure 1.9a. From the free body diagram of Figure 1.11a,
Figure 1.11 Free body diagram of a beam element in torsional vibration by the various sign conventions defined in Figure 1.9: (a) sign convention 1 and (b) sign convention 2.
Simplifying Eq. (1.38a),
Applying Newton’s second law for rotational motion,
where is the torsional deflection, is the volume mass density of the shaft, and is the polar moment of inertia of the cross-sectional area.
Simplifying Eq. (1.39) gives
From the mechanics of materials and the sign convention of Figure 1.9a (Hibbeler 2017), the internal resistant torque is related to the torsional deflection by
where is the torsional rigidity of a beam whose cross section is rotationally symmetric. For a rotationally asymmetric cross section, an adjustment coefficient is normally needed for obtaining an equivalent torsional rigidity.
Substituting Eq. (1.41) into Eq. (1.40) gives the equation of motion for torsional vibration of a beam whose cross section is rotationally symmetric,
Next, consider the sign convention of Figure 1.9b. From the free body diagram of Figure 1.11b,
Simplifying Eq. (1.43a),
Applying Newton’s second law for rotational motion,
Simplifying Eq. (1.44) gives
For a beam whose cross section is rotationally symmetric, from the mechanics of materials and the sign convention of Figure 1.9b, the internal resistant torque is related to the torsional deflection by
Substituting Eq. (1.46) into Eq. (1.45) gives the same equation of motion for torsional vibration as Eq. (1.42). The relationships between torques and deflections by sign conventions 1 and 2 are described in Eqs. (1.41) and (1.46), respectively.
In summary, the equations of motion are not sign convention dependent; however, the relationships between moments/forces/torques and deflections are.
References
Cowper G. R. The Shear Coefficient in Timoshenko’s Beam Theory,
Journal of Applied Mechanics
, 33, 335–340 (1966).
Hibbeler R. C.
Statics and Mechanics of Materials
, Pearson, Prentice Hall (2017).
2 Longitudinal Waves in Beams
Similar to the propagation of acoustical waves and electro-magnetic waves through gaseous, liquid, and solid media, mechanical vibrations can be described as waves that propagate in a solid medium (Graff 1975; Cremer et. al. 1987; and Doyle 1989). From the wave standpoint, vibrations in a distributed or continuous structure can be viewed as waves that propagate along uniform waveguides and are reflected and transmitted at structural discontinuities (Mace 1984). The propagation relationships of waves are governed by the equations of motion of a beam for free vibration, and the reflection and transmission relationships are determined by the equilibrium and continuity at a structural discontinuity. Assembling these propagation, reflection, and transmission relationships provides a concise and systematic approach for vibration analysis of a distributed or continuous structure.
In this chapter, fundamental concepts related to longitudinal waves are introduced, such as the propagation coefficient of longitudinal vibration waves along a uniform beam (the waveguide) and the reflection coefficient of longitudinal vibration waves at either a classical or non-classical boundary (the discontinuity). Natural frequencies, modeshapes, as well as steady state frequency responses, are obtained, with comparison to experimental results. MATLAB scripts for numerical simulations are provided.
2.1 The Governing Equation and the Propagation Relationships
The governing equation of longitudinal vibrations based on the elementary theory is
where x is the position along the beam axis, t is the time, A(x) is the cross-sectional area, is the longitudinal deflection of the centerline of the beam, is the externally applied longitudinal force per unit length, E is the Young’s modulus, and is the volume mass density of the beam.
For a uniform beam subjected to no external force, the differential equation for free longitudinal vibration is
